bluuebunny/crossref_metadata_2025 huggingface dataset:
High-Temperature Properties of Random Spin Systems
Authors: Gallavotti G.
Published: February 2004

The free energy and the correlation functions are proved to be analytic functions of the inverse temperature β, the external magnetic field H, and the impurity probability p, provided the temperature is high enough and the spin-spin interaction has finite range. In particular, this excludes spontaneous magnetization at high temperatures.
Absence of Phase Transitions in Hard-Core One-Dimensional Systems with Long-Range Interactions
Authors: Gallavotti G. , Miracle-Sole S.
Published: February 2004

We prove the impossibility of phase transitions for a class of infinite-range potentials extending recent analogous results. We prove also a cluster property for the equilibrium state γ̄ and apply some collateral results to describe, in the case of finite-range interactions, the state γ̄ in terms of a family of density distributions, and to verify a general variational property of γ̄.
Bases for Irreducible Representations of the Unitary Group in the Symplectic Group Chain
Authors: Devi V. Syamala
Published: February 2004

A method of obtaining the highest weight polynomials of irreducible representations (λ) of Sp2n occurring in a reduction of an irreducible representation (k) of U2n is described. The highest weight polynomials of equivalent representations (λ) are labeled by means of parameters which occur naturally from the Littlewood's theorem to determine the branching rules for the representations of the unitary group with respect to the symplectic subgroup, when supplemented by modification rules. The results are given explicitly for U4⊃Sp4 and U6⊃Sp6, the former being a canonical chain and the latter a noncanonical chain.
Energy-Dependent Boltzmann Equation in the Fast Domain
Authors: Nicolaenko Basil
Published: February 2004

This work presents some aspects of the static energy-dependent Boltzmann equation in plane geometry using a continuous-energy formulation. In a first part, solutions are found for a class of synthetic separable (but nondegenerate) energy-transfer kernels. Such kernels are representative, for instance, of neutron inelastic slowing down. In a second part, the same problem is considered with the addition of a projection kernel (typical of neutron fission); it is shown that the solutions split into space-energy separable components and nonseparable ``slowing-down transients.''
On the Theory of Quantum Corrections to the Equations of State and to the Particle-Distribution Functions
Authors: Nienhuis G.
Published: February 2004

The first-order quantum corrections to the equations of state of an almost-classical N-particle system are calculated to all orders in the particle density by expanding the normalized Wigner distribution function in powers of h2. In this way one avoids the expansion of the partition function, which has the unsatisfactory property that the correction terms diverge in the thermodynamic limit. Similarly, the first-order quantum correction to the pair distribution function is derived.
Functional Formulation of the Coherence Theory
Authors: Zardecki Andrzej
Published: February 2004

A functional formalism for coherence theory is developed in terms of the (quasi) probability functional of the radiation field. The formalism is closely connected to the P representation of the density operator. For the fields confined within a spatial volume of finite size, this connection is expressed as a multidimensional Fourier integral. A certain type of correspondence, illustrated by explicit examples, between the classical and the quantum mechanical description of the field statistics is established. The recent results of Titulaer and Glauber concerning the (1, 1)th-order coherent fields are analyzed from the point of view of the functional formulation.
Sine-Gordon Equation
Authors: Rubinstein Julio
Published: February 2004

The equation φtt − φxx + m2 sin φ = 0 is presented as a model field theory and studied in detail. It exhibits discrete conserved quantities and extended particle states, with the proper behavior regarding covariance, stability, etc. Those particles do not interact with small oscillations, and we show that this (plus a few reasonable requirements) defines the model uniquely. A connection with the Korteweg-de Vries equation, which has similar properties, is established.
Anisotropy of Scattering in Neutron Transport Theory by the Two-Sided Laplace Integral Method
Authors: Boffi V. C.
Published: February 2004

In this paper the effects of arbitrarily anisotropic scattering in establishing the eigenvalue spectrum of the operator associated with the stationary monoenergetic neutron transport equation in a medium of infinite extent are investigated through an extensive application of the two-sided Laplace integral theory. How the contribution of the discrete eigenvalues imbedded in the continuous part of the spectrum can explicitly be evaluated when inverting the bilateral Laplace transform of the sought distribution is shown by resorting to the Plemelj formulas, which are in order in the theory of the Cauchy integrals.
Generalized Young Tableaux and the General Linear Group
Authors: King R. C.
Published: February 2004

A generalization of the Young tableau is defined, and use is made of this in the study of some of the properties of the irreducible representations (IR's) of each of the linear groups in n dimensions induced in a space defined by mixed tensors without recourse to lowering or raising of indices. A formula for the dimensions of any IR of Ln is given. Procedures are derived for the reduction of the outer product of such IR's and for the decomposition of these IR's into IR's of some subgroups of interest in the theory of elementary particles.
Asymptotic Theory of Transition Radiation
Authors: Lewis Robert M. , Cohen Jack K.
Published: February 2004

We treat the electromagnetic fields arising from the passage of a point source through an interface separating two dissimilar dispersive media. Since the media are dispersive, they are partially characterized by a characteristic frequency λ. Our expansions are valid for λa/c » 1, where a is a characteristic length and c is the speed of light.
Relativistically Dynamic Elastic Dielectrics
Authors: Bragg Lincoln E.
Published: February 2004

A relativistically exact theory of elastic dielectrics is developed, plus incidentals useful in a wider context. The most general form of a relativistically objective elastic response function is found in terms of Bressan's deformation gradient and of arbitrary tensors on the body manifold depending on the Born strain tensor. A form for constitutive relations is postulated for dielectrics which involves objective response functions, physical laws are stated in terms of integrals of exterior forms on the world manifold, and the resultant differential equations of motion are derived. Facts about algebra and calculus of exterior forms are developed which will be useful in continuum physics whenever one introduces laws involving exterior forms on manifolds. The relation to historical theories of dielectrics is indicated.
Entropy Principle for the Derivation of a New Cluster Expansion
Authors: Ramanathan G. V. , Dawson J. M. , Kruskal M. D.
Published: February 2004

A product expansion for the s-particle distribution function (s arbitrary) is derived by means of a variational principle involving the entropy associated with the (s + 1)-particle distribution function.
Spinor Structure of Space-Times in General Relativity. II
Authors: Geroch Robert
Published: February 2004

Spinor fields can only be defined on a space-time which has been given a spinor structure. A number of conditions (some sufficient, others necessary and sufficient) for the existence of a spinor structure are derived. By applying one or another of these conditions, it is shown that many well-known solutions of Einstein's equations do have spinor structure. The question of the existence of spinor structure depends only on the topology of the underlying manifold, not on the (time- and space-oriented) metric. It is shown that, nonetheless, a certain ``threshold'' of curvature must be exceeded before there can be even the possibility of a space-time's having no spinor structure.
Charge and Pole: Canonical Coordinates without Potentials
Authors: Kerner Edward H.
Published: February 2004

For particles having both magnetic and electric charge it is shown that (a) in the nonrelativistic many-particle problem where only Coulomb and Biot-Savart fields need be considered and (b) in the one-particle relativistic problem (orbital pole-charge moving around a fixed pole-charge), the well-set classical dynamics can be reduced directly from the equations of motion to Hamiltonian form without the introduction of potentials and Dirac strings. The Lie-Koenigs theorem, which can give Hamiltonian format to any dynamics, is invoked for this. The essential feature is that canonical coordinates cannot be physical particle coordinates. For (a) and (b), suitable canonical variables are explicitly constructed. Using only Bohr-Sommerfeld quantization, the Schwinger charge-pole quantum condition is obtained for pure-charge-pure-pole interactions; but when Coulomb forces are additionally considered, no quantum restriction on charge and pole strength is required.
Null Electromagnetic Fields in General Relativity Admitting Timelike or Null Killing Vectors
Authors: Banerjee A.
Published: February 2004

The paper presents a study of stationary null electromagnetic fields coexisting with a dust distribution. Two exact solutions corresponding to pure radiation fields with the propagation vector as the Killing vector are presented.
Fourier Expansions of Functions of the Distance between Two Points
Authors: Sourour A. R. , Ashour A. A.
Published: February 2004

The expansion of any function f(r) of the distance r between two points is given as a Fourier series. This is a generalization of results given earlier by Ashour for rn and log r. The Fourier-series expansion for the product rNeiMθ is also given, where now r = (r, θ) denotes the sum of r1 and r2, the coordinate vectors of the two points. This is further generalized for the product of a function f(r) of r and a circular harmonic. Expansions of similar products involving spherical harmonics have been given earlier by Sack. Special cases when f(r) is a Bessel function or a modified Bessel function are considered.
Existence and Uniqueness of Crossing Symmetric N/D-Type Equations Corresponding to the Klein-Gordon Equation
Authors: Cornille H.
Published: February 2004

N/D-type equations satisfying crossing symmetry are established inside the relativistic wave-mechanics formalism (Klein-Gordon equations). We show that, for a superposition of exponential potentials with finite zeroth moment, the N/D-type equations have unique solutions. Furthermore, for sufficiently weak couplings the solutions are physically available.
Vector Spherical Harmonic Expansion in the Time Domain of the Retarded Hertz Vector for a Distributed, Transient Source-Current Configuration
Authors: Clapp Roger E. , Huang Luc , Li H. Tung
Published: February 2004

An equation giving the retarded Hertz vector II(r, t) in terms of a source-current distribution J(r1, t1) is derived. Both vector functions are written as expansions in vector spherical harmonics, with the expansion coefficients containing the dependence upon radial distance and time, while the angular dependence is kept within the harmonic functions. After integration over two angles, expressions are obtained giving the expansion coefficients for the Hertz vector, which depend upon (r, t), in terms of the corresponding coefficients for the current, which depend upon (r1, t1). The original four-dimensional problem is thus reduced to two dimensions, but with the four-dimensional causality requirements satisfied at each step of the analysis.
Noncommutation Requirements and Field Dependence of Sources of Tensor Fields
Authors: Raman K.
Published: February 2004

Starting with the noncommutation requirements among different components of a conserved source tensor for a massive spin-2 field, we discuss the resulting noncommutation requirements between the source and the field variables, and the field dependence of the source that this necessitates.
Relative Position Representation for a Relativistic Two-Particle System
Authors: McDonald S. C.
Published: February 2004

Two-particle angular momentum states are constructed which are localized with respect to the magnitude of the relative position in the rest system and which have arbitrary 3-momentum dependence. The associated relative position operator is constructed, and a quantum-mechanical analog of the classical impact parameter is identified. Two-particle angular momentum states are constructed, which are also localized with respect to the ``mean-position'' of the 2-particle system, and the associated ``mean-position'' operator is seen to be a generalization of the 1-particle Newton-Wigner position operator.
Pure Thermodynamical Phases as Extremal KMS States
Authors: Emch Gérard G. , Knops Hubert J. F.
Published: February 2004

We compare the dynamical characterization of pure thermodynamical phases as extremal KMS states and their characterization as extremal time- or space-invariant states. We find that, for a class of Weiss-Ising models with periodic potentials, the extremal KMS states coincide exactly with the solutions of the self-consistency equations familiar from molecular field methods. We show that the models considered are not η-asymptotically Abelian in time. We conclude that the characterization of pure thermodynamical phases as extremal KMS states is the only correct one for these models. We pay special attention (in particular, in the decomposition of an arbitrary KMS state into its extremal KMS components) to the fact that the time evolution is not an automorphism of the C*-algebra of the quasilocal observables.
Boundary-Value Problems of Linear-Transport Theory—Green's Function Approach
Authors: Kanal Madhoo
Published: February 2004

Case's technique utilizing Green's functions for dealing with boundary-value problems of the neutron linear-transport theory is exploited. We show that the Fourier coefficients of the Green's function over the Case spectrum are precisely the normal modes. In particular, if we assume that the scattering kernel is rotationally invariant (which indeed we do assume) and approximate it by a degenerate kernel consisting of spherical harmonics, the set of modes is deficient for problems lacking azimuthal symmetry. We also show that the expansion of the scattering kernel, in terms of spherical harmonics (or any set of orthogonal functions for that matter), permits the linear factorization of the Fourier coefficients of the Green's function in terms of the lowest element, with the proportionality functions consisting of complete orthogonal polynomials. As a consequence of this attribute of Fourier coefficients, the eigenfunctions (continuum and discrete) also factorize, which then permits decoupling of the appropriate singular integral equations. To illustrate our idea, we solve half-space and slab problems. However, the basic procedure is kept sufficiently general so that the extension to problems involving other geometrics remains straightforward.
Polynomial Algebras
Authors: Raghavacharyulu I. V. V. , Menon Nalini B.
Published: February 2004

The present work is concerned with what are called polynomial algebras as an extension of the work of Ramakrishnan and his colleagues on the algebras of matrices satisfying conditions like Lm = I and Lm = Lk. Assuming Lm to be an m-dimensional linear space, we generate a class of associative algebras called polynomial algebras by requiring that every element L of Lm satisfy a polynomial equation Ln + P1Ln−1 + ⋯ + Pn = 0. We show that some very important algebras which physicists have found useful can be obtained by various restrictions on the polynomial. A few general properties of these algebras are established.
Extension of the Riemann-Lebesgue Lemma
Authors: Kantor Paul B.
Published: February 2004

We show that, in the limit of large λ, integrals of the form H(λ)≡ ∫ abf(x)dxu(x)+eiλxare essentially given by ∫R′[f(x)/u(x)]dx where the region R′ is the union of all those subintervals in which |u| ≥ 1. The corrections to this expression are of two kinds: terms O(1/λ) which depend on the details of averaging to remove logarithmic singularities in H(λ) and terms O[(ln λ)/λ]. Some examples are given. If |u| ≤ 1, the leading term in H vanishes and H(λ) is bounded by (ln λ)/λ.
Use of the WKB Quantization Condition for Calculating Expectation Values
Authors: Siebert E. , Krieger J. B.
Published: February 2004

A systematic technique is developed for the calculation of expectation values for 1-dimensional problems by a direct use of the WKB quantization rule including the higher-order integrals without the introduction of wavefunctions. As an example of the technique we calculate the moments 〈rp〉 for the Coulomb potential for 2 ≥ p ≥ −3, for all bound states of the system. Our results are exact except for p = −3, in which case the inclusion of the higher-order integrals substantially improves the accuracy of the calculation.
Three-Colorings of the Square Lattice: A Hard Squares Model
Authors: Baxter R. J.
Published: February 2004

The evaluation of the partition function of the 2-dimensional ice model is equivalent to counting the number of ways of coloring the faces of the square lattice with three colors so that no two adjacent faces are colored alike. In this paper we solve a generalized problem in which activities are associated with the colors. If one of the colors is regarded as a particle and the others as forming a background, then the model is reminiscent of the hard-square lattice gas. It is found to undergo a phase transition with infinite compressibility at the density ρ = 1/3.
Radiation Transport along Curved Ray Paths
Authors: Lau Chau-Wa , Watson Kenneth M.
Published: February 2004

The transport of radiation in a turbulent, refracting medium is studied. It is shown that the conventional transport equation must be generalized. Path integrals are taken along curved ray trajectories. When these ray paths have torsion, a rotation of the polarization vectors needs to be taken into account. Two derivations of the transport equation are given. One is phenomenological and one is based on Maxwell's equations. Some discussion is given of cross polarization of radar backscatter.
Structure of the Point Spectrum of Schrödinger-Type Tridiagonal Operators
Authors: Ifantis Evangelos K.
Published: February 2004

The main result in the present work concerns a criterion on the existence and the structure of proper values in a class of bounded operators (Schrödinger-type tridiagonal operators) on an abstract separable Hilbert space. The realization of these operators in the space of square summable sequences l2(1, ∞) represents a boundary-value problem of difference equations of the following form: f(n + 1) + f(n − 1) + a(n)f(n) = Ef(n). In our case f(n) ∈ l2(1, ∞), and the condition f(2) + a(1)f(1) = Ef(1) must hold. The approach followed is based on the reduction by Deliyannis and Ifantis [J. Math. Phys. 10, 421 (1969)] of the above boundary-value problem to an abstract operator form, which makes possible the application of the methods of functional analysis. It is shown that for every monotonically convergent and real-valued sequence a(n) ≠ 0, n = 1, 2, ⋯, there exist proper values, the greatest of which can be determined by the Ritz approximation method.
New Approach to the Motion of a Pole-Dipole Particle
Authors: Newman E. T. , Young Roger
Published: February 2004

A new approach to the equations of motion of the pole-dipole particle in linearized general relativity is presented. It is based on earlier work for the monopole particle [E. T. Newman and R. Posadas, Phys. Rev. Letters 22, 1196 (1969); Phys. Rev. 187, 1784 (1969)]. Though the results are not new, the novelty of the method holds out the serious hopes that the work can be extended to the full theory.
Fluids of Particles with Short-Range Repulsion and Weak Long-Range Attractive Interaction. II. The Two-Particle Distribution Function
Authors: Jalickee John B. , Siegert Arnold J. F. , Vezzetti David J.
Published: February 2004

In Paper I, we presented an expansion of the pressure and density in grand canonical form and corrections to the Maxwell rule for a system of particles with short-range repulsion and weak long-range attraction. These expansions can be ordered in powers of γ, the inverse range of the attractive potential. It was assumed that the thermodynamic functions and the molecular distribution functions of the reference system, i.e., the system with only the repulsive interaction, are given. In the present paper we have calculated the γ expansion of the pair distribution function, under the same assumption. The result is obtained by functional differentiation of the series for the pressure and presented in the form of a series of diagrams. The dominant order in γ of each diagram is the same as the order of that diagram in the series for the pressure, from which it is derived.
Complete Sets of Functions on Homogeneous Spaces with Compact Stabilizers
Authors: Anderson Robert Leonard , Wolf Kurt Bernardo
Published: February 2004

We formulate and solve the problem of determining a complete set of generalized functions for a wide class of homogeneous spaces with compact stabilizers. This allows us to say precisely what unitary irreducible representations can be realized on a given homogeneous space. The techniques are applied to the n-dimensional orthogonal and unitary groups.
Coefficients Connecting the Stark and Field-Free Wavefunctions for Hydrogen
Authors: Tarter C. Bruce
Published: February 2004

A general expression for the coefficients connecting the Stark (parabolic coordinates) and field-free (spherical coordinates) wavefunctions for hydrogen is obtained. The result, which involves a generalized hypergeometric function, has been numerically evaluated through principal quantum number n = 10.
The Proper Vibration of the Space
Authors: Kulhánek J.
Published: February 2004

The proper vibrations of homogeneous and isotropic space are examined on the basis of the equations of the de Broglie wave field (field equations). The time-dependent part of the wavefunction, which is a solution of the Klein-Gordon equation, satisfies the differential equation which coincides with the differential equation derived from field equations for the time-dependent part of the Robertson-Walker metric.
Calculation of Correlation Functions of Solutions of a Stochastic Ordinary Differential Equation
Authors: Morrison J. A.
Published: February 2004

In this paper we use the ``smoothing method'' to calculate the correlation functions of the solutions of the equation d2udz2+β02[1+ηN(z)]u=0,satisfying nonstochastic initial conditions, where N(z) is a real, wide-sense stationary stochastic process with zero mean and β0 and η ≪ 1 are positive constants. It is shown that an appropriate application of the smoothing method leads to the exact results in the case when N(z) is the random telegraph process. Moreover, under appropriate conditions on the general process N(z), approximate expressions are obtained for the correlation functions in terms of the first- and second-order moments of the solutions, and approximate expressions are given for these moments.
Mixed-Basis D Functions and Clebsch-Gordan Coefficients of Noncompact Groups
Authors: Klink William H.
Published: February 2004

Mixed-basis D functions are introduced as a tool for deriving Clebsch-Gordan coefficients of induced representations of semisimple groups. The Clebsch-Gordan coefficients of SU(1, 1) and SL(2, C) are computed as examples.
Realization of the Lie Group G(0, 1) by the Function of Landau Levels
Authors: Yanagawa S. , Moriya T.
Published: February 2004

The irreducible basis of the Lie group G(0, 1) are obtained in connection with the quantum physical problem: a motion of a free electron in a magnetic field. The differential operators are shown to be the infinitesimal operators defined by Brown.
Regge Trajectories for the Inverse Square Potential
Authors: Ferreira Erasmo , Sesma Javier
Published: February 2004

The analytic properties of the S function in the complex angular momentum plane for regular potentials with inverse square tails are discussed. Special attention is given to the determination of the poles of S in the limits of low and high energies. Two soluble examples are considered in detail.
Electromagnetic Sources in Moving Simple Medium
Authors: Johannsen Günther
Published: February 2004

A retarded potential tensor (4-vector) is derived in an arbitrary system of inertia for an arbitrary electromagnetic source in a moving homogeneous, isotropic, nondispersive, lossless dielectric. The velocity is uniform, and the result is relativistic correct.
Expansion of an n-Point Function as a Sum of Commutators
Authors: Zagier Don B.
Published: February 2004

We write the n-point function of currents as a sum over nested commutators, a form more suitable for certain current algebra calculations.
General Formula to Derive Branching Rules
Authors: van Daele A.
Published: February 2004

We consider an irreducible representation of a semisimple Lie algebra L. When restricted to a semisimple subalgebra K of L, this representation can be reduced with respect to K. We derive a general formula for the multiplicity of a certain irreducible representation of K, which occurs in it. The result is an extension of Kostant's formula for the multiplicity of a weight, where the subalgebra K is the Cartan subalgebra of L. Using Kostant's formula, we write down a set of equations, containing the required multiplicity, completely analogous with the usual formula involving the characters. We rewrite these equations using some properties of the partition function (used in Kostant's formula) and of the Weyl groups. Finally we solve them with the help of an ``orthogonality property.'' We illustrate the applicability by working out two nontrivial examples.
Cell Model Field Theories
Authors: Grimmer Hans
Published: February 2004

We investigate a class of nonrelativistic, self-interacting model field theories, assuming the existence of field and momentum operators that satisfy the canonical commutation relations (CCR's). The large symmetry that we assume permits us to determine explicitly all the relevant CCR representations and Hamiltonians. For each irreducible representation there exists only one Hamiltonian, which describes no interaction. Hence every model with interaction requires a reducible CCR representation. We expect a nontrivial S operator to exist for many of these models. We examine how close our models can come to relativistic theories.
Energy-Momentum Spectrum and Vacuum Expectation Values in Quantum Field Theory
Authors: Glimm James , Jaffe Arthur
Published: February 2004

We consider nonlinear boson self-interactions with a periodic spatial cutoff. We prove that the energy-momentum spectrum lies in the forward light cone. A momentum cutoff does not influence this result. For theories with finite-field strength renormalization, we obtain bounds on the vacuum expectation values of products of the φt's and ∇φ's. These bounds are uniform in the volume (and possible momentum) cutoff.
Semiclassical Perturbation Theory of Molecular Collisions. III. Graphical Angular Momentum Methods and the nth Order
Authors: Rabitz H. A. , Gordon R.
Published: February 2004

A review of graphical angular momentum methods is presented. As an example, the graphical methods are applied to the coupling problem in the semiclassical first and second Born approximations with an electric multipole potential. The general nth-order Born term is then considered. A simple expression is derived for the matrix elements of the anisotropic terms in the potential. The evaluation of the remaining integrals over time is discussed.
Spinor Treatment of Stationary Space-Times
Authors: Perjés Z.
Published: February 2004

A generalized SU(2) spinor calculus is established on the ``background space'' V3 of the stationary space-time. The method of spin coefficients is developed in three dimensions. The stationary field equations can be put to a form which in V3 is analogous to the Newman-Penrose equations. A V3 filling family of curves is determined by the gravitational field and is called the eigenray congruence. Stationary space-times may be characterized by the geometric properties of eigenrays. The relation of this classification to the algebraic ones is discussed. The method of solving the equations obtainable for various classes is illustrated on the case of nonshearing geodetic eigenrays. Assuming asymptotic flatness, we obtain the Kerr metric.
Newman-Penrose Constants and Their Invariant Transformations
Authors: Glass Edward N. , Goldberg Joshua N.
Published: February 2004

The origin and significance of the Newman-Penrose (N-P) constants of the motion are examined from the point of view that constants of the motion generate invariant transformations. Here the calculation makes use of a generalization of Green's theorem to a situation applicable to the coupled Einstein-Maxwell fields in general relativity. One finds strictly electromagnetic constants generated by an incoming electromagnetic shock wave, with dipole symmetry, at future null infinity. The gravitational constants contain an admixture from the electromagnetic field. They are generated by an incident quadrupole gravitational shock wave at future null infinity (J+). Both the electromagnetic dipole field and the gravitational quadrupole field behave like linearized fields at J+. All higher-multipole fields do not uncouple from the nonlinear corrections induced by the self-coupling of the gravitational field and its coupling with the electromagnetic field. It is shown that the gravitational constants are related to the rate of shear of the null rays near J+ a result which suggests a connection with sources. Finally, the supertranslation invariance of the result is discussed.
Diagrammatic Technique for Constructing Matrix Elements
Authors: Gilmore Robert
Published: February 2004

A diagrammatic technique is presented for computing the matrix elements of the generators of the unitary, orthogonal, symplectic, and symmetric groups (An, Dn, Bn, Cn, and Sn) within any of their unitary irreducible representations. Examples are worked out.
Unique Hamiltonian Operators via Feynman Path Integrals
Authors: Kerner Edward H. , Sutcliffe William G.
Published: February 2004

The old problem of how to represent uniquely a prescribed classical Hamiltonian H as a well-defined quantal operator Ĥ is shown to have a clear answer within Feynman's path-integral scheme (as expanded by Garrod) for quantum mechanics. The computation of Ĥ involves the momentum Fourier transform of a coordinate average of H. A differential equation for a reduced form of the Feynman propagator giving Ĥ from H is found; and the example of polynomial H worked out to give the Born-Jordan ordering rule for Ĥ in this case.
Bethe-Salpeter Equation and the Goldstein Problem
Authors: Keam R. F.
Published: February 2004

The Goldstein problem for the ladder-approximation Bethe-Salpeter equation for a spin-½ fermion-antifermion system bound to zero total mass by particle exchange is re-examined. It is suggested that the problem arises because attention has previously been focused on the wrong Dirac-space sectors of the equation. Criteria limiting the acceptable behavior of solutions are investigated and it is shown that discrete spectra for the values of the coupling constant are allowed to exist for the T-A sector and S-V sector solutions. Continuous spectra are always excluded.
Calculation of High-Field Distribution Functions in Semiconductors
Authors: Vassell M. O.
Published: February 2004

The relation between various methods proposed recently for calculating the hot-carrier distribution function in semiconductors is discussed. In particular, it is shown that the method deduced heuristically by Rees from considerations of the stability of the steady state is an iterative prescription for solving a suitably chosen integral form of the Boltzmann equation. It is shown that this method is essentially an adaptation of Kellogg's method with additional sufficiency conditions imposed to guarantee the existence of a positive solution and the convergence of the iterative process. The essential ingredient of these conditions is that the kernel of the integral equation be positive. It is further pointed out that the self-scattering process introduced by Rees belongs to a larger class of operators that ensure the required positivity of the kernel.
Ising-Model Spin Correlations on the Triangular Lattice. IV. Anisotropic Ferromagnetic and Antiferromagnetic Lattices
Authors: Stephenson John
Published: February 2004

A detailed discussion of pair correlations ω2(r) = 〈σ0σr〉 between spins at lattice sites 0 and r on the axes of anisotropic triangular lattices is given. The asymptotic behavior of ω2(r) for large spin separation is obtained for ferromagnetic and antiferromagnetic lattices. The axial pair correlation for the ferromagnetic triangular lattice has the same qualitative behavior as that for the ferromagnetic rectangular lattice: There is long-range order below the Curie point TC and short-range order above. It is shown that correlations on the anisotropic antiferromagnetic triangular lattice must be given separate treatment in three different temperature ranges. Below the Néel point TN (antiferromagnetic critical point), the completely anisotropic lattice exhibits antiferromagnetic long-range order along the two lattice axes with the strongest interactions. Spins along the third axis with the weakest interaction are ordered ferromagnetically. Between TN and a uniquely located temperature TD, there is antiferromagnetic short-range order along the two axes with the strongest interactions, and ferromagnetic short-range order along the other axis. TD is named the disorder temperature because it divides the short-range-order region TN < T < TD from the region TD < T < ∞, in which the axial pair correlations have exponential decay with temperature-dependent oscillatory envelope. There is no singularity in the partition function at TD, so there are only two thermodynamic phases: ordered below the Néel point, and disordered above. Correlations at TD decay exponentially. Finally, special consideration is given to the anisotropic antiferromagnetic lattice when the two weakest interactions are equal, and TN = TD = 0. The single disordered phase exhibits exponential correlation decay with oscillatory envelope for T > 0. The exact values of the axial pair correlations at T = 0 are calculated. For large spin separation r along the strong interaction axis, ω2 = (−1)r, and along the weak (equal) interaction axes ω2∼212E2·r−12cos (12πr)[1−(8r2)−1+…],where 212E2=0.588352663…,and E is a decay constant relating to pair correlations at the Curie point of a square lattice.
Domain of Dependence
Authors: Geroch Robert
Published: February 2004

The various properties of the domain of dependence (Cauchy development) which have been found particularly useful in the study of gravitational fields are reviewed. The basic techniques for constructing proofs and counterexamples are described. A new tool—the past and future volume functions—for treating certain global properties of space-times is introduced. These functions are used to establish two new theorems: (1) a necessary and sufficient condition that a space-time have a Cauchy surface is that it be globally hyperbolic; and (2) the existence of a Cauchy surface is a stable property of space-times.
Evaluation of Some Fermi-Dirac Integrals
Authors: Wasserman A. , Buckholtz T. J. , DeWitt H. E.
Published: February 2004

The evaluation of Fermi-Dirac integrals is discussed for cases in which the Sommerfeld method fails. Such cases occur when the integrand has a singularity at the Fermi surface and when the integrand is a rapidly oscillating function. As examples, the first-order exchange integral for electrons and the free-energy integral of the noninteracting electron gas in a magnetic field are evaluated. The method uses a contour-integral representation of the Fermi function (previously mentioned by Dingle), supplemented by Mittag-Leffler type expansions.
Parametrization of Crossing Symmetric Amplitudes
Authors: Roskies Ralph Z.
Published: February 2004

The crossing equations of Balachandran and Nuyts are solved to give the most general expansion of the π0-π0 amplitude consistent with crossing symmetry and in which the partial wave expansion is manifest. We find that crossing symmetry imposes 2L + 1 constraints on the Lth partial wave if the lower waves are known. The simplest such constraint (in the units were 4mπ2=1) is ∫ 01(1−s)(3s−1)f0(s) ds=0,where f0(s) is the s-wave π0-π0 amplitude.
Random Walks on Lattices with Traps
Authors: Rosenstock Herbert B.
Published: February 2004

We consider random walks on simple cubic lattices containing two kinds of sites: ordinary ones and ``traps'' which, when stepped on, absorb the walker. We study two related problems: (a) the probability of returning to the origin and (b) the situation in which the particle can meet its end, not only by absorption at a trap, but also by a process, called spontaneous emission, which has a constant probability per step. In problem (b), we ask for the probability that emission, rather than absorption, occurs. The solution to (a) is known for 1 dimension, and given here for the 3-, 4-, … dimensional cases; the 2-dimensional case remains unsolved. The solution to (b) is known for the 1-, 3-, 4-, … dimensional cases; we give it for 2-dimensional case.
On the Inverse Problem for the Klein-Gordon s-Wave Equation
Authors: Degasperis Antonio
Published: February 2004

The s-wave Klein-Gordon equation of a particle in an electriclike, spherically symmetrical, local, and energy-independent potential is studied with the aim of deriving exact relationships between the potential and the experimental data. We obtain the value of the potential and of all its derivatives at the origin as explicit functionals of the phase shift of the particle and the antiparticle and of their bound-state parameters. Furthermore, we find a set of infinitely many nonlinear sum rules for the phase shift and the bound-state parameters of both the particle and the antiparticle; these sum rules embody the requirement that the potential be electriclike. This restriction on the potential is also briefly discussed in the framework of the Gel'fand-Levitan approach to the inverse problem.
Orthogonality of a Set of Polynomials Encountered in Neutron-Transport and Radiative-Transfer Theories
Authors: Inönü Erdal
Published: February 2004

The properties of the polynomials hl(ν) which appear in the spherical-harmonics expansion of the eigen-solution φν(μ)=∑l=0∞12(2l+1)Pl(μ)hl(ν) in plane-symmetric one-speed transport problems with anisotropic scattering are reviewed and further investigated. These polynomials are shown to be orthogonal in the Stieltjes sense with a weight distribution which contains a continuous as well as a discrete portion. Some further properties of the hl(ν) are listed, taken from the Tchebycheff theory of orthogonal polynomials.
Alternative Method for Deriving the Fock Currents
Authors: Ott R. H.
Published: February 2004

Fock's principle is extended to apply to the fields near the shadow boundary on a parabolic cylinder, illuminated by a plane wave propagating at an arbitrary angle but normal to the cylinder axis. By solving the wave equation in parabolic coordinates, with an impedance boundary condition at the surface and a radiation condition at infinity, we find that the fields thus obtained agree very well with those predicted by Fock's principle, as long as the observation point is near where the wave propagation vector is tangent to the cylinder. When the propagation vector is tangent to the cylinder's apex, Fock's principle gives the exact currents everywhere on the cylinder. These results are more general than those of Jones, which are limited to perfectly conducting cylinders.
Stress-Tensor Commutators in Nonlinear Electrodynamics
Authors: Deser S. , Morrison L. K.
Published: February 2004

Equal-time stress-tensor commutators are shown to have a simple form for general systems of spin zero and one. Electrodynamics with arbitrary self-coupling is treated in detail. Ordering and other problems of a purely quantum nature are not discussed.
Correlations in Plasma. I
Authors: Dasgupta Brahmananda , Dasgupta Padmanabha
Published: February 2004

We derive for nonequilibrium isotropic plasma the correlations of fluctuation in electric-field intensity and current density. The calculation is based on the ring-approximation results of Balescu and Taylor. The results reduce to the results obtained by Rostoker in the special case when the velocity-distribution function is Maxwellian at the initial time.
Quantum Theory on a Network. I. A Solvable Model Whose Wavefunctions Are Elementary Functions
Authors: Montroll Elliott W.
Published: February 2004

A network model is developed for electrons in molecules and crystals. Wavefunctions and energy levels can be calculated exactly. A network which is topologically equivalent to that of the molecule or crystal of interest is constructed. One-dimensional bonds connect various node points. The potential V(x) = −V0 sech2 γx is postulated to act along each bond so that a potential well exists at each node point (atom) of the network. For some values of V0 the wavefunctions are elementary while generally they are hypergeometric functions. When the parameter γ is very large, the model is essentially the tight binding model and, when it is very small, one has the free-electron network model investigated by Ruedenberg and Scherr, Griffith, Coulson, and others. The density of states, energy band structure, and other features are discussed.
Functional Random-Walk Model of the Many-Particle System
Authors: Hosokawa Iwao
Published: February 2004

By Fourier-transforming the author's recently proposed state functional formalism for the BBGKY hierarchy, a new perspective of nonequilibrium statistical mechanics is given: the basic equation is formally very close to the Fokker-Planck equation and may readily be modified to a universal master equation (with irreversibility) by a slight change. Hence, the problem reduces to one of a generalized random-walk such that the stochastic quantity to be considered is the particle-number density in the 1-body phase space. A general solution is formulated for the weak interaction case.
Pair Occupation, Fermi Condensation, and Phase Transitions in Many-Fermion Systems
Authors: Girardeau M. D.
Published: February 2004

For a system of 2n fermions it is shown that the occupation number nα of any fermion-pair state φα(x1x2) is nα=n∫φα*(x1x2)ρ2(x1x2,x1′x2′)φα(x1′x2′) dx1 dx2 dx1′ dx2′, where ρ2, assumed normalized to unity, is the two-particle density matrix. The known upper bound on the largest eigenvalue of ρ2 implies that nα ≤ ½ in the thermodynamic limit n → ∞, equality being approached, with suitable φα, only for certain limiting BCS states ψBCS. Bose condensation of fermion pairs, in the sense of macroscopic occupation of any nα, is impossible. Fermi condensation into φα is defined to be present if nα > 0 in the thermodynamic limit. It occurs, for example, for suitable BCS states, but for a normal Fermi system all nα are of order n−1 or smaller. It is argued that the physical criterion for ψBCS exhibiting Fermi condensation is that the pair state from which it is constructed must have a bound-state component of range ≲kF−1, where kF is the Fermi momentum (3π2ρ)⅓. The maximally-occupied pair state is the eigenfunction of ρ2 belonging to the largest eigenvalue, associated with off-diagonal long-range order of the type defined by Yang. A formula for n0 for the original BCS state is derived. Some remarks are made concerning the interpretation of the Fermi condensation as a superconducting transition. The analysis is generalized to occupation of l-fermion states. It is conjectured that, when Fermi condensation first sets in at a given even l, this is associated with the formation of bound states of l fermions, and a formula for the maximal occupation of such states is exhibited. The implications, for the theory of liquid 4He, of the fact that a helium atom contains electrons are examined. It is shown that Bose condensation into a single-4He-atom state is impossible, but a Fermi condensation similar in some respects to that in a superconductor can and probably does occur. It is argued that the mechanism preventing Bose condensation in superconductors and liquid 4He lies in the effect of collisions, acting via the exclusion principle, in causing virtual internal excitations.
Solution of the Lee Model in All Sectors by Dynamical Algebra
Authors: Fivel Daniel I.
Published: February 2004

A potential theory canonically equivalent to the Lee model in all sectors is deduced with algebraic techniques. For the Vθ sector this potential is free of disconnected graph difficulties and so is soluble in closed form. For higher sectors the problem can be reduced to Fredholm equations.
Application of Iteration Technique to the Integral Equations of Scattering Theory
Authors: Doleschall P. , Révai J.
Published: February 2004

An iterative method based on the generalization of the Fredholm reduction is developed for the solution of the integral equations of the scattering theory.
Irreducible Cartesian Tensors. II. General Formulation
Authors: Coope J. A. R. , Snider R. F.
Published: February 2004

A general formulation is given of a method of reduction of Cartesian tensors, by Cartesian tensor operations, to tensors irreducible under the three-dimensional rotation group. The criterion of irreducibility is that a tensor be representable as a traceless symmetric tensor, its reduced or natural form, invariantly embedded in the space of appropriate order. The general formulation exploits the properties of invariant linear mappings between tensor spaces. Considered abstractly, such mappings bring out the structure of the theory and illuminate the relation to spherical tensor theory. On the other hand, any linear invariant mapping between tensor spaces is equivalent to a combination of operations with the elementary invariant tensors U and ε. The general abstract formation therefore has a direct operational representation in terms of the ordinary tensor operations of contraction and permutation of indices. An analogous formulation is given for spinors, and the relations between spinors, Cartesian tensors, and spherical tensors is discussed in the language of the present formalism. Lastly, several examples are given as to how the general formalism may be applied to groups other than the rotation group.
On Some Classes of Solutions of the Wave Equation ∂t2f−Δf=0
Authors: Kastrup H. A. , Mayer D. H.
Published: February 2004

We investigate certain classes of solutions of the wave equation for which Rf(x) = −(1/x2)f(+x/x2) is a well-defined C∞ solution for all points of the Minkowski space, if f(x) is a C∞ solution. We mainly exploit the fact that the transformation R is a generalized Hankel transformation in momentum space. We make use of several recent results obtained by Zemanian in connection with the Hankel transformation and construct a self-adjoint representation of the Lie algebra of the group O(2, 4) which is an invariance group of the wave equation. Finally, we construct and discuss the eigenfunctions of R in Minkowski space.
Extremum Principles for the Föppl-Hencky Equation
Authors: Anderson N. , Arthurs A. M.
Published: February 2004

Maximum and minimum principles are obtained for the nonlinear Föppl-Hencky differential equation which arises in the theory of elasticity. Approximate solutions are obtained from a simple class of trial functions.