# Bulk Local Operators, Conformal Descendants, and Radial Quantization

Authors: Zhao-Long Wang; Yi Yan
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

We establish a construction of the bulk local operators in AdS by considering CFT at finite energy scale. Without assuming any prior knowledge about the bulk, the solution to the bulk free field equation automatically appears in the field theory arguments. In the radial quantization formalism, we find a properly regularized version of our initial construction. Possible generalizations beyond pure AdS are also discussed.


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## Body

## 1. Introduction

The AdS/CFT correspondence [1–4] implies a duality between the quantum gravity in d+1-dimensional anti-de Sitter space and the d-dimensional conformal field theory which is defined on the boundary of AdSd+1. The AdS metric in the Poincaré patch is given by(1)ds2=1z2dz2+ημνdxμdxν,where the boundary is the d-dimensional flat space at z=0. The relations between the boundary data of AdS and the CFT quantities have been well established in [4] by the field-operator correspondence. That is, the correlators of a conformal primary operator O(x) in the CFT are reproduced by the asymptotical data of a bulk field ϕ near the boundary. However, the explicit CFT construction of the bulk local degree of freedoms ϕ(x,z) inside the AdS space is not well understood yet. The earlier attempts [5–7] suggested to reconstruct ϕ(x,z) by propagating the bulk modes from the bulk to the boundary, and then [8–10] showed that it is equivalent to the smearing operator construction. In this letter, we suggest a different construction based on almost purely CFT arguments. In Section 2.1, we establish the construction by considering CFT at finite energy scale. The possible divergence and the prescription of regulator are discussed in Section 2.2. Then in Section 3, we find that our construction can get improved in the radial quantization formalism. We summarize our main results in Section 4, and a possible way of generalizing the construction beyond pure AdS is also proposed there.


## 2. CFT Construction of Bulk Local Operators

### 2.1. Renormalized Primary at Finite Energy Scale

It has been pointed out qualitatively [1–4, 11] that the bulk radial direction z is related to the energy scale in the dual field theory. In order to reconstruct ϕ(x,z), the first candidate is to consider in CFT the renormalized primary operator O(x,μ) which is defined at a finite energy scale μ. On the other hand, the behaviors of a primary operator under the conformal transformation have already been encoded in its conformal family. Thus it is natural to expect, at least in the leading order, that the renormalization of the primary operator at a finite energy scale will lead to a mixing between the primary operator and its descendants(2)Ox,μ=Zμ,∂Ox.For simplicity, we will only consider the scalar operator from now on. It is also natural to require that the renormalized primary operator recovers the Lorentz properties and the scaling dimension of the original primary. Then we find it can only be in the following form:(3)Ox,μ=Zμ-2□Ox.Is it possible to fix the explicit form ofZμ-2□ by imposing certain renormalization condition? The idea is to give the word “primary” a renormalized meaning. In the usual CFT language, the definition of a primary operator is equivalent to requiring that it transforms as a tensor under conformal transformations. We also notice that the renormalization scale μ will transform nontrivially under conformal transformations. Thus a direct guess is that the proper renormalization condition should be the following.The renormalized primary transforms as a tensor under the generalized conformal transformations including the energy scale.To address the generalized conformal transformations including the energy scale, let us firstly review the realization of conformal algebra on thex-space. Acting on the coordinates xμ, the conformal generator can be expressed as follows:(4)Pμ∘=-i∂μ,Mμν∘=-ixμ∂ν-xν∂μ,D∘=-ixμ∂μ,Kμ∘=-ix2∂μ-2xμxν∂ν.It implies the standard conformal algebra(5)Pμ,Pν=0,Mμν,Mρσ=iημρMνσ+ηνσMμρ-ηνρMμσ-ημσMνρ,Mμν,Pρ=iημρPν-ηνρPμ,Mμν,Kρ=iημρKν-ηνρKμ,D,Mμν=0,D,Kμ=-iKμ,D,Pμ=iPμ,Kμ,Kν=0,Kμ,Pν=-2iMμν-2iημνD.To include the energy scale (for different approaches of introducing the finite energy scale, see  [12, 13].), a straightforward way is to add ∂/∂μ as well as μ-dependent coefficients into the realization (4). For latter convenience, we define z≡1/μ and equivalently consider z instead. From the fact that the energy scale is Poincaré invariant, we conclude that the forms of Pμ and Mμν remain intact. The scaling dimension of energy scale is obviously 1; thus we can easily write down the following generalized form of dilatation D:(6)D∘=-iz∂z+xμ∂μ.For the special conformal generator, the strategy is to take its most general ansatz(7)Kμ∘=-ix2δμν-2xμxν+fμνz,x∂ν-igμz,x∂zand then try to find the explicit form which satisfies the conformal algebra.From[Kμ,Pν]=-2iMμν-2iημνD, we get(8)2ημνz∂z=-∂νfμρz,x∂ρ-∂νgμz,x∂z.It implies that(9)fμνz,x=δμνfz,gμz,x=-2xμz.From [D,Kμ]=-iKμ, we further get(10)fz=αz2,where α is an arbitrary constant. Finally, we can check that the above results satisfy [Kμ,Kν]=0. In conclusion, we have(11)Pμ∘=-i∂μ,Mμν∘=-ixμ∂ν-xν∂μ,D∘=-iz∂z+xμ∂μ,Kμ∘=-ix2+αz2∂μ-2xμxρ∂ρ-2xμz∂z.In fact, this is exactly the isometry generator of the AdS space when α>0 and it suggests to identify αz here with the standard AdS radial coordinate. We also notice that it corresponds to dSd+1 when α is negative, but we will only concentrate on the AdS case in this paper.Given the generalized conformal transformation including the energy scale (11), we can try to decide the form of Z(z2□) by our renormalization condition on primary. For a scalar in the {x,z} space, we can expand it by powers of z(12)Φz,x=zΔ∑n=0∞znΦnx.The scalar transformation rule(13)Φ~z~,x~=Φz,ximplies that the terms appearing in the power expansion should transform as follows:(14)Pμ,Φnx=i∂μΦnx,Mμν,Φnx=ixμ∂ν-xν∂μΦnx,D,Φnx=iΔ+n+xμ∂μΦnx,Kμ,Φ0x=ix2∂μ-2xμxν∂ν-2xμΔΦ0x,Kμ,Φ1x=ix2∂μ-2xμxν∂ν-2xμΔ+1Φ1x,Kμ,Φnx=iα∂μΦn-2x+ix2∂μ-2xμxν∂ν-2xμΔ+nΦnxn>1.Now the task is to construct Φn(x) by the primary O and its scalar descendants □mO. From the conformal transformation rules of the primary we can deduce that(15)Pμ,□nOx=i∂μ□nOx,Mμν,□nOx=ixμ∂ν-xν∂μ+ΣνρO□nOx,D,□nOx=iΔ+2n+xμ∂μ□nOx,Kμ,Ox=ix2∂μ-2xμxν∂ν-2xμΔ-2xνΣμνOOx,Kμ,□nOx=2nd-2Δ+ni∂μ□n-1Ox-4nΣμνO∂ν□n-1Ox+ix2∂μ-2xμxρ∂ρ-2xμΔ+2n-2xρΣμρO□nOxn≥1.Comparing with (14), it implies the unique identification(16)Φ2n+1=0,Φ2n=-1nαnΓΔ-d/2+14nn!ΓΔ-d/2+n+1□nO.In conclusion, up to an overall constant, our arguments show that the renormalized primary at energy scale μ=1/z is given by(17)Ox,z=Zz2□Ox=F01;Δ-d2+1;-αz24□Ox,and zΔO(x,z) corresponds to a bulk scalar field ϕ(x,z). In the α=0 limit, it actually comes back to the usual language of CFT.We notice thatZ(z2□) obtained in (17) is nothing but the Fourier transformation of the solutions to the bulk free field equation with Φ(z→0)~zΔ behavior at the boundary. This construction is different from the one suggested in [5–10]. The approach there encountered only the k2=k→2-w2<0 part of the bulk Φ(z→0)~zΔ modes, and thus it cannot be generalized to the Euclidean AdS case. Instead, our construction encounters all the bulk Φ(z→0)~zΔ modes since it is the honest Fourier transformation. Obviously, (17) is applicable for both signatures.


### 2.2. Two-Point Correlators: The Divergent Regime and the Regulator

As a consistency check, let us use (17) to recover the well-known bulk-boundary propagator. We find(18)zΔOz,xOx′=zΔF01;Δ-d2+1;-αz24□OxOx′=zΔx-x′2Δ∑n=0∞ΔΔ+1⋯Δ+n-1n!-αz2x-x′2n.In the regime x-x′2>αz2, the series is convergent and gives rise to the expected form of the bulk-boundary propagator(19)Kx,z;x′=zαz2+x-x′2Δ.However, in the regime |x-x′|2<αz2, the series (18) is divergent. In fact, this result is not surprising. The Z(z2□) given in (17) is just the Φ(z→0)~zΔ modes of the bulk solution, while the Fourier transformation of (19) is a linear combination of the Φ(z→0)~zΔ modes and the Φ(z→0)~zd-Δ modes [14] which regulate the divergence of (18). The existence of the Φ(z→0)~zd-Δ constituent in (19) can be easily seen from the z→0 limit [15](20)limz→0zαz2+x-x′2Δ~zd-Δδx-x′.Although both the Φ(z→0)~zΔ and Φ(z→0)~zd-Δ modes diverge exponentially as z→∞, the combination is well-behaved in the interior since the two divergences cancel with each other. The explicit computation of the corresponding Fourier transformations is performed in Appendix.In order to understand the above issue better, let us recall a simple fact in field theory. That is, the correlation function for composite operators always has zeroth-order UV divergence due to its composite natural. For example, consider the composite operatorO≕ϕaϕa:. The two-point correlator 〈OO〉 receives zeroth-order UV divergence from the following loop diagram even in the free theory.In the coordinate space, the corresponding divergence takes the following form:(21)fΛ,μ,□δdx-x′,where μ is the renormalization scale and Λ is the cut-off scale. This divergence cannot be canceled by any local counterterm in the original action. Instead, we need to define the regularized two-point function directly and remove it by hand. Or equivalently speaking, we need to add a local counterterm in the free energy W[J](22)eWJ=eJO,WJ⟶WRJ=WJ-∫ddx∫ddx′cx-x′JxJx′,where c(x-x′)=c(Λ,μ,□)δd(x-x′). In principle, after the cancelation of the divergent part, a possible remnant term in the form R(μ,□)δd(x-x′) would still be there, and its explicit form depends on the prescription of the regularization.We notice that theΦ(z→0)~zd-Δ modes of (19) in the coordinate space are given by(23)zd-ΔF01;d2-Δ+1;-αz24□δdx-x′.It is right in the form of R(μ,□)δd(x-x′) appearing above. This fact suggests that one can understand it as the possible remnant term. The only special point is that there is an infinite order derivative operator acting on δd(x-x′). Thus it is no longer a local function but a quasi-local term which is identically vanishing in the outer region |x-x′|2>αz2. Adding such a term does not affect the result (18) in the region |x-x′|2>αz2, and it is possible to cancel the divergence in the region |x-x′|2<αz2. If we take the continuity at |x-x′|2=αz2 as the prescription of the regularization of the two-point function, it will pick the correct ratio between the Φ(z→0)~zΔ modes and the Φ(z→0)~zd-Δ modes and then recovers (19) everywhere. This prescription is equivalent to the momentum space IR regularity condition used in the literatures [14]. Since it is natural to expect that effective operators defined at finite energy scale have some ambiguity in probing the distance shorter than its typical scale, the dependence on the prescription of regularization above is actually acceptable.One can also check that the bulk-bulk propagator can be recovered by computing(24)zΔz′ΔOz,xOz′,x′.Again, there is a divergent regime at short distance. If the continuity prescription is imposed, it implies that one should take the following regulator:(25)zMd-ΔzmΔF01;d2-Δ+1;-αzM24□F01;Δ-d2+1;-αzm24□δdx-x′,where zM=max⁡{z,z′} and zm=min⁡{z,z′}.


## 2.1. Renormalized Primary at Finite Energy Scale

It has been pointed out qualitatively [1–4, 11] that the bulk radial direction z is related to the energy scale in the dual field theory. In order to reconstruct ϕ(x,z), the first candidate is to consider in CFT the renormalized primary operator O(x,μ) which is defined at a finite energy scale μ. On the other hand, the behaviors of a primary operator under the conformal transformation have already been encoded in its conformal family. Thus it is natural to expect, at least in the leading order, that the renormalization of the primary operator at a finite energy scale will lead to a mixing between the primary operator and its descendants(2)Ox,μ=Zμ,∂Ox.For simplicity, we will only consider the scalar operator from now on. It is also natural to require that the renormalized primary operator recovers the Lorentz properties and the scaling dimension of the original primary. Then we find it can only be in the following form:(3)Ox,μ=Zμ-2□Ox.Is it possible to fix the explicit form ofZμ-2□ by imposing certain renormalization condition? The idea is to give the word “primary” a renormalized meaning. In the usual CFT language, the definition of a primary operator is equivalent to requiring that it transforms as a tensor under conformal transformations. We also notice that the renormalization scale μ will transform nontrivially under conformal transformations. Thus a direct guess is that the proper renormalization condition should be the following.The renormalized primary transforms as a tensor under the generalized conformal transformations including the energy scale.To address the generalized conformal transformations including the energy scale, let us firstly review the realization of conformal algebra on thex-space. Acting on the coordinates xμ, the conformal generator can be expressed as follows:(4)Pμ∘=-i∂μ,Mμν∘=-ixμ∂ν-xν∂μ,D∘=-ixμ∂μ,Kμ∘=-ix2∂μ-2xμxν∂ν.It implies the standard conformal algebra(5)Pμ,Pν=0,Mμν,Mρσ=iημρMνσ+ηνσMμρ-ηνρMμσ-ημσMνρ,Mμν,Pρ=iημρPν-ηνρPμ,Mμν,Kρ=iημρKν-ηνρKμ,D,Mμν=0,D,Kμ=-iKμ,D,Pμ=iPμ,Kμ,Kν=0,Kμ,Pν=-2iMμν-2iημνD.To include the energy scale (for different approaches of introducing the finite energy scale, see  [12, 13].), a straightforward way is to add ∂/∂μ as well as μ-dependent coefficients into the realization (4). For latter convenience, we define z≡1/μ and equivalently consider z instead. From the fact that the energy scale is Poincaré invariant, we conclude that the forms of Pμ and Mμν remain intact. The scaling dimension of energy scale is obviously 1; thus we can easily write down the following generalized form of dilatation D:(6)D∘=-iz∂z+xμ∂μ.For the special conformal generator, the strategy is to take its most general ansatz(7)Kμ∘=-ix2δμν-2xμxν+fμνz,x∂ν-igμz,x∂zand then try to find the explicit form which satisfies the conformal algebra.From[Kμ,Pν]=-2iMμν-2iημνD, we get(8)2ημνz∂z=-∂νfμρz,x∂ρ-∂νgμz,x∂z.It implies that(9)fμνz,x=δμνfz,gμz,x=-2xμz.From [D,Kμ]=-iKμ, we further get(10)fz=αz2,where α is an arbitrary constant. Finally, we can check that the above results satisfy [Kμ,Kν]=0. In conclusion, we have(11)Pμ∘=-i∂μ,Mμν∘=-ixμ∂ν-xν∂μ,D∘=-iz∂z+xμ∂μ,Kμ∘=-ix2+αz2∂μ-2xμxρ∂ρ-2xμz∂z.In fact, this is exactly the isometry generator of the AdS space when α>0 and it suggests to identify αz here with the standard AdS radial coordinate. We also notice that it corresponds to dSd+1 when α is negative, but we will only concentrate on the AdS case in this paper.Given the generalized conformal transformation including the energy scale (11), we can try to decide the form of Z(z2□) by our renormalization condition on primary. For a scalar in the {x,z} space, we can expand it by powers of z(12)Φz,x=zΔ∑n=0∞znΦnx.The scalar transformation rule(13)Φ~z~,x~=Φz,ximplies that the terms appearing in the power expansion should transform as follows:(14)Pμ,Φnx=i∂μΦnx,Mμν,Φnx=ixμ∂ν-xν∂μΦnx,D,Φnx=iΔ+n+xμ∂μΦnx,Kμ,Φ0x=ix2∂μ-2xμxν∂ν-2xμΔΦ0x,Kμ,Φ1x=ix2∂μ-2xμxν∂ν-2xμΔ+1Φ1x,Kμ,Φnx=iα∂μΦn-2x+ix2∂μ-2xμxν∂ν-2xμΔ+nΦnxn>1.Now the task is to construct Φn(x) by the primary O and its scalar descendants □mO. From the conformal transformation rules of the primary we can deduce that(15)Pμ,□nOx=i∂μ□nOx,Mμν,□nOx=ixμ∂ν-xν∂μ+ΣνρO□nOx,D,□nOx=iΔ+2n+xμ∂μ□nOx,Kμ,Ox=ix2∂μ-2xμxν∂ν-2xμΔ-2xνΣμνOOx,Kμ,□nOx=2nd-2Δ+ni∂μ□n-1Ox-4nΣμνO∂ν□n-1Ox+ix2∂μ-2xμxρ∂ρ-2xμΔ+2n-2xρΣμρO□nOxn≥1.Comparing with (14), it implies the unique identification(16)Φ2n+1=0,Φ2n=-1nαnΓΔ-d/2+14nn!ΓΔ-d/2+n+1□nO.In conclusion, up to an overall constant, our arguments show that the renormalized primary at energy scale μ=1/z is given by(17)Ox,z=Zz2□Ox=F01;Δ-d2+1;-αz24□Ox,and zΔO(x,z) corresponds to a bulk scalar field ϕ(x,z). In the α=0 limit, it actually comes back to the usual language of CFT.We notice thatZ(z2□) obtained in (17) is nothing but the Fourier transformation of the solutions to the bulk free field equation with Φ(z→0)~zΔ behavior at the boundary. This construction is different from the one suggested in [5–10]. The approach there encountered only the k2=k→2-w2<0 part of the bulk Φ(z→0)~zΔ modes, and thus it cannot be generalized to the Euclidean AdS case. Instead, our construction encounters all the bulk Φ(z→0)~zΔ modes since it is the honest Fourier transformation. Obviously, (17) is applicable for both signatures.


## 2.2. Two-Point Correlators: The Divergent Regime and the Regulator

As a consistency check, let us use (17) to recover the well-known bulk-boundary propagator. We find(18)zΔOz,xOx′=zΔF01;Δ-d2+1;-αz24□OxOx′=zΔx-x′2Δ∑n=0∞ΔΔ+1⋯Δ+n-1n!-αz2x-x′2n.In the regime x-x′2>αz2, the series is convergent and gives rise to the expected form of the bulk-boundary propagator(19)Kx,z;x′=zαz2+x-x′2Δ.However, in the regime |x-x′|2<αz2, the series (18) is divergent. In fact, this result is not surprising. The Z(z2□) given in (17) is just the Φ(z→0)~zΔ modes of the bulk solution, while the Fourier transformation of (19) is a linear combination of the Φ(z→0)~zΔ modes and the Φ(z→0)~zd-Δ modes [14] which regulate the divergence of (18). The existence of the Φ(z→0)~zd-Δ constituent in (19) can be easily seen from the z→0 limit [15](20)limz→0zαz2+x-x′2Δ~zd-Δδx-x′.Although both the Φ(z→0)~zΔ and Φ(z→0)~zd-Δ modes diverge exponentially as z→∞, the combination is well-behaved in the interior since the two divergences cancel with each other. The explicit computation of the corresponding Fourier transformations is performed in Appendix.In order to understand the above issue better, let us recall a simple fact in field theory. That is, the correlation function for composite operators always has zeroth-order UV divergence due to its composite natural. For example, consider the composite operatorO≕ϕaϕa:. The two-point correlator 〈OO〉 receives zeroth-order UV divergence from the following loop diagram even in the free theory.In the coordinate space, the corresponding divergence takes the following form:(21)fΛ,μ,□δdx-x′,where μ is the renormalization scale and Λ is the cut-off scale. This divergence cannot be canceled by any local counterterm in the original action. Instead, we need to define the regularized two-point function directly and remove it by hand. Or equivalently speaking, we need to add a local counterterm in the free energy W[J](22)eWJ=eJO,WJ⟶WRJ=WJ-∫ddx∫ddx′cx-x′JxJx′,where c(x-x′)=c(Λ,μ,□)δd(x-x′). In principle, after the cancelation of the divergent part, a possible remnant term in the form R(μ,□)δd(x-x′) would still be there, and its explicit form depends on the prescription of the regularization.We notice that theΦ(z→0)~zd-Δ modes of (19) in the coordinate space are given by(23)zd-ΔF01;d2-Δ+1;-αz24□δdx-x′.It is right in the form of R(μ,□)δd(x-x′) appearing above. This fact suggests that one can understand it as the possible remnant term. The only special point is that there is an infinite order derivative operator acting on δd(x-x′). Thus it is no longer a local function but a quasi-local term which is identically vanishing in the outer region |x-x′|2>αz2. Adding such a term does not affect the result (18) in the region |x-x′|2>αz2, and it is possible to cancel the divergence in the region |x-x′|2<αz2. If we take the continuity at |x-x′|2=αz2 as the prescription of the regularization of the two-point function, it will pick the correct ratio between the Φ(z→0)~zΔ modes and the Φ(z→0)~zd-Δ modes and then recovers (19) everywhere. This prescription is equivalent to the momentum space IR regularity condition used in the literatures [14]. Since it is natural to expect that effective operators defined at finite energy scale have some ambiguity in probing the distance shorter than its typical scale, the dependence on the prescription of regularization above is actually acceptable.One can also check that the bulk-bulk propagator can be recovered by computing(24)zΔz′ΔOz,xOz′,x′.Again, there is a divergent regime at short distance. If the continuity prescription is imposed, it implies that one should take the following regulator:(25)zMd-ΔzmΔF01;d2-Δ+1;-αzM24□F01;Δ-d2+1;-αzm24□δdx-x′,where zM=max⁡{z,z′} and zm=min⁡{z,z′}.


## 3. Radial Quantization

### 3.1. Radial Quantization in CFT

In Section2, we have constructed the bulk local operator and also explained its divergent regime with the regularization prescription there. However, the present formula is not convenient in discussing the bulk physics since the regulator should always be added by hand. It will be pretty nice if one can find a smart formula in which the regulator has been automatically built in. To achieve such a formula, let us discuss the radial quantization in usual CFT language firstly.The radial quantization for CFT was detailedly reviewed in [16, 17]. We will equivalently reexpress the results there by introducing the radial expansion for the operators. Again, we will just consider scalar primary here. The radial expansion of a scalar primary operator O(x) is given by(26)Ox=∑m=0∞1m!xμ1⋯xμmOμ1⋯μm+∑n=0∞1n!xˇ2Δxˇν1⋯xˇνnOˇν1⋯νn,where xˇ is the inversion of x(27)xˇμ=I∘xμ=xμx2.In a unitary theory, its Hermitian conjugation is induced by the inversion(28)O†x=Oˇ-x=1x2ΔO-xˇ=∑n=0∞-1nn!xν1⋯xνnOˇν1⋯νn+∑m=0∞-1mm!xˇ2Δxˇμ1⋯xˇμmOμ1⋯μm.In terms of the component operators, it is given by(29)Oμ1⋯μn†=-1nOˇμ1⋯μn,Oˇμ1⋯μn†=-1nOμ1⋯μn.The vacuum |1〉 is defined by(30)Oˇν1⋯νn1=0,1Oμ1⋯μm=0.It is equivalent to requiring that the state O(x)|1〉 and all its descendants are regular at x=0, while the conjugated state 〈1|Oˇ(x) and all its descendants are also regular at x=0. One can check that the vacuum |1〉 defined in (30) is actually conformal invariant. The conformal transformation rules of the component operators can be deduced from the standard rules for primary O(x). We find(31)Pμ,Oμ1⋯μm=iOμμ1⋯μm,Pμ,Oˇν1⋯νn=inn-1η(ν1ν2Oˇν3⋯νn)μ-2nΔ+n-1ημ(ν1Oˇν2⋯νn),Mμν,Oμ1⋯μm=imημ(μ1Oμ2⋯μm)ν-ην(μ1Oμ2⋯μm)μ,Mμν,Oˇν1⋯νn=inημ(ν1Oˇν2⋯νn)ν-ην(ν1Oˇν2⋯νm)μ,D,Oμ1⋯μm=iΔ+mOμ1⋯μm,D,Oˇν1⋯νn=-iΔ+mOˇν1⋯νn,Kμ,Oμ1⋯μm=imm-1η(μ1μ2Oμ3⋯μm)μ-2mΔ+m-1ημ(μ1Oμ2⋯μm),Kμ,Oˇν1⋯νn=iOˇμν1⋯νn.Given the input data(32)O′0∣O0=1Oˇ0′O01=CO′O,one can decide the inner product between the states |∂μn⋯∂μ1O(0)〉=Oμ1⋯μn|1〉 by using (31) and the conformal invariance of the vacuum. The result is(33)1Oˇνn⋯ν1′Oμ1⋯μm1=δmnCO′O∑k=0n/2-1n-k2nn!2ΓΔ+n-kn-2k!k!2!2kΓΔδ(ν1(μ1⋯δνn-2kμn-2kηνn-2k+1νn-2k+2⋯ηνn-1νn)ημn-2k+1μn-2k+2⋯ημn-1μn).Now we can reproduce the well-known two-point correlator by the “silly” computation(34)1OxO′x′1=∑m=0∞∑n=0∞1m!n!xμ1⋯xμmx′ν1⋯x′νnx2Δ+m1Oˇμ1⋯μmOν1⋯νn′1=CO′O∑n^=0∞∑k=0∞-1n^+k2n^ΓΔ+n^+kn^!k!ΓΔx·x′n^x′2kx2Δ+n^+k.The convergence of the series requires that x>x′. (Strictly speaking, the convergence argument is accurate only for the Euclidean case. For the Lorentzian case, proper analytical continuations are needed as what usually happened in the quantum field theory computations.) It means that the usual CFT correlator is reproduced by the radial ordered function 1R^OxO′(x′)1. In the radial quantization where the dilatation operator D is treated as the Hamiltonian, the radial ordered function is the natural analogy of the time ordered function in the usual quantum field theory.


### 3.2. Radial Quantization at Finite Energy Scale

Now let us consider the CFT radial quantization in the presence of the finite energy scaleμ. A direct idea is acting upon the radial expansion (26) with the Z(μ-2□) in  (17). We get(35)F01;Δ-d2+1;-αz24□Ox=∑s=0∞∑k=0∞-αz2kΓΔ-d/2+122kk!s!ΓΔ-d/2+k+1xν1⋯xνsOμ1⋯μkμ1⋯μkν1⋯νs+∑s=0∞∑l=0∞-αz2lΓΔ-d/2+122ll!s!ΓΔ-d/2+l+1F10Δ+2l+s;;-αz2x2xν1⋯xνsx2Δ+2l+sOˇμ1⋯μlμ1⋯μlν1⋯νsμ1⋯μlμ1⋯μl.We notice that when |x|2>αz2(36)1x2Δ+2l+sF10Δ+2l+s;;-αz2x2=1x2+αz2Δ+2l+s,and it is divergent in the regime |x|2<αz2. The structure of the divergent regime is quite similar to what we have seen in Section 2. Thus it is natural to expect the following as the regularized radial expansion at finite energy scale(37)Ox,z=∑m=0∞1m!xμ1⋯xμmOμ1⋯μmz+∑n=0∞1n!xν1⋯xνnx2+αz2Δ+nOˇν1⋯νnzˇ,where the component operator at finite energy scale is a linear combination of the original ones(38)Oμ1⋯μmz=∑k=0∞-αz2kΓΔ-d/2+122kk!ΓΔ-d/2+k+1Oν1⋯νkν1⋯νkμ1⋯μm,Oˇμ1⋯μmzˇ=∑k=0∞-αzˇ2kΓΔ-d/2+122kk!ΓΔ-d/2+k+1Oˇν1⋯νkν1⋯νkμ1⋯μmν1⋯νkν1⋯νk,and the inversion of x and z in the generalized scene is given by(39)xˇμ=I∘xμ=xμx2+αz2,zˇ=I∘z=zx2+αz2.Correspondingly, the inversion of O(x,z) is(40)Oˇx,z=1x2+αz2ΔOxˇ;zˇ=∑n=0∞1n!xν1⋯xνnOˇν1⋯νnz+∑m=0∞1m!xμ1⋯xμmx2+αz2Δ+nOμ1⋯μmzˇ.From (29), we can see that the Hermitian conjugation relation keeps intact at finite energy scale(41)O†x,z=Oˇ-x,z.Simply by using (31), one can write down the conformal transformation rule for the component fields at finite energy scale directly as follows:(42)Pμ,Oμ1⋯μmz=iOμμ1⋯μmz,Pμ,Oˇμ1⋯μmz=iOˇμ2⋯μm)αz2Oˇμ1⋯μmμz+mm-1η(μ1μ2Oˇμ3⋯μm)μz-2mΔ+m-1+z∂zημ(μ1Oˇμ2⋯μm)z,Mμν,Oμ1⋯μmz=imημ(μ1Oμ2⋯μm)νz-ην(μ1Oμ2⋯μm)μz,Mμν,Oˇμ1⋯μmz=imημ(ν1Oˇν2⋯νm)νz-ην(ν1Oˇν2⋯νm)μz,D,Oμ1⋯μmz=iΔ+m+z∂zOμ1⋯μmz,D,Oˇμ1⋯μmz=-iΔ+m+z∂zOˇμ1⋯μmz,Kμ,Oμ1⋯μmz=iOμ2⋯μm)αz2Oμ1⋯μmμz+mm-1η(μ1μ2Oμ3⋯μm)μz-2mΔ+m-1+z∂zημ(μ1Oμ2⋯μm)z,Kμ,Oˇμ1⋯μmz=iOˇμμ1⋯μmz.Similar to the arguments in Section 2.1, we can show that a bulk scalar field Φ(x,z) expanded as(43)Φx,z=∑m=0∞zΔm!xμ1⋯xμmΦμ1⋯μmz+∑n=0∞1n!zΔxν1⋯xνnx2+αz2Δ+nΦˇν1⋯νnzˇindeed requires the transformation rules (42) for its components. Thus, the regularized version (37) of primary at finite energy scale satisfies our basic renormalization condition for primary operators.Parallel with (34), the standard bulk-boundary as well as bulk-bulk propagator can be reproduced by computing the radial ordered function 1R^Ox,zO′x′,z′1. In our present case where energy scales are introduced, the radial order is defined by x2+αz2. Providing this radial order, there is no ambiguity everywhere in reproducing the bulk-boundary and bulk-bulk propagator. Therefore, (37) is indeed the smart formula which we are looking for.Since our construction directly comes back to the standard CFT language in theα→0 limit, it is possible that (37) will not suffer from the problem about bulk locality appearing in the smearing operator construction [18]. In order to address it properly, one should generalize the standard results about OPE to the cases with finite energy scales. We hope to report on this issue in a future work.Formula (37) also suggests that one could define the finite energy scale effective Hilbert space Hz by acting on the vacuum |1〉 with Oμ1⋯μm(z). An interesting observation is that Hz actually contains less information than the UV Hilbert space H=Hz=0. For example, considering the scalar sector, one can construct the following state:(44)λ=∑n=0∞λnΓΔΓΔ-d/2+1Γd/224nn!ΓΔ+nΓΔ-d/2+n+1Γn+d/2On1,where we denote On(z)=Oν1⋯νnν1⋯νn(z). It is a well-defined state in H since the norm is finite(45)λ∣λ=CO∑n=0∞λnΓΔΓΔ-d/2+1Γd/224nn!ΓΔ+nΓΔ-d/2+n+1Γn+d/2.The inner product between the state |λ〉 and the states in Hz is given by(46)λOmz1=COλm∑k=0∞-αλz2kΓΔ-d/2+122kk!ΓΔ-d/2+k+1.Since the F01 function has infinite number of zeros, the inner product will be zero for infinite many λ’s providing z≠0. Therefore, there are an infinite number of states in H perpendicular to the finite energy scale effective Hilbert space Hz≠0. Although this observation is something one could expect for effective descriptions at finite energy scale, it may have some possible advantages in discussing the c-theorem and the entanglement entropy.


## 3.1. Radial Quantization in CFT

In Section2, we have constructed the bulk local operator and also explained its divergent regime with the regularization prescription there. However, the present formula is not convenient in discussing the bulk physics since the regulator should always be added by hand. It will be pretty nice if one can find a smart formula in which the regulator has been automatically built in. To achieve such a formula, let us discuss the radial quantization in usual CFT language firstly.The radial quantization for CFT was detailedly reviewed in [16, 17]. We will equivalently reexpress the results there by introducing the radial expansion for the operators. Again, we will just consider scalar primary here. The radial expansion of a scalar primary operator O(x) is given by(26)Ox=∑m=0∞1m!xμ1⋯xμmOμ1⋯μm+∑n=0∞1n!xˇ2Δxˇν1⋯xˇνnOˇν1⋯νn,where xˇ is the inversion of x(27)xˇμ=I∘xμ=xμx2.In a unitary theory, its Hermitian conjugation is induced by the inversion(28)O†x=Oˇ-x=1x2ΔO-xˇ=∑n=0∞-1nn!xν1⋯xνnOˇν1⋯νn+∑m=0∞-1mm!xˇ2Δxˇμ1⋯xˇμmOμ1⋯μm.In terms of the component operators, it is given by(29)Oμ1⋯μn†=-1nOˇμ1⋯μn,Oˇμ1⋯μn†=-1nOμ1⋯μn.The vacuum |1〉 is defined by(30)Oˇν1⋯νn1=0,1Oμ1⋯μm=0.It is equivalent to requiring that the state O(x)|1〉 and all its descendants are regular at x=0, while the conjugated state 〈1|Oˇ(x) and all its descendants are also regular at x=0. One can check that the vacuum |1〉 defined in (30) is actually conformal invariant. The conformal transformation rules of the component operators can be deduced from the standard rules for primary O(x). We find(31)Pμ,Oμ1⋯μm=iOμμ1⋯μm,Pμ,Oˇν1⋯νn=inn-1η(ν1ν2Oˇν3⋯νn)μ-2nΔ+n-1ημ(ν1Oˇν2⋯νn),Mμν,Oμ1⋯μm=imημ(μ1Oμ2⋯μm)ν-ην(μ1Oμ2⋯μm)μ,Mμν,Oˇν1⋯νn=inημ(ν1Oˇν2⋯νn)ν-ην(ν1Oˇν2⋯νm)μ,D,Oμ1⋯μm=iΔ+mOμ1⋯μm,D,Oˇν1⋯νn=-iΔ+mOˇν1⋯νn,Kμ,Oμ1⋯μm=imm-1η(μ1μ2Oμ3⋯μm)μ-2mΔ+m-1ημ(μ1Oμ2⋯μm),Kμ,Oˇν1⋯νn=iOˇμν1⋯νn.Given the input data(32)O′0∣O0=1Oˇ0′O01=CO′O,one can decide the inner product between the states |∂μn⋯∂μ1O(0)〉=Oμ1⋯μn|1〉 by using (31) and the conformal invariance of the vacuum. The result is(33)1Oˇνn⋯ν1′Oμ1⋯μm1=δmnCO′O∑k=0n/2-1n-k2nn!2ΓΔ+n-kn-2k!k!2!2kΓΔδ(ν1(μ1⋯δνn-2kμn-2kηνn-2k+1νn-2k+2⋯ηνn-1νn)ημn-2k+1μn-2k+2⋯ημn-1μn).Now we can reproduce the well-known two-point correlator by the “silly” computation(34)1OxO′x′1=∑m=0∞∑n=0∞1m!n!xμ1⋯xμmx′ν1⋯x′νnx2Δ+m1Oˇμ1⋯μmOν1⋯νn′1=CO′O∑n^=0∞∑k=0∞-1n^+k2n^ΓΔ+n^+kn^!k!ΓΔx·x′n^x′2kx2Δ+n^+k.The convergence of the series requires that x>x′. (Strictly speaking, the convergence argument is accurate only for the Euclidean case. For the Lorentzian case, proper analytical continuations are needed as what usually happened in the quantum field theory computations.) It means that the usual CFT correlator is reproduced by the radial ordered function 1R^OxO′(x′)1. In the radial quantization where the dilatation operator D is treated as the Hamiltonian, the radial ordered function is the natural analogy of the time ordered function in the usual quantum field theory.


## 3.2. Radial Quantization at Finite Energy Scale

Now let us consider the CFT radial quantization in the presence of the finite energy scaleμ. A direct idea is acting upon the radial expansion (26) with the Z(μ-2□) in  (17). We get(35)F01;Δ-d2+1;-αz24□Ox=∑s=0∞∑k=0∞-αz2kΓΔ-d/2+122kk!s!ΓΔ-d/2+k+1xν1⋯xνsOμ1⋯μkμ1⋯μkν1⋯νs+∑s=0∞∑l=0∞-αz2lΓΔ-d/2+122ll!s!ΓΔ-d/2+l+1F10Δ+2l+s;;-αz2x2xν1⋯xνsx2Δ+2l+sOˇμ1⋯μlμ1⋯μlν1⋯νsμ1⋯μlμ1⋯μl.We notice that when |x|2>αz2(36)1x2Δ+2l+sF10Δ+2l+s;;-αz2x2=1x2+αz2Δ+2l+s,and it is divergent in the regime |x|2<αz2. The structure of the divergent regime is quite similar to what we have seen in Section 2. Thus it is natural to expect the following as the regularized radial expansion at finite energy scale(37)Ox,z=∑m=0∞1m!xμ1⋯xμmOμ1⋯μmz+∑n=0∞1n!xν1⋯xνnx2+αz2Δ+nOˇν1⋯νnzˇ,where the component operator at finite energy scale is a linear combination of the original ones(38)Oμ1⋯μmz=∑k=0∞-αz2kΓΔ-d/2+122kk!ΓΔ-d/2+k+1Oν1⋯νkν1⋯νkμ1⋯μm,Oˇμ1⋯μmzˇ=∑k=0∞-αzˇ2kΓΔ-d/2+122kk!ΓΔ-d/2+k+1Oˇν1⋯νkν1⋯νkμ1⋯μmν1⋯νkν1⋯νk,and the inversion of x and z in the generalized scene is given by(39)xˇμ=I∘xμ=xμx2+αz2,zˇ=I∘z=zx2+αz2.Correspondingly, the inversion of O(x,z) is(40)Oˇx,z=1x2+αz2ΔOxˇ;zˇ=∑n=0∞1n!xν1⋯xνnOˇν1⋯νnz+∑m=0∞1m!xμ1⋯xμmx2+αz2Δ+nOμ1⋯μmzˇ.From (29), we can see that the Hermitian conjugation relation keeps intact at finite energy scale(41)O†x,z=Oˇ-x,z.Simply by using (31), one can write down the conformal transformation rule for the component fields at finite energy scale directly as follows:(42)Pμ,Oμ1⋯μmz=iOμμ1⋯μmz,Pμ,Oˇμ1⋯μmz=iOˇμ2⋯μm)αz2Oˇμ1⋯μmμz+mm-1η(μ1μ2Oˇμ3⋯μm)μz-2mΔ+m-1+z∂zημ(μ1Oˇμ2⋯μm)z,Mμν,Oμ1⋯μmz=imημ(μ1Oμ2⋯μm)νz-ην(μ1Oμ2⋯μm)μz,Mμν,Oˇμ1⋯μmz=imημ(ν1Oˇν2⋯νm)νz-ην(ν1Oˇν2⋯νm)μz,D,Oμ1⋯μmz=iΔ+m+z∂zOμ1⋯μmz,D,Oˇμ1⋯μmz=-iΔ+m+z∂zOˇμ1⋯μmz,Kμ,Oμ1⋯μmz=iOμ2⋯μm)αz2Oμ1⋯μmμz+mm-1η(μ1μ2Oμ3⋯μm)μz-2mΔ+m-1+z∂zημ(μ1Oμ2⋯μm)z,Kμ,Oˇμ1⋯μmz=iOˇμμ1⋯μmz.Similar to the arguments in Section 2.1, we can show that a bulk scalar field Φ(x,z) expanded as(43)Φx,z=∑m=0∞zΔm!xμ1⋯xμmΦμ1⋯μmz+∑n=0∞1n!zΔxν1⋯xνnx2+αz2Δ+nΦˇν1⋯νnzˇindeed requires the transformation rules (42) for its components. Thus, the regularized version (37) of primary at finite energy scale satisfies our basic renormalization condition for primary operators.Parallel with (34), the standard bulk-boundary as well as bulk-bulk propagator can be reproduced by computing the radial ordered function 1R^Ox,zO′x′,z′1. In our present case where energy scales are introduced, the radial order is defined by x2+αz2. Providing this radial order, there is no ambiguity everywhere in reproducing the bulk-boundary and bulk-bulk propagator. Therefore, (37) is indeed the smart formula which we are looking for.Since our construction directly comes back to the standard CFT language in theα→0 limit, it is possible that (37) will not suffer from the problem about bulk locality appearing in the smearing operator construction [18]. In order to address it properly, one should generalize the standard results about OPE to the cases with finite energy scales. We hope to report on this issue in a future work.Formula (37) also suggests that one could define the finite energy scale effective Hilbert space Hz by acting on the vacuum |1〉 with Oμ1⋯μm(z). An interesting observation is that Hz actually contains less information than the UV Hilbert space H=Hz=0. For example, considering the scalar sector, one can construct the following state:(44)λ=∑n=0∞λnΓΔΓΔ-d/2+1Γd/224nn!ΓΔ+nΓΔ-d/2+n+1Γn+d/2On1,where we denote On(z)=Oν1⋯νnν1⋯νn(z). It is a well-defined state in H since the norm is finite(45)λ∣λ=CO∑n=0∞λnΓΔΓΔ-d/2+1Γd/224nn!ΓΔ+nΓΔ-d/2+n+1Γn+d/2.The inner product between the state |λ〉 and the states in Hz is given by(46)λOmz1=COλm∑k=0∞-αλz2kΓΔ-d/2+122kk!ΓΔ-d/2+k+1.Since the F01 function has infinite number of zeros, the inner product will be zero for infinite many λ’s providing z≠0. Therefore, there are an infinite number of states in H perpendicular to the finite energy scale effective Hilbert space Hz≠0. Although this observation is something one could expect for effective descriptions at finite energy scale, it may have some possible advantages in discussing the c-theorem and the entanglement entropy.


## 4. Discussions

In the previous sections, we suggest a CFT construction of the bulk local operators in pure AdS space. The construction is based on considering CFT at finite energy scale. The basic result is that bulk operator is given by acting upon the original CFT primary with an infinite order differential operatorZAdS(μ,∂). Although we do not assume any knowledge about the bulk in advance, our arguments automatically show that ZAdS(μ,∂) should be the Fourier transformation of the Φ(z→0)~zΔ solution to the bulk free field equation. We also discuss the relation between the regulator of the two-point function and the Φ(z→0)~zd-Δ modes. In Section 3, based on the radial quantization in CFT, we find an improved formula of our construction in which the regulator is automatically built in.The next challenge is how to generalize our construction to geometries beyond pure AdS. A naive guess is that the bulk local operator is also effectively given by acting upon the original CFT primary with the infinite order differential operatorZGeom(μ,∂), which is the Fourier transformation of the bulk Φ(z→0)~zΔ modes in the corresponding geometry. On the other hand, our CFT arguments in Section 2.1 seem state independent. Thus, it suggests that the bulk local operator should always be given by ZAdS(μ,∂) for all asymptotic AdS geometries which are basically very heavy excited states in the CFT. We conjecture that these two possibilities are actually complementary to each other. The explicit proposal [19] for the underlying mechanism can be summarized as the following.Bulk geometries are actually dual to the coherent states|Geom〉=F(Tμν)|1〉 which is created by acting on the vacuum |1〉 with certain function F(Tμν) of stress tensor and its descendants.The bulk correlators of the dual fieldϕ(x,z) can be reproduced by computing (47)Geom⋯ZAdSμ,∂Ox⋯Geom=1F†Tμν⋯ZAdSμ,∂Ox⋯FTμν1.On the other hand, by using the local conformal Ward identity [20–22], one may convert (at least in the two-dimensional CFT) the effects of F(Tμν) to a differential operator F^[F] acting on the operator ZAdS(μ,∂)O(x) as follows: (48)1F†Tμν⋯ZAdSμ,∂Ox⋯FTμν1=1⋯F^FZAdSμ,∂Ox⋯1.The new differential operator F^[F]ZAdS(μ,∂) is expected to be exactly ZGeom(μ,∂) of the corresponding geometry. For the black hole geometry, the horizon is the position where the series in ZGeom(μ,∂) becomes ill-defined. However, everything could be still well defined after coming back to the Geom description and one can explore the black hole interior in this formalism.Finally, it is also possible that multiple states with differentF(Tμν) give rise to the same differential operator ZGeom(μ,∂). Thus it could be a dual CFT way to explain the entropy of AdS black hole. If indeed so, it means that all the black hole microstates should correspond exactly to the same geometry, and thus one does not need to take any average over different microgeometries. This picture seems different from what people usually expected for quantum gravity and may offer new possibilities to the discussions of the black hole firewall problem [23, 24].As the early version of this work was drawing a conclusion, [25] appeared with results which partially overlap with Section 2.1 in this manuscript. We also realized that the two-dimensional version of their results has already appeared in [26].



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*Source: 8185690-2017-10-02.xml*