bluuebunny/crossref_metadata_2025 huggingface dataset:
Models of Zermelo Frankel set theory as carriers for the mathematics of physics. II
Authors: Benioff Paul A.
Published: March 2003

This paper continues the study of the use of different models of ZF set theory as carriers for the mathematics of quantum mechanics. The basic tool used here is the construction of Cohen extensions of ZFC models by use of Boolean valued ZFC models [C=axiom of choice]. Let M be a standard transitive ZFC model. Inside M, B (ℋM) is the algebra of all bounded linear operators over some Hilbert space ℋM. It is shown that with each state ρ in B (ℋM) and projection operator ο in B (ℋM) one can associate a unique Boolean valued ZFC model Mℬρο. ℬρο is the algebra of all Borel subsets of {0,1}ω, the set of all infinite 0–1 sequences, modulo sets of Pρο =⊗pρο measure zero with pρο({1}) =Trρο in M. Let ΨM and ΦM be respective maps from the sets of state preparation and question measuring procedures into B (ℋM). Let M=M0, the minimal standard transitive ZFC model. It is then shown that with each state preparation procedure s - Dom(ΨM0) and each question measuring procedure q - Dom(ΦM0) and with each infinite repetition (tsq) of doing s and q at times t (0), t (1),..., if the definition of randomness is sufficiently strong, one can associate the Cohen extension M0[ψtsq] of M0 by ψtsq. ψtsq is the random outcome sequence associated with (tsq). A third condition, in addition to the two given in the previous paper, is then given which must be satisfied if a ZFC model M is to serve as a carrier for the mathematics of quantum mechanics. In essence it says that for each pair (tsq) and (wuk) of distinct infinite repetitions of doing s and q and of doing u and k with s, u - Dom(ΨM) and q, k - Dom(ΦM), the two outcome sequences ψtsq and ψwuk are mutually statistically independent. It is then shown that for a strong definition of independence, corresponding to the definition of randomness used previously, no Cohen extension M0[ψtsq] of M0 can serve as the carrier for the mathematics of quantum mechanics.
Conformal transformation for the Coulter–Weinberg form of the equations for mass zero spin-2 field
Authors: Galles Carlos D. , Zandron Oscar S.
Published: March 2003

In this work we show that it is impossible to introduce a third-rank tensor potential that preserves the conformal covariance of the mass zero spin-2 field equations in the Coulter–Weinberg scheme.
Charge quantization and canonical quantization
Authors: Miller J. G.
Published: March 2003

Dirac’s charge quantization condition is derived by means of a canonical quantization procedure of an enlarged classical phase space in which the interaction constant is a dynamical variable. The charge quantization condition follows by imposing a superselection rule. The method avoids string singularities and does not depend on spherical symmetry. The charge quantization condition is due solely to the topology of the enlarged classical configuration space.
Conditioning of states
Authors: Deliyannis Platon C.
Published: March 2003

A system of axioms for the state space of a quantum system is proposed which, together with the concept of conditioning a state by the occurrence of an event, leads to the construction of the standard orthomodular events system.
A class of states on the boson–fermion algebra
Authors: Rocca F. , Sisson P. N. M. , Verbeure A.
Published: March 2003

On the tensor product C*-algebra of bosons and fermions a class of states determined by the two-point functions is proved to exist. It is indicated how they induce a class of states on the CCR algebra which are not quasifree, but determined by their two- and four-point functions.
U(5) ⊃ O(5) ⊃ O(3) and the exact solution for the problem of quadrupole vibrations of the nucleus
Authors: Chacón E. , Moshinsky M. , Sharp R. T.
Published: March 2003

Over twenty years ago A. Bohr discussed the quantum mechanical problem of the quadrupole vibrations in the liquid drop model of the nucleus. States of definite angular momentum L could not be obtained exactly except when L=0,3. In the present paper we indicate how we can determine states for arbitrary angular momentum L and definite number of quanta ν in terms of polynomials of the creation operators characterized by irreducible representation (IR) of the chain of groups U(5)⊃O(3). We furthermore characterize the states by a definite IR λ of O(5) by replacing the creation operators by traceless ones. These states are fully determined by an extra label μ that gives the number of triplets of traceless creation operators coupled to angular momentum zero. We show then how all the wavefunctions of the problem discussed by Bohr can be obtained in a recursive fashion and briefly discuss some of their applications.
Asymptotic solutions and conservation laws for the nonlinear Schrödinger equation. I
Authors: Segur Harvey , Ablowitz Mark J.
Published: March 2003

In the absence of solitons, the nonlinear Schrödinger equation has an asymptotic solution which decays in time as t−1/2, and contains two arbitrary functions (in the amplitude and phase, respectively). For appropriate initial data, the amplitude function is uniquely determined in terms of the initial data by the conservation laws; the other function is undetermined. This method determines the leading two terms in each of the asymptotic expansions for the amplitude and phase, but no more. The method makes no direct use of th Marchenko integral equations.
Asymptotic solutions and conservation laws for the nonlinear Schrödinger equation. II
Authors: Segur Harvey
Published: March 2003

We find the dominant asymptotic behavior of the solution of the nonlinear Schrödinger equation when there is one soliton and decaying oscillations. The solution behaves like the soliton near the soliton, and like the solution found in the preceding paper (I) elsewhere. The method of solution uses the conservation laws, rather than the integral equations.
Quantum numbers for particles in de Sitter space
Authors: Patera J. , Winternitz P. , Zassenhaus H.
Published: March 2003

All subalgebras of the Lie algebra of the de Sitter group O(4,1) are classified with respect to conjugacy under the group itself. The maximal continuous subgroups are shown to be O(4), O(3,1), D⧠E(3) (the Euclidean group extended by dilatations), and O(2) ⊗O(2,1). Representatives of each conjugacy class are shown in the figures, also demonstrating all mutual inclusions. For each subalgebra we either derive all invariants (both polynomial and nonpolynomial ones) or prove that they have none. The mathematical results are used to discuss different possible sets of quantum numbers, characterizing elementary particle states in de Sitter space (or the states of any physical system, described by this de Sitter group).
Geometry of hyperspace. I
Authors: Kucha? Karel
Published: March 2003

Hyperspace is heuristically defined as an (infinitely dimensional) manifold of all spacelike hypersurfaces embedded in a given Riemannian spacetime. The Riemannian structure (ℳ,g) of spacetime induces a rich geometrical structure in hyperspace. Part of that structure, especially the moving normal frames in hyperspace, Lie derivatives, and symmetrical ∇ and asymmetrical ∇* covariant hyperderivatives, are studied in detail. The formalism introduced in this paper sets the stage for the geometrical study of hypersurface kinematics and dynamics of general tensor fields with derivative gravitational coupling, and of the Dirac–ADM geometrodynamics with such tensor sources, in the following papers.
A characteristic glimpse of the renormalization group
Authors: Klauder John R.
Published: March 2003

Characteristic functions for arbitrary physical systems are discussed in general terms with special emphasis on the space of allowed test variables for which the characteristic function is defined. A fundamental change in the probability measure (at a phase boundary, for example) is reflected in a change of allowed test variables, generally, some new ones added and some old ones dropped. Physically motivated transformations on the space of test variables (implicitly introducing block spins, for example) are designed, after repeated operation, to accentuate or even to isolate the fundamental changes. Several examples of such renormalization-group type transformations are given.
Bäcklund transformations and the equation z
x
y=F (x,y,z)
Authors: Byrnes S. G.
Published: March 2003

It will be shown that the only equations of the form zxy=F (x,y,z) which posses Bäcklund transformations to take one solution of this equation into another solution of the same equatin are either linear or else can be obtained from the sin(h) -Gordon equation (or zxy=ez) by a simple change of scale and/or a displacement of the dependent variable.
The smooth-path topology for curved space–time which incorporates the conformal structure and analytic Feynman tracks
Authors: Göbel Rüdiger
Published: March 2003

It is obvious that the usual (Riemannian) topology on a space–time has no natural relation with the (pseudo-Riemannian) metric of general (or special) relativity. Therefore, several new topologies on a space–time were proposed in recent years in order to overcome this ’’classical’’ disharmony. In this paper an infinite variety of more physical topologies is investigated (including the most interesting known topologies). It turns out that one of the candidates ’’𝒫ω*’’ has all desired physical advantages: (1) It is defined in a very physical way by tests with particles of mass ≳0. (2) It carries intrinsically all information about the smooth conformal structure of space–time: Its homeomorphism group is the conformal group. (3) The (only) 1–1 continuous curves are the analytic Feynman tracks. It turns out that this topology 𝒫ω* is strictly finer than the topology suggested by S. W. Hawking, A. R. King, and P. J. McCarthy. During this investigation a conjecture of E. C. Zeeman (1967; for Minkowski space) will be proved for all strongly causal space–times (including Minkowski space).
High frequency approximations for elastic surface waves propagating along cylinders of general cross section
Authors: Morrison J. A.
Published: March 2003

The propagation of high frequency elastic surface waves along the generators of a homogeneous isotropic cylinder of general cross section is considered. The boundary surface is stress-free and the surface waves, or Rayleigh waves, are disturbances whose amplitudes decay rapidly with depth into the cylinder. An approximate equation, and a refined one, are derived which describe the high frequency behavior of the surface wave modes. These approximate equations lead to the asymptotic results derived earlier by Wilson and the author for the case of an open boundary curve for which the curvature attains its algebraic maximum at a single point, and in fact they permit a more complete analysis of the higher order modes. Moreover, the refined approximate equation describes the behavior of the surface wave modes in the transition region, at high frequencies, between the case of cross-sectional boundary curves of nonconstant (and not ’’almost’’ constant) curvature, for which the modes are localized, and the case of constant curvature, for which they are not localized. Some particular examples are considered.
Derivation of an exact quantum-mechanical formula for the second term in the activity expansion of the correlation function S (Q,ω)
Authors: Johnson R. E. , Pope N. K.
Published: March 2003

We derive an exact expression for the second term in the activity expansion of S (Q,ω), the Fourier transform of Van Hove’s correlation function, for a quantum system of bosons or fermions (spin neglected). The results for both the self and distinct parts of this function are given in terms of standard two-particle T matrices. The detailed-balance condition and zeroth and first moment relations are confirmed. An expression for the second term in the activity expansion of S (Q), the structure factor, is obtained. The derivation follows directly from the definition of S (Q,ω) using formal operator techniques; in addition, we show that the conserving T approximation for two-particle Green’s functions yields the same result.
A class of conservative diffusion processes with delta function initial conditions
Authors: Lehnigk Siegfried H.
Published: March 2003

A class of conservative one-dimensional diffusion processes is discussed which satisfy a delta function initial condition. A similarity relationship exists between the space and time variables. This class contains as special cases three well-known diffusion processes and another one that very recently became of interest in the theory of superradiant emission.
Coulombian asymptotic states
Authors: van Haeringen H.
Published: March 2003

We define and study Coulombian asymptotic states, denoted by ‖k∞〉, which are useful and important for stationary Coulombian scattering theory. Using these asymptotic states, we are able to derive from the off-shell T matrix the physical on-shell T matrix for Coulomb plus short-range potentials. This function is proportional to the physical scattering amplitude. A generalized Lippmann–Schwinger equation and related equations are derived. Many other applications of the asymptotic states exist. We show the precise connection of ‖k∞〉 with the time-dependent Coulomb scattering theory.
Uniform bounds of the pressures of the λφ43 field model
Authors: Park Yong Moon
Published: March 2003

We show that the pressures of the λφ43 field model with free and with periodic boundary conditions are bounded uniformly in space cutoff.
Almost singular potentials
Authors: Schwartz Charles
Published: March 2003

Explicit solutions are constructed for the lowest bound states of the Schrödinger equation with an attractive potential that behaves typically as r−2+ε at the origin. The energy levels and wavefunctions, which depend on the small parameter ε in a nonanalytic way, show some interesting properties; and some relations between this model and aspects of elementary particle physics are noted.
The Riemann solution and the inverse quantum mechanical problem
Authors: Coz Marcel , Coudray Christiane
Published: March 2003

Systematic use is made of the Riemann function to find the conditions for the existence of the kernel of the inverse problem at fixed value of the angular momentum. When the reference potential is the centrifugal one, only an l-dependent condition on the potential must be required in the Marchenko case; in the Gel’fand–Levitan case, the condition is l-independent. When the Coulomb potential is included in the reference potential, an exponential decrease of the potential is needed in both instances.
Partial differential matrix equations for the inverse problem of scattering theory
Authors: Coz M. , Rochus P.
Published: March 2003

Sufficient conditions for the existence of a continuous translation operator are found in the case of a system of differential equations in which the matrix potential has the singularity of the centripetal term. The sufficient conditions are found in terms of moments of the nuclear potential. The method used employs the Riemann Green’s function. Threshold energies introduce a threshold energy dependence into the translation kernel and lead to a requirement of an exponential decrease for terms of the matrix potential.
A characterization of the Einstein tensor in terms of spinors
Authors: Anderson Ian M. , Lovelock David
Published: March 2003

All tensors of contravariant rank two which are divergence-free on one index, concomitants of a spinor field σiAX′ together with its first two partial derivatives, and scalars under spin transformations are constructed. The Einstein and metric tensors are the only candidates.
Nonlinear differential–difference equations and Fourier analysis
Authors: Ablowitz M. J. , Ladik J. F.
Published: March 2003

The conceptual analogy between Fourier analysis and the exact solution to a class of nonlinear differential–difference equations is discussed in detail. We find that the dispersion relation of the associated linearized equation is prominent in developing a systematic procedure for isolating and solving the equation. As examples, a number of new equations are presented. The method of solution makes use of the techniques of inverse scattering. Soliton solutions and conserved quantities are worked out.
Bivariational bounds in a complex Hilbert space, and correction terms for Padé approximants
Authors: Barnsley M. F. , Baker George A.
Published: March 2003

Let A be a closed linear transformation from a complex Hilbert space б into itself, with D (A), the domain of the operator, dense in б. Suppose that A satisfies 〈Aφ,Aφ〉⩾a2〈φ,φ〉 for all φ - D (A), where a is a positive constant. Then for the case where the equation Aφ=f, with f ∈б given, possesses a solution φ∈ D (A), bivariational functionals are presented which yield upper and lower bounds for both the real and imaginary parts of 〈g,φ〉 for any given g∈б. The case A= (1+zK) with z a complex parameter is then considered. With appropriate choices for the trial functions involved, the functionals yield the [N+J−1/N−J] Padé approximants (J=0, 1,⋅⋅⋅,N; N=1,2,⋅⋅⋅,) to F (z) =〈g,φ〉 together with correction terms such that bounds on F (z) ensue. Finally, an arbitrary analytic function G (z), regular within a closed contour C and continuous within and on C, is considered. The above theory leads to bounds on G (z) in the form of [N+J−1/N−J] Padé approximants together with correction terms. The results are then generalized for arbitrary [L/M] Padé approximants to G (z) with the aid of the Hermite error formula.
Quantization as deformation theory
Authors: Bloore F. J. , Assimakopoulos M. , Ghobrial I. R.
Published: March 2003

We show that any rule for quantization of functions on the phase space of a dynamical system is equivalent to a deformation of the Poisson bracket, in the sense of Lie algebras. All Schrödinger-type quantizations are equivalent mathematically but not physically. We examine whether a quantization rule exists in which the Poisson bracket of the free Hamiltonian and any function not more than quadratic in momentum gets quantized into the commutator of the quantized versions of these two observables. We show that if the (Riemannian) configuration manifold is irreducible, this condition uniquely specifies the quantization of these functions, in the cases when it can be satisfied. However, it can only be satisfied for systems whose configuration manifolds are one-dimensional, or have vanishing Ricci tensor, or are spaces of constant curvature. Thus the condition will not serve as a general physical principle which could be invoked to fix a quantization rule.
Some new ensembles of random matrices
Authors: McDonald J. F.
Published: March 2003

In this article, we introduce ensembles of random matrices in which not all of the matrix elements are functionally independent. The functional dependence is defined by restricting the class of unitary similarity transformations which will diagonalize a member of the ensemble. The statistical properties of the eigenvalues are then studied for some of these ensembles.
Wave operators for multichannel scattering by long-range potentials
Authors: Zachary W. W.
Published: March 2003

The methods of Alsholm in single channel scattering are developed so as to apply to multichannel scattering. For a large class of long-range potentials we prove the existence of partially isometric wave operators, intertwining relations, and the orthogonality of the ranges of these wave operators for distinct channels.
Excitation of horizontally polarized waves in critical-coupling regions where the permittivity gradient approaches zero—Full wave solutions
Authors: Bahar E.
Published: March 2003

Maxwell’s equations for inhomogeneous isotropic media are transformed into coupled ordinary differential equations for the wave amplitudes. To facilitate the solutions of these equations for dielectric layers with critical coupling regions, a generalized WKB approach is used. To this end, sets of auxiliary functions that are solutions to the wave equation for homogeneous or linearly varying permittivity profiles are employed. By introducing an additional set of auxiliary, parabolic cylindrical functions, the generalized WKB approach is extended to obtain suitable solutions for critical coupling regions of the dielectric layer where the gradient of the permittivity profile also approaches zero. Expressions for the reflection and transmission coefficients and the characteristic surface impedance for an inhomogeneous dielectric layer are derived as functions of the transverse wavenumber. Realizability and reciprocity relationships are also derived.
Self-gravitating fluids with cylindrical symmetry
Authors: Ray Dipankar
Published: March 2003

In a recent paper [P. S. Letelier, J. Math. Phys. 16, 1488 (1975)], the problem of self-gravitating fluid with cylindrical symmetry with p = ρc2 has been reduced to a single equation. The present note solves the equation and also rectifies an oversight in the above-mentioned paper.
Fundamental canonical realizations of connected Lie groups
Authors: Martinez Alonso Luis
Published: March 2003

An important class of transitive canonical realizations of connected Lie groups is studied by means of a general formalism. We give a simple method for the classification and the construction of these realizations.
Approximate form of the Tomimatsu–Sato δ=2 solution near the poles x=1, y=±1
Authors: Economou John E.
Published: March 2003

A consideration of the line element for the Tomimatsu–Sato δ=2 solution near the poles x=1, y=±1 reveals the existence of a metric which in nonsingular there. The further study of this metric indicates that it corresponds to a vacuum, type D gravitational field, and, as such, it is among those type D vacuum solutions specified by W. Kinnersley. Reasons are given in support of the belief that the derived metric is a valid approximation to the exact T–S solution close to the poles.
The solution to an inverse problem in stratified dielectric media
Authors: Gopinath B.
Published: March 2003

We solve the problem of determining the average dielectric constant and thickness of the layers that constitute a stratified dielectric medium from measurements of transmitted power at a single frequency. Each sample of the medium that is available for measurement is modeled as a stack of ’’n’’ layers of dielectrics of thicknesses li and dielectric constants Ki (see Fig. 1). We assume that n, li, Ki, i=1,2,..., are all independent random variables and their values, of course, depend on the particular sample and the layer indexed by ’’i’’. Furthermore, it is assumed that the li are identically distributed with some exponential distribution and that the Ki are identically distributed. There are no other constraints or assumptions about these distributions except the following which are made more precise in the text: (1) The various averages of interest are finite. (2) Distributions which precipitate certain singular conditions (which is not a problem in ’’almost all cases’’) are excluded. Then the method described in this paper determines uniquely: (1) the average of the thicknesses of the layers Eli; (2) the average of the dielectric constants EKi from measurements of transmitted power and without any further knowledge of the distributions of li, Ki, or n. We remark that the theory presented here applies also to acoustic or mechanical systems with the appropriate interpretation of the physical parameters.
Angular momentum of systems of electric and magnetic charges and of singular flux surfaces
Authors: Barut A. O. , Schneider H.
Published: March 2003

The angular momentum of a system consisting of an electric charge e and a magnetic charge g is, as is well known, (eg). We derive general formulas for systems consisting of arbitrary electric and magnetic charges, dyons and of singular magnetic flux lines or surfaces of arbitrary integrable topological shapes. The total angular momentum is then quantized and related to the quantized flux.
Canonical symmetrization for unitary bases. I. Canonical Weyl bases
Authors: Patterson Chris W. , Harter William G.
Published: March 2003

It is shown that the Weyl basis formed by the canonical symmetrization of an n-dimensional, p-rank tensor space with canonical projection operators of Sp is a Gel’fand basis of U(n). This basis may easily be generated using standard projection operator techniques.
Uniform bounds of the Schwinger functions in boson field models
Authors: Park Yong Moon
Published: March 2003

We study the lattice and space cutoff boson field models with periodic boundary condition in d-dimensional space–time (d-N+). We prove that if the pressures of the interaction (and also the pressures of the interaction with linear external fields) under consideration are bounded uniformly in the cutoffs, the corresponding Schwinger functions are also bounded uniformly in the cutoffs. By applying the above result we prove the uniform bounds of the space cutoff Schwinger functions for the (λφ4−σφ2−μφ)3 model and the lattice and space cutoff Schwinger functions for the exponential type interactions in d-dimensional space–time.
Critique of the generalized cumulant expansion method
Authors: Fox Ronald Forrest
Published: March 2003

The method of ordered cumulants is presented for the solution of multiplicative stochastic processes. The relationship between this method and the irreducible cluster integral method of Mayer, which is used in the theory of the imperfect gas, is elucidated. The cluster property of ordered cumulants is proved. A critique of the literature in this area is presented which exposes some errors in the formulas. Several examples are indicated for the application of the ordered cumulant method.
Exponential decay and regularity properties of the Hartree approximation to the bound state wavefunctions of the helium atom
Authors: Benci V. , Fortunato D. , Zirilli F.
Published: March 2003

The exponential decay and regularity properties of the Hartree approximation to the bound state wavefunctions of the helium atom are proved.
Boundary conditions and singular potentials in diffusion theory
Authors: Narnhofer H. , Klauder J. R.
Published: March 2003

The Feynman–Kac integral and its associated stochastic differential equations are generalized to incorporate the analog of arbitrary boundary conditions in the equivalent differential equation formulation of diffusion problems. The influence of additional singular potentials is considered, especially the interplay between the boundary conditions and various regularizations of the singular potential. The basic results are translated into d-dimensional processes and compared with standard results known for differential operators.
A supereigenvalue problem for the solution of the generator coordinate integral equation
Authors: Lathouwers L.
Published: March 2003

The unconventional Griffin–Hill–Wheeler integral equation is replaced by a classical Fredholm eigenvalue problem which is the continuous analog of Löwdin’s supersecular equation.
Prolongation structures of nonlinear evolution equations. II
Authors: Estabrook F. B. , Wahlquist H. D.
Published: March 2003

The prolongation structure of a closed ideal of exterior differential forms is further discussed, and its use illustrated by application to an ideal (in six dimensions) representing the cubically nonlinear Schrödinger equation. The prolongation structure in this case is explicitly given, and recurrence relations derived which support the conjecture that the structure is open—i.e., does not terminate as a set of structure relations of a finite-dimensional Lie group. We introduce the use of multiple pseudopotentials to generate multiple Bäcklund transformation, and derive the double Bäcklund transformation. This symmetric transformation concisely expresses the (usually conjectured) theorem of permutability, which must consequently apply to all solutions irrespective of asymptotic constraints.
Number of independent missing label operators
Authors: Peccia A. , Sharp R. T.
Published: March 2003

When the bases for irreducible representations of a semisimple group are reduced according to a semisimple subgroup the number of functionally independent missing label operators available is just twice the number of missing labels. The argument presented suggests that the result holds for any Lie group.
Hertzian gravitational potentials for type D space–times
Authors: Güven R.
Published: March 2003

Hertzian gravitational potentials for type D vacuum space–times are defined. The Geroch–Held–Penrose formalism is expressed in the language of exterior forms, and by using its formal simplifications, the existence of Hertzian potentials for this class of nonlinear gravitational fields is demonstrated. The explicit form of these potentials possessing the spin and boost weights and the self-dual property of the field 2-form is constructed. The gauge freedom associated with these potentials is discussed, and the procedure is found to resemble the Debye reduction of the electromagnetic Hertzian potentials in that it enables us to construct a complex scalar superpotential from the Weyl spinor.
A gauge invariant formulation of time-dependent dynamical symmetry mappings and associated constants of motion for Lagrangian particle mechanics. I
Authors: Katzin Gerald H. , Levine Jack
Published: March 2003

In this paper (part I of two parts), which is restricted to classical particle systems, a study is made of time-dependent symmetry mappings of Lagrange’s equations (a) Λi(L) =0, and the constants of motion associated with these mappings. All dynamical symmetry mappings we consider are based upon infinitesimal point transformations of the form (b) χi=xi+δxi [δxi≡ξi(x,t) δa] with associated changes in trajectory parameter t defined by (c) t̄=t+δt [δt≡ξ0(x,t) δa]. The condition (d) δΛi(L) =0 for a symmetry mapping may be represented in the equivalent form (e) Λi(N) =0, where (f) Nδa≡δL+Ld (δt)/dt. We consider two subcases of these symmetry mappings which are referred to as R1, R2 respectively. Associated with R1 mappings [which are satisfied by a large class of Lagrangians including all L=L (χ̇,x)] is a time-dependent constant of motion (g) C1≡ (∂N/∂χ̇i) χ̇i −N+(∂/∂t)[(∂L/∂χ̇i) ξi−Eξ0]+γ1(x,t), where γ1 is determined by R1. The R2 subcase is the familiar Noether symmetry condition and hence has associated with it the well-known Noether constant of motion which we refer to as C2. For symmetry mappings which satisfy both R1 and R2 it is shown that (h) C1=∂C2/∂t+γ1. The various forms of symmetry equations and constants of motion considered are shown to be invariant under the Lagrangian gauge transformation (i) L→L′=L+dψ (x,t)/dt.
A new functional equation in the plasma inverse problem and its analytic properties
Authors: Szu H. H. , Carroll C. E. , Yang C. C. , Ahn S.
Published: March 2003

In the one-dimensional form of the plasma inverse problem, reflection of transverse electromagnetic waves is used to determine the electron density in a cold, collisionless, unmagnetized plasma. We extend the applicable Gel’fand–Levitan integral equation so that it is valid for all times. Laplace transformation of the extended equation gives a linear functional equation containing the complex reflection coefficient. We solve the functional equation analytically in special cases, and classify reflection coefficients by their analytic properties.
Bogolubov–Parasiuk theorem in the α-parametric representation
Authors: Bergere M. C. , Lam Yuk-Ming P.
Published: March 2003

A renormalized Feynman amplitude expressed in the α-parameters is defined by introducing a subtraction operator acting directly upon the α-integrand. Different forms of this subtraction operator are discussed. We define the isotropic and nonisotropic normal products and we give a more general oversubtraction rule which ensures both the absolute convergence of the amplitude and the Bogolubov, Parasiuk and Hepp recurrence. The proof of absolute convergence of the amplitude is performed using Hepp’s sectors and equivalence classes of nests.
Thermodynamics
Authors: Zeleznik Frank J.
Published: March 2003

The fundamental structure of thermodynamics is purely algebraic, in the sense of atopological, and it is also independent of partitions, composite systems, the zeroth law, and entropy. The algebraic structure requires the notion of heat, but not the first law. It contains a precise definition of entropy and identifies it as a purely mathematical concept. It also permits the construction of an entropy function from heat measurements alone when appropriate conditions are satisfied. Topology is required only for a discussion of the continuity of thermodynamic properties, and then the weak topology is the relevant topology. The integrability of the differential form of the first law can be examined independently of Caratheodory’s theorem and his inaccessibility axiom. Criteria are established by which one can determine when an integrating factor can be made intensive and the pseudopotential extensive and also an entropy. Finally, a realization of the first law is constructed which is suitable for all systems whether they are solids or fluids, whether they do or do not exhibit chemical reactions, and whether electromagnetic fields are or are not present.
Eigenvalues of the Casimir operators of the orthogonal and symplectic groups
Authors: Nwachuku C. O. , Rashid M. A.
Published: March 2003

Eigenvalues of the Casimir operators of the orthogonal and the symplectic groups are obtained in closed and simple form by diagonalizing directly the matrices introduced by Perelomov and Popov. This method unifies the treatment of the problem for the semisimple Lie groups.
Cross sections for screened potentials
Authors: Semon Mark D. , Taylor John R.
Published: March 2003

We consider scattering by screened Coulomb and screened short-range potentials. We prove, in Born approximation, that the cross section for a sharply cutoff Coulomb potential does not converge to the Rutherford cross section as the screening radius ρ→∞. On the other hand, for certain ’’smooth’’ screening functions we show that the screened Coulomb cross section does approach the Rutherford cross section as ρ→∞. In the case of short-range potentials, we prove (exactly, not just in Born approximation) that the screened cross section approaches the unscreened cross section as ρ→∞, whether the screening function is smooth or not.
The state labeling problems for SO(N) in U(N) and U(M) in Sp(2M)
Authors: Green H. S. , Hurst C. A. , Ilamed Y.
Published: March 2003

It is shown that, in a boson representation, the operators whose eigenvalues serve to label representations of SO(N) in U(N) also serve to label representations of U(M) in Sp(2M). The problem of labeling U(2) in Sp(4) is considered in detail, and it is shown how to find labeling operators with rational eigenvalues, depending, however, on the representation. The solution of this problem is shown to provide a solution of the equivalent problem of the labeling of SO(3) in U(3).
Particle permutation symmetry of multishell states. I. Two shells
Authors: Williams S. A. , Pursey D. L.
Published: March 2003

A method is developed for constructing N-particle states of definite symmetry from n1-particle and n2-particle states of definite symmetry where N=n1+n2. A canonical resolution of the attendant multiplicity question is given. The results, which are a first step toward the construction of appropriate coefficients of fractional parentage, do not rely upon any particular form for the N-particle Hamiltonian. Rather, the results are based entirely upon properties of the symmetric groups Sn1, Sn2, and SN. The group theoretic problem which is the construction of irreducible representations of SN from those of Sn1 ×Sn2 is solved using induced representation theory together with projection operator techniques.
Relativistic multiple scattering of electromagnetic waves
Authors: Berntsen Svend , Johannsen Günther
Published: March 2003

A general, relativistically exact formulation of scattering of electromagnetic waves by two or more objects in relative, uniform motion is presented. Coupled and separated integral equations are obtained for far-field scattering amplitudes of bodies within a system in terms of known far-field scattering amplitudes of the isolated obstacles. The integral equations are accessible for iteration analogous with the case of nonmoving objects. Explicit expressions and numerical results are given for the two-dimensional problem of two parallel, perfectly conducting cylinders.
Nonspreading solutions to a class of differential equations
Authors: Chang S. C. , Janis Allen I.
Published: March 2003

Explicit nonspreading (i.e., characteristically propagating) solutions to a certain class of two-dimensional, hyperbolic, differential equations are found. This class of equations is a generalization of equations that arose in a study of radiation in cosmological backgrounds.
Radiation in cosmological backgrounds
Authors: Chang S. C. , Janis Allen I.
Published: March 2003

The purpose of this investigation is to find the conditions for characteristic propagation of multipole radiation in Friedmann backgrounds. The radiation fields studied are Klein–Gordon scalar fields, conformally invariant scalar fields, electromagnetic fields, and gravitational fields. The behavior of electromagnetic and conformally invariant scalar radiation is similar to that of the corresponding radiation in flat space-time, since both fields satisfy conformally invariant equations and the Friedmann backgrounds are conformally flat. Thus characteristically propagating solutions are possible for both fields in any Friedmann background. For the Klein–Gordon and gravitational fields, it is found that characteristic propagation is possible only for special Friedmann backgrounds. Two physically important Friedmann backgrounds, those for which P=0 and P=ρ/3 (where P is pressure and ρ is density), are among these special backgrounds for both types of radiation. In the course of this study, all Friedmann backgrounds for which P=αρ, where α is an arbitrary constant, are found; the methods used and the resulting solutions are much simpler than those previously given by Tauber.
Fredholm determinants and multiple solitons
Authors: Case K. M.
Published: March 2003

The discrete inverse scattering problem in one dimension is considered. Exact solutions are obtained using elementary algebraic tools. Expressions found involve determinants of infinite-dimensional matrices. A simple, heuristic, limiting process yields the solution for the continuous problem. When the reflection coefficients do not contribute (the general N soliton case), the determinants reduce to those of given N×N matrices.
A continuum theory of deformable ferrimagnetic bodies. II. Thermodynamics, constitutive theory
Authors: Maugin Gérard A.
Published: March 2003

In order to close the system of differential field equations developed in Paper I, this article proposes a rational development of the relevant macroscopic thermodynamics and of a constitutive theory. In particular, by following Coleman’s thermodynamics, exact nonlinear constitutive equations for thermoelastic antiferromagnetic insulators are formulated. According to the deductive scheme adopted in Paper I, the important case of elastically isotropic antiferromagnets with a magnetic easy axis, and possibly endowed with the property of weak ferromagnetism, is developed in detail by using approximations. In order to supplement the description of thermodynamically recoverable processes and in accordance with the Onsager–Casimir theory of irreversible processes, the constitutive equations governing phenomena such as viscosity, electric and heat conduction, and spin relaxation, the latter either for strong or weak damping, are obtained. Regarding the latter effect, it is shown, thanks to the formalism adopted in Paper I, that both viscosity and spin relaxation participate in the Cauchy equations. The relaxation term of Gilbert is thus generalized to the case of deformable antiferromagnets.
Deformations of Poisson brackets, Dirac brackets and applications
Authors: Flato M. , Lichnerowicz A. , Sternheimer D.
Published: March 2003

After a short review of results which we recently obtained on deformations of Lie algebras associated with symplectic manifolds, we discuss physical applications and treat some examples with deformed Poisson brackets. We make explicit a connection between classical and quantum mechanics, and the theory of Dirac brackets for second class constraints, from the viewpoint of deformation theory. Finally we discuss the general Dirac constraints formalism.
Critical length of a transport process in rod geometry
Authors: Pao C. V.
Published: March 2003

In this paper, we study a two-point boundary-value problem which governs the transport of n different type of particles in a rod of finite length. Through the construction of an upper solution we establish a simple relation between the rod length and the physical parameters of the transport medium under which the maximal and the minimal sequences obtained in an earlier paper converge, respectively, to a maximal and a minimal solution of the problem. This relation leads to a lower bound for the critical length of the rod when fission occurs in the system. The convergence of the constructed sequences gives a mathematical justification for the existence of a physically meaningful solution to the system. It is also shown, under a slightly stronger condition on the rod length, that the maximal solution coincides with the minimal solution and the physical system is subcritical. In addition, an explicit recursion formula for the calculation of approximate solutions is given.
Classification of all simple graded Lie algebras whose Lie algebra is reductive. II. Construction of the exceptional algebras
Authors: Scheunert M. , Nahm W. , Rittenberg V.
Published: March 2003

The exceptional simple graded Lie algebras whose existence is suggested by the results of the preceding paper are explicitly constructed. In this way the classification of all simple graded Lie algebras whose Lie algebra is reductive is completed.
Canonical parameters of the 3j coefficient
Authors: Lockwood Loren A.
Published: March 2003

The 3j coefficient is expressed as a function of five new parameters which have unique properties. They are completely independent, satisfy simple validity criteria, and display the symmetry properties of the function in a particularly transparent manner. By means of the new parameters, the known 72-element symmetry group is reduced to an eight-element group, and the absolute symmetries are separated in a clear way from those which contain a phase factor.
Quantum two-particle scattering in fuzzy phase space
Authors: Prugove?ki Eduard
Published: March 2003

The concepts of configuration and momentum representation space for state vectors are generalized to that of fuzzy-phase-space representation spaces L2(Γs), 0<s<∞, which are interpolated in between these two standard representations. It is shown that the wavepacket in L2(Γs) displays the familiar evanescence property from any region Ks × Ms in the fuzzy phase space Γs if that region is bounded in its configuration part Ks; also, that the probability of detecting the system in Ks × Ms has a finite asymptotic time limit if Ks is a (fuzzy) cone. For scattering states the existence of free states that are asymptotic in Γs is established, and a formula for differential cross section in Γs is derived.
Quantum mechanical soft springs and reverse correlation inequalities
Authors: Ellis Richard S. , Newman Charles M.
Published: March 2003

Various properties of one-dimensional Schrödinger operators with ’’soft spring’’ potentials are derived as a consequence of the fact that the GHS and other correlation inequalities are reversed for certain general Ising modules.
On the matrix representation of unbounded operators
Authors: Epifanio G.
Published: March 2003

It is shown that a matrix representation with properties analogous to the ones that hold for the bounded operators in Hilbert space is possible also for important sets of unbounded operators. These sets consist of the --algebras CD of the linear operators on any noncomplete scalar product space D, which have an adjoint in D. (These algebras have already been studied by the author, in collaboration with others, in previous papers.) Specifically it is proved that for these operators a matrix representation is possible with respect to an arbitrary orthonormal basis within D, in contrast to the situation that has been found by von Neumann for the unbounded closed symmetric operators. The matrix representation of the operators considered here also allows the usual algebraic operations. Besides, the changes of basis induced by automorphisms of D are allowed.
General formalism solving linear recursion relations
Authors: Antippa Adel F. , Phares Alain J.
Published: March 2003

We give the general solution of multiterm linear homogeneous recursion relations with nonconstant coefficients, by introducing a new class of special combinatorics functions based on the partition of an interval into classes. This provides power series solutions, whenever they exist, of ordinary linear differential equations, of any order, with polynomial coefficients.
Lie theory and the wave equation in space–time. I. The Lorentz group
Authors: Kalnins E. G. , Miller W.
Published: March 2003

In this article we begin a study of the relationship between separation of variables and the conformal symmetry group of the wave equation ψtt −Δ3ψ=0 in space–time. In this first article we make a detailed study of separation of variables for the Laplace operator on the one and two sheeted hyperboloids in Minkowski space. We then restrict ourselves to homogeneous solutions of the wave equation and the Lorentz subgroup SO(3,1) of the conformal group SO(4,2). We study the various separable bases by using the methods of integral geometry as developed by Gel’fand and Graev. In most cases we give the spectral analysis for these bases, and a number of new bases are developed in detail. Many of the special function identities derived appear to be new. This preliminary study is of importance when we subsequently study models of the Hilbert space structure for solutions of the wave equation and the Klein–Gordon equation ψtt−Δ3ψ=λψ.
The Wegner approximation of the plane rotator model as a massless, free, lattice, Euclidean field
Authors: Bricmont J. , Debacker-Mathot F.
Published: March 2003

The approximate plane rotator model proposed by Wegner can be reinterpreted as a massless, free, lattice, Euclidean field model. Using this approach, we compute the two-point correlation function and the magnetization for the spin model as functions of the covariance of the fields.
The positive energy conjecture and the cosmic censor hypothesis
Authors: Jang Pong Soo , Wald Robert M.
Published: March 2003

We show that a positive energy argument of Geroch can be modified to rule out a possible class of counterexamples to the cosmic censor hypothesis proposed by Penrose.
Symmetry and exact dyon solutions for classical Yang–Mills field equations
Authors: Hsu J. P. , Mac E.
Published: March 2003

We show that the generalized electromagnetic field tensor F̄μν and the magnetic and electric charges in non-Abelian gauge theories have little to do with the Higgs scalars and/or the dynamics of the Lagrangian. They are consequences of a symmetry in the theory. We present several exact static dyon solutions to the nonlinear classical field equations in both massless and massive Yang–Mills theories, which possess both electric and magnetic charges. The implications of F̄μν are also discussed.
A classical perturbation theory
Authors: Schwartz Charles
Published: March 2003

A compact formula is found for the perturbation expansion of a general one-dimensional Hamiltonian system in classical mechanics. The technique is also applied to the mathematical problem of functional inversion.
Canonical parameters of the 6j coefficient
Authors: Lockwood Loren A.
Published: March 2003

A set of six canonical parameters is derived for the 6j coefficient in a way analogous to that used previously for the 3j coefficient. The procedure involves implicitly a twelve-to-one homomorphism which reduces the known 144-element symmetry group to a 12-element group. The symmetries are completely and elegantly described by only five of the six parameters. A further property of the canonical parameters is their complete independence.