bluuebunny/crossref_metadata_2025 huggingface dataset:
Classification of Finite-Dimensional Irreducible Representations of Connected Complex Semisimple Lie Groups
Authors: Bose A. K. , Patera J.
Published: February 2004

Using the results of Malcev and Dynkin, the irreducible representations of connected components of complex semisimple Lie groups are classified. The simple Lie algebras A1, Bn, Cn, G2, F4, E7, and E8 have no outer automorphism and, for the Lie algebra D2k, the automorphism corresponding to the contragredient transformation is inner. Hence, all the irreducible representations of all simple Lie algebras except An (n > 1), D2k+1, and E6 are self-contragredient. For An (n > 1), D2k+1, and E6, an irreducible representation is self-contragredient, provided its highest weight possesses the symmetry to reduce the outer automorphism of the algebra into an inner one for the representation. All the self-contragredient irreducible representations are classified into orthogonal and symplectic types by reducing the problem to the case of the angular-momentum algebra.
Canonically Conjugate Pairs, Uncertainty Relations, and Phase Operators
Authors: Garrison John C. , Wong Jack
Published: February 2004

Apparent difficulties that prevent the definition of canonical conjugates for certain observables, e.g., the number operator, are eliminated by distinguishing between the Heisenberg and Weyl forms of the canonical commutation relations (CCR's). Examples are given for which the uncertainty principle does not follow from the CCR's. An operator F is constructed which is canonically conjugate, in the Heisenberg sense, to the number operator; and F is used to define a quantum time operator.
O(4) Symmetry and the BS Equation for a Spin-1 + Spin-0 System
Authors: Abo-Shaeer Muhsin A. R.
Published: February 2004

A set of vector functions analogous to the 3-dimensional vector spherical harmonics is defined in four dimensions. These functions are employed to separate the Bethe-Salpeter (BS) equation for a spin-1 + spin-0 system at vanishing total 4-momentum K, where O(3) symmetry degenerates into O(4) symmetry. The angular parts of the matrix elements of all possible tensor operators, symmetry conserving and otherwise, are reduced. Finally, the BS equation at small K is briefly discussed.
Block Diagrams and the Extension of Timelike Two-Surfaces
Authors: Walker Martin
Published: February 2004

The work of Finkelstein, Kruskal, Graves and Brill, Carter, and Boyer and Lindquist on the extension and schematic representation of 2-dimensional metrics is systematized and generalized. As a result, a number of extensions may be found by inspection. Some well-known examples are given, and the technique is applied to the ``soluzioni oblique'' of Levi-Civita.
Relation of the Inhomogeneous de Sitter Group to the Quantum Mechanics of Elementary Particles
Authors: Aghassi J. J. , Roman P. , Santilli R. M.
Published: February 2004

A relativistic dynamical group recently introduced by Aghassi, Roman, and Santilli for the quantum mechanics of elementary particles is briefly reviewed. It is shown in detail that the algebra of this group can be obtained by contracting the Lie algebra of the inhomogeneous de Sitter group ISO(3, 2). Some crucial concepts of the proposed new group are shown to appear in a new light when viewed in the context of a de Sitter world. The emergence of proper time as an additional kinematical variable is discussed in some detail.
Eigenvalues and Eigenfunctions of the Spheroidal Wave Equation
Authors: Hodge D. B.
Published: February 2004

A technique is presented for the calculation of the oblate and prolate spheroidal wave equation eigenvalues and eigenfunctions. The eigenvalue problem is cast in matrix form and a tridiagonal, symmetric matrix is obtained. This formulation permits the immediate calculation of the eigenvalues to the desired accuracy by means of the bisection method. The eigenfunction expansion coefficients are then obtained by a recursion method. This technique is quite simple to program, and the computation speed is rapid enough to allow its use as a function subroutine where values not previously tabulated or large numbers of values are required.
New Theorems about Spherical Harmonic Expansions and SU(2)
Authors: Beers Brian Lee , Dragt Alex J.
Published: February 2004

Despite the heroic efforts of Laplace, Legendre, Maxwell, Hobson, and company, there are still new chapters to be written in the theory of spherical harmonics. This paper describes some of the special windfalls that result when one expands analytic functions. In technical language, we are concerned with Fourier series of analytic functions on SU(2).
Application of the Smoothing Method to a Stochastic Ordinary Differential Equation
Authors: Morrison J. A. , McKenna J.
Published: February 2004

We study the use of the ``smoothing method'' to calculate the second-order moments of solutions of stochastic, ordinary, linear differential equations. We consider in detail the equation d2udz2+β02[1+ηN(z)]u=0,where N(z) is a real, zero mean, wide-sense stationary stochastic process and β0 and η ≪ 1 are positive constants. We show that, for one choice of N(z), the conventional use of the smoothing method yields correct first-order moments of the solutions, but badly incorrect second-order moments. We develop what we believe is a better way to use the smoothing method to calculate second-order moments. For the special choice of N(z), this method yields exact results. The method can be extended to the calculation of moments of all orders for arbitrary stochastic, ordinary, linear differential equations.
Singular Integral Operator Encountered in Scattering Theory
Authors: Pearson D. B.
Published: February 2004

The domain of an unbounded singular integral operator occurring in scattering theory is investigated and is proved to be everywhere dense. An application is made to a simple scattering system, for which the interaction Hamiltonian is of rank one, and the existence of the wave operator is proved by a ``time-dependent'' method which does not assume ``smoothness'' of the element of L2(− ∞, ∞) from which the interaction is constructed.
Complex 2-Dimensional Internal Space in General Relativity
Authors: Oliveira Colber G.
Published: February 2004

A calculus of vectors in 2-dimensional symplectic spaces is developed from the concept of existence of local basis systems. The similarities, as well as the differences, of this calculus with the tetrad formulation of 4-dimensional curved spaces are discussed. The affinity and curvature of the symplectic space are derived and its relationships with the affinity and curvature of the usual spinor formalism are given. A system of hybrid geometrical objects displaying a tensor and a spinor index take over the role of the usual Hermitian matrices σμKṀ(x).
Solution of the Initial-Value Neutron-Transport Problem for a Slab with Infinite Reflectors
Authors: Newman Perry A. , Bowden Robert L.
Published: February 2004

The initial-value transport problem of monoenergetic neutrons migrating in a thin slab surrounded by infinitely thick reflectors is solved for isotropic scattering by using the normal-mode expansion technique of Case. The results obtained indicate that the reflector may give rise to a branch-cut integral term typical of a semi-infinite medium while the central slab may contribute a summation over discrete residue terms. Exact expressions are obtained for these discrete time eigenvalues, and sample numerical results are presented showing the behavior of real time eigenvalues as a function of the material properties of the slab and reflector. In the limit of purely absorbing reflectors or a bare slab, the present solution has the properties which have been previously reported by others who used the approach of Lehner and Wing.
Correlation Inequalities for Heisenberg Ferromagnets
Authors: Hurst C. A. , Sherman S.
Published: February 2004

First, for a classical discrete approximation to the Heisenberg ferromagnet, in a Euclidean space of dimension 2k where k is a positive integer, the correlation is nonnegative and a monotonic increasing function of interactions. Second, for the quantum mechanical Heisenberg ferromagnet itself, the correlation is nonnegative but not necessarily a monotonic increasing function of interaction. Both results allow for long-range interactions. Third, a family of inequalities on the number of connected components in graphs is deduced.
Equal-Time Commutation Relation for the Energy-Momentum Tensor
Authors: Bergström K.
Published: February 2004

Using a method outlined in a previous paper, we derive an equal-time commutation relation for the energy-momentum tensor. We make no restrictions on the external gravitational field gμν that couples to the energy-momentum tensor. As a special case of the equal-time commutation relation derived, we obtain Schwinger's commutation relation for the energy density.
Error Bounds for Asymptotic Approximations of Zeros of Hankel Functions Occurring in Diffraction Problems
Authors: Hethcote Herbert W.
Published: February 2004

Error bounds on the asymptotic approximation for large argument of the zeros of the Hankel function of the first kind as a function of its order are found. These zeros occur as residue poles in the problem of scalar diffraction of high-frequency plane waves by a totally reflecting sphere.
Is the Group R4 an Approximate Symmetry for Many-Electron Theory?
Authors: Butler P. H. , Wybourne B. G.
Published: February 2004

The 4-dimensional rotation group R4 is used to analyze the structure of the (2s + 2p)N shell and to determine the relevance, if any, of the R4 group as an approximate symmetry group for many-electron theory. It is found that while the R4 symmetrized states of the (2s + 2p)N complex approximately diagonalize the Coulomb energy matrix for interactions within the complex, the inclusion of the inner 1s2 shell effects leads to a large symmetry violating term, which invalidates the R4 group as an approximate symmetry for many-electron theory.
Quantum Theory on a Network. II. A Solvable Model Which May Have Several Bound States per Node Point
Authors: Mills R. G. J. , Montroll E. W.
Published: February 2004

This is a further development of a network model for electrons in solids. It is postulated that the electrons are restricted to move along 1-dimensional lines between atoms. The network for a given solid corresponds to that which one would use in a wire and ball model to exhibit the crystal structure. A potential of the form V = −V0 sech2γx is associated with each ``atomic'' node point. The atom can have 1, 2, 3, or more electronic bound states, depending on V0 and γ. The wavefunctions have the usual Bloch form and involve hypergeometric functions. The density of states is plotted. Localized defect energy levels are also given for various types of atomic defects. An especially interesting feature of the model is that all the above-mentioned calculations are made exactly, without resorting to perturbation theory.
Theory of Formal Series with Applications to Quantum Field Theory
Authors: Dubois-Violette M.
Published: February 2004

The theory of formal series is developed in such a way that it includes conventional formal series as well as Volterra's series (generating functionals) as special cases. A method for expressing solutions of nonlinear equations is developed.
Electromagnetic Wave Propagation through Slowly Varying Plasma
Authors: Chakraborty B.
Published: February 2004

We solve the second-order linear inhomogeneous wave equation by the WKB method. Then a sequence of successive substitutions are developed, by the help of which solutions correct up to any higher order are reduced to the WKB solution of an equation like the original equation. This procedure is adopted to study propagation of electromagnetic wave obliquely through slowly but continuously varying plasma layers. Across a plane where N (the equilibrium plasma concentration) is continuous, but its first- and second-order space rate of variation along the normal are discontinuous, we have solved the boundary value problems for wave propagation. The characteristics of the reflected and transmitted fields depend on these derivatives of N such that the reflected wave vanishes in the limit of constancy of N. We find that the direction of flow of electromagnetic energy is deviated from its straight course in the uniform region. Expressions for relative shift of the wave vector along and perpendicular to the direction in the uniform medium, with respect to the distance of wave advancement in that medium in a given time, have been deduced. The deviation of the direction of propagation is slightly more for the field of transverse magnetic (TM) polarization than for the transverse electric (TE) polarization, but both these directions lie in the plane of incidence and are opposite to one another with respect to the direction in the uniform medium, where by transverse polarizations are meant those with respect to the plane of incidence. Hence, the original wave is split into two waves. Deflection is increased and, consequently, speed of propagation decreased for the TM field, whereas deflection is decreased and speed of propagation increased for the TE component if the wave propagates into a plasma of increasing concentrations; consequently, the TM field will be reflected back earlier than the TE part. Besides these results, it is hoped that our treatment may help in attacking problems of coupling of different types of plasma waves due to slow variation of equilibrium parameters occuring from more complete set of equations describing plasma behavior.
Topological Analysis of a Nonlinear Field Theory
Authors: Williams J. G.
Published: February 2004

This paper is concerned with a class of nonlinear field theories that exhibit conserved particlelike structures. An example of such a theory is quoted. Fields belonging to this theory are classical fields, single-valued under the action of the rotation group. It is explained how, when the theory is quantized, the quantum mechanical states which correspond to 1-particle systems in the unquantized theory may have properties similar to fermion states. In particular, it is shown that it is possible for wavefunctionals to exist which are double-valued under 2π rotations. This is the first known example in which a single-valued field theory can give rise to a quantized theory with half-odd spin properties.
Characters of the Poincaré Group
Authors: Fuchs Gérard , Renouard Pierre
Published: February 2004

We calculate the characters of the Poincaré group P̄ as solutions of differential equations in a way which is valid for a large class of Lie groups. We discuss the solutions of linear differential equations on an analytic manifold in a space of distributions. Then, we calculate the characters of SL(2, R) and the central distributions on P̄. Finally, we give the characters of all unitary irreducible representations of P̄, including mass 0, and some expressions which may be ``characters'' of nonunitary representations.
Mathematics of the N/D Method
Authors: Lyth D. H.
Published: February 2004

A complete mathematical account is given of the N/D method, subject to certain restrictions on the given functions L and ρ (left-hand cut contribution and phase-space factor). The restrictions are the weakest possible if the phase shift is to be Hölder continuous. Any asymptotic behavior ρ ∼ xβ, 0 < β < 2, is allowed. No analyticity is assumed for L. Exhaustive existence and uniqueness theorems are given: Any allowed function F (= eiδ sin δ/ρ) possesses an N/D decomposition, and the integral equation for N acts in a space of L2 functions; this equation satisfies the Fredholm alternative theorems [though its kernel has a continuous spectrum if δ(∞)/π is not integral]; any L2 solution of the equation yields an allowed function F; all allowed solutions may be obtained by varying the CDD parameters (which enter linearly); each solution has a uniqueness index κ; there will usually be a κ parameter infinity of solutions with each κ ≥ 0, and none with κ < 0, but precise conditions on L and ρ are given in order that there will be a negative κ solution.
Global Singularities and the Taub-NUT Metric
Authors: Feinblum David A.
Published: February 2004

Several examples of singular behavior which are not apparent in the line element are discussed. It is shown that the presence of such singularities depends on the manner in which coordinate patches are assembled to form the entire manifold. It is shown that the usual connection of a coordinate patch with the Taub metric to a patch with the NUT metric is singular in the sense of this particle.
Radiative Transfer in a Rayleigh-Scattering Atmosphere with True Absorption
Authors: Schnatz T. W. , Siewert C. E.
Published: February 2004

The singular-eigenfunction-expansion technique is used to solve the equation of transfer for partially polarized light in a Rayleigh-scattering atmosphere with true absorption. The normal modes for the considered nonconservative vector equation of transfer are established; two discrete eigenvectors and two linearly independent continuum solutions are thus derived. Further, the necessary full-range completeness and orthogonality theorems are proved, so that all expansion coefficients can be determined explicitly, and, in order to illustrate the technique, an exact analytical solution for the infinite-medium Green's function is developed. Finally, a numerical tabulation of the required discrete eigenvalue, as a function of the single-scatter albedo, is given.
Relationship between Geometrical-Optical and Full-Wave Solutions to the Problem of Propagation over a Nonparallel Stratified Medium
Authors: Bahar E.
Published: February 2004

The problem of propagation over a wedge-shaped overburden has been analyzed using a full-wave-solution approach. The relationship between this solution and an earlier one employing the compensation theorem is discussed in detail. The compensation theorem method makes use of the simplifying concept of of the surface impedance which is determined by a geometrical-optical approach. The applicability and limitations of the latter approach are carefully studied for both dissipative and nondissipative over-burdens.
Kruskal's Perturbation Method
Authors: Stern David P.
Published: February 2004

A general method for eliminating the angle variable from the equations of a perturbed periodic motion and for deriving an ``adiabatic invariant'' J has been given by Kruskal and, for a special class of Hamiltonian systems, McNamara and Whiteman have shown (to order ε2) that J is related to a set of invariants I obtained from the expansion of Poisson-bracket relations. In this work, an order-by-order algorithm for Kruskal's method is introduced, and a new set of invariants Z1 is obtained. It is shown that these invariants bear a close relation to those obtained from the Poisson-bracket expansion and, in the special case investigated by McNamara and Whiteman, the relation between I and Z1 may be brought to the same form as the relation between I and J derived by those authors. Finally, the relationship between Z1 and J is examined, and arguments are presented that in certain cases the two are equal to all orders.
Direct Canonical Transformations
Authors: Stern David P.
Published: February 2004

Some of the perturbation methods in classical Hamiltonian mechanics lead to near-identity transformations of the variables, with the new variables explicitly given as functions of the old ones. Two methods are used for identifying and characterizing the subclass of all such transformations which are also canonical: one approach is related to the conventional method of generating canonical transformations, while the other one uses the properties of Poisson brackets and is related to an operator method of Lie. Either of the methods may be used to derive certain steps in a perturbation method devised by Lacina, inadvertently omitted by that author.
Second Quantized Atomic Wavefunctions with Definite Unitary and Rotational Symmetry
Authors: Cunningham M. J.
Published: February 2004

Some atomic wavefunctions for equivalent electrons in the group scheme SU2×(U2l+1⊃R2l+1⊃R3) are constructed in terms of electron fermion creation and annihilation operators. The concept of semiconjugacy is defined and shown to reduce the number of states that must be explicitly calculated. The states for the d shell are calculated and tabulated.
Ray Theory of Diffraction by Open-Ended Waveguides. I. Field in Waveguides
Authors: Lee S. W.
Published: February 2004

This paper presents an extension of Keller's ray method to problems involving two or more parallel plates. The approach used here is to solve a canonical problem (two staggered plates) in a rigorous manner, and then extract its dominant asymptotic terms so that the result admits ray interpretation. It is found that the coupling effect between two plates may be accounted for by introducing two multiplicative factors G+(k cos θ0) and G+(k cos θ) to the conventional half-plane diffraction coefficients, where (θ0, θ) are the directions of incoming and outgoing rays at the edge. The function G+(α) is the ``plus part'' of the transformed Green's function G(α) = 1 − exp [−2b(α2 − k2)½] in the Wiener-Hopf technique, and b is the guide width. A table for G+(k cos θ) is provided so that the calculation of the modified field on the rays can be accomplished even with a slide rule. An application of the present ray method to the problem of radiation from an unflanged open-ended parallel-plate waveguide recovers the exact solution obtainable by the Wiener-Hopf technique. In Paper I of this series of papers, we present the main theory and discuss the field in waveguide, while in Paper II the field in free space, particularly that on the shadow boundary, will be examined.
Propagation of Electromagnetic Waves over a Smooth Multisection Curved Earth—An Exact Theory
Authors: Wait James R.
Published: February 2004

In this paper, we present a self-contained exposition of ground-wave propagation over a smooth inhomogeneous earth model. A factorization procedure is used to obtain an exact solution for a 2-dimensional cylindrical model which may have any number of homogeneous sections. It is demonstrated that, under most conditions in practice, the solution for the mode conversion coefficients can be simplified to a form which was derived earlier by an application of the compensation theorem. Formulas are given which permit the refinement of earlier approximate calculations if such accuracy is required.
Unitary Irreducible Representations of SL(n, C) · R(n2). I
Authors: Angelopoulos E.
Published: February 2004

The family of Lie groups SL(n, C) · R(n2) is studied from the mathematical point of view, i.e., their structure and unitary irreducible representations. The first steps in the study of representations are based on the Wigner-Mackey-Bruhat method. The homogeneous spaces and little groups are determined by introducing a (generally) pseudo-Hermitian metric on vector spaces of finite dimension. However, the presence of null vectors does not allow this approach to be an exhaustive one; there are cases where the stabilizer is a semidirect product of a reductive group by a nilpotent one. In such cases, results are available if the reductive subgroup is compact, but it seems that these results can be generalized. The connection is also examined, both in structure and in representations, between this family and creation-annihilation operators. A complete list of inequivalent classes of representations is given in terms of orbits and little groups for n = 2, 3, 4, 6.
Stationary States for a Nonlinear Wave Equation
Authors: Berger Melvyn S.
Published: February 2004

Global variational methods are used to derive mathematically rigorous results on the existence and nonexistence of stationary states for some nonlinear wave equations (e.g., the nonlinear Schrödinger equation and the complex nonlinear Klein-Gordan equation). A priori estimates useful in the study of stability of stationary states are also derived.
Positivity Constraints on Crossing-Symmetric Partial-Wave Expansions
Authors: Roskies Ralph
Published: February 2004

Based on positivity of the absorptive parts, constraints on the parameters appearing in Roskies' crossing-symmetric parametrization of the π0π0 partial waves below threshold are derived. These are then expressed as constraints involving integrals over the π0π0 s wave below threshold.
Extremum Principles for Irreversible Processes
Authors: Keller Joseph B.
Published: February 2004

A general problem is considered for the determination of the generalized fluxes and the conjugate generalized forces describing irreversible processes in nonequilibrium thermodynamics. It is shown that for linear systems the solution of this problem is also the solution of a minimum problem and of a maximum problem. In certain cases the functional which is minimized is the rate of entropy production, while in other cases the functional which is maximized is minus this rate. Thus, in these cases Prigogine's principle of the minimum rate of entropy production is valid. For certain dynamical systems it is also shown that the functional in the minimum problem is a decreasing function of time, while for other systems the functional in the maximum problem is an increasing function of time. The results are applied to a system of chemical components in which various chemical reactions, diffusion, and heat conduction are occurring. Similar results are obtained for special nonlinear systems of this kind and for certain linear systems in which convection is also occurring. The results are based, in part, upon the theory of reciprocal variational problems, which is shown to follow from Fenchel's duality theorem involving convex functions.
Suggestive Approximate Dynamical Symmetry of an Electron Moving in the Field of Many Stationary Nuclei
Authors: Ray Supriya
Published: February 2004

A group-theoretical study has been employed in an effort to uncover the suggestive operator, if it exists, of the form L2 + Ô which commutes with the Hamiltonian of an electron moving in the many-nucleus Coulomb field. The Casimir operator L2 may easily be identified as the square angular-momentum operator, while the operator Ô may involve the internuclear separations and operators of the Lie algebra. Finally, however, the analysis has revealed the fact that such operator or dynamical invariant does not, in general, exist for any arbitrary stationary nuclei except in few special cases which, however, have their special geometrical symmetry.
Stochastic Processes and Their Representations in Hilbert Space
Authors: Hall F. G. , Collins R. E.
Published: February 2004

Beginning with an intuitive consideration of sequences of measurements, we define a time-ordered event space representing the collection of all imaginable outcomes for measurement sequences. We then postulate the generalized distributive relation on the event space and examine the class of measurements for which this relation can be experimentally validated. The generalized distributive relation is shown to lead to a σ-additive conditional probability on the event space and to a predictive and retrodictive formalism for stochastic processes. We then show that this formalism has a predictive and a retrodictive representation in a separable Hilbert space H, which has no counterpart in unitary quantum dynamics.
Powers of a Matrix and the Generalized Lucas Polynomials
Authors: Ilamed Yehiel
Published: February 2004

The functions Lnk(r)(Φ1,⋯,Φn) are defined by Xr=∑k=1nLnk(r)Xn−k, where X is an indeterminate n × n matrix and Φ1, ⋯, Φn are the invariants of X (basic symmetric functions in the eigenvalues of X). In this paper the generalized Lucas polynomial Ln1(r) is expressed explicitly as a determinant of order r − n + 1 or as a ratio of two determinants of order n.
Normalization of Certain Higher-Order Phase-Integral Approximations for Wavefunctions of Bound States in a Potential Well
Authors: Yngve Staffan
Published: February 2004

Furry has given a formula for the normalization integral of a physically acceptable solution of the time-independent 1-dimensional Schrödinger equation corresponding to a bound state. Considering the wavefunction of a bound state in a single-well potential and using Furry's formula and the connection formulas discussed by N. Fröman, we derive the normalization factor for the higher-order phase-integral approximations introduced earlier by N. Fröman. For the special case of the first-order JWKB approximation, the approximate normalization factor thus obtained is equivalent to that given by Pauli and Furry.
Conformal Charge Conjugation
Authors: Hristev Anatole P.
Published: February 2004

Conformal spinor calculus for an arbitrary Euclidean space En, n even, n = 2v, is developed, and fundamental spin tensors for Clifford algebra Cln+2 of the space of representation En+2 are calculated. It is found that the conformal charge conjugate of a 2v-semispinor ψc± differs from the relativistic, conventional φc±, by the permutation of the semispinors and the factor γn+1=1n!it−ν|g|12γ[1⋯γn],  t=number of timelike dimensions of En;namely, ψc±=∓γn+1φc∓  for ν even, t odd. This is related to the Pauli-Gürsey isospin group. The transformation laws for spinors and conjugate spinors under conformal group are studied. Conformal identities for matrix elements and bilinear covariants are indicated.
Photon Spin Formalism
Authors: Epstein Kenneth J.
Published: February 2004

A new approach to quantum electrodynamics is considered, in which photon spin terms are incorporated phenomenologically into the equations of classical electrodynamics in such a way as to yield quantized equations in the Heisenberg picture, for which relativistic requirements are satisfied. Photon momentum and energy analyses in the new formalism are then compared to the corresponding analyses in conventional field-theoretic formalisms, and questions regarding mathematical and physical consistency are discussed. Different notations are used to distinguish between physical and Hilbert space vectors, noting that the Hilbert space of the vector wavefunction has a 3-dimensional subspace resembling physical space, mathematically, in a way that allows interesting results, which can be described by two formalisms when auxiliary notation is necessary to analyze relationships involving both kinds of vectors.
Can Local Gauge Transformations Be Implemented?
Authors: Dell'Antonio G. F.
Published: February 2004

One studies conditions under which a gauge transformation can be implemented by a unitary operator in some representations of the canonical (anti-) commutation relations; an application is then given to local gauge transformations in field theory.
Bifurcate Space-Times
Authors: Hajicek P.
Published: February 2004

The notion of space-time is generalized in order to include the non-Hausdorff manifolds that do not admit any sort of pathological bifurcate curves. By means of these space-times, a geodesically complete extension of Taub and NUT space is constructed.
Extremum Principles for the Equation ∇2φ = φ − φ3
Authors: Robinson Peter D.
Published: February 2004

It has been conjectured that the Lagrangian functional G(Φ)=∫ [12Φ(−∇2+1)Φ−14Φ4] drprovides an upper bound for G(φ0), where φ0 is the ground-state eigenfunction of the nonlinear field equation −∇2φ + φ − φ3 = 0, provided that G(Φ) is constrained to be stationary with respect to variations in the amplitude of Φ. In this paper we demonstrate that this conjecture is true. The effect of a stationary-scale constraint on G(Φ) is also shown to guarantee an upper bound. Complementary functionals arising from an Euler-Hamilton approach to the problem are investigated. Unfortunately, these do not (as in more favorable circumstances) provide lower bounds for G(φ0), but merely alternative upper bounds. With a very simple trial function, a complementary bound is closer to G(φ0) than is G(Φ).
Tidal Gravitational Accelerations near an Arbitrary Timelike Geodesic in Schwarzschild Space
Authors: Finley J. D.
Published: February 2004

It is suggested that a ``physical'' definition of a singularity in a space-time manifold might be that it is a point where the relative accelerations of nearby timelike geodesics become infinite. Along an arbitrary timelike geodesic in Schwarzschild space, we construct an orthonormal tetrad of 4-vectors which are used to define ``elevator coordinates'' in a neighborhood of the geodesic. We use these coordinates to determine the tidal gravitational accelerations near the geodesic, and we point out that these accelerations are finite (and continuous) at r = 2m, the ``Schwarzschild surface,'' although they are unbounded as r approaches 0.
Field-Theoretic Description of Massless Particles with Higher Spin and Definite Parity. II. Half-Integer Spin
Authors: Cukierda T.
Published: February 2004

The S + ½ equivalent theories describing a free massive field with half-integer spin S in the limit m → 0 are investigated. It is shown that for m = 0 they are not equivalent and describe different massless particles, with helicities H= ±12, ±32,⋯, ±S. The inequivalence is caused by the possibility of having different gauge transformations for different massless spinor-tensors G⋯σ⋯[μν]A(k,S).
Wigner Coefficients for the Group E(2)
Authors: Gottlieb H. P. W.
Published: February 2004

The Wigner coefficients for the faithful unitary representations of the 2-dimensional Euclidean group are derived from two identities involving Bessel functions. Since the multiplicity in the decomposition of a direct product is two, we find two sets of coefficients which are real and mutually orthogonal and symmetric and antisymmetric, respectively, under interchange of constituent representations. Properties of the coefficients at the ends of the decomposition spectrum are discussed.
Class of Perturbation Theories of Ordinary Differential Equations
Authors: Kummer Martin
Published: February 2004

A class of perturbation theories of ordinary differential equations is studied in a systematic and rigorous way. This class contains the perturbation theory by Kruskal [J. Math. Phys. 3, 806 (1962)] and its generalization discussed by Coffey [J. Math. Phys. 10, 426 (1969)] as well as the formal aspects of the perturbation theory for quasiperiodic solutions by Moser [Math. Ann. 169, 136 (1967)]. It is shown that the systematic generalization of some algebraic ideas by Sternberg [J. Math. Mech. 10, 451 (1961)] and Moser provides a framework in which many structures of these perturbation theories become more transparent. Especially, two questions raised by Coffey are answered. Finally, we touch upon the question of convergence of the formal expansions.
High-Temperature Expansion of the Spin-½ XY Model
Authors: Lee M. Howard
Published: February 2004

The high-temperature expansion of the spin-½ XY model is shown to be particularly simple when the Hamiltonian is written in the second-quantization form. By aid of a few simple rules, the partition function and susceptibility are easily evaluated to high orders.
Introduction to Multiple Asymptotic Series with an Application to Elastic Scattering
Authors: Shere Kenneth D.
Published: February 2004

The concept of asymptotic power series is extended to multiple series of the form ∑ m=0∞  ∑ n=0∞am,nx−ne−λmx. This theory is then applied to elastic scattering from the Yukawa potential. An asymptotic series for the solution at large distances is obtained which is valid for arbitrary wavenumber and angular momentum.
Relaxation Spectrum of Phonons: A Solvable Model
Authors: Claro Francisco H. , Wannier Gregory H.
Published: February 2004

The approach towards equilibrium of a slightly disturbed phonon gas is analyzed in terms of its normal decay modes. It is shown that a simplified model permits a complete solution of the linearized phonon Boltzmann equation. The model considers 3-phonon normal collisions as the relaxing process and assumes a nondispersive isotropic elastic continuum. In analogy with the Maxwellian molecule model, the coupling term is assumed independent of the frequency. The spectrum of relaxation rates found has no discrete modes except for one trivial solution of rate zero. Above this value there is a gap and then a continuum extending to infinity.
Dimensionality of Space-Time
Authors: Mariwalla K. H.
Published: February 2004

It is shown that, for a local Poincaré covariant quantum field theory in which the usual relation between the TCP symmetry and spin-statistic holds, the dimensionality of space-time should be even. Further plausibility arguments are presented in favor of the observed dimensionality, on basis of the following factors: (1) simplicity, (2) relativistic local quantum field theory, (3) general relativity, (4) the nature of some observed interactions, (5) classical mechanical force concept.
Relaxation of Local Thermal Deviations from Equilibrium
Authors: Emch Gérard G. , Radin Charles
Published: February 2004

We give precise conditions sufficient to guarantee that an infinite system will act as a thermal reservoir for any of its finite parts. In particular we show that these conditions are satisfied for the X-Y model. Further results on the ergodic behavior and general relaxation properties of the systems considered are also obtained directly from the C*-algebraic methods used in the main body of the paper.
A Convergent Expansion for the Resolvent of: φ4: 1 + 1
Authors: Federbush Paul
Published: February 2004

For the model theory, N + λ ∫ : φ4 : dx in a box, a convergent expansion of the resolvent is exhibited. This also provides another proof of boundedness below for the model.
On the Decomposition SO(p,1)⊃SO(p−1,1) for Most Degenerate Representations
Authors: Boyer Charles P. , Ardalan Farhad
Published: February 2004

We discuss the problem of decomposing the matrix elements for degenerate representations of SO(p, 1) according to the noncompact subgroup SO(p − 1, 1). The overlap functions which obtain this decomposition are calculated explicitly and their properties are used to discuss the SO(p, 1) generalized Regge expansion.
Another Bound on the Absorptive Part of Elastic Scattering Amplitudes
Authors: Savit R. , Blankenbecler R. , Einhorn M. B.
Published: February 2004

Via Lagrange multipliers for inequality constraints, an upper bound on the absorptive part of the elastic scattering amplitudes is derived assuming unitarity, a fixed total and elastic cross section, and the condition that the partial waves decrease monotonically with increasing angular momentum. Numerical results are given.
Exact Nearest-Neighbor Statistics for Two-Dimensional Rectangular Lattices
Authors: McQuistan Richmond B.
Published: February 2004

Expressions are developed which describe exactly the ensemble average of the number of nearest, next-nearest, and third-nearest-neighbor pairs per particle, when indistinguishable particles are arranged on a two-dimensional rectangular lattice.
On the Relation between Master Equations and Random Walks and Their Solutions
Authors: Bedeaux Dick , Lakatos-Lindenberg Katja , Shuler Kurt E.
Published: February 2004

It is shown that there is a simple relation between master equation and random walk solutions. We assume that the random walker takes steps at random times, with the time between steps governed by a probability density ψ(Δt). Then, if the random walk transition probability matrix M and the master equation transition rate matrix A are related by A = (M − 1)/τ1, where τ1 is the first moment of Ψ(t) and thus the average time between steps, the solutions of the random walk and the master equation approach each other at long times and are essentially equal for times much larger than the maximum of (τn/n!)1/n, where τn is the nth moment of ψ(t). For a Poisson probability density ψ(t), the solutions are shown to be identical at all times. For the case where A ≠ (M − 1)/τ1, the solutions of the master equation and the random walk approach each other at long times and are approximately equal for times much larger than the maximum of (τn/n!)1/n if the eigenvalues and eigenfunctions of A and (M − 1)/τ1 are approximately equal for eigenvalues close to zero.
The Korteweg—de Vries Equation: Transition Period
Authors: Shih L. Y.
Published: February 2004

A general form of the nonlinear wave equation with dispersive and dissipative terms involving small coefficients has been treated for the transition period when the time is around the breakdown time. In the first part of this paper, some relations have been established in order to see how the classical overtaking phenomenon is eventually prevented when the curve has steepened sufficiently. In the second part of this paper, the growth and damping of solitary waves due to the effects of dispersion and dissipation have been analyzed.
Representations of O(3)
Authors: Tripathy Kishor C.
Published: February 2004

We analyze the various classes of local irreducible representations (both finite and infinite-dimensional) of O(3) realized on a linear complex vector space. The well-known finite-dimensional unitary representations are obtained when the scalar product is positive definite. The infinite-dimensional representations are realized on an indefinite metric space. The possible applications of these new classes of representations are briefly discussed.
An Algebraic Realization of Spin-Wave Theory
Authors: Liu C. J. , Chow Yutze
Published: February 2004

The Heisenberg Hamiltonian of a ferromagnet with Zeeman term is treated by the method of algebraic realization of the spin algebra. The general features of this approach are first analyzed. A detailed calculation with a particular algebraic realization is then carried out by use of two boson operators; one of the operators is interpreted as the observable spin waves and the other as ``spurions'' which carry only spin quantum number without energy. The Bethe-Salpeter integral equation is solved in the long-wavelength region for the bound states of spin waves. A perturbation calculation yields a leading T4 correction to the usual T3/2 behavior of the magnetization.
Second Sound in a Low-Temperature Weakly Interacting Bose Gas
Authors: Ma Shang-keng
Published: February 2004

Second sound in a weakly interacting Bose gas at low temperatures is described from a purely microscopic viewpoint. The simplicity of the model, which is specified by a single scattering length, allows a precise field theoretic analysis within the framework of equilibrium statistical mechanics in the weak coupling limit. The dispersion formula ν = c2q − iτq2 + O(q3) for the frequency of the second sound ν in terms of its wavenumber q is established. The leading terms in the low-temperature limit of the constants c2 and τ are evaluated exactly.
Multiple Scattering Theory of Radiative Transfer in the Light of Singular Modes
Authors: Kanal Madhoo
Published: February 2004

The multiple scattering theory of radiative transfer, in the light of Case's normal modes, is investigated. In particular, it is shown that in the Neumann expansion of the total intensity the expansion coefficients are related to the normal modes in the form of certain integrals over the continuum Case spectrum. In the general formulation of the theory, the phase function is kept arbitrary. Two types of boundary-value problems are considered which involve semi-infinite and bounded media. To illustrate the structure of the general formulation, the isotropic case is treated in detail.
Remarks on the Existence of Solutions of the Two-Particle Lippmann-Schwinger Equation. II
Authors: Shastry C. S. , Rajagopal A. K.
Published: February 2004

We prove the following: The kernel Ks(l) corresponding to the lth partial wave of the modified partial wave Lippmann-Schwinger equation having symmetric kernel Ks=V1/2G0+V1/2 belongs to the Hilbert-Schmidt (L2) class for spherically symmetric potential V such that V(r) → r−δ, δ < 2 as r → 0; V(r) → r−η, η > 1 as r → ∞. For V(r) = g/rα, 1 <α<2, the kernel Ks of the modified Lipmann-Schwinger equation satisfies Tr{(Ks†)mKsm}<∞, if α > 1 + 1/m, m = 2, 3, ⋯. The above results are valid even for positive real energy. It is also shown that for potentials for which both Ks and the kernel K=G0+V of the usual Lippmann-Schwinger equation belong to L2 class, the corresponding off-shell twoparticle T matrix 〈k|T(E+iε)|k′〉 has a unique limit as ε → 0, E > 0. Some important consequences of these results are discussed.
Asymptotic Form of the Wavefunction for Neutral Particle-Hydrogen Atom Scattering
Authors: Doolen G. , Nuttall J.
Published: February 2004

The asymptotic form of the wavefunction for the scattering of a neutral particle on a hydrogen atom bound state is discussed, with particular reference to the case where the two charged particles may be close together in the asymptotic region.
The Algebras of Representations and Classes of Finite Groups
Authors: de B. Robinson G.
Published: February 2004

In a previous paper the author proposed a representation r for the algebra of representations of a finite group G. Here a corresponding representation c for the algebra of classes of G is established, and the relations between the two representations are investigated. In particular, the idempotents of r and c are studied in some detail.
Explicit Decomposition of a General Two-Body Hamiltonian into Irreducible Symplectic Tensors
Authors: Li Shou-yung
Published: February 2004

The seniority classification of a general two-body Hamiltonian according to the group chain U(N)⊃Sp(N)⊃R(3) is considered. It is found that by using the standard contraction and symmetrization process, one can explicitly decompose such a Hamiltonian into irreducible tensors with respect to the symplectic group. An example is given where the explicit angular momentum coupled form of these tensors are also worked out.
T-Matrix Analyticity Using Fredholm Theory
Authors: Shaffer Gedaliah R.
Published: February 2004

The behavior of the Regge pole residues for the T matrix analyzed into partial waves is studied as a function of E for scattering from a Yukawa potential. Using a variational formulation of Fredholm theory, we explicitly show that the residues on the energy shell have no branch cuts in the left-hand E plane.
Motion and Structure of Singularities in General Relativity. II
Authors: Newman Ezra T. , Posadas Roger
Published: February 2004

This, the second paper in a series on the motion and structure of a class of (elementary) singularities, extends the earlier work on ``free'' singularities to the interaction of these singularities with ``background'' gravitational and electromagnetic fields. The principle result is an unusual derivation of the Lorentz-Dirac equations of motion for a charged particle. The results do not depend on any ad hoc assumptions nor upon any renormalization, but follow uniquely from the Einstein-Maxwell equations applied to the elementary singularities. A concomitant result is the time evolution of the structure of the singularity.
The Phase Shift. I. As a Bounded Functional
Authors: Frank William M.
Published: February 2004

Classes of potentials are defined by the finiteness of certain norms. For potentials in any of these classes, the phase shift for any partial wave is shown to possess a norm-dependent bound as a function of energy. Two norms are adduced in which the phase shift is uniformly bounded for all energies. A number of theorems are proven concerning the high-energy behavior of the phase shifts corresponding to potentials in these classes.
Coordinates and Democracy in the N-Body Problem
Authors: Galbraith Harold W.
Published: February 2004

It is shown that the construction of ``democratic'' subgroups of O3N−3 in the N-body problem is greatly facilitated by the ``proper'' choice of relative vectors in the center of mass frame. The word ``proper'' is taken to mean that the set of (N − 1) relative vectors forms a basis for reduced representations of the corresponding democracy subgroup of SN. The imposition of this requirement easily leads us to the reduction chains O6⊃SU3⊃SO3rot and O9⊃SO31×SO32×SO33⊃SO3rot in the three- and four-body problems, respectively, and to O3N−3⊃SON−1×SO3rot in the N > 4 case.