bluuebunny/crossref_metadata_2025 huggingface dataset:
Moment-theory approximations for nonnegative spectral densities
Authors: Corcoran C. T. , Langhoff P. W.
Published: March 2003

Moment-theory approximations constructed from finite numbers of spectral power moments are described for continuous, nonnegative spectral densities and associated Stieltjes integrals. Derivatives of the mean (Stieltjes) values of the nth-order Tchebycheff bounds on nondecreasing distributions provide the appropriate approximations to the associated spectral densities. The nth-order Tchebycheff density so defined is shown to be real, nonnegative, and continuous on the real axis, to have 2n−4 continuous derivatives there, and to support 2n-2 positive-integer power moments. Related approximations to the associated Stieltjes integral are obtained from corresponding principal-value quadratures. The Tchebycheff densities are convergent in the limit of large numbers of spectral moments for determined moment problems, but they are not solutions of reduced moment problems of appropriate finite order. An illustrative application in the case of normal-mode lattice vibrations in a diatomic chain indicates that the Tchebycheff densities are suitably convergent, and provide faithful images of the forbidden band gap and Van Hove singularities present.
Some examples of transparent potentials in the classical approximation
Authors: Cuer Michel
Published: March 2003

We give classes and examples of potentials which give a classical deflection function equal to zero (mod 2nπ) for any value of the impact parameter at a fixed energy. Thus these potentials are completely transparent for a classical collision at this energy.
Free electromagnetic fields on a compact Lie group manifold
Authors: Henry Leighton L.
Published: March 2003

Maxwell’s equations for free fields are studied on the underlying C∞ manifold of a compact Lie group. The formulation is in terms of exterior differential forms as given by Wheeler. It is found that on a compact connected Lie group there are no free electromagnetic fields. The results obtained are essentially a physical interpretation of the well-known theorem that the second Betti number of a compact semisimple Lie group is zero.
Generating exactly soluble nonlinear discrete evolution equations by a generalized Wronskian technique
Authors: Chiu S. -C. , Ladik J. F.
Published: March 2003

By means of generalized Wronskian relations an operator formulation is developed allowing one to generate a class of discrete evolution equations which are soluble by inverse scattering. This class includes nonlinear difference–difference equations as well as nonlinear differential–difference equations. The development of the method exhibits the significance of the dispersion relation of the associated linearized equation. The generalized Wronskian technique also permits one to obtain generalized Bäcklund transformations.
Solutions of nonlinear network equations by the inverse scattering method
Authors: Ladik John F. , Chiu Shiu-Chu
Published: March 2003

Two coupled equations for a nonlinear network are solved by the inverse-scattering method. N-soliton solutions are obtained. The effect of the continuum spectrum is investigated.
Rotating steady state configurations in general relativity
Authors: Glass E. N.
Published: March 2003

Spacetimes with axial and timelike Killing fields ημ and ξμ respectively are studied. A natural basis is constructed for the steady state configurations of matter with 4-velocity uμ =αξμ+βημ. It is shown that the rate of shear tensor always has an eigenvector with zero eigenvalue lying in the 2-space of Killing orbits. A natural coordinate system is constructed on the 2-space of Killing orbits for rigidly rotating perfect matter. Gravitational isorotation is proved for the case of a perfect fluid with a one-parameter equation of state. Locally nonrotating frames are constructed for purely stationary spacetimes.
Generalized invariants for the time-dependent harmonic oscillator
Authors: Günther N. J. , Leach P. G. L.
Published: March 2003

A generalized class of invariants, I (t), for the three-dimensional, time-dependent harmonic oscillator is presented in both classical and quantum mechanics. For convenience a simple notation for types of harmonic oscillator is introduced. Two interpretations, one in terms of angular momentum and the other employing a canonical transformation, are offered for I (t). An invariant symmetric tensor, Imn(t), is constructed and shown to reduce to Fradkin’s invariant tensor for time-independent systems. The usual SU(3) (compact) or SU(2,1) (noncompact) is shown to be a noninvariance group for the time-dependent oscillator with S{U(2) ⊗U(1) } as the invariance subgroup. Extensions to anisotropic systems and the singular quadratic perturbation problem are discussed.
Superfields as an extension of the spin representation of the orthogonal group
Authors: Winnberg J. O.
Published: March 2003

We show that a superfield can be interpreted as a spinor belonging to the spin representation of a Clifford algebra. A subset of this algebra is connected by a linear automorphism to the orthogonal group. With this interpretation it should be possible to give a physical meaning to Grassmann variables.
Schrödinger equation with Yukawa potential, a differential equation with two singular points
Authors: Bühring Wolfgang
Published: March 2003

The Schrödinger radial equaion with Yukawa potential is treated analytically be means of a double contour integral representation for the solution. Standard solutions are defined relative to each of the singular points of the differential equaion. Convergent expressions are obtained for the connection coefficients which occur in the linear relations persisting between any three of the standard solutions. These expressions are double series the terms of which are hypergeometric functions multiplied by factors which can be calculated recursively. As an application, the expression for the S matrix, which is simply related to the connection coefficients, is considered with regard to its convergence properties.
A criterion for reducibility of a relativistic wave equation
Authors: Sudarshan E. C. G. , Khalil M. A. K. , Hurley W. J.
Published: March 2003

In general when one writes a relativistic wave equation of the form (−iΓ⋅∂+m) ψ (x) =0, that transforms covariantly under some representation Λ→T (Λ) of SL(2,ℂ), it is nontrivial to determine whether or not the equation is irreducible or to avoid ending up with a reducible equation; especially if T (Λ) contains repeat ing irreducible r epresentations. In this paper a simple (st) criterion is given by which one can determine whether or not an equation is irreducible. It is shwon that if Λμ have any invariant subspace at all, then that subspace must be a representation space of some combination of SL(2,ℂ) representations in T (Λ). Knowing this it is shown that a wave equaiton is reducible if an only if there exists some idempotent projector P̃ such that (1−P̃) Γ0P̃=0 other than P̃=O or I. A method for constructing all possible addmissable P̃’s is given. A simple example of the technique is given. A simple example of the technique is also given.
Classical systems of infinitely many noninteracting particles
Authors: Pulè J. V. , Verbeure A.
Published: March 2003

It is proved that the C*-algebra of observables of an infinite classical system is isomorphic to the group algebra on the test function space 𝒟. The physical dynamical system consisting of infinitely many noninteracting particles is studied. A particular class of states, called the quasifree states, is exhibited and their properties are studied. Some results on the spectral properties of monoparticle evolutions are obtained. Finally we give explicitly a solution of the classical KMS condition for these evolutions.
A systematic investigation of the Petrov G4 types
Authors: Ray J. R. , Zimmerman John Charles
Published: March 2003

In order to investigate exact solutions in general relativistic cosmology, one usually assumes the spacetime possesses symmetry. Here, we study exact solutions for the Petrov four-parameter Lie groups G4, acting on nonnull hypersurfaces where the energy-momentum tensor is that of a pressureless perfect fluid (a so-called ’’dust’’). We find that the preponderance of solutions are for a spacelike dust and, in several cases, are able to give their explicit forms. Among these are spacelike versions of previously known timelike matter cosmological models.
Phase-integral calculation of quantal matrix elements without the use of wavefunctions
Authors: Fröman Nanny , Fröman Per Olof
Published: March 2003

Simple phase-integral formulas for the calculation, without the use wavefunctions, of quantal matrix elements of multiplicative and differential operators are given for the case of bound states in a single-well potential. The matrix elements are obtained to within the accuracy corresponding to any conveniently chosen order of the kind of phase-integral approximations used.
Energy forms, Hamiltonians, and distorted Brownian paths
Authors: Albeverio Sergio , Ho/egh-Krohn Raphael , Streit Ludwig
Published: March 2003

We study the Hamiltonians for nonrelativistic quantum mechanics—and for the heat equation—in terms of energy forms ∫∇f∇fd’gm, where dμ is a positive, not necessarily finite measure on Rn. We cover the cases of very singular interactions (e.g., N particles in R3 interacting by two-body ’’δ potentials’’). We also exhibit, on the other hand, regularity conditions for μ in order that H be realized as a perturbation of the Laplacian by a measurable or generalized functon. The Hamiltonians defined by energy forms alwasy generate Markov semigroups, and the associated processes are symmetric homogeneous strong Markov diffusion Hunt processes with continuous paths realizations. Ergodicity, transiency, and recurrency are also discussed. The associated stochastic differential equaiton is discussed in the situaion were μ is finite but the drift coefficient is only restricted to be l (Rr,dμ). These results provide a large class of examples where solutions of the heat equaion can be expressed by averages with respect to the constructed Hut processes, rather than with respect to Brownian motion. This is discussed in relation to recent work of Ezawa, Klauder, and Shepp, as well as of Hida and Streit.
Instability of the continuous spectrum: The N-band Stark ladder
Authors: Avron J. E. , Zak J. , Grossmann A. , Gunther L.
Published: March 2003

It is shwn that the energy spectrum of the Bloch electron in an external field is continuous. Furthermore, it is shown that all approximations which take into account interband coupling whithin groups of finite number of bands (the N-band approximation) lead to a pure-point spectrum of intertwining Wannier-stark ladders. This instability of the continuous spectrum under the N-band approximation is related to a theorem due to Weyl and von Neumann. Approximation methods for dealing with interband coupling within a group of finite number of bands are given.
Covariant canonical formulation and centers of mass and motion for a relativistic two-body problem
Authors: Fahnline Donald E.
Published: March 2003

Fokker has given a multitime action principle yielding the equations of motion for a relativistic two-body system in which the first particle responds to the retarded Liénard–Wiechert field of the second, while the second responds to the advanced field of the first. The present paper exhibits a single time parameter Lagrangian which avoids the no-interaction theorems and leads to a manifestly covariuent and canonical description of this system via Dirac’s generalized dynamics. Within this formulation the system possesses a manifestly covariant and canonical center of mass and a separate manifestly covariant but noncanonical ’’center of motion’’ which moves with the constant 4-velocity corresponding to the conserved total 4-momentum. In the one-dimensional case the two centers coincide, and the differential equation fo the internal motion reduces to the classical Kepler problem. In the general case the motion lies in a plane, and the center of mass moves in a circle about the center of motion.
Asymptotic behavior of Feynman integrals with zero mass particles
Authors: Manoukian Edward B.
Published: March 2003

The asymptotic behavior of Feynman integrals when some (or all) of their external momenta become small and/or some (or all) become large in Euclidean space nonexceptionally (and the power counting theorem) are derived for integrals involving, in general, zero mass particles as well as a subset of their masses.
Symmetries of the Hamilton–Jacobi equation
Authors: Boyer C. P. , Kalnins E. G.
Published: March 2003

We present a detailed discussion of the infinit esimal symmetries of the Hamilton-Jacobi equation (an arbitrary first order partial equation) Our presentation clucidates the role played by the characteristic system in determining the symmetries. We then specialize to the case of a free particle in one space and one time dimension, and study of local Lie group of point transformations locally isomorphie to O(3,2). We show that the separation of variables of the corresponding Hamilton-Jacobi equation n the form of a sum is related to orbits in the Schrödinger subalgebra of c(3,2). The remaining orbits of o(3,2) yield symmetry related solutions which separate in more complicated product forms. Finally some connections with the primordial equation of hydrodynamics (without force terms) are made.
Spatial coherence of acoustic signals in randomly inhomogeneous waveguides—A multiple-scatter theory
Authors: Sutton G. R. , McCoy J. J.
Published: March 2003

A normal mode theory is formulated for estimating the loss of spatial coherence of acousitc signals propagating through in randomly inhomogeneous waveguide. The theory is applicable for experiments in which multiple-scatter effects are important provided the rate at which energy is scattered with increasing range is limited. Other limitations on the theory require that intermodal scattering between propagating and nonpropagating modes be negligible and that the intramodal scatter of energy within a single mode be narrow-angled, when measured in the waveguide plane. The scattering of acoustic signals, of significant frequency content ranging upwards from tens of Hertz, by large scale interval wave fields, i.e., wavelengths of the order of 1 kilometer or greater, satisfy the conditions for applicability of the theory.,
Painlevé functions of the third kind
Authors: McCoy Barry M. , Tracy Craig A. , Wu Tai Tsun
Published: March 2003

We explicitly construct the one-parameter family of solutions, η (ϑ;ν,λ), that remain bounded as ϑ→∞ along the positive real ϑ axis for the Painlevé equation of third kind ww′′= (w′)2−ϑ−1ww′+2νϑ−1(w3−w) +w4−1, where, as ϑ→∞, η ∼ 1−λΓ (ν+1/2)2−2νϑ−ν−1/2e−2ϑ. We further construct a representation for ψ (t;ν,λ) =−ln[η (t/2;ν,λ)], where ψ (t;ν,λ) satisfies the differential equation ψ′′+t−1ψ′= (1/2)sinh(2ψ)+2νt−1 sinh(ψ). The small-ϑ behavior of η (ϑ;ν,λ) is described for ‖λ‖<π−1 by η (ϑ;ν,λ)  ∼ 2σBϑσ. The parameters σ and B are given as explicit functions of λ and ν. Finally an identity involving the Painlevé transcendent η (ϑ;ν,λ) is proved. These results for the special case ν=0 and λ=π−1 make rigorous the analysis of the scaling limit of the spin–spin correlation function of the two-dimensional Ising model.
Dilations and interaction
Authors: Buchholz Detlev , Fredenhagen Klaus
Published: March 2003

As a consequence of the geometrical features of dilations massless particles do not interact in a local, dilationally invariant quantum theory. This result also holds in models in which dilations are only a asymmetry of the S matrix.
Synchronized solitons
Authors: Newell Alan C.
Published: March 2003

We find that, under certain conditions, two solitons of an integrable system can form a synchronized bound state when the system is perturbed.
Paraboson uniqueness for infinitely many degrees of freedom
Authors: Robbins Steven
Published: March 2003

As with the boson field, the creation and annihilation operators for the paraboson field are unbounded, and the usual paraboson relations therefore, cannot be regiorously satisfied. There are no Weyl relations for parabosons, and so the paraboson relations must be treated in unbounded form. A new approach for treating such unbounded relations is developed, and it is found that the paraboson field is determined by the requirement that the appropriate operator is the one-dimensional number operator. A similar result is shown th hold for the boson field.
Quantum theory and measures on Hilbert space
Authors: Krips H.
Published: March 2003

It is shown that merely the decision to represent the fundamental events of quantum theory by vectors in a Hilvert space uniquely determines the form of any probability measure over the set of events. The proof proceeds by showing that any nonnegative meausre on the vectors of a Hilbert space, for which the sum of the measures on the vectors of any complete orthonormal set is 1, takes the usual form damanded by quantum theory. This result represents a considerable strengthening of one of the consequences of Gleason’s theorem.
Spaces of analytic functions on a complex cone as carriers for the symmetric tensor representations of SO(n)
Authors: Bargmann V. , Todorov I. T.
Published: March 2003

We study the space P of all polynomial functions on the complex cone Kn={𝒵= (𝒵1⋅⋅⋅𝒵n) ε Cn, 𝒵2=𝒵21+ ⋅⋅⋅+𝒵2n=0} (n=3,4,⋅⋅⋅). Its subspaces Kl (=Kln) of homogeneous polynomials of degree l (=0,1,2,⋅⋅⋅) provide a convenient realization of the carrier spaces for the symmetric tensor representations of the real orthogonal group SO(n). The multiplication operator 𝒵μ (μ=1,...,n) maps Kl into Kl+1. We define its adjoint as an interior differential operator on Kn which maps Kl+1 into Kl and transforms as an n-vector. We show that the lowest order differential operator with this property is proportional to Dμ= (n/2−1+√∂) ∂μ −(1/2) 𝒵μΔ. We define a scalar product in P with respect to which the operators 𝒵μ and Dμ are Hermitian adjoint to each other and consider the Hilbert space completion Kn of P with respect to this scalar product. The spaces Kn are imbedded for all n (=3,4,⋅⋅⋅) in the Fock type spaces Bn, studied earlier by Bargmann. The space Kn possesses a reproducing kernel that allows us to define a (unique) harmonic extension of every analytic function in Kn. It is shown that the spaces K3 and K4 can be imbedded isometrically in the Hilbert spaces B2 and B4 associated with the representations of SU(2) and SU(2) ×SU(2) [⊇SU(4)].
Conformal properties of a class of exactly solvable N-body problems in space dimension one
Authors: Barucchi G. , Regge T.
Published: March 2003

An algebra of collective variables in the generalized (Calogero–Sutherland) N-body classical one-dimensional model is introduced, and their transformation properties under the conformal group are discussed in detail.
Kerr’s theorem and the Kerr–Schild congruences
Authors: Cox D.
Published: March 2003

A simple flat spacetime derivation of the Kerr–Schild congruences is presented starting from Kerr’s theorem.
Exact nearest neighbor statistics for λ-bell particles on a one-dimensional lattice
Authors: McQuistan R. B.
Published: March 2003

Relationships are developed which describe exactly the degeneracies of those states specified by the number of either the occupied, mixed, or vacant nearest neighbor pairs which exist when indistinguishable λ-bell particles (occupying λ contiguous sites) are distributed on a one-dimensional lattice space.
A stochastic derivation of the Klein–Gordon equation
Authors: Lehr William J. , Park James L.
Published: March 2003

Several years ago Nelson succeeded in deriving the nonrelativistic Schrödinger equation within a stochastic model which included Newton’s second law as the fundamental dynamical rule. Unfortunately, the relativistic extension of Nelson’s work is not so straightforward as might at first be supposed. This paper examines the difficulties inherent in such a relativization and proposes supplemental axioms which resolve those difficulties. A stochastic derivation of the Klein–Gordon equation is then presented.
Variational methods for chemical and nuclear reactions
Authors: Crawford Oakley H.
Published: March 2003

All the variational functionals are derived which satisfy certain criteria of suitability for molecular and nuclear scattering, below the threshold energy for three-body breakup. The existence and uniqueness of solutions are proven. The most general suitable functional is specialized, by particular values of its parameters, to Kohn’s tanη, Kato’s cot(η−ϑ), the inverse Kohn coη, Kohn’s S matrix, our S matrix, Lane and Robson’s functional, and several new functionals, an infinite number of which are contained in the general expression. Four general ways of deriving algebraic methods from a given functional are discussed, and illustrated with specific algebraic results. These include equations of Lane and Robson and of Kohn, the fundamental R matrix relation, and new equations. The relative configuration space is divided as in the Wigner R matrix theory, and trial wavefunctions are needed for only the region where all the particles are interacting. In addition, a version of the general functional is presented which does not require any division of space.
Eigenfrequency density oscillations and Walfisz lattice sums
Authors: Baltes H. P. , Steinle B.
Published: March 2003

We show that recent results on the distributions of eigenfrequencies for the scalar and electromagnetic wave equations in a cube-shaped domain were anticipated by the work of Walfisz on the number of lattice points in a sphere.
Relativistically rotating dust cylinders
Authors: Vishveshwara C. V. , Winicour J.
Published: March 2003

A previously derived method for obtaining the general differentially rotating dust metric is applied to the cylindrically symmetric case. The dust metric, the exterior vacuum metric, the matching conditions at the boundary, and the asymptotic behavior at infinity are worked out explicitly. As a special case, the solutions include van Stockum’s rigidly rotating models. Inertial dragging effects are found to be enhanced when the angular velocity decreases outward so that the angular momentum is concentrated near the axis. In the extreme relativistic case, the null cones emanating from points on the axis develop a caustic surface at which they turn around and eventually refocus at the axis. However, this is associated with acausal behavior rather than horizons.
On the integration of the differential equations of five-parametric double-hypergeometric functions of second order
Authors: Olsson P. O. M.
Published: March 2003

Solutions of the differential equations associated with Appell’s hypergeometric function F2 (a,b1,b2,c1,c2; x1,x2) are obtained by considering them as a Laplace transform of a product of two confluent hypergeometric functions. Since these differential equations may be transformed into the equations associated with Appell’s function F3(a,b1,b2,c1,c2; x1,x2) and Horn’s function H2(a,b,c,d,e; x1,x2), these equations are solved simultaneously. A set of 36 distinct solutions in terms of six types of series is given. This set is the smallest set which accounts for the general behavior of any of the three double-hypergeometric second order functions F2, F3, and H2. Various representations of the solutions in terms of series and integrals are given as well as connections between the solutions which continue the functions analytically.
Algebraic solutions to some spin-1 problems
Authors: Weaver D. L.
Published: March 2003

It is shown how the algebraic difficulties arising in some spin-1 problems may be reduced to (almost) Pauli matix algebra.
Error bounds on some approximate solutions of the Fredholm equations and applications to potential scattering and bound state problems
Authors: Singh S. R. , Turchetti G.
Published: March 2003

One may produce sequences that converge to the eigenvalues of a compact operator and the solution of an inhomogeneous equation involving this operator by the Bubnov–Galerkin method. We produce easily computable, converging and accurate error bounds on these approximations, provided that the operator belongs to a fairly wide subclass of the compact operators. These bounds in turn enable one to compute error bounds, having the same properties, on the approximations to the scattering parameters and the bound state energies obtained by the Schwinger variational method for some two-body interactions. This provides a method of obtaining upper and lower bounds to these quantities.
King and domino polynomials for polyomino graphs
Authors: Motoyama Akiko , Hosoya Haruo
Published: March 2003

For the purposes of treating several enumeration problems of lattice dynamics, king and domino polynomials are defined for a chessboard, polyomino, or square lattice of arbitrary size and shape. These polynomials are shown to be closely related to the partition function of the dimer statistics, the number of Kekulé structures, or maximum matching number. Several recursion formulas are found. Interpretation of these newly proposed quantities is given, and the possibility of extending them to the important physical models is discussed.
Maxwell’s equations in axiomatic quantum field theory. I. Field tensor and potentials
Authors: Bongaarts P. J. M.
Published: March 2003

An approach to the investigation of the Maxwell field in the framework of axiomatic quantum field theory is presented which employs Borchers’ algebraic reformulation of Wightman theory in a modified form adapted to the special features of the electromagnetic field. This makes it possible to clarify the relation between tensor and potential field operators, the meaning and properties of different gauges, the sense in which field equations hold and the properties of state spaces with their special subspaces.
Matrix elements of vector and axialvector operators in an algebraic formulation of baryon decays
Authors: Böhm A. , Teese R. B.
Published: March 2003

Properties of the matrix elements of baryon transition operators are derived from a minimal set of assumptions about the algebra of observables. Poincaré invariance, SU(3) classification, and V–A transitions are assumed. SU(3) is treated as a spectrum generating group rather than a symmetry group, and the usual form factors are expressed in terms of form factors which are invariant with respect to the spectrum generating SU(3). As a particular case, it is shown that the condition that the form factors be first class with respect to the spectrum generating SU(3) does not lead to a vanishing pseudotensor contribution.
Representation of the entropy functional for a grand canonical ensemble in classical statistical mechanics
Authors: Forte B. , Sastri C. C. A.
Published: March 2003

A representation theorem for the entropy functional of a system of an arbitrary but finite number of particles is proved. This theorem is a generalization of the main result of a previous paper of ours [J. Math. Phys. 16, 1453 (1975)], which gives a characterization of the entropy functional on the set of all probability densities f on Rn s.t. flogf is integrable. As may be expected when n is a random variable, the expression for entropy consists of two parts; one which arises from the ignorance about n and another which is the average, over n, of the conditional entropy given the number of particles. As in the above-mentioned paper, the expression includes the term corresponding to chemical reactions (with n replaced by the average number of particles) and the continuous analog of the Hartley entropy. It is conjectured that this last term might be of some significance in physics.
Energy–momentum tensors in the theory of electromagnetic fields admitting electric and magnetic charge distributions
Authors: Rund Hanno
Published: March 2003

When the field tensor of an electromagnetic field admitting both electric and magnetic charge distributions is expressed in terms of a Clebsch representation, the extended Maxwell equations in the presence of a given gravitational field are derivable from an invariant variational principle in which the Clebsch potentials play the role usually assumed by the classical 4-potentials. The corresponding Lagrange density gives rise in a unique manner to a symmetric tensor density Thj, which displays some of the properties normally associated with the energy–momentum tensor density of the electromagnetic field. However, this interpretation may be in conflict with the generally accepted expression for the modified Lorentz force. Accordingly an alternative energy–momentum tensor density ϑhj is derived which does not suffer from this drawback. However, when a generalized variational principle for the simultaneous determination of the behavior of both the electromagnetic and the dynamical gravitational fields is introduced, the resulting Euler–Lagrange equations give rise to extended Einstein–Maxwell equations which involve the density Thj. On the other hand, the alternative Einstein–Maxwell equations, obtained by the replacement of Thj by ϑhj, are not derivable from a variational principle. The solutions of the two Einstein–Maxwell equations, for the case of a spherically symmetric metric and static electromagnetic field, predict distinctly different effects of the magnetic charges on the gravitational field.
Group contraction in a fiber bundle with Cartan connection
Authors: Drechsler W.
Published: March 2003

A contraction of the structural group with respect to the stability subgroup is performed in a fiber bundle with Cartan connection. The relation of the connections in the original and in the contracted bundle is examined. As an example interesting for physics the contraction of the SO(4,1) de Sitter bundle over space–time to the affine tangent bundle over space–time is discussed with the latter bundle possessing the Poincaré group as structural group.
Static stars : Some mathematical curiosities
Authors: Collins C. B.
Published: March 2003

The equations of structure of static Newtonian and general relativistic stars are investigated. By using Lie group theory, it is shown that, in each case, the condition that there should exist a simple ’’homologous’’ family of similar solutions necessitates precisely those equations of state for the stellar matter that are usually invoked by means of extraneous physical arguments. In the relativistic case, a diagram which depicts these families is drawn, using the qualitative theory of differential equations. This vividly exhibits the nature of the general soutions, and the exceptional character of the Misner–Zapolsky solution. This diagram is contrasted with similar ones obtained by Chandrasekhar in the Newtonian case.
Complete integrability conditions of the Einstein–Petrov equations, type I
Authors: Brans Carl H.
Published: March 2003

The post-Bianchi equations, defined as the integrability conditions of the Bianchi equations, are explicitly stated for the algebraically general (type I) Einstein–Petrov vacuum equations. A computer analysis of these equations has shown that they constitute a completely integrable set. Hence all conditions imposed by the Einstein equations of this type on the derivatives of the dependent variables are now known.
Orbits of the rotation group on the density matrices of spin-1 particles
Authors: Daumens M. , Perroud M.
Published: March 2003

The SO(3) orbits of the spin-one mixed states, contained in each SU(3) orbit are shown to be characterized by the squares (defined by means of a symmetric bilinear form) of the eigenvectors of the density matrices. The orbits of matrices diagonalizable by a rotation in a spherical basis are deduced, and quadrupole matrices are considered as a special case.
Generating functions for the eigenvalues of the Casimir operators of the orthogonal and symplectic groups
Authors: Nwachuku C. O. , Rashid M. A.
Published: March 2003

By constructing the appropriate generating functions, the eigenvalues of the Casimir operators for the orthogonal and the symplectic groups are expressed in terms of power sums which are formally the same for the O(2n), Sp(2n), O(2n+1) groups as for the U(n) groups. The results for the O(2n), Sp(2n), and the O(2n+1) groups are written as the corresponding results for the U(n) groups plus very simple correction terms. This approach unifies the treatment of the problem for the semisimple Lie groups. Explicit evaluation of the eigenvalues of the Casimir operators becomes very simple.
Hydrostatic density distribution of fluids in steep field gradients and near critical points
Authors: Gates D. J.
Published: March 2003

A new perturbation expansion for the hydrostatic density distribution of a fluid in a steep field gradient is presented. It is a power series in the scale γ of an external potential ψ (γx), and is asymptotically valid as γ→0, although convergence is not proved. Significant corrections to the conventional hydrostatic equations are found near the critical point of a fluid, even in the gravitational field.
Inclusions of arbitrary shape in an elastic medium
Authors: Chen F. C. , Young K.
Published: March 2003

The problem of elasticity where a homogeneous linear solid of arbitrary shape and elastic constants is embedded in an infinite homogeneous isotropic medium is discussed in a manner which is a straightforward generalization of Eyges’ work on the Laplace equation. An exact integral equation is recast into an infinite system of algebraic equations and a systematic scheme of approximation is obtained by truncation. The lowest order approximation is discussed in detail, and its accuracy is shown to be quite good even for large differences in moduli between the two phases. Application to the effective moduli of composite materials is briefly discussed.
Geometrization of configuration space
Authors: Eichinger B. E.
Published: March 2003

Projective geometry provides the means to geometrize configuration space, granted three elementary postulates: (1) It is possible to describe physical systems as collections of a countable number n of particles; (2) the mass-inertial tensor of such a system is positive definite; (3) the Galilei group acting on elastic bodies can be rationally extended to the whole of GL(4,R). From these it is possible to deduce that any of the three symmetric spaces O(4,n)/O(4) ×O(n), U(2,n)/U(2) ×U(n), and Sp(1,n)/Sp(1) ×Sp(n) represents the configuration space. These three spaces are Einstein manifolds of constant sectional curvature. Several geometrical theorems are given for all three spaces; the metric on each is given in both Cartesian and polar forms; infinitesimal generators of the Lie algebras so(4,n), su(2,n), and sp(1) ×sp(n) are also obtained. Physical quantities are presumed to be obtained from eigenfunctions of the Laplace–Beltrami operator on the three spaces, and partial solutions for such functions are obtained on Sp(1,n)/Sp(1) ×Sp(n). These solutions fall into two classes: Those with integral values of a quantum number l⩾0 form the sequence 2(l+1)2; those with half-integral l⩾1/2 form a series with (2l+1)(l+3/2) members, and possibly represent leptons, quarks, etc.
Product integrals and the Schrödinger equation
Authors: Dollard John D. , Friedman Charles N.
Published: March 2003

A brief introduction to product integration is given. The theory developed is used to give a simple and rigorous analysis of the asymptotic behavior (r→∞) of positive-energy solutions of the radial Schrödinger equation. Absence of positive-energy bound states is proved for various classes of potentials. It is shown that E=1 is the only positive energy for which the Wigner–von Neumann potential can have a positive-energy bound state. The results proved imply (as will be shown in a later publication) existence of the Mo/ller wave matrices for the potential V (R) = (sinr)/r and various related potentials. A brief discussion is given to justify the WKB approximation which gives the wavefunction asymptotically for large positive values of the energy E.
Derivation of quantum mechanics from stochastic electrodynamics
Authors: de la Peña-Auerbach L. , Cetto A. M.
Published: March 2003

From the equation of motion for a radiating charged particle embedded in the zero-point radiation field we construct a stochastic Liouville equation which serves to derive, by a smoothing process, a Fokker–Planck-type equation with infinite memory. We show that an exact alternative form of this phase-space equation is the Schrödinger equation in configuration space, with radiative corrections. In the asymptotic, radiationless limit (when the radiative corrections became negligible), the phase-space density reduces to Wigner’s distribution, thus confirming Weyl’s rule of correspondence. We briefly discuss several other implications of stochastic electrodynamics which are relevant for quantum theory in general.
Correlation functions in the spherical and mean spherical models
Authors: Kac Mark , Thompson Colin J.
Published: March 2003

A transformation is obtained relating spherical and mean and spherical averages. The kernel of the transformation is the probability density of N−1ΣNi=1xi2 in the mean spherical model. The transformation is inverted to obtain a simple method for computing spherical averages from mean spherical averages. Averages in the two ensembles are identical except in zero field below the critical temperature.
Quantum axiomatics, representation theorem, and communicability of observations in physics
Authors: Miklavc Adolf
Published: March 2003

We show how a fundamental assumption in the Dirac formulation of quantum mechanics, namely that the states of a physical system at a particular time are mathematically represented by unit vectors in Hilbert space, can be deduced from certain aspects of our experimental procedures and of the observed outcome of quantum mechanical experiments. Our assumptions have clear empirical meaning and the results hold true for any dimensionality of the system, without anomalies in low dimensions which exist in the two well-known axiomatic approaches to quantum mechanics. The propositional logic approach of Birkhoff and von Neumann does not work for quantum systems of dimension less than four and requires an assumption which does not have an empirical basis. Jordan algebra axioms, on the other hand, also lead to anomalies in low dimensions and, moreover, are formal and cannot be directly physically interpreted. In our work it was possible to avoid these shortcomings.
Group-theoretical foundations of classical and quantum mechanics. I. Observables associated with Lie algebras
Authors: Martinez Alonso L.
Published: March 2003

This paper is a first attempt to explore the relationship between classical and quantum mechanics from a group-theoretical point of view. We deal here with the algebraic aspects of the sets of classical and quantum observables in the framework of the algebraic structures associated with finite-dimensional Lie algebras. In particular, we investigate the canonical structure of the quotient fields predicted by the Gel’fand–Kirillov and Vergne conjectures in order to study the types of observables that emerge from a given Lie algebra.
Geometrodynamics with tensor sources. IV
Authors: Kucha? Karel
Published: March 2003

We develop the Hamiltonian hypersurface dynamics of the gravitational field derivatively coupled to general tensor sources. The closing of constraints follows from the independence of the hypersurface action on the path in the space of embeddings. The derivative coupling breaks the DeWitt supermetric in Riem (𝓂).
Invariant solutions for a class of diffusion equations
Authors: Humi Mayer
Published: March 2003

We show that a class of diffusion equations, related to superradiant emission, forms a subset of a wider class of equations which are invariant with respect to some one parameter group of transformations. This property gives rise to solutions which are invariant with respect to the aforementioned group. In particular we show that a solution found by L. H. Narducci et al. [Phys. Rev. A 11, 1354 (1975)] for one of these equations is an invariant solution.
A remark on the construction of quantum fields from Markov fields
Authors: Hai Yao Te
Published: March 2003

We give a proof of the existence of φ0(g) as a continuous map from 𝒮 (Rd−1) to ℒ (ℋ̃k′, ℋ̃−k′), for k′ sufficiently large, where ℋ̃k′ belongs to the scale associated with a certain Hilbert space ℋ̃ and a certain self-adjoint operator (positive) ℋ̃ in ℋ̃, φ0(g) being a sharp time field in the process of constructing a quantum field from an Euclidean field over 𝒮 (Rd).
Classes of solvable Volterra integral equations
Authors: Luban Marshall , Deutsch Moshe
Published: March 2003

We display several classes of Volterra equations which can be transformed to the so-called faltung form and are thus solvable in terms of Laplace transforms.
Lie theory and the wave equation in space–time. 5. R-separable solutions of the wave equation ψt
t−Δ3ψ=0
Authors: Kalnins E. G. , Miller W.
Published: March 2003

A detailed classification is made of orthogonal coordinate systems for which the wave equation ψtt−Δ3ψ=0 admits an R-separable solution. Only those coordinate systems are given which are not conformally equivalent to coordinate systems that have been found in previous articles. We find 106 new coordinates to give a total of 367 conformally inequivalent orthogonal coordinates for which the wave equation admits an R-separation of variables.
Continuum calculus. II. The heterogeneous continuous functional differentiation applied to the Feynman path integral
Authors: Lee L. L.
Published: March 2003

The continuum calculus proposed previously [L. L. Lee, J. Math. Phys. 17, 1988 (1976)] is here extended to the study of continuous differentiation of a functional. The result, called HCFD, is shown to be the inverse operation of the functional integration for Feynman path integrals, in analogy to the case in ordinary differential calculus. The class of ’’separable’’ functionals is defined, which are useful in the derivation of the theory, playing a role similar to that of the characteristic functions in the Lebesgue theory of integration. A Radon–Nikodym type derivative is introduced in the definition of the continuous derivative for a general Banach algebra. This development constitutes a functional calculus of the continuum type. Comparisons with other types of functional derivatives are also made.
The topology of Euclidean Higgs fields
Authors: Woo G.
Published: March 2003

It is proved that in a class of Euclidean Higgs theories, a single index serves to describe both the gauge field and the Higgs field topologies.
Boost matrix elements and Clebsch–Gordan coefficients of the homogeneous Lorentz group
Authors: Wong M. K. F. , Yeh Hsin-Yang
Published: March 2003

It is shown that the boost matrix elements of SO(3,1) obtained by Smorodinskii and Shepelev can be written as a sum of two Fourier series, whose coefficients are Clebsch–Gordan coefficients of SO(3) with complex angular momenta and magnetic quantum numbers. Moreover, the second term is equal to zero for all representations of the principal series except the most degenerate case. The connection between our expression and the spherical functions of Dolginov and other authors is explained. It is also noted that there are two ways of writing the boost matrix elements of SO(3,1), differing from each other by a phase factor. A proof for the orthogonality relation of the representation functions of SO(3,1) is given in the Appendix. The Clebsch–Gordan coefficients of SO(3,1) for the principal series in the general case (σ1ν1) × (σ2ν2) → (σ3ν3) are obtained as X functions with complex angular momenta. An integral representation for these X functions is obtained. It is shown that the CG coefficients of SO(3,1) have a multiplicity-two problem. A solution to the multiplicity-two problem is presented.
Conformal Killing tensors in reducible spaces
Authors: Weir G. J.
Published: March 2003

It is shown that the dimension of the vector space of second order, trace-free conformal Killing tensors (CKT’s) in a Riemannian space of dimension n (?3) is bounded above by (1/12)(n−1)(n+2)(n+3)(n+4) and that this is attained in flat space. The discussion is eventually restricted to four-dimensional spaces which admit a two-dimensional, Abelian, orthogonally transitive symmetry group, as well as one nonredundant CKT. A sufficient condition is given for an empty space to be Type D.
Instantaneous Cauchy surfaces, topology change, and exploding black holes
Authors: Gowdy Robert H.
Published: March 2003

Instantaneous Cauchy surfaces are defined and several of their properties are given. Instantaneous Cauchy surfaces are achronal surfaces whose Cauchy development interiors are maximal on the set of all achronal surfaces. It is shown that topology changes in such surfaces always result in nonempty future Cauchy horizons and departures from global hyperbolicity. It is also shown that the structure which is usually assumed for the background spacetime of a self-consistent exploding black hole implies a topology change in instantaneous Cauchy surfaces.
Spin-weighted angular spheroidal functions
Authors: Fackerell Edward D. , Crossman Robert G.
Published: March 2003

The analytic properties of the spin-weighted angular spheroidal functions introduced by Teukolsky are investigated by means of a series involving Jacobi polynomials. This approach facilitates the numerical determination of eigenvalues, particularly in the case of complex frequencies.
Exact solution for the linear response of a hydrogenic atom to an external electromagnetic field. I. Frequencies below photoelectric threshold
Authors: Luban Marshall , Nudler-Blum Batya
Published: March 2003

We obtain an exact closed-form expression for the first-order correction δΨ (r,t) of the wavefunction of a hydrogenic atom of atomic number Z, initially in the ground state, due to a linearly polarized, monochromatic electromagnetic field treated in the dipole approximation. This has been obtained by employing the method of integral transformations to solve the inhomogeneous differential equation governing δΨ. We find, for angular frequencies ω below the ionization frequency ωc, that δΨ (r,t) is characterized by an exponential decrease for large r with a frequency-dependent range parameter (a0/Z)(1− ω/ωc)−1/2, where a0=h/2/me2 is the Bohr radius. As an application, starting with our result for δΨ, we have calculated the frequency dependent polarizability α (ω).
Phase space path integrals, without limiting procedure
Authors: Mizrahi Maurice M.
Published: March 2003

This paper defines path integrals in phase space without using a time-division approach followed by a limiting process, thereby generalizing a similar procedure used in configuration space. This is useful since the path integral approach cannot always be formulated in configuration space (e.g., when the Hamiltonian is arbitrary) but can always be formulated in phase space. The most general Gaussian measure, absorbing the quadratic portion of the functional to be integrated, is constructed, and large classes of path integrals are evaluated with respect to it. Applications are given to the perturbation expansion and the semiclassical (WKB) expansion for arbitrary Hamiltonians.
Generalization of Dirac’s monopole to SU2 gauge fields
Authors: Yang Chen Ning
Published: March 2003

Dirac’s monopole is generalized to SU2 gauge fields in five-dimensional flat space or four-dimensional spherical space. The generalized fields have SO5 symmetry. The method used is related to the concept of orthogonal gauge fields which is developed in an appendix. The angular momenta operators for the SO5 symmetrical fields are given.