bluuebunny/crossref_metadata_2025 huggingface dataset:
Group representations in the Liouville representation and the algebraic approach
Authors: Strand Michael P. , Berry R. Stephen
Published: March 2003

Group representations on the Liouville representation spaces are considered. It is shown that the state space ℐ1(H) of trace class operators on Hilbert space H and the observable space ℒ(H) of bounded operators are completely reducible under physically induced representations of compact Hausdorff groups when appropriate topologies are used. For state space ℐ1(H) both the norm topology and the weak topology lead to complete reducibility, while for observable space ℒ(H) the weak-* topology—but not the norm topology—suffices. This leads to conservation laws, selection rules, and Wigner–Eckart theorems for the Liouville representation. It is shown that serious difficulties are encountered when a similar theory is attempted on the observable space and state space used in the algebraic approach.
Maximum of the spin-flip cross section from unitarity and three constraints
Authors: Sakmar I. A.
Published: March 2003

A sharper upper bound on the spin-flip cross section is found by applying the variational calculus with three equality constraints and unitarity. These are the total cross section, elastic cross section, and the forward slope. Unitarity of the partial waves provides the inequality constraints.
Building things in general relativity
Authors: Geroch Robert
Published: March 2003

In the context of general relativity, there is a sense in which certain objects cannot be constructed, using reasonable matter, from normal initial conditions. An attempt is made to capture this sense as a definition. The implications of such a definition, along with some related results and open questions, are discussed.
Relationship between the inverse scattering techniques of Belinskii–Zakharov and Hauser–Ernst in general relativity
Authors: Cosgrove Christopher M.
Published: March 2003

We make a quantitative comparison between the pure-nonsoliton part of the inverse scattering method of Belinskii and Zakharov (BZ) and the homogeneous Hilbert problem of Hauser and Ernst (HE), these being two independent representations of an infinite-dimensional subgroup 𝒦 of the Geroch group K of invariance transformations for spacetimes with two commuting Killing vectors. An explicit formula for the BZ representing matrix function G0(λ) in terms of the HE representing matrix function u(t) is derived. It is shown how certain solution generating techniques (e.g., Harrison’s Bäcklund transformation, HKX transformation, generation of Weyl solution from flat space, generation of n-Kerr–NUT solution from n-Schwarzschild) can be derived directly from the BZ formalism, including the soliton part in some cases, thereby bringing our understanding of the BZ formalism up to the level of the more fully developed HE formalism. A technical point which needed to be resolved along the way was how to analytically continue the complex matrix potential F(t) across a quadratic branch cut and onto the second Riemann sheet. Finally, we consider how the subgroup 𝒦⊆K represented by the BZ and HE formalisms can be enlarged either by simple limiting transitions or by relaxing boundary conditions.
Geometric quantization and UIR’s of semisimple Lie groups. II. Discrete series
Authors: Sitaram B. R. , Tripathy K. C.
Published: March 2003

The Auslander–Kostant induction scheme is extended to yield the discrete series UIR’s of G (G semisimple) in the L2-cohomology group ℋi0 (ℒλ), where i = (1/2)(dim𝓀−rankℊ) or i = (1/2)dim 𝓅. We show that this involves a choice of complex structure in each Weyl chamber which is intimately connected with the choice of a positive polarization at that point. We illustrate our results with the example of Spin (4,2).
Application of a new functional expansion to the cubic anharmonic oscillator
Authors: Fliess Michel , Lamnabhi-Lagarrigue Françoise
Published: March 2003

A new representation of causal functionals is introduced which makes use of noncommutative generating power series and iterated integrals. This technique allows the solutions of nonlinear differential equations with forcing terms to be obtained in a simple and natural way. It generalizes some properties of Fourier and Laplace transforms to nonlinear systems and leads to effective computations of various perturbative expansions. Illustrations by means of the cubic anharmonic oscillator are given in both the deterministic and the stochastic cases.
Densities of covariant observables
Authors: Cattaneo U.
Published: March 2003

Positive-operator-valued observables in a complex Hilbert space H based on a topological homogeneous space G/H and G-covariant with respect to a strongly continuous unitary representation on H of a locally compact group G are investigated. The goal is to determine whether they are the weak integrals of some positive-operator-valued densities relative to a G-quasi-invariant measure μ on G/H. To this end, the notion of a kernel Hilbert space is generalized to spaces of equivalence classes with respect to μ of mappings of G/H into a complex Hilbert space K.
Generalized completeness relations in the theory of resonant scattering
Authors: Zohni O.
Published: March 2003

Generalized completeness relations involving resonance states are constructed within the framework of analytically continued symmetrized scattering kernels into the unphysical sheet of the complex-energy plane. The bases states utilized are identified with complex-energy generalized eigenvectors over an extended or rigged Hilbert space. The resulting relations are uniquely defined and do not exhibit the usual divergence problems encountered with the regularization methods.
Existence theorem for solutions of Witten’s equation and nonnegativity of total mass
Authors: Reula Oscar
Published: March 2003

We prove that given an asymptotically flat (in a very weak sense) initial data set, there always exists a spinor field that satisfies Witten’s equation and that becomes constant at infinity. Thus we fill a gap in Witten’s arguments on the nonnegativity of the total mass of an isolated system, when measured at spatial infinity. We also include a review of Witten’s argument.
Complex orthogonal and symplectic matrices depending on parameters
Authors: Patera J. , Rousseau C.
Published: March 2003

Versal deformations of elements of complex orthogonal and symplectic Lie algebras are studied. For a general element M of o(n,C) or sp(2n,C), a normal form MA is found which, unlike the Jordan normal form MJ, depends holomorphically on M and on the similarity transformation MA = gMg−1 from the corresponding group. Orthogonal and symplectic cases are treated simultaneously in order to underline their close relation. Bundles of matrices of low codimension are listed and bifurcation diagrams of two-parameter families of orthogonal matrices are shown. Finally, versal deformations of all elements of o(6,C) are explicitly shown.
A theorem on A-proper mappings and its application in scattering theory
Authors: Kröger Helmut , Perne Rainer
Published: March 2003

We propose a projectionally complete scheme yielding an approximate solution of the functional equation Bf = g in a Hilbert space. We prove that B is an A-proper mapping. The result is applied to an integral equation with a kernel appearing in multichannel scattering theory.
Bargmann transform, Zak transform, and coherent states
Authors: Janssen A. J. E. M.
Published: March 2003

It is well known that completeness properties of sets of coherent states associated with lattices in the phase plane can be proved by using the Bargmann representation or by using the kq representation which was introduced by J. Zak. In this paper both methods are considered, in particular, in connection with expansions of generalized functions in what are called Gabor series. The setting consists of two spaces of generalized functions (tempered distributions and elements of the class S*) which appear in a natural way in the context of the Bargmann transform. Also, a thorough mathematical investigation of the Zak transform is given. This paper contains many comments and complements on existing literature; in particular, connections with the theory of interpolation of entire functions over the Gaussian integers are given.
Valleys and fall lines on a Riemannian manifold
Authors: Rowe D. J. , Ryman A.
Published: March 2003

The concepts of fall lines, ridges, and general stationary paths are defined for a potential energy function on a Riemannian manifold. Some theorems governing their properties and relationships are derived. These concepts are of interest in the classical mechanics of constrained systems and in the theory of collective motions in many-body quantum mechanics.
Hamiltonian perturbation theory in noncanonical coordinates
Authors: Littlejohn Robert G.
Published: March 2003

The traditional methods of Hamiltonian perturbation theory in classical mechanics are first presented in a way which clearly displays their differential-geometric foundations. These are then generalized to the case of noncanonical in phase space. In the new method the Hamiltonian H is treated, not as a scalar in phase space, but as one component of the fundamental form p dq−Hdt. The perturbation analysis is applied to this entire form, in all of its components.
Electromagnetic multipole propagation in a homogeneous conducting wholespace
Authors: Weir G. J.
Published: March 2003

The electric and magnetic field components are expanded in terms of spin weighted spherical harmonics, thereby defining a multipole structure whose propagation through a homogeneous conducting wholespace is obtained. For current sources contained inside the unit sphere, the resultant exterior electromagnetic fields are uniquely associated with either currents from electrodes placed inside the unit sphere, or from insulated wires inside the unit sphere. The former fields have transverse magnetic fields, while the latter (with the exception of the magnetic monopole) have transverse electric fields. When displacement currents are ignored, the fields from step function current sources are described by the incomplete gamma functions, while the exact solutions containing the hyperbolic contributions are obtained from convolutions with integer order modified Bessel functions and functions dependent on the temporal behavior of the current source.
Nonrecurrence of the stochastic process for the hydrogen atom problem in stochastic electrodynamics
Authors: Claverie Pierre , Soto Francisco
Published: March 2003

It is shown, following a criterion borrowed from Khas’minskii, that the stochastic process associated with the (approximate) Fokker–Planck equation of the hydrogen atom problem in stochastic electrodynamics (SED) is nonrecurrent and therefore also nonergodic. The demonstration of this nonrecurrence property does not use any explicit solution. The property implies, among other things, that all the invariant measures of the process will be nonfinite. Some remarks concerning the consequences for SED are made.
Pseudoscalar interaction of coupled quantum-mechanical oscillators with independent Fermi systems
Authors: Battle III Guy A.
Published: March 2003

Using the techniques of constructive quantum field theory we analyze the dynamics of a cubic lattice of quantum mechanical oscillators with nearest-neighbor coupling that interact with a corresponding lattice of finite-volume truncations of independent relativistic Fermi fields. Since the model is nonrelativistic, we rely on a nonrelativistic version of the Osterwalder–Schrader (OS) reconstruction theorem. Also, the absence of Nelson’s symmetry in the (un)Euclidean picture is not serious because the transfer matrix in a given space direction is simple enough to make the verification of spatial OS positivity easy. After establishing for our model many of the basic results that hold for the more standard models, we give a proof of the Fortuin–Kasteleyn–Ginibre (FKG) inequality that is essentially independent of the dimension of the Fermi systems.
Unrenormalized Schwinger–Dyson equations and dynamical mass generation
Authors: Yoshimura Tetz
Published: March 2003

In dynamical mass generation of fermion fields by fermion-spin-1-boson interactions, anomalous magnetic moment-like terms in three-point functions play an essential role, so that the Schwinger–Dyson equations do not admit a trivial solution, i.e., a solution without dynamically generated mass. Applicability of Altman’s and Lika’s versions of nonlinear operator theory to unrenormalized Schwinger–Dyson equations is discussed and algorithms for construction of approximate solutions are proposed.
Higher Legendre transforms and their relationship to Bethe–Salpeter kernels and r-field projectors
Authors: Cooper Alan , Feldman Joel , Rosen Lon
Published: March 2003

We analyze the structure of the higher Legendre transforms Γ(r){A}(r?1) of the generating functional G of the connected Green’s functions Gn in Euclidean boson field theories. In addition to the vertex functions, Γ(r) generates a variety of objects of interest for their r-irreducibility in certain channels, e.g., r-irreducible expectations, rth order Bethe–Salpeter kernels, and r-field projectors. Our analysis is independent of perturbation theory, our definition of r-irreducibility being based on Spencer’s idea of t-lines. We derive formulas for ∂ntΓ(r){A;t} (in terms of either δnAΓ(r){A;t} or the Gn’s) to be used as input in the proofs of r-irreducibility. For the case of the weakly coupled P(φ)2 model, we establish the existence of the moments δnAΓ(r){0;t} and their regularity in t.
On the topological meaning of magnetic charge quantization
Authors: Quirós M. , Mittelbrunn J. Ramírez , Rodríguez E.
Published: March 2003

In this paper, all inequivalent principal fiber bundles with base space S2 and fiber S1, corresponding to Milnor classification, are constructed. Electromagnetism with magnetic monopoles is geometrically described in terms of these fiber bundles. In each of them, the coincidence of winding number and Chern number is established. The Dirac quantization condition is shown to have a topological origin. In fact, magnetic monopoles with charge g = (h/c/2e)n correspond, in a topological sense, to the fiber bundle with winding number equal to n.
The eikonal equation: some results applicable to computer vision
Authors: Bruss Anna R.
Published: March 2003

In this paper we investigate certain first order partial differential equations which formulate the relationship between the light reflected from a surface and its shape. Particular emphasis is given to eikonal equations. Two results are presented. First, we prove that a special type of eikonal equation has only one convex and positive C2 solution in some neighborhood of a singular point. Using this result, we show that a restricted form of this equation has exactly two solutions. These results have application in scanning electron microscopy.
Quasicontinual representation of a field given on a finite lattice. Interpolation properties
Authors: Garczyński Wl/odzimierz , Stelmach Jerzy
Published: March 2003

A general theory is systematically presented for handling sets of discrete quantities distributed over finite lattice–lattice functions. This is a more general case than we treated before and in the limit when a lattice becomes infinite our previous results are recovered. We find a one-to-one correspondence between lattice functions which interpolate between discrete lattice values. The uniqueness of interpolations of lattice functions with some entire analytic functions, forming the quasicontinua QC(a,b) and QC(b,a)*, is a key result used.
On the determination of πN phase shifts from isospin constraints and fixed t analyticity
Authors: Stefanescu I. Sabba
Published: March 2003

This paper treats the problem of the unique and constructive determination of πN scattering amplitudes by means of fixed t dispersion relations and isospin invariance from data on the differential cross section and polarization of the elastic and charge exchange reactions. The conclusion is that the amplitudes can be uniquely constructed in a neighborhood (−α, a) of t = 0 if they are continuous functions of both real s and t for s on the cuts and 4μ2≳t≳−α0, α0≳α and there exists ν0 such that, for ‖ν‖?ν0, both ‖A+(ν,t)‖/‖A+(ν,t)+νB+(ν,t)‖ <const, t∈(−α0,4μ2) (as indicated by spin rotation measurements) and ‖Re(A++νB+)‖/Im(A++νB+)<const, t∈(−α0,0). This result is specific for πN scattering, as it depends on a small number of experimental facts concerning this process, which are enumerated in the text.
Scattering of electromagnetic waves from random media with multiple scattering included
Authors: Fante Ronald L.
Published: March 2003

Closed-form results which include all orders of multiple scattering have been obtained for the field and intensity when an arbitrary electromagnetic wave is incident on a randomly inhomogeneous medium. In obtaining the results it is assumed that the magnitude of the permittivity fluctuations is small in comparison with unity and the spatial extent of the fluctuations is large in comparison with the wavelength. Simplified approximations, which are valid outside the backscatter cone, are then obtained. These approximations are found to depend only on the second moments of the field in the medium, and are valid if the mean free path between photon scatterings is large in comparison with the size of the inhomogeneities.
An upper bound on the number of eigenvalues of an infinite dimensional Jacobi matrix
Authors: Geronimo Jeffrey S.
Published: March 2003

An upper bound is given on the number of eigenvalues of a class of infinite dimensional Jacobi matrices. The theorem presented is a discrete analog of the celebrated result of V. Bargmann on the number of bound states of the Schrödinger equation.
Permutational properties of the generalized Clebsch–Gordan coefficients
Authors: Chatterjee R. , Lulek T.
Published: March 2003

The properties of the permutational symmetry of the 3Γγ symbols for an arbitrary compact group have been generalized to a multiple Kronecker product of irreducible representations. It has been shown that these properties of the corresponding NΓγ symbols under a permutation of their columns are related to a choice of the coupling schemes for the appropriate polyads Γ1⋅⋅⋅ΓN. The simplest permutational symmetry, i.e., associated with a Young diagram for the symmetric group ΣN, is, in general, incompatible with schemes having definite intermediate representations and, in the case of mixed symmetry, does not preserve the absolute value of a NΓγ symbol. The case of N = 4 for SU(2) has been studied in detail.
Variational calculation of the multipole potential from an arbitrary localized charge distribution
Authors: Csavinszky P.
Published: March 2003

Very recently Painter has developed a method for solving Poisson’s equation as a set of finite-difference equations for an arbitrary localized charge distribution ρ(r) that is expanded in a partial-wave representation as ρ(r) = 𝒥LρL(r)YL(r̂), where L denotes l and m. In the present work a variational principle is established, and a possible approach is outlined, for obtaining approximate partial-wave coefficients VL(r) of the potential V(r) = 𝒥LVL(r)YL(?).
A nonstandard infinite dimensional vector space approach to Gaussian functional measures
Authors: Pecora Louis M.
Published: March 2003

Nonstandard analysis is used to apply the usual concepts of finite dimensional vector spaces in order to define various Gaussian functional measures without using a limiting process. The approach is to define matrices on the infinite dimensional space and use straightforward techniques to determine the properties of the functions the measures are concentrated on. The power of nonstandard analysis allows one to work directly with infinite and infinitesimal quantities and ’’visualize’’ certain sets upon which the basic Gaussian form is or is not infinitesimal. The relation of the choice of Gaussian to the function (process) properties remains heuristic since a proof of the Holder continuity of the Weiner paths is not complete, but remains at a local (infinitesimal) level. However, given the Holder continuity the nonbounded variation property easily follows. Higher derivative Gaussian measures are also easily developed and their analytic properties displayed along with their covariances. By using Fourier analysis on finite abelian groups transferred to the nonstandard universe and applied to hyperfinite abelian groups a rigorous transformation from the discrete form of the Weiner measure to a Fourier series form is accomplished. It is shown here that the functions (processes) are infinitesimally close to those of the discrete version of the measure. The Fourier approach is also extended to more general measures. Finally, some speculations show directions in which this direct approach to Gaussian functional measures can be extended and generalized.
Quantum theory as a probability theory on Hilbert space
Authors: Cyranski John F.
Published: March 2003

It is shown how quantum theory (QT) can be expressed as a probability theory on Hilbert space, treated as a measure space. The approach generalizes the work of Bach and clarifies the ’’generalized trace’’ of Langerholc. It permits the description of both bounded and unbounded observables as measurable functions. Dynamical evolution can be described in terms of stochastic processes.
A probabilistic formulation of quantum theory. II
Authors: Bach Alexander , Wenning Thomas
Published: March 2003

An integral representation of the quantum mechanical expectation which recently has been set up for bounded operators is generalized to unbounded operators by means of representation and integration in Gel’fand triplets. For Gaussian measures it turns out that a time translation invariant measure exists which, contrary to classical statistical mechanics, is singular w.r.t. the measures which represent the states.
Invariant *-quantization associated with the affine group
Authors: Huynh T. V.
Published: March 2003

Invariant *-quantization means quantization with the introduction of invariant *-products on C∞ functions on phase space. A systematic method of constructing invariant *-products for dynamical Lie groups is presented. It involves the notions of invariant Wigner correspondence and *-representation. As an illustration, all possible invariant *-products, in an integral form, are constructed for the affine group of the real line. The Moyal product is recovered as a special case.
Para-Bose and para-Fermi operators as generators of orthosymplectic Lie superalgebras
Authors: Palev Tchavdar D.
Published: March 2003

It is shown that the relative commutation relations between n pairs of para-Fermi operators and m pairs of para-Bose operators can be defined in such a way that they generate the simple orthosymplectic Lie superalgebra B(n,m). In a case of ordinary statistics this leads to mutually anticommuting Bose and Fermi fields.
Scattering by periodic surfaces
Authors: Sáenz A. W.
Published: March 2003

If Ω⊆ Rν (ν?2) is an exterior domain containing a half-space and contained in a half space, we prove that the wave operators W± = s-limt→±∞ exp(itH) Jexp(−itH0) are partial isometries and that the invariance principle W±(φ) = W± holds for suitable real functions φ on R (’’admissible’’ functions). Here H0 is the negative (distributional) Laplacian in L2(Rν) (ν?2); H is HD or HN , the negative Dirichlet or Neumann Laplacians in ℋ = L2(Ω), respectively; J is an appropriate identification operator; and W± (φ) are defined as were W± , but with H0 and H replaced by φ(H0) and φ(H), respectively. Suppose, in addition, that Ω has a suitable periodicity property and that when H = HN it has a certain mild local compactness property. Then we prove: (1) that W± are asymptotically complete, in the sense that Ran W± = ℋscatt(H) = ℋ○ℋsurf(H) ; (2) that ℋscatt(φ(H)) = ℋscatt(H) and ℋsurf (φ(H)) = ℋsurf(H) for each ’’admissible’’ function φ, and hence that W± (φ) are asymptotically complete in a similar sense for each such φ. Here, for any self-adjoint operator A in ℋ, ℋscatt(A), and ℋsurf(A) are naturally defined, mutually orthogonal subspaces of scattering and surface states of ℋ, respectively. No smoothness assumptions on ∂Ω are made in this paper. Its results entail the asymptotic completeness, in a physically very satisfactory sense, of wave operators describing acoustic and certain types of electromagnetic scattering by a very wide class of periodic surfaces.
Spinor sources in cosmology
Authors: Jantzen Robert T.
Published: March 2003

An alternative Hamiltonian formulation is presented for the spatially homogeneous Einstein–Dirac system which in the nondegenerate case enables the number of gravitational degrees of freedom to be explicitly reduced to two.
Self-dual Kerr–Schild metrics and null Maxwell fields
Authors: Tod K. P.
Published: March 2003

A construction for self-dual solutions of Einstein’s equations with metric in Kerr–Schild form from null electromagnetic fields in Minkowski space is described. Two examples are given.
The metric dependence of four-dimensional formulations of electromagnetism
Authors: Birss R. R.
Published: March 2003

Three-dimensional and four-dimensional formulations of electromagnetism are compared with particular regard to whether it is necessary to postulate a metric locally (i.e., to define an inner product in the tangent space). In three dimensions, a metric-independent formulation is possible but in four dimensions the required set of fundamental differential forms can not be defined by experiment on a differential manifold without a metric.
Stochastic optimal control and quantum mechanics
Authors: Papiez Lech
Published: March 2003

Classical mechanics is formulated as a kind of deterministic optimal control. The simplest, from the optimal control point of view, stochastic generalization of classical mechanisms is submitted. The connections between stochastic mechanics and quantum mechanics are shown, thanks to the stochastic optimal control method.
On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian systems
Authors: Fokas A. S. , Anderson R. L.
Published: March 2003

We present an algorithmic method for obtaining an hereditary symmetry (the generalized squared-eigenfunction operator) from a given isospectral eigenvalue problem. This method is applied to the n×n eigenvalue problem considered by Ablowitz and Haberman and to the eigenvalue problem considered by Alonso. The relevant Hamiltonian formulations are also determined. Finally, an alternative method is presented in the case two evolution equations are related by a Miura type transformation and their Hamiltonian formulations are known.
Electrovac generalization of Neugebauer’s N = 2 solution of the Einstein vacuum field equations
Authors: Guo Dong-sheng , Ernst Frederick J.
Published: March 2003

We show that the N = 2 Neugebauer solution of the Einstein vacuum field equations is easily reproduced by employing two successive Kinnersley–Chitre (K–C) transformations of a type considered earlier by I. Hauser. Furthermore, by employing two successive K–C transformations of a type considered recently by C. Cosgrove we are able to produce a new electrovac generalization of the N = 2 Neugebauer solution. In principle, an analogous approach could be employed for the explicit construction of electrovac generalizations of Neugebauer solutions corresponding to higher values of N.
New exact static solutions to Einstein’s equations for spherically symmetric perfect fluid distributions
Authors: Ibañez J. , Sanz J. L.
Published: March 2003

New exact solutions to Einstein’s equations are given which are spherically symmetric and static with perfect fluid distributions satisfying a linear equation of state p = nρ and n∈(0,1]. Heintzmann’s generating method is then used to build up a family of new solutions for each value of n.
Determination of basis for the irreducible representations of the unitary group for U( p+q)↓U( p)⊗U( q)
Authors: Sarma C. R.
Published: March 2003

A direct procedure is outlined for determining the basis spanning finite dimensional irreducible representations of U( p+q) adapted to the subgroup U( p)⊗U( q). Using a tableau based analysis, it is shown that the realization of the semimaximal states follows readily from a knowledge of the matrix elements of the generators Ei+1 of U( p), U( q)⊆U( p+q).
Branching rules for the supergroup SU(N/M) from those of SU(N+M)
Authors: Balantekin A. Baha , Bars Itzhak
Published: March 2003

The decomposition of representations of supergroups into representations of subgroups is needed in practical applications. In this paper we set up and exploit a fruitful one-to-one correspondence between the Lie group branching SU (N+M)⊇SU(N)⊗SU(M)⊗U(1) and the supergroup branchings SU(N/M)⊇SU(N)⊗SU(M)⊗U(1) and SU(N1+N2/M1+M2)⊇SU(N1/M1) ⊗SU(N2/M2)⊗U(1). A simple and useful prescription is discovered for obtaining the SU(N/M) branching rules from those of SU(N+M) for any representation. A large class of examples, sufficient for many physical applications we can foresee, are explicitly worked out and tabulated.
New sum rule for products of Bessel functions with application to plasma physics
Authors: Newberger Barry S.
Published: March 2003

In our investigations of the linear theory of the stability of relativistic beam-plasma systems immersed in a magnetic field we have been led to consider sum rules for an infinite series of products of Bessel functions of the form 𝒥∞n = −∞(n jJ2n)/ (n+μ). In this work we report on the sum of this series treated as a special case of a more general infinite series. We also mention the extension of the results beyond the range of the parameters for which formulae are explicitly given and indicate how intermediate results obtained may be useful in their own right. Finally, an additional application of our result is indicated.
Theory of nonbijective canonical transformations in mechanics: Application to the Coulomb problem
Authors: Boiteux M.
Published: March 2003

It is shown that the study of nonbijective transformation requires a fiber bundle formulation of mechanics. The conditions upon which nonbijective canonical point transformations can be defined are given. Then, as an example, we apply that theory to the study of the Coulomb problem in two and three dimensions. The Hopf fibration leads to an inverse harmonic oscillator problem. Since the completion of this work, a paper by G. H. Ringwood and J. T. Devreese has been published in J. Math. Phys. 21, 1390 (1980), dealing with the same problem. Their work is based on the construction of propagators in quotient spaces. The identity between the propagator prescriptions and nonlinear canonical transformations is not automatically fulfilled. Therefore it seems that the reliability of their results is not due to their method, which in general is not correct, but to an underlying property of the transformation used, namely the Kustaanheimo-Stiefel map (see our results).
Continued fraction theory of the rotating harmonic oscillator
Authors: Singh Virendra , Biswas S. N. , Datta K.
Published: March 2003

We study the rotating-vibrating system, consisting of the rotating harmonic oscillator, using the analytic theory of continued fractions. We prove that there is a convergent continued fraction representation of the Green’s function which is analytic in the complex coupling constant plane, except for a cut along the negative real axis. The perturbation series for the Green’s function is unambiguously defined by the continued fraction but diverges on account of an essential singularity at the origin. An infinite but incomplete set of exact solutions for certain specific valeus of the coupling follows from the representation of the Green’s function as a continued fraction. Finally, we use Worpitzky’s theorem in continued fraction theory to show that in the strong coupling limit α→0+ (α being the inverse of the coupling parameter), there exists a lower bound to all energy eigenvalues for a given value of l, the orbital angular momentum.
Explicit solutions of the conformal scalar equations in arbitrary dimensions
Authors: Cerveró José M.
Published: March 2003

Solutions of the equations of motion derived from the scalar conformal invariant Lagrangian in arbitrary dimensions are found. The solutions are invariant under the maximal compact subgroup of the corresponding conformal group. They have finite energy and action. In the case N = 2, we also find noticeable topological properties of the solutions.
Relativistic field theories in three dimensions
Authors: Binegar Birne
Published: March 2003

We provide a complete classification of the unitary irreducible representations of the (2+1)-dimensional Poincaré group. We show, in particular, that only two types of ’’spin’’ are available for massless field theories. We also construct generalized Foldy–Wouthuysen transformations which connect the physical UIR’s with covariant field theories in three dimensions.
The nonlinear Schrödinger equation as a Galilean-invariant dynamical system
Authors: Alonso L. Martínez
Published: March 2003

The invariance of the nonlinear Schrödinger equation under the Galilei group is analyzed from the point of view of the inverse scattering transform. It is shown that this group induces an infinite-dimensional nonlinear canonical realization which is locally equivalent to a direct product of the two well-known Galilean actions describing classical particles and the free Schrödinger equation.
An analysis of Taylor’s theory of toroidal plasma relaxation
Authors: Faber V. , White A. B. , Wing G. M.
Published: March 2003

The Taylor theory of toroidal plasma relaxation is considered as a nonlinear eigenvalue problem. The analysis is rigorous and applies to quite general toroidal cross sections. Emphasis is placed on the symmetric state case, where the existence of field reversal and flux free states is demonstrated. Certain anomalies are revealed by the mathematical treatment and their significance is studied. Existence of a solution to the Taylor problem in the symmetric state is proved and the location of the eigenvalue of this solution state relative to other states is examined. The question of the existence of helical states is not resolved, but it is shown that in many respects any helical states behave like the anomalous cases in the symmetric problem and hence do not significantly affect the theory.
Finite sum approximations to Brillouin zone integrals with symmetrized plane waves
Authors: Folland N. O. , Osman Junaidah
Published: March 2003

Techniques are presented for expanding a periodic function of cubic translational symmetry and arbitrary rotational symmetry in a finite set of symmetry-adapted plane waves. Results for all cubic lattices are tabulated in a form convenient for use in computation. The role of ’’special points’’ in the sense of Chadi and Cohen is discussed in the extended context of symmetry-adapted plane waves.
Structure and matrix elements of the degenerate series representations of U(p+q) and U(p,q) in a U(p)×U(q) basis
Authors: Klimyk Anatoli U. , Gruber Bruno
Published: March 2003

The representations of the most degenerate series of the group U(p,q) which are induced by the representations of the maximal parabolic subgroup are considered in this article. By making use of the infinitesimal operators of these representations in the U(p)×U(q) basis the conditions are derived which are necessary and sufficient for irreducibility. For the reducible representations we describe their structure (composition series). We select from among the irreducible representations which are obtained in this article all representations of U(p,q) which admit unitarization. As a result we obtain the principal degenerate series, the supplimentary degenerate series, the discrete degenerate series, and the exceptional degenerate series of unitary representations of U(p,q). The U(p)×U(q) spectrum of the representations of U(p+q) with highest weights (λ1, 0,..., 0, λ2) is defined. We obtain the integral representation for the matrix elements of the degenerate representations of U(p,q) in the U(p)×U(q) basis. The matrix elements of the irreducible representations of U(p+q) with highest weights (λ ,0,...,0), (0,...,0, λ) are evaluated in the U(p)×U(q) basis.
Congruence classes of finite representations of simple Lie superalgebra
Authors: Lemire F. W. , Patera J.
Published: March 2003

The concept of congruence of representations of Lie algebras is generalized and applied to the finite-dimensional representations of Lie superalgebras.
Supersymmetry and Lie algebras
Authors: Levine R. Y. , Tomozawa Y.
Published: March 2003

Starting from the standard supersymmetry algebra, an infinite Lie algebra is constructed by introducing commutators of fermionic generators as members of the algebra. From this algebra a finite Lie algebra results for fixed momentum analogous to the Wigner analysis of the Poincaré algebra. It is shown that anticommutation of the fermionic charges plays the role of a constraint on the representation. Also, it is suggested that anticommuting parameters can be avoided by using this infinite Lie algebra with fermionic generators modified by a Klein transformation.
General superposition of solitons and various ripplons of a two-dimensional nonlinear Schrödinger equation
Authors: Nakamura Akira
Published: March 2003

We consider the two-dimensional nonlinear Schrödinger equation of Benney–Roskes. It is shown that the equation admits superposition solutions of solitons and various ripplons.
Korteweg-de Vries surfaces and Bäcklund curves
Authors: Kupershmidt B. A.
Published: March 2003

It is shown that every point w(ε) on the curve γr (ε) representing a 1-parameter family of integrable equations containing a given rth Korteweg–de Vries (KdV) equation w(0), also belongs to a different integrable curve Γr(ε,ν) . Symmetries of the resulting surface make it possible to construct a curve of Bäcklund transformations, that is, infinitesimal automorphisms, of points on γr (ε) starting with the usual infinitesimal automorphisms of w(0). In addition, we obtain four new Bäcklund transformations of the second order for all higher KdV equations.
Interior and exterior solutions for boundary value problems in composite elastic media. I. Two-dimensional problems
Authors: Jain D. L. , Kanwal R. P.
Published: March 2003

We study two-dimensional problems of elasticity when a homogeneous and isotropic solid of an arbitrary shape is embedded in an infinite homogeneous isotropic medium of different properties. Solutions are obtained both inside the guest and host media. These solutions are derived by first transforming the boundary value problems to the equivalent integral equations. The interior displacement field is obtained by a simple method of truncation. By this method the integral equations are recast into an infinite number of algebraic equations and a systematic scheme of solutions is constructed by an appropriate truncation. The exterior solutions are obtained by substituting the interior solutions in the integral equations valid for the entire medium. The boundaries considered are rectangular cylinder, equilateral triangular prism, and elliptic cylinder and its limiting configurations. It emerges that the solutions for the elliptic cylinder and its limiting configurations are exact.
Time-dependent scattering by a bounded obstacle in three dimensions
Authors: Boström Anders
Published: March 2003

In the prsent paper we introdue a new method of treating the scattering of transient fields by a bounded obstacle in three-dimensional space. The method is a generalization to the time domain of the null field approach first given by Waterman. We define new sets of time-dependent basis functions, and use these to expand the free space Green’s function and the incoming and scattered fields. The scattering problem is then reduced to the problem of solving a system of ordinary differential equations. One way of solving these equations is by means of Fourier transformation, and this leads to an efficient way of obtaining the natural frequencies of the obstacle. Finally, we have calculated the natural frequencies numerically for both a spheroid and a peanut-shaped obstacle for various ratios of the axes.
Feynman path integrals of operator-valued maps
Authors: Parthasarathy K. R. , Sinha K. B.
Published: March 2003

Feynman integral is defined for operator-valued maps which are Fourier transforms of an operator-valued measure of bounded variation. Such an integral is then used to describe perturbation of certain unitary groups by certain cocycles.
Dynamical importance of vorticity and shear in the universe
Authors: Sanz J. L.
Published: March 2003

We study the dynamical importance of vorticity, ω2/ρ, assuming different upper limits on the relative shear, σ/ϑ, for a general relativistic model with a content represented by a perfect fluid distribution with a linear equation of state. Adopting a very conservative point of view with respect to the values of the Hubble constant and the density parameters of matter and radiation, we obtain that either σ/ϑ≳7% or there was a bounce at some point in the past during the matter era if the present-day relative vorticity is (ω/ϑ)0≳4%. Taking into account the latest results on singularities, the possibility of a bounce must be regarded from a local point of view.
The Melnick–Tabensky solutions have high symmetry
Authors: Bonnor W. B. , MacCallum M. A. H.
Published: March 2003

Melnick and Tabensky recently gave a class of static perfect fluid solutions of Einstein’s equations in which the metric (in co-moving coordinates) takes the conformastat form. In this paper we point out that all these solutions have spherical or plane or pseudospherical symmetry.
Manifestly covariant equations of motion for a particle in an external field
Authors: Lim P. H.
Published: March 2003

A relativistic variational principle for a particle in an external field is developed both in flat spacetime and in curved spacetime. In flat spacetime Kalman’s equations follow from the variational principle, and their relationship to four-dimensional Euler–Lagrange equations is clarified. It is shown that Kalman’s equations are uniquely defined and that they may be recast into a generalized Hamiltonian formalism. The equations of motion arising from the curved-spacetime variational principle are shown to be uniquely defined.
Conditional symmetries in parametrized field theories
Authors: Kuchař Karel
Published: March 2003

In parametrized field theories, spacelike hypersurfaces and fields which they carry are evolved by a Hamiltonian which is a linear combination of the super-Hamiltonian and supermomentum constraints. We say that a dynamical variable K generates a conditional symmetry of the Hamiltonian when it is linear both in the hypersurface and the field momenta and its Poisson bracket with the Hamiltonian vanishes by virtue of the constraints. Generators are classified by their dependence on the momenta: P-restricted generators depend only on the hypersurface momenta, π-restricted generators depend only on the field momenta, while mixed generators depend on both kinds of momenta. Conditional symmetries in a parametrized Hamiltonian theory are then linked either with ordinary symmetries (isometries, conformal motions, or homothetic motions) of the spacetime background, or with internal symmetries of the fields. In particular, we prove that a generic field with nonderivative gravitational coupling and a quadratic energy density has a P-restricted conditional symmetry if and only if the spacetime background has a Killing vector, while a field with a trace-free energy–momentum tensor has a P-restricted conditional symmetry if and only if the background has a conformal Killing vector. An algorithm allowing us to enumerate all possible mixed conditional symmetries in a given parametrized field theory is explained on an example of the Klein–Gordon field. These results complement our previous proof that canonical geometrodynamics does not possess any conditional symmetry.
Spatial topology and Yang–Mills vacua
Authors: Isham C. J. , Kunstatter G.
Published: March 2003

We study the canonical vacuum structure of Yang–Mills theories defined on an arbitrary (nonsimply connected) three-space. We find that the presence of flat Yang–Mills connections with a nontrivial (discrete) holonomy group has profound consequences at the quantum level. In particular, such connections may lead to either an increase or a decrease in the number of quantum vacuum sectors. Our method consists of finding a representation for the space of classical zero-energy field configurations in terms of a function space 𝒟. A simple assumption concerning the physical equivalence of these classical configurations then permits a formal classification of the quantum vacua by the zeroth homotopy set π0(𝒟). Significant progress is made in the analysis of π0(𝒟) for arbitrary three-spaces and gauge groups, and several specific questions concerning the vacuum states and their diagonalization are answered.
Boson operator representation of Brownian motion
Authors: Steiger Ulrich R. , Fox Ronald F.
Published: March 2003

In the framework of the classical theory of Brownian motion the time evolution of the distribution function in the full phase space of a particle immersed in a fluid is governed by a Fokker–Planck equation. The reduced distribution function in coordinate space fulfills the Smoluchowski equation in first approximation. This work improves previous derivations by including higher order corrections and by using an expansion which permits the discussion of the size of the error made by truncating the infinite series. The derivation is based on the adaption of a powerful mathematical tool used in quantum field theory: The Fokker–Planck equation is written in terms of boson operators. Conditional equilibrium averages of operators are defined which play the role of vacuum expectation values. The time-ordered cumulant expansion is used to calculate the formal diffusion operator in terms of conditional equilibrium averages of powers of the ’’Liouville operator in the interaction picture.’’ It is shown that all these averages can be obtained from a Gaussian generating functional which is explicitly calculated using the time-ordered version of Glauber’s theorem. The resulting diffusion equation, a fourth order partial differential equation in the position space, is obtained by calculating the cumulant expansion up to sixth order. Conditions on the potential are established which guarantee that these equations are dissipative and it is shown that all solutions approach the Boltzmann distribution as t→∞. Curvilinear, non-Euclidean coordinates are introduced in order to interpret these diffusion equations. Nonlinear diffusion equations and their application regarding the self-avoiding random walk are discussed.
Scalar interactions of supersymmetric relativistic spinning particles
Authors: Van Alstine Peter , Crater Horace
Published: March 2003

We introduce scalar interactions for the relativistic spinning particle in such a way as to preserve a supersymmetry that leaves a special position-variable invariant. This generates systems of particles in scalar interaction with a supersymmetry for each spinning particle. For two-particle systems the supersymmetry eliminates all spin complications and reduces consistency problems to those of a purely bosonic system. Once the latter are disposed of, our approach leads to consistent systems of quantum mechanical wave equations.
Measurement systems and Jordan algebras
Authors: Faulkner John R.
Published: March 2003

An axiomatization of the measuring process leads to a Jordan algebra structure on the observables. The novel features in this development include a proof of the existence of the sum of observables, a proof of the quadratic nature of the square of an observable, the lack of a finite dimensionality assumption, and the exploitation of the change in the measuring process due to a change in the counting observable.