bluuebunny/crossref_metadata_2025 huggingface dataset:
High density expansions in the Thomas-Fermi approximation
Authors: Nauenberg Michael
Published: February 2004

Two expansions are discussed for the solution of the Thomas-Fermi equation for a degenerate electron gas at high density. In the first case, the boundary conditions appropriate to the Wigner-Seitz sphere are imposed, and terms up to fourth order are calculated explicitly to show the appearance of a logarithmic singularity in the density. This singularity accounts for the divergence difficulties in higher order encountered by previous authors, who assumed the existence of a power series expansion in the density. In the second case, we apply the boundary conditions appropriate to electron screening of an ion in a neutral plasma at high density, and an expansion is obtained which gives the nonlinear corrections to the Debye-Hückel approximation.
On Litvin's lemma
Authors: Butler P. H.
Published: February 2004

We produce a counterexample to a lemma given by D. B. Litvin on the decomposition of direct products of irreducible representations [J. Math. Phys. 13, 1386 (1972)]. The error in his proof is noted.
On matrix superpropagators. II
Authors: Ashmore J. , Delbourgo R.
Published: February 2004

The techniques developed in a previous paper (I) to compute the T product 〈φNαβ(x), φNγδ(o)〉 for arbitrary N are extended to cover the case when φ(x) is a Hermitian matrix-valued field in ν dimensions. We obtain a closed expression which is used to determine superpropagators like 〈[expκφ(x)]αβ, [expκφ(o)]γδ〉, which occur in strong interaction physics when φ is an SU(3) field, say.
Generalized Fermi transport in generalized gravitation theories
Authors: Enosh M. , Kovetz A.
Published: February 2004

If geodesics in space-time can be classified as timelike, null, and spacelike, the affine connection must be of the form Γjki={jki}+2giaejka−(δjiδka+δkiδja−gjkgia)da, with da an arbitrary vector and eijk a tensor satisfying e(ijk)=e[ij]k= 0. It is possible to generalize Fermi's law of transport to this affine connection. The requirement that any observer be able to construct and maintain a nonrotating orthogonal space triad along his world line by the bouncing photon experiment implies the Weyl's geometry of paths.
Linear adiabatic invariants and coherent states
Authors: Malkin I. A. , Man'ko V. I. , Trifonov D. A.
Published: February 2004

The Born-Fock adiabatic theorem is extended to all orders for some quadratic quantum systems with finitely or infinitely degenerate energy spectra. A prescription is given for obtaining adiabatic invariants to any order. For any quadratic quantum system with N degrees of freedom there are 2N linear adiabatic invariant series, which correspond to the 2N exact invariants. The exact quantum mechanical solution for any nonstationary quadratic quantum system is also constructed by making use of the coherent-state representation: The Green's function, coherent states, transition amplitudes and probabilities and their generating functions are obtained explicitly. Two particular systems, the N-dimensional time-dependent general oscillator and charged particle motion in a varying and uniform electromagnetic field, are considered in greater detail as examples.
A formula for the pressure in statistical mechanics
Authors: Kac M. , Luttinger J. M.
Published: February 2004

A new expression for the pressure in terms of irreducible distribution functions is given. A derivation for both the classical and quantum mechanical case and a brief discussion of the limits of validity of the formula are given.
Generalized isoperimetric inequalities
Authors: Luttinger J. M.
Published: February 2004

New inequalities for certain Green's functions are given. They may be interpreted physically in many ways, for example, as applying to the quantum mechanical motion of a particle in a potential or to diffusion in the presence of absorbers. These inequalities involve a symmetrization process very closely related to Steiner symmetrization used in the theory of isoperimetric inequalities. The usual geometrical and physical isoperimetric inequalities are very special cases of our general inequality (3.9), arising when the potential is taken to be a characteristic function of a bounded domain and the ``time'' in the Green's function is allowed to get very large or very small.
A discrete version of the inverse scattering problem
Authors: Case K. M. , Kac M.
Published: February 2004

A discrete version of the inverse scattering problem is considered. The Schrödinger equation with a potential is replaced by a difference equation. The problem is to determine the coefficients given the phase shift. Application is mainly pedagogical. The motivation for the principal steps becomes obvious and the mathematics is elementary. At any stage one can pass, by a limiting procedure, to the usual Schrödinger problem-and reobtain the classical results.
Weak quantization
Authors: Popp Fritz A.
Published: February 2004

The present study shows the significance of weak quantization within the electric field dependence of optical absorption. This phenomenon is responsible for effects which cannot be derived from application of ordinary perturbation theory. It may play an important role in biochemical processes which occur in electric fields. In this paper the problems of how to treat higher-order radiation effects and how to include excited states in a molecule are mentioned.
On the supplementary series of SO0(p, 1)
Authors: Boyer Charles P.
Published: February 2004

We construct multiplier representations of the unitary irreducible supplementary series of SO0(p,1) on the sphere Sp − 1 with the aid of a certain bilinear functional. The matrix elements of these representations are shown to be the analytic continuation of the matrix elements of the principal series. The decomposition of such representations with respect to the noncompact subgroup SO0(p − 1,1) is then performed by the analytic continuation of the ``overlap'' functions. Furthermore, the expansion of the relevant bilinear functional on the hyperboloid Hp − 1 is performed, and the connection is made with the previous decomposition.
The explicit determination of the linear boson transformation coefficients
Authors: Tanabe Kosai
Published: February 2004

The problem of determining the wavefunction is solved for the boson Bogoliubov transformation. The method of series expansion is applied to derive the general expression for the coefficients which connect the new Fock states of an arbitrary number of quasiparticles with the states before the transformation.
On the relationship between bifurcation points of the Kirkwood-Salsburg equation and phase transitions
Authors: Cheng Ing-Yih S. , Kozak John J.
Published: February 2004

In this paper we explore the Vlasov conjecture on the relationship between bifurcation points and phase transitions. Because of the availability of the exact results of Ruelle, we focus our attention on the Kirkwood-Salsburg hierarchy, and recognizing that the first equation in this hierarchy is of the form of a Lichtenstein-Lyapunov nonlinear operator equation, we use a fundamental theorem of Krasnosel'skii to determine, under a suitable closure, bifurcation points for the same system considered by Ruelle. A special example is treated—that of a one-dimensional system of hard rods— and our main conclusion follows from the results of this study: namely, that ``in this one-dimensional system'' the bifurcation point does not seem to be related to the onset of a phase transition.
Generation of stationary Einstein-Maxwell fields
Authors: Kinnersley William
Published: February 2004

The Einstein-Maxwell field equations in the presence of one Killing vector are shown to possess covariance under an eight-parameter group of linear substitutions in the field variables. This internal symmetry group is isomorphic to SU(2,1). Three of the degrees of freedom correspond to gage transformations, but the remaining ones allow us to generate a five-parameter family of solutions given a single solution.
Canonical formalism for degenerate Lagrangians
Authors: Shanmugadhasan S.
Published: February 2004

A Lagrangian is degenerate when the Hessian matrix whose elements consist of all the second-order derivatives of the Lagrangian with respect to the generalized velocities has (for simplicity) a constant singular rank everywhere in the space of the arguments of the Lagrangian. This singularity entails a definite number of first-order Lagrange equations, which act as subsidiary conditions on the coordinates and velocities. Consistency of these subsidiary conditions with the Langrange system requires them to be an invariant system with respect to the second-order Lagrange equations. An invariant system is analogous to a system of first integrals except that absolute constants appear where arbitrary constants characterize first integrals. The usual definitions of momenta and of Hamiltonian make the Hamiltonian a function of the functionally dependent canonical variables only. Introduction of the momentum variables into the subsidiary conditions on the coordinates and velocities yields under certain circumstances additional subsidiary conditions on the canonical variables only. All the subsidiary conditions on the canonical variables are determined before setting up the multiplier rule for the canonical equations of motion. The multiplier rule is exploited to deduce the invariant system among the subsidiary conditions, the explicit modifications of the canonical equations by the other susidiary conditions, and Dirac's formula for the corresponding modified Poisson brackets. The modifications are caused by the reduction in the number of independent canonically conjugate pairs. A canonical transformation adapted to the subsidiary conditions, which is found with the help of Lie's theory of function groups, transforms the canonical system from the multiplier rule into a canonical system in terms of physical variables. The invariant system is then used to reduce the order of the resulting canonical system by following Levi-Civita. This reduced canonical system is suitable for integration or quantization.
Canonical transformations and accidental degeneracy. II. The isotropic oscillator in a sector
Authors: Louck J. D. , Moshinsky M. , Wolf K. B.
Published: February 2004

In this paper we discuss the accidental degeneracy in the problem of a particle in two dimensional oscillator potential constrained to move in a sector of angle π/q,q integer. The degeneracy is due to both the Hamiltonian and the boundary conditions. The symmetry Lie group of canonical transformations is suggested by the explicit form of a complete nonorthonormal set of states expressed in terms of the creation operators. This group is complex and the corresponding representation in quantum mechanics is nonunitary. We discuss briefly the appearance of complex canonical transformations in physical problems.
Propagation of frequency-modulated pulses in a randomly stratified plasma
Authors: Besieris Ioannis M. , Tappert Frederick D.
Published: February 2004

Quantitative predictions are made of the effects of randomness and dispersion on the spreading, attenuation, and modulation coefficient of a frequency-modulated wave packet propagating in a randomly stratified isotropic plasma. In the absence of random fluctuations, the recently reported findings of Millman and Bell are recovered.
Scattering into cones. II. n-body problems
Authors: Dollard John D.
Published: February 2004

An n-body scattering process is studied in the framework of nonrelativistic time-dependent scattering theory. The probability is calculated that the n bodies emerge in cones C1,…,Cn with apices at the origin of coordinates. Results are given for short-range and for Coulomb potentials. The results corroborate the usual interpretation of time-dependent scattering theory.
Maximal foliations of extended Schwarzschild space
Authors: Reinhart Bruce L.
Published: February 2004

Smooth families of spherically symmetric maximal surfaces which are spacelike except at r = 2m are explicitly constructed in Schwarzschild space. Such surfaces should be useful in the study of initial value problems.
Theory of nonlocal electromagnetic elastic solids
Authors: Eringen A. Cemal
Published: February 2004

A continuum theory of nonlocal electromagnetic elastic solids is proposed. Nonlocal and local balance laws and jump conditions are obtained. Through the use of an extension of Clausius-Duhem inequality, encompassing nonlocal effects, E-M momentum and constitutive equations are derived and restricted.
C
∞-perturbations of a cosmological model
Authors: White Paul C.
Published: February 2004

It is shown that the so-called ``moment condition'' imposed by R. K. Sachs and A. M. Wolfe in their general treatment of the pertrubation problem for k = 0 Robertson-Walker universes both exclude physically reasonable solutions and are unnecessary. This restriction is removed by establishing that the Sachs-Wolfe solution gives the most general C∞ solution provided the potentials are C∞.
A space-time calculus based on pairs of null directions
Authors: Geroch R. , Held A. , Penrose R.
Published: February 2004

A formalism is presented for the treatment of space-times, which is intermediate between a fully covariant approach and the spin-coefficient method of Newman and Penrose. With the present formalism, a pair of null directions only, rather than an entire null tetrad, is singled out at each point. The concept of a spin- and boost-weighted quantity is defined, the formalism operating entirely with such quantities. This entails the introduction of modified differentiation operators, one of which represents a natural extension of the definition of the operator ð which had been introduced earlier by Newman and Penrose. For suitable problems, the present formalism should lead to considerable simplifications over that achieved by the standard spin-coefficient method.
Isospinors
Authors: Maia M. D.
Published: February 2004

A spinor structure which is covariant under a representation of the groups of isometries in general relativity is defined. The relationship of isospinors with space-time objects is investigated and their use in describing elementary particles in general relativity is discussed.
Multiple scattering theory of radiative transfer in inhomogeneous atmospheres
Authors: Kanal Madhoo
Published: February 2004

In this paper we treat the multiple scattering theory of radiative transfer in plane-parallel inhomogeneous atmospheres. The treatment presented here may be adopted to model atmospheres characterized by any optical depth dependent coherent scattering phase function. For the purpose of illustration we consider the semi-infinite medium in which the absorption property of the atmosphere is characterized by an exponential function. The methodology employed here is the extension of the case treated previously by the author for homogeneous atmospheres.
Local observables in the Dirac theory
Authors: Hestenes David
Published: February 2004

By a new method, the Dirac electron theory is completely reexpressed as a set of conservation laws and constitutive relations for local observables, describing the local distribution and flow of mechanical quantities. The coupling of the electromagnetic field to the electron is shown to be determined by the definitions of the observables rather than by the Dirac equation. Planck's constant appears in the equations only in connection with the electron spin. The equations are most readily interpreted by assuming that the electron is a structureless point charge, the spin and magnetic moment arising from the dynamics of electron motion.
On discrete inverse scattering problems. II
Authors: Case K. M.
Published: February 2004

A discrete version of the inverse scattering problem of the Schrödinger equation with a potential is discussed. The approach is via the Marchenko equation. Interest is primarily pedagogical. All steps are elementary and relatively obvious. Passage to the continuous case as a limit is heuristically straightforward. An example shows ``how'' the formalism does produce the potential from scattering data.
Equivalent Lagrangians in generalized mechanics
Authors: Anderson D.
Published: February 2004

A correspondence is established between differential equations and equivalence classes of Lagrangians containing higher derivatives. This suggests that a disputed problem of quantization in generalized mechanics is due to an inconvenient choice of Lagrangians characterizing the variational problems. These general concepts are then applied to the set of Lagrangians bilinear in the field variable and its derivatives.
Scattering function of a two-component fully ionized gas
Authors: Cheng Chiang S.
Published: February 2004

The effects of collisions on the scattering function of a two-component fully ionized gas are discussed, using an improved Brownian motion type collision model. In the collision model, the particle numbers as well as the energy and the linear momentum are conserved; and the electron-ion collisions are taken into account. The dispersion relations are discussed in detail, and explicit calculations are carried out for the scattering function and the energy-loss function in the long wavelength approximation when the wavelength is greater than the mean free path and the Debye length. The damped plasma oscillations dominate the energy-loss function; while for the scattering function thermal diffusions and sound propagations dominate the low frequency region and contribute most to the total scattering cross section. The contribution from the damped plasma oscillations to the scattering becomes important only for high frequencies near the plasma oscillation frequency.
Multichannel stationary scattering theory in two-Hilbert space formulation
Authors: Prugovečki E.
Published: February 2004

Two-Hilbert space versions of the basic integral-representation formulas for wave and transition operators are derived within the framework of time-independent scattering theory. Multichannel scattering theory for state vectors as well as for statistical operators is presented in a two-Hilbert space formulation to which these formulas become applicable. This formulation stays valid when long-range interactions, for which renormalized wave operators exist, are present.
Calculation of reflection and transmission coefficients for a class of one-dimensional wave propagation problems in inhomogeneous media
Authors: Baños Alfredo
Published: February 2004

Making use of an extension of Langer's method, we have solved the one-dimensional Helmholtz equation w″+k02g(z)w(z)=0, k0 = 2π/λ0, in which λ0 is the free space wavelength and in which g(z) represents any one of the three characteristic symmetric profiles. The main objective of this paper is the computation of the reflection and transmission coefficients for such a family of profiles making use of one and the same method throughout. First, we introduce our modification of Langer's method that reduces the problem to the solution of a comparison differential equation of the parabolic cylinder type, which we choose to solve in terms of confluent hypergeometric functions. Then, we set up a solution w+(z) valid for 0 ≤ z which, as z → ∞, becomes the transmitted plane wave. At this juncture we apply the method of analytic continuation, as z is changed into − z, and we obtain a linear combination w−(z) = wr(z) + wi(z) valid for z ≤ 0 which, as z → − ∞, yields the superposition of the incident and reflected waves. The reflection and transmission coefficients then ensue readily upon division by the coefficient of exp(i k0z) in the incident wave. The body of the work is first carried out for symmetric profiles, which we then proceed to extend to nonsymmetric profiles. Because we are using Langer's method in lowest approximation, we examine in detail the conditions underlying the applicability of the method and compare our results with the exact results derived from symmetric Epstein profiles. Although this paper is concerned mainly with the solution of the reflection-transmission problems, we have in fact set up all the necessary mathematical machinery for the computation of the actual wavefunctions in the vicinity of the origin. We have undertaken to do so, for specific parameter values, in a companion paper to appear shortly.
Group structure of the relativistic equivalent oscillator
Authors: Chaffin Eugene F.
Published: February 2004

The Dirac Hamiltonian with an equivalent oscillator potential introduced by Swamy is discussed from the point of view of group theory. It is shown to possess an SO(4) × SU(2) invariance group and an SO(4,1) × SU(2) noninvariance group.
Quantization by Feynman's path integration method
Authors: Cheng Kuo-Shung
Published: February 2004

The uniqueness of the quantization of a quantum mechanical system by Feynman's path integration method is discussed.
Uniqueness of the time coordinate for certain singular space-times in general relativity
Authors: Goorjian Peter
Published: February 2004

Many singular space-times are described locally in geodesic normal coordinates with the singularity described by the time coordinate equal to zero. Under appropriate conditions the time coordinate in this description is uniquely determined. Furthermore, every timelike curve in the coordinate patch is inextendible and has the same limit point as exactly one time coordinate line. The proof of uniqueness of the time coordinate employs methods from the theory of first order partial differential equations in a new way.
Rigorous results for Ising ferromagnets of general spin with degeneracy or symmetric potentials
Authors: Suzuki Masuo
Published: February 2004

The circle theorem of Lee and Yang is proved for Ising ferromagnets of general spin with degeneracy or symmetric potentials φj(−Sj) = φj(Sj); i.e., all the zeros of the partition function Ξ = Tr exp (ΣKijSiSj + βλΣjφj(Sj) + hΣjSj) lie on the unit circle of the complex fugacity plane (z = e−h) for Kij ≥ 0, λ ≥ 0, and for ``nondecreasing functions'' φj(Sj); φj(S) ≥ φj(S − 1) ≥ … ≥ φj(1/2) [or φj(0)], including the proof by Griffiths for the usual Ising model of arbitrary spin. The analyticity of the limiting free energy of such a generalized Ising ferromagnet and the absence of a phase transition are thereby established for all (real) nonzero magnetic field. Griffiths-Kelly-Sherman inequalities on spin correlations and Baker's inequalities on critical exponents are discussed in connection with the above model and also in a more general case.
Static gravitational fields. II. Ricci rotation coefficients
Authors: Das A.
Published: February 2004

Investigations of static gravitational fields in paper I are continued further. Instead of the system of second order partial differential equations in I, the equivalent first order system involving the Ricci rotation coefficients is dealt with here. The algebraic and the differential dependences among the equations have been sorted out to prove that the system is determinate. Then an attempt towards classifying this system of partial differential equations has been made. Except for ``the potential equation'' which is obviously elliptic, the remaining system is hyperbolic for which the characteristic surfaces have been determined. To obtain some exact solutions the lead of Newman-Penrose is followed in constructing the complex linear combinations of the equations. The class of static universes where ``gravitational lines of force'' are geodesics have been found. In this class one subclass is transformable to the conformastat metric and the remaining one reduces to a new metric involving the gravitational field of an arbitrary number of parallel infinite plates clamped at infinity. The source at infinity corresponds to that of the Newtonian potential φ =(1/2)[(m + 1/2)(x1)2 −(1/2) {m + 1/2 − (m2 − 1/4)1/2} (x2)2 − (1/2) {m + 1/2 + (m2 − 1/4)1/2} (x3)2]. This metric belongs to the nondegenerate Type I of Petrov's classification scheme. Next the class of static universes with ``shearfree lines of force'' is obtained. Here too one subclass goes over to the conformastat metric and the remaining one reduces to a Weyl-type universe.
Constants of motion and degeneration in Hamiltonian systems
Authors: Onofri E. , Pauri M.
Published: February 2004

A rigorous formulation of the connection between nonergodicity (degeneration) of the motion of a Hamiltonian system and existence of global constants of motion (isolating integrals) is proposed. Necessary and sufficient conditions for the occurrence of a properly defined kind of complete degeneration are given. Finally, a wide-spread opinion is criticized about the mutual implications between complete degeneration and separability of the Hamilton-Jacobi equation in more than one coordinate systems.
On the noniterative solution of integral equations for scattering of electromagnetic waves
Authors: Kouri Donald J.
Published: February 2004

The scattering of electromagnetic waves is considered using the integral equation form of Maxwell's equations for the electric field. These equations are analogous to but not identical to the Lippmann-Schwinger equation in quantum mechanical scattering theory. A transformation procedure discovered by Drukarev (and independently by Sams and Kouri) is employed to obtain Volterra integral equations of the second kind for ``modified electric field functions.'' In deriving the Volterra equations, one is led by analogy with quantum mechanical scattering to define a ``Jost matrix'' associated with electromagnetic scattering. A simple quadrature procedure is suggested for obtaining numerical solutions to the Volterra integral equations.
Minimal extensions as generalized semidirect products
Authors: Herbut F. , Vujičić M. , Šijački Dj.
Published: February 2004

An extension of a two-element symmetry group Z2 (defined, e.g., by C, P, or T) by another symmetry group K is called a minimal extension (ME). From the general theory of group extensions it follows that a ME is determined by an element φ of K and an automorphism F in K which are related in a certain way. It is shown that every ME can be expressed as a generalized semidirect product (GSP): (K [circled tau] H)/K′0, where the homomorphism τ is defined by F, and K′0,H are cyclic groups of order m and 2m, respectively, m being the order of φ. The simplest GSP form of any ME is obtained depending on φ being outside or inside the center of K (in particular, φ may be equal to the unit element), and F being an inner or an outer automorphism. A complete classification of inequivalent ME's is presented, and its possible significance for magnetic space and point groups is indicated. The usefulness of the GSP form for finding the irreducible representations of a ME is pointed out.
Symmetry transformations of the classical Kepler problem
Authors: Rogers Harold H.
Published: February 2004

The finite, canonical symmetry transformations of the negative energy motions of the classical Kepler problem are constructed by solving the fundamental differential equations of the dynamical invariance group. The geometric interpretation of the transformations is discussed.
A new basis for the representations of the rotation group. Lamé and Heun polynomials
Authors: Patera J. , Winternitz P.
Published: February 2004

The representation theory of the rotation group O(3) is developed in a new basis, consisting of eigenfunctions of the operator E = −4(L12 + rL22), where 0 < r < 1 and Li are generators. This basis |Jλ〉 is shown to be a unique nonequivalent alternative to the canonical basis (eigenfunctions of L3). The functions |Jλ〉 are constructed as linear combinations of canonical basis functions and are shown to fall into four symmetry classes, distinguished by their behavior under reflections of the inidividual space axes. Algebraic equations for the eigenvalues λ of E are derived. When realized in terms of functions on an O(3) sphere, the basis |Jλ〉 consists of products of two Lamé polynomials, obtained by separating variables in the corresponding Laplace equation in elliptic coordinates. When realized in a space of functions of one complex variable, |Jλ〉 are Heun polynomials. Applications of the new basis in elementary particle, nuclear, and molecular physics are pointed out, due in particular to the symmetric form of |Jλ〉 as functions on a sphere and to the fact that they are the wavefunctions of an asymmetrical top.
Nonnegativity of the Yukawa Hamiltonian
Authors: Hemmer P. C.
Published: February 2004

The Yukawa Hamiltonian H = − Δ − r−1 e−μr is shown to be nonnegative for μ ≥ [1+6 log 2−(9/2)log3]1/3 ≈ (1.67)−1.
Covariance of the conserved current equations and transformations of Noether's current
Authors: Bayen F. , Hillion P.
Published: February 2004

The covariance of the conserved current equations is first proved in a direct way without using the transformation laws of Noether currents. Then the transformation of these currents is investigated and we find, when the action integral is invariant up to a divergence, a result which does not seem to have been noted previously.
Statistical mechanics of parafermi systems
Authors: Richard J. L. , Sudarshan E. C. G.
Published: February 2004

The statistical mechanics of a collection of parafermi harmonic oscillators considered as a canonical ensemble is examined. The factorization property of the corresponding partition function selects the representations of the parafermi ring allowed to furnish the energy levels of the oscillators. While the representations corresponding to Green's ansatz are in an obvious way perfect candidates, it is shown however that, contrary to expectations, they do not exhaust all the possibilities. The general conditions on the representations are given and special families of representations are outlined.
Stieltjes summability and convergence of the Padé approximants for the vacuum polarization by an external field
Authors: Graffi Sandro
Published: February 2004

It is proven that the divergent perturbation expansion of the exactly solvable vacuum polarization by an external constant electromagnetic field is Stieltjes summable. This implies the convergence of the Padé approximants to the solution. The convergence of the approximation method based on Borel summability is also proven.
Local products as operators
Authors: Ruskai Mary Beth , Klauder John R.
Published: February 2004

It is shown for a large class of domains that the bilinear forms associated with formal expressions of the type ∫h(x)asp †(x).…as1 †(x)at1(x)…atq(x)dx define closable operators if and only if p, qε{0, 1}.
On a certain class of transformations in statistical mechanics
Authors: Shen C. Y.
Published: February 2004

In statistical mechanics, the probability densities and the correlation functions are related by a pair of linear operators T, U, which are defined on a suitable subspace of the space of integrable functions and are inverse to each other. Our main result gives the conditions under which they commute with the members of another class F of transformations which includes the usual exp and log mappings. Furthermore, the compatibility property possessed by the probability densities of an infinite system and the extension property possessed by the corresponding correlation functions are preserved, in some instances, by the members of F.
On nonrelativistic field theory: Interpolating fields and long-range forces
Authors: Klein James R. , Zinnes Irving I.
Published: February 2004

A field-theoretic version of nonrelativistic, multichannel scattering theory is presented in which the particles interact via two-body potentials, which can include both long-range (Coulomb, etc.) and short-range potentials. By utilizing known first-quantized results, in and out fields and interpolating fields are constructed both for simple and composite particles, and an asymptotic condition using strong convergence is proved. Complexities present when long-range potentials are involved are discussed.
Related integral theorem. II. A method for obtaining quadratic constants of the motion for conservative dynamical systems admitting symmetries
Authors: Katzin Gerald H.
Published: February 2004

By use of Lie derivatives, symmetry mappings for conservative dynamical systems are formulated in terms of continuous groups of infinitesimal transformations within the configuration space. Such symmetry transformations, called trajectory collineations, may be interpreted as point mappings which drag along coordinates and geometric objects as they map trajectories into trajectories. It is shown that if a conservative dynamical system admits a trajectory collineation, then in general a new quadratic (in the velocity) constant of the motion will result from the deformation of a given quadratic constant of the motion under such a symmetry mapping. The theory is applied to obtain the group of symmetry transformations and concomitant constants of the motion associated with the deformations of the energy integral for the Kepler problem and for the isotropic simple harmonic oscillator. The Runge-Lenz vector of the Kepler problem and the symmetric tensor constant of the motion for the three-dimensional oscillator are readily obtained by this method. The trajectory collineation group for the Kepler problem is a seven-parameter projective collineation group which is isomorphic to the similitude group in three-dimensional Euclidean space. For the oscillator the trajectory collineation group is the nine-parameter affine group. This dynamical symmetry group contains an eight-parameter subgroup which is shown to have the structure of SU3.
Charges as generators of unitary symmetry groups
Authors: Snellman H.
Published: February 2004

We show, that in a local field theory of the Wightman type, with an irreducible set of finitely many, relatively local field operators φk, 1≤k≤n, and a set of local, locally conserved symmetric current operators Jiu(x), the charge operators Gi that are associated with these currents are essentially self-adjoint on the domain of strictly localized states, provided the Gi:s satisfy [Gi,φk(g)]=Σl=1nTkliφl(g), g∈D(R4), where the Ti:s are finite dimensional matrices. Furthermore when the Ti:s close to a faithful representation of a Lie algebra g, then g is finite-dimensional and compact and its representation can be integrated to a unitary representation of corresponding Lie group.
Canonical transformations applied to the free Landau electron
Authors: Boon M. H. , Seligman T. H.
Published: February 2004

It is shown that the problem of a free eletron in a uniform magnetic field provides an interesting example for the use of recently developed canonical transformation methods in quantum mechanics. The Schrödinger equation is immediately solved by a unitary transformation corresponding to a finite linear symplectic transformation in phase space. The natural invariance and dynamical groups involve affine and not only linear symplectic transformations, and the Weyl group turns out to be an appropriate invariance group to account completely for the infinite degeneracies.
The Clebsch-Gordan coefficients of SU(3) and the orthogonalization problem
Authors: Majumdar S. Datta , Basu Biplab Kumar
Published: February 2004

The Clebsch-Gordan coefficients of the group SU(3) are determined by integrating the product of three matrix elements of finite transformations belonging to three irreducible representations of the group. Compact expressions involving a single or a double sum over products of 3-j and 6-j symbols of SU(2) are obtained for several different classes of coefficients by suitably restricting the initial states but keeping the final states of the matrix elements arbitrary. To orthogonalize the CG coefficients, a linear combination of several integrals with the same final but different initial states is taken. The coefficients of the linear combination are determined by the Schmidt procedure and are found to be expressible in terms of integrals of the same type.
The decay of the continuous spectrum for solutions of the Korteweg-deVries equation
Authors: Ablowitz Mark J. , Newell Alan C.
Published: February 2004

The asymptotic behavior of the solution u(x, t) of the Korteweg-deVries equation ut + uux + uxxx = 0 is investigated for the class of problems where the initial data does not give rise to an associated discrete spectrum. It is shown that the behavior is different in the three regions (i) x≫t1/3, (ii) x = 0(t1/3), (iii) x≪− t1/3. Asymptotic solutions in each of these regions are found which match at their respective boundaries. One of the Berezin-Karpman similarity solutions is the asymptotic state in (ii) and u(x, t) decays like 1/t2/3 in this region. For region (i) there is exponential decay, whereas in region (iii) the structure of u(x, t) is highly oscillatory and the amplitudes of the oscillations decay as 1/t1/2 when − x/t is of order unity. For x/t large and negative these amplitudes decay at least as 1/(− x/t)1/4t1/2.
A geometric generalization of Hooke's law
Authors: Glass Edward N. , Winicour Jeffrey
Published: February 2004

A geometrical version of Hooke's law for prestressed materials is established using techniques suggested by general relativistic elasticity theory. The concept of dragged (D) tensor fields is used to present a simple coordinate independent form of Hook's law for prestressed materials: P̄ij = D Pij + Cijkl ηkl, where Pij is the initial stress, Cijkl are the second order elastic coefficients, and ηkl is the strain. The effects of initial stresses on Hooke's law have recently been given empirical significance by Wallace and the work here agrees with and simplifies his formalism. The elastic behavior of neutron stars, which requires a general relativistic treatment, is a case in which large prestresses of the type considered here are as important as the elastic modulii themselves. The version of Hooke's law developed here is the classical limit of the general relativistic theory of stellar elasticity. A simple derivation of the seismic response of a Newtonian self-gravitating body to elastic perturbations is presented.
Large coupling expansions for eigenenergies and Regge trajectories of the Yukawa potential
Authors: Müller-Kirsten H.J.W. , Vahedi-Faridi N.
Published: February 2004

Simple and straightforward procedures are developed for deriving complete large coupling expansions for the wavefunctions, eigenenergies, and Regge trajectories of the generalized Yukawa potential. The procedures may easily be applied to similar problems.
Renormalization and Ward identities using complex space-time dimension
Authors: Speer Eugene R.
Published: February 2004

Complex-dimensional renormalization is defined for an arbitrary Feynman amplitude and shown to be equivalent to BPH renormalization. Using quantum electrodynamics as an example, Ward identities are proved; here Carlson's theorem extends the identities from integer to complex dimension. Both complex dimensional and analytic regularization are necessary at intermediate stages.
Scattering of scalar waves from a random irregular interface
Authors: Zipfel G. G.
Published: February 2004

The problem of a scalar wave scattering from a random interface between two isotropic homogeneous media is solved in terms of a diagrammatic series expansion. It is assumed that the interface is single valued and that it has multivariate Gaussian statistics. Partial summation of the series in terms of linear integral equations is briefly discussed.
Extra selection rules
Authors: Backhouse N. B.
Published: February 2004

The recognition of antiunitary symmetry in a quantum mechanical system is considered in relation to selection rules. It is shown that the interaction of unitary symmetry and the anti-invariance of matrix elements under antiunitary operators can produce extra selection rules in magnetic systems. The criteria obtained for the occurrence of these additional selection rules are presented in the form of group character tests, and in a special case provide an application for the symmetrized and antisymmetrized squares of group representations.
On the exponential superpropagator
Authors: Bollini C. G. , Giambiagi J. J.
Published: February 2004

Series developments for distributions of the type Raexp( − κ2/R) (with R = r2 − t2) are discussed in the light of the Gel'fand-Shilov method for the definition of causal distributions. It is shown that there is no arbitrariness in the series. In Sec. I the concept of causal distribution is reviewed, in Sec. II the series developments are given, fixing the values of the constants in the □nδ terms. In Sec. III the Fourier transforms are computed. In Sec. IV we obtain the series development for the Fourier transform of exp( − κ2/R)∂μ ∂ν(R−1).
One-dimensional model of the rearrangement process and the Faddeev equations
Authors: Lipszyc K.
Published: February 2004

In the paper we study the Faddeev-Lovelace equations for the one-dimensional system of three particles interacting via zero-range potentials. The half-off-shell rearrangement and elastic amplitudes are constructed explicitly for the particular case of the model, namely for the system consisting of three particles with equal masses and with two-body potentials of equal strength. These exact amplitudes are shown to satisfy the modified form of the Faddeev-Lovelace equations.
A comment on the paper ``Lower Bounds on the Energy Eigenvalues of Systems Containing Identical Particles''
Authors: Calogero F. , Marchioro C.
Published: February 2004

It is pointed out that the extension by Yeh [J. Math. Phys. 13, 227 (1972)] of the result by Calogero and Marchioro [J. Math. Phys. 10, 562 (1969)] is invalid. Counterexamples are indicated; the fault hidden in the proof of the extended result is spotted.
Debye potentials in Riemannian spaces
Authors: Stephani H.
Published: February 2004

By means of Debye potentials it is possible to get all solutions of source-free Maxwell equations in vacuum from a single scalar equation. Sufficient conditions for the existence of Debye potentials in a given four-dimensional Riemannian space have been found. Some examples of metrics are given, including plane gravitational waves, metrics with spherical symmetry, and cosmological models. The method is generalized to Maxwell fields with sources and Maxwell fields in dielectric media.
Scattering of photons by an external current
Authors: Reents G.
Published: February 2004

Considering the interaction of a quantized electromagnetic field with a prescribed external c-number current, we show that, for each current, there is a representation of the field operators in a separable Hilbert space and a unitary S operator transforming the free incoming field into the free outgoing one. The representation, a generalized product representation, depends on the current so that, in general, for two different currents the corresponding representations are unitarily inequivalent. Photon states and asymptotic observables are defined for the case of an infrared divergent current.
Cluster coefficients for the one-dimensional Bose gas with point interactions
Authors: Dodd L. R. , Gibbs A. M.
Published: February 2004

Exact expressions for the cluster coefficients b2 and b3 for the one-dimensional Bose gas with repulsive δ-function interactions are found. The calculation depends only on scattering information in the form of two- and three-body scattering wavefunctions. Although not in agreement with a previous calculation by Servadio, our results are shown to be consistent with the work of Yang and Yang which involves the use of periodic rather than scattering boundary conditions.
Exact calculation of the energy and heat capacity for the triangular lattice with three different coupling constants
Authors: Cunningham Gordon W. , Meijer Paul H. E.
Published: February 2004

Calculation of the internal energy and heat capacity of the general anisotropic triangular Ising lattice is derived from the double integral form of the partition function. The principal result is the reduction of the elliptic integrals to a standard form for three arbitrary coupling constants. Both the standard form and the method of reduction are due to Legendre. The method of reduction is one involving two linear transformations. A straightforward reduction of elliptic integrals to standard form could not be used in this application. This is because of the functional dependence of the two linear transformations on the many combinations and permutations of the signs and relative magnitudes of the coupling energies of the lattice. A relatively simple formulation is presented in which the many combinations and permutations previously mentioned are reduced to only two distinct cases. An independent numerical solution was calculated directly from the partition function as a means of verifying the formulation presented in this paper.
Asymptotic solution of neutron transport problems for small mean free paths
Authors: Larsen Edward W. , Keller Joseph B.
Published: February 2004

A method is presented for solving initial and boundary value problems for the energy dependent and one speed neutron transport equations. It consists in constructing an asymptotic expansion of the neutron density ψ(r, v, τ) with respect to a small parameter ε, which is the ratio of a typical mean free path of a neutron to a typical dimension of the domain under consideration. The density ψ is expressed as the sum of an interior part ψi, a boundary layer part ψb, and an initial layer part ψ0. Then ψi is sought as a power series in ε, while ψb decays exponentially with distance from a boundary or interface at a rate proportional to ε−1. Similarly ψ0 decays at a rate proportional to ε−1 with time after the initial time. For a near critical reactor, the leading term in ψi is determined by a diffusion equation. The leading term in ψb is determined by a half-space problem with a plane boundary. The initial and boundary conditions for the diffusion equation are obtained by requiring ψ0 and ψb to decay away from the initial instant and from the boundary, respectively. The results are illustrated by specializing them to the one speed case. The method may make it possible to treat more realistic and more complex problems than can be handled by other methods.
Dimensional regularization for zero-mass particles in quantum field theory
Authors: Capper D. M. , Leibbrandt G.
Published: February 2004

We show by a suitable redefinition of momentum-space integrals that the technique of dimensional regularization can be extended consistently to include nonnormally ordered theories describing zero-mass particles.
On a conjecture by 't Hooft and Veltman
Authors: Capper D. M. , Leibbrandt G.
Published: February 2004

The conjecture by 't Hooft and Veltman that ∫ d2ωq (q2) λ − 1 = 0 (ω, λ complex), within the context of dimensional regularization, is proven in the case of zero-mass particles for any ω when λ = 1,2,… and for λ = 0 in the limit ω → 2.
The mathematical theory of the R matrix. II. The R matrix and its properties
Authors: Narcowich Francis J.
Published: February 2004

In this paper, it is shown that Wigner's R matrix, for a certain class of unbounded potentials which may be nonlocal or have Coulomb-type singularities, exists, is a compact operator, and that the expansions associated with the R-matrix converge. For the same class of potentials, a perturbation theory is constructed and conditions are given for the convergence of the resulting Born-type expansions.
The Clebsch-Gordan problem and coefficients for the three-dimensional Lorentz group in a continuous basis. III
Authors: Mukunda N. , Radhakrishnan B.
Published: February 2004

Along the lines of two previous papers, the Clebsch-Gordan problem for products of representations of SU(1, 1) of the form D+⊗ C is related to the properties of the Lorentz group O(3, 1). The structure of the Clebsch-Gordan series for this case is understood in a new way as being due to the properties of O(3, 1) spherical harmonics on the timelike and spacelike hyperboloids in Minkowski space. The Clebsch-Gordan coefficients in a continuous basis are then evaluated.
Orbits of the rotation group on spin states
Authors: Bacry H.
Published: February 2004

A simple theorem on projective spaces generalizes the concept of the Riemann sphere. This leads us to a generalized interpretation of the ray space associated with a finite-dimensional Hilbert space. An application is given about the way the rotation group acts on states of given spin j.