bluuebunny/crossref_metadata_2025 huggingface dataset:
Structure of the Phonon Propagator for an Anharmonic Crystal
Authors: Maradudin A. A. , Ipatova I. P.
Published: February 2004

The phonon propagator for an arbitrary crystal is the analytic continuation to the complex z plane of the Fourier coefficient of the imaginary time correlation function Djj′(k;u)=〈TAkj(u)Akj′+(0)〉, where Akj is the field operator for phonons with wavevector k and polarization or branch index j. Considering D(k; u) as a 3r × 3r matrix whose elements are labeled by j and j′ (j,j′ = 1, 2, …, 3r), where r is the number of atoms in a primitive unit cell of the crystal, the restrictions imposed on the form of this matrix by the symmetry and structure of the crystal are determined here. In particular, it is proved that the element Djj′(k; u) vanishes unless j and j′ label normal modes of vibration which transform according to the same row of the same irreducible multiplier representation of the point group of the wavevector k, G0(k). As a corollary to this result it follows that if no two modes labeled by the wavevector k exist whose frequencies are different, but whose associated eigenvectors transform according to the same irreducible multiplier representation of G0(k), the matrix Djj′(k; u) is diagonal in j and j′.
Zero-Mass Representations of the Poincaré Group in an O(3, 1) Basis
Authors: Mukunda N.
Published: February 2004

Unitary irreducible representations of the Poincaré group, corresponding to zero mass and finite helicity, are reduced with respect to the subgroup of homogeneous Lorentz transformations. The action of the energy-momentum operators in the basis suited to this reduction is examined.
Current Formalism. II. The S Matrix in Perturbation Theory
Authors: Wray J. G.
Published: February 2004

This work has accomplished in the context of asymptotic quantum field theory the following objectives. (1) The S-matrix equations of Pugh [R. E. Pugh, Ann. Phys. (N.Y) 23, 335 (1961)] and their generalization in the manner of Chen [T. W. Chen, Ann. Phys. (N.Y.) 42, 476 (1967)] are derived without the aid of an interacting field. (2) A diagrammatic representation of these integrodifferential equations is demonstrated. (3) The problem of boundary conditions for a self-interacting system is solved in perturbation theory. This leads to a finite, divergence free, no cutoff expansion in the physical coupling constant. For renormalizable interactions, the only additional parameter is the physical mass, whereas for nonrenormalizable interactions, uniqueness of the expansion requires additional parameters with increasing order of expansion. (4) The success of the perturbation expansion serves as a posteriori justification of the formulation in CF.I., [J. G. Wray, J. Math. Phys. 9, 537 (1968)] upon which the present work is built. The Π or ``generalized step'' function and its associated algebra plays the principal technical role in facilitating this work. The crucial Π-ordering relations and theorems developed in CF.I. are reviewed here.
Point Transformation of Classical Hard-Core Potential
Authors: Cooper Martin J.
Published: February 2004

The method of an extended canonical point transformation is used to reformulate the singular repulsions in a classical hard-sphere gas as equivalent velocity-dependent interactions. The approach provides a Hamiltonian in which the repulsions appear as nonlocal potential interactions between the particles and may therefore be treated within any of the conventional perturbation methods of many-body analysis. Application of the technique to obtain a kinetic equation for a hard-sphere gas is outlined.
Green's Function of a Particle in a Uniform External Field
Authors: Eftimiu C. , Thompson K. W.
Published: February 2004

The exact propagator of a spinless charged particle in a uniform electromagnetic field with |E| = |H| and E·H = 0 is shown to lack the pole corresponding to the mass of the particle.
Gravitational Radiation in an Expanding Universe
Authors: Hawking S. W.
Published: February 2004

Asymptotic expansions are used to study outgoing gravitational radiation in an expanding, dust-filled Friedmann universe of negative curvature. It is found that the interaction with the matter modifies the ``peeling-off'' behavior. A quantity is defined which is interpreted as the total mass of the source and the disturbance, and which monotonically decreases as gravitational radiation is emitted. The group of coordinate transformations that preserve the asymptotic form of the metric and the boundary conditions is the same as the isometry group of the undisturbed Friedmann model. This may indicate that no physical significance attaches to the extra transformations of asymptotically flat space which are not contained in the inhomogeneous Lorentz group.
Screening in the Schrödinger Theory of Scattering
Authors: Dollard John D.
Published: February 2004

The time-dependent Schrödinger theory of scattering is studied rigorously with the potential V(x) replaced by e−ε|x|V(x). Sufficient conditions are given that the Mo/ller wave matrices Ω± be obtainable from this theory as ε → 0. The conjecture that this theory can be used to define a reasonable S matrix when the Ω± do not exist is false for the Coulomb potential.
Comments on Separability Operators, Invariance Ladder Operators, and Quantization of the Kepler Problem in Prolate-Spheroidal Coordinates
Authors: Kiefer H. M. , Fradkin D. M.
Published: February 2004

The Schrödinger equation for the hydrogen atom separates in three coordinate systems: spherical, parabolic, and prolate spheroidal. The separability operators associated with the separation constants for these three systems are exhibited and discussed. Also, for these systems, the invariance ladder operators which transform a simultaneous eigenfunction of the separability operators into a different simultaneous eigenfunction of the same energy are discussed with reference to the elements of the O4 Lie algebra. Quantization of the Kepler problem in terms of prolate spheroidal coordinates is accomplished and discussed.
Existence, Uniqueness, and Convergence of the Solutions of Models in Kinetic Theory
Authors: Cercignani Carlo
Published: February 2004

A recently proved theorem of existence and uniqueness for the linearized Boltzmann equation is extended to two- and three-dimensional domains and general boundary conditions. The proof is valid for collision operators having a finite collision frequency, which can arise either from an angular or a radial cutoff or by assuming a model equation. Finally, convergence of the solutions of kinetic models to solutions of the actual Boltzmann equation is shown to hold for the boundary-value problems considered in this paper.
Dimers on a Rectangular Lattice
Authors: Baxter R. J.
Published: February 2004

A set of matrix equations is derived which yields the statistical mechanical properties of a system of monomers and dimers on a rectangular lattice in the thermodynamic limit. As the matrices are strictly of infinite dimensionality, the equations cannot be solved directly, but if they are restricted to be of finite and quite small dimensionality, very good approximations to the thermodynamic properties are obtained.
O(5) Polynomial Bases
Authors: Sharp R. T. , Pieper S. C.
Published: February 2004

Polynomial bases are derived for the irreducible representations of the group O(5). The matrix elements of the infinitesimal generators are given.
Stability of Matter. II
Authors: Lenard A. , Dyson Freeman J.
Published: February 2004

The stability of a system of charged point particles is proved under the assumption that all negatively charged particles are fermions. A lower bound on the energy is found to be −Aq⅔Nme4ℏ−2, where q is the number of distinct negative species, N the total number of negative particles, m an upper bound for their mass, e an upper bound for the absolute value of the charge on both negative and positive particles, and A is a numerical constant.
An Integral Equation for the Scattering Operator
Authors: Wilner M.
Published: February 2004

An integral equation, originally derived for a perturbation expansion of the n-point scattering function (Pugh's equation), whose (finite) solution is the renormalized perturbation result, is derived here as an exact ``strong'' equation for the nth operator derivative of the scattering operator, whose vacuum expectation value is the n-point function.
Field-Theoretic Formulation of Quantum Statistical Mechanics
Authors: Alexanian Moorad
Published: February 2004

The notions of strong convergence of state vectors, introduced by Haag in his formalism of axiomatic quantum field theory, are extended to the case of vectors with an infinite number of particles but finite densities. Some general properties of nonequilibrium distribution functions are derived without the use of power series expansions or any other simplifying assumption. An integral representation is obtained for the distribution functions which makes it possible to discuss their behavior for small and large energies and to obtain some information about the singularities of these functions when continued analytically.
Quantum Corrections to the Pair Distribution Function of a Classical Plasma
Authors: Diesendorf Mark , Ninham Barry W.
Published: February 2004

Quantum-mechanical corrections to the pair distribution function of a plasma at high temperature and low density are calculated to order e2 in the interaction, using standard diagram perturbation techniques. Both the effects due to quantum statistics (exchange) and the finite size of a wave packet (dynamic screening), are considered.
Solution of the Transport Equation with Anisotropic Scattering in Slab Geometry
Authors: Bowden Robert L. , McCrosson F. Joseph , Rhodes Edgar A.
Published: February 2004

Some systematics which exist between eigenfunctions and adjoint singular integral equations arising in the solution of the transport equation in slab geometry are illustrated. The transport equation is shown to obey a singular integral equation and its relationship to the eigenfunction expansion-method solution is shown. A new method for solving for the expansion coefficients in the eigenfunction expansion method is illustrated by solving Milne's problem. The role adjoint singular integral equations play in finding appropriate weight functions for use in orthogonality relations between the eigenfunctions of the transport equation is briefly discussed.
Two-Dimensional Measure-Preserving Mappings
Authors: Greene John M.
Published: February 2004

A particular area-preserving mapping of a plane onto itself has been studied in detail with the aid of a digital computer. A large number of fixed points, finite sets of points that transform into each other, were located and classified as elliptic or hyperbolic depending on the nature of the linearized mapping in the neighborhood. A quantity called the residue was calculated for each fixed point. This quantity can be used to predict whether other nearby fixed points are elliptic or hyperbolic. The results showed that there are considerable regions in which almost all the fixed points are hyperbolic. Further calculations were made to estimate the area enclosed by the invariant curves whose existence has been established by Moser. The boundary of this region appeared to coincide with the boundary of the region in which almost all the fixed points are hyperbolic.
Weyl Transform in Nonrelativistic Quantum Dynamics
Authors: Leaf Boris
Published: February 2004

The Weyl transform is applied in quantum dynamics to derive and extend Moyal's statistical theory of phase-space distributions for noncommuting coordinate and momentum operators. The distinction is made between Weyl transforms in Schrödinger and Heisenberg pictures; the general case of time-dependent Hamiltonians is considered. The Wigner function for the probability distribution in a phase space of Cartesian coordinates Q and momenta K propagates according to a conditional probability P(t, Q, K | t0, Q0, K0), which is exhibited as a Feynman path integral in phase space. Properties of P(t, Q, K | t0, Q0, K0) are developed; it is expressed in terms of the quantum generalization of the classical Liouville operator. The Weyl transform of a Heisenberg operator propagates according to P(t, Q0, K0 | t0, Q, K) which is also given as a Feynman path integral. An equation for the time evolution of Weyl transforms of Heisenberg operators is obtained, according to which the transform of Heisenberg coordinate and momentum operators obey a quantum form of Hamilton's equations of motion. If the initial density operator of the system commutes with the coordinate operator, then the state of the system is a mixture of pure coordinate states; the spectrum of the density operator in this case is continuous. For a Heisenberg operator AH(t) with Weyl transform AH(t, Q, K), the function A(t,Q)=∫dKAH(t,Q,K) is the expectation at time t of the dynamical property for a quantum system initially in a pure state of coordinate Q; it is the quantum-mechanical generalization of the dynamical property of the system along the classical trajectory in configuration space at time t. The probability amplitude for the time dependence of A(t,Q) can be expressed as a Feynman path integral with a Heisenberg Lagrangian. The amplitude of the conditional probability P(t, Q | t0, Q0) considered by Feynman is expressed as a path integral with a Schrödinger Lagrangian. The velocity in the Heisenberg Lagrangian is the negative of that in the Schrödinger Lagrangian of Feynman, but it agrees with the velocity appearing in the Hamiltonian equations. It is the Heisenberg Lagrangian that is the Lagrangian of classical dynamics. For a particle whose potential energy is a function of position, a quantum form of Newton's second law is obtained. An extension of the formalism to non-Cartesian coordinate systems is given.
Properties of Higher-Order Commutator Products and the Baker-Hausdorff Formula
Authors: Eriksen Erik
Published: February 2004

The element z = log exey, which is known to be an element of the Lie-algebra generated by x and y, is expressed as a commutator series in x and y with coefficients given in terms of certain fixed polynomials. The result is given explicitly to sixth order. Useful recurrence relations are obtained. The method is based on certain properties of higher-order commutator products, particularly their idempotent character.
Two- and Three-Particle Coulomb Systems
Authors: Nutt Gerald L.
Published: February 2004

The Coulomb T matrix is examined in a two-body and a three-body Hilbert space. The purpose is to demonstrate how this function may be used to construct the Faddeev three-body kernel. In the process the apparent paradox of the indefiniteness of the Coulomb T matrix on the energy shell is resolved. The question of off-shell unitarity is discussed.
Improved Method for Quantum-Mechanical Three-Body Problem. III. Use of Sturmian Functions
Authors: Eyges Leonard , Jasperse John R.
Published: February 2004

We extend previous work on the ground state of the symmetric three-body problem by expanding the two-body orbitals φ(k, κ) or φ(k, κ) (one and three dimensions, respectively) in a set of Sturmian functions. This retains the advantages of the previous expansions, and gains several new ones as well. Among these are a simplification of the equations and a transparent way of estimating convergence. The one-dimensional problem is reduced to an infinite set of coupled integral equations in one variable and the three-dimensional one to a doubly infinite set. As an application and a test of convergence we have solved the one-dimensional equations numerically in successive truncations. We find that keeping only the first term of the set yields results typically accurate to a fraction of a percent.
Inhomogenizations and Complex Representations
Authors: Berzi Vittorio , Gorini Vittorio
Published: February 2004

Real Lie-algebra inhomogenizations are considered for complex defining representations and some emphasis is devoted to the case when the defining representation is irreducible. A theorem is given concerning the conditions under which nonequivalent complex representations give rise to isomorphic inhomogenizations and a classification is made of the complex inhomogenizations of semisimple Lie algebras.
Some Total Invariants of Asymptotically Flat Space-Times
Authors: Winicour J.
Published: February 2004

The total energy, momentum, supermomentum, and angular momentum of asymptotically flat space-times are calculated in terms of coordinate and conformally invariant expressions by taking the limit in an invariant way of the asymptotic symmetry linkages through a sequence of finite closed two-spaces which converge to a sphere at null infinity. The resulting expressions consist of integrals over the sphere at null infinity of cordinate and conformally invariant quantities. In the case of energy and momentum these integrals may be reduced to expressions previously proposed by Penrose.
Electric-Field Penetration into a Plasma with a Fractionally Accommodating Boundary
Authors: Mason Rodney J.
Published: February 2004

Landau's field penetration study is extended to a plasma with a boundary that reflects a fraction σ of the incident electrons specularly and the remainder diffusely. Exact solutions for specular and diffuse reflection, and series solutions for fractional accommodation are obtained. At great depths in the plasma the field is found to exhibit negligible σ dependence for ω near ωp and weak dependence through a factor (1 + σ) at other frequencies.
Generalized Ornstein-Zernike Approach to Critical Phenomena
Authors: Green Melville S.
Published: February 2004

A generalization of the Ornstein-Zernike integral equation is derived and suggestions are made about a possible application to an improved theory of critical phenomena. A fundamental maximum principle of statistical mechanics is used to place the generalized equation in the context of phase transitions and critical points. The equation is a relationship between a generalized correlation matrix by means of which the average fluctuation product of any two sum functions may be expressed and a generalized direct-correlation matrix which is the second functional derivative of the functional in the maximum principle. The existence of a critical eigenvector of the direct-correlation matrix is proposed and three physical meanings of this vector are given. An explicit formula for the direct-correlation matrix is given and is used to derive two asymptotic properties. This formula exhibits an unexpected relationship between the generalized Ornstein-Zernike equation and the Percus-Yevick equation.
Angular Momentum and the Kerr Metric
Authors: Cohen Jeffrey M.
Published: February 2004

For Kerr's rotating metric, it is shown that −ma is the angular momentum of the body where m is the mass and a is the rotation parameter. This is true even for large m and a.
Diffuse Transmission of Light from a Central Source through an Inhomogeneous Spherical Shell with Isotropic Scattering
Authors: Bellman R. E. , Kagiwada H. H. , Kalaba R. E. , Ueno S.
Published: February 2004

A partial-differential-integral equation is derived in this paper for the angular distribution of the radiation which is diffusely transmitted through an inhomogeneous, isotropically scattering, spherical shell when there is a constant net flux of radiation normally incident on the inner surface. An equation is also derived for the strength of the diffusely reflected radiation when the shell is illuminated at each point on the outer surface by constant isotropic incident radiation.

The equations obtained appear to lend themselves well to numerical solution. Astrophysically, the situation corresponds to determining the brightness of a spherical planetary nebula. As far as is known, the equations are new and exact.
Possible Interpretation of Quantum Mechanics
Authors: de la Peña-Auerbach L. , García-Colín Leopoldo S.
Published: February 2004

It is shown that to Schrödinger's equation, one may associate a Markoff process in the Smoluchowski approximation with an external force acting on the system which is a measure of its interaction with the vacuum. This interaction is, in this scheme, responsible for the stochastic character of the motion. The physical interpretation of the usual momentum and energy operators emerges in a natural way from the theory. Thus, we are immediately led to Heisenberg's uncertainty relations. However, this interpretation of quantum mechanics is valid only within the limits of validity of Smoluchowski's equation. As a simple example, we treat the one-dimensional harmonic oscillator.
Weak-Graph Method for Obtaining Formal Series Expansions for Lattice Statistical Problems
Authors: Nagle John F.
Published: February 2004

A unified exposition of the weak-graph method for obtaining formal series expansions for lattice statistical problems is presented. The prototype of this method is the derivation of the hyperbolictangent high-temperature expansion for the spin-½ Ising model. Also, recent expansions of the monomer-dimer problem and various hydrogen-bonded problems have been treated by essentially the same method. In this paper the method is further illustrated by obtaining series expansions for the low-temperature spin-½ Ising problem, the low-density hard-core lattice-gas problem, the high-temperature spin-1 Ising problem, the k-color problem, and two new model problems, the ramrod model and a special ternary model. The weak-graph method enables one to obtain especially useful series expansions for a certain class of problems, including the spin-½ Ising problem and the monomer-dimer problem, which have essentially a binary nature.
Radiating Spheres in the Scalar-Tensor Theory of General Relativity
Authors: Kaufmann William J.
Published: February 2004

Using Bondi's ``radiation coordinates,'' the Brans-Dicke field equations describing the interior of a sphere consisting of a perfect fluid and a radiation field, part of which is purely isotropic and the remainder of which is purely radial, are stated. A method for seeking solutions to these equations in the radiation-filled space outside the sphere is given.
Bases for the Representations of U4 in the Chain U4⊃U2+U2
Authors: Devi V. Syamala , Venkatarayudu T.
Published: February 2004

The highest-weight polynomials of irreducible representations of U3 occurring in the reduction of the direct product of two irreducible representations are constructed from Young diagrams. The different irreducible representations in the product are labeled by a parameter which distinguishes their multiplicities also. The method of this paper can be easily extended to any Un.
Spherical Model as an Instance of Eigenvalue Degeneracy
Authors: Thompson Colin J.
Published: February 2004

It is shown that the free energy of the spherical model can be expressed in terms of the largest eigenvalue of an integral equation. In three dimensions the spectrum of the integral equation becomes degenerate as the critical point is approached from the high-temperature side, thus heralding the onset of long-range order.
3-Sphere ``Backgrounds'' for the Space Sections of the Taub Cosmological Solution
Authors: Matzner Richard A.
Published: February 2004

The cosmological solution due to Taub resembles a radiation filled Robertson-Walker solution which contains long-wavelength gravitational radiation rather than electromagnetic radiation. This paper presents several schemes for obtaining 3-sphere ``backgrounds'' for the t = const space slices M(t) in the Taub solution. [M(t) is characterized by being homogeneous for each t.] The reason for finding such backgrounds is to present models for defining wave-background separation in general. Unspecified averaging methods as suggested by Isaacson work for short-wavelength gravitational radiation. We attempt here to specify a method which works even when the radiation is of long wavelength. One background is obtained by averaging the metric by Lie transport along certain invariantly defined vector fields on M(t). This background is compared with two other (different) plausible definitions for a background, and reasons are given to suggest that the Lie-transport average is the preferable definition.
Construction of Orthonormal Angular-Momentum Operators
Authors: Corio P. L.
Published: February 2004

The problem of constructing an orthonormal basis for the (2j + 1)2-dimensional space of angular-momentum operators is reduced to the problem of constructing a sequence of orthogonal polynomials of a discrete real variable. A number of properties of the polynomials are obtained, including a pure recurrence formula and several mixed recurrence relations. Three explicit representations of the polynomials are given, together with a table of orthonormal angular-momentum operators.
Solutions of Einstein's Equations for a Fluid Which Exhibit Local Rotational Symmetry
Authors: Stewart J. M. , Ellis G. F. R.
Published: February 2004

All solutions of Einstein's equations for pressure-free matter which exhibit local rotational symmetry were classified in an earlier paper by one of us. This paper extends the earlier theory to the case of a general fluid, with an electromagnetic field possibly present. A classification of these solutions for a perfect fluid is given, and assuming a physically reasonable equation of state, some exact solutions of cosmological interest are obtained. Finally, the difficulties encountered when extending the treatment to a general fluid are discussed; the same general classification can be made.
Symmetric Representation for Three-Body Problems. II. Motion in Space
Authors: Whitten R. C. , Smith F. T.
Published: February 2004

The symmetric representation of three bodies is extended to the case of motion in spece. It is found that the generalized angular-momentum eigenfunctions are linear combinations of products of elements of the ``kinematic-rotation'' group with elements of the spatial-rotation group. Recurrence relations among the expansion coefficients are derived; values are tabulated for λ ≤ 4. Finally, coupling coefficients for the eigenfunctions are derived.
Mean Entropy of States in Quantum-Statistical Mechanics
Authors: Lanford Oscar E. , Robinson Derek W.
Published: February 2004

The equilibrium states for an infinite system of quantum mechanics may be represented by states over suitably chosen C* algebras. We consider the problem of associating an entropy with these states and finding its properties, such as positivity, subadditivity, etc. For the states of a quantum-spin system, a mean entropy is defined and it is shown that this entropy is affine and upper semicontinuous.
Parameter Differentiation of Exponential Operators and the Baker-Campbell-Hausdorff Formula
Authors: Lutzky M.
Published: February 2004

Parameter differentiation of exponential operators is used to derive a method for obtaining the Baker-Campbell-Hausdorff coefficients in a more explicit form than is available from the standard Hausdorff recursion formula. In passing, a derivation of the Hausdorff recursion formula is given which is simpler than the proof usually presented.
Cauchy Problem in the Scalar-Tensor Gravitational Theory
Authors: Cocke W. J. , Cohen Jeffrey M.
Published: February 2004

We show that the scalar-tensor gravitational field equations can be split up into a set of initial-value equations and a set of time-evolution equations and that the initial-value equations preserve their form as they propagate forward in time. The proper amount of freedom for the stress-energy tensor is maintained, and the Cauchy problem is thus posed reasonably well.
Special Functions and the Complex Euclidean Group in 3-Space. II
Authors: Miller Willard
Published: February 2004

This paper is the second in a series devoted to the derivation of identities for special functions which can be obtained from a study of the local irreducible representations of the Euclidean group in 3-space. A number of identities obeyed by Jacobi polynomials and Whittaker functions are derived and their group - theoretic meaning is discussed.
Korteweg-de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion
Authors: Miura Robert M. , Gardner Clifford S. , Kruskal Martin D.
Published: February 2004

With extensive use of the nonlinear transformations presented in Paper I of the series, a variety of conservation laws and constants of motion are derived for the Korteweg-de Vries and related equations. A striking connection with the Sturm-Liouville eigenvalue problem is exploited.
Properties of the S Matrix for Velocity-Dependent Potentials
Authors: Ferreira Erasmo M. , Guillén Nilo , Sesma Javier
Published: February 2004

Solutions of the Schrödinger equation with a velocity-dependent potential are discussed. The analytic behavior of the S matrix as a function of the momentum is investigated, and some properties of general validity are demonstrated. The configuration of the poles of the scattering matrix in the complex-momentum plane is described in detail for the case of a spherically symmetric potential.
Extension of the Factorization Method
Authors: Humi Mayer
Published: February 2004

We show that it is possible to extend the formalism of the factorization method for any displacement in the spectrum space of any second-order differential equation. Following this, we show that we can extend, at least formally, the formalism for some nth-order ordinary differential equations.
``Lorentz Basis'' of the Poincaré Group
Authors: Chakrabarti Amitabha , Levy-Nahas Monique , Seneor Roland
Published: February 2004

An explicit derivation is given for the matrix elements of the translation generators Pμ of the Poincaré algebra with respect to the ``Lorentz basis,'' namely, in terms of states which diagonalize the two Casimir operators of the homogeneous Lorentz group (HLG). The results are given for the cases mass μ > 0 and μ = 0 and, for the latter, for discrete and continuous spin. The transforms connecting the momentum and Lorentz bases are discussed, a detailed derivation being given for the zero-mass discrete-spin case. The matrix elements of Gμ = i[(N2 − M2), Pμ] are considered and several interesting aspects of the algebras generated by N, M′, and Pμ′=(ε1Pμ+ε2Gμ) are discussed for the cases of positive as well as zero rest mass.
Random Spin Systems: Some Rigorous Results
Authors: Griffiths Robert B. , Lebowitz J. L.
Published: February 2004

Several general results are obtained for a system of spins on a lattice in which the various lattice sites are occupied at random, and the spins, if present, interact via a general Heisenberg or Ising interaction decreasing sufficiently rapidly with distance. It is shown that the free energy per site exists in the limit of an infinite system, is a continuous function of concentration, and has the usual convexity (stability) properties. For Ising systems with interactions of finite range, the free energy is an analytic function of concentration and magnetic field for a suitable range of these variables. The random Ising ferromagnet on a square lattice (or simple cubic lattice) with nearest-neighbor interactions is shown to exhibit a spontaneous magnetization at sufficiently high concentrations and low temperatures.
Current Distribution in a Thin Superconducting Strip
Authors: Swan G. W.
Published: February 2004

A mathematical model of the alternating-current distribution for a long superconducting strip with small thickness is analyzed. The interesting feature of the model is that it is soluble. The current distribution is found from the analytic solution to a singular integral equation, and from this solution it is possible to predict the behavior of the distribution throughout a complete cycle. Previous (unpublished) work showed that purely physical arguments did not give the correct predictions for this behavior at the end of each half-cycle. The present model overcomes this deficiency.
One- and Two-Center Expansions of the Breit-Pauli Hamiltonian
Authors: Fontana Peter R. , Meath William J.
Published: February 2004

The orbit-orbit, spin-spin, and spin-orbit Hamiltonians of the Breit-Pauli approximation are expressed in terms of irreducible tensors. One- and two-center expansions are given in a form in which the coordinate variables of the interacting particles are separated. In the one-center expansions of the orbit-orbit and spin-orbit Hamiltonians the use of the gradient formula reduces some of the infinite sums to finite ones. Two-center expansions are discussed in detail for the case of nonoverlapping charge distributions. The angular parts of the matrix elements of these Hamiltonians are evaluated for product wavefunctions.
Antisymmetric Projection in the Approximation of No Spin-Orbit Coupling
Authors: Sullivan John J.
Published: February 2004

The explicit form of the projection operator for constructing antisymmetric wavefunctions for N fermions in the approximation of no spin-orbit coupling is developed. Projection is applied within the one particle approximation. It is shown that if the orbitals associated with the minority spin can be completely expanded in terms of the orbitals associated with the majority spin, then the projected Hartree-Fock scheme is completely equivalent to unprojected Hartree-Fock theory. In the unrestricted case, deviations from this condition are not expected to be large, and integral properties such as energies calculated in the projected scheme should not be significantly different from unprojected results. However, for such properties as spin density at the nucleus in atoms or ions with nominally closed s shell, there may be significant differences between projected and unprojected schemes.
Scalar Product for Harmonic Functions of the Group SU(2)
Authors: Pandres Dave , Jacobson Donald A.
Published: February 2004

A scalar product is defined which results in the single- and double-valued spherical harmonics spanning a seminormed linear vector space that carries all of the irreducible unitary representations of the group SU(2). The possibility of defining such a scalar product was indicated in a previous paper. A Hilbert space is derived from the seminormed space through a further construction involving equivalence classes of vectors.
``Haag Theorem'' for the Point-Coupling Relativistic Lee Model
Authors: Ynduráin F. J.
Published: February 2004

We prove that the free and total (renormalized) Hamiltonians exist as essentially self-adjoint operators in the relativistic Lee model without cutoff. Nevertheless, their domains of definition only have the zero vector in common and thus the interaction Hamiltonian is meaningless (``Haag's theorem'').
Generalized Second-Order Relativistic Wave Equations. I
Authors: Rafanelli Kenneth
Published: February 2004

A covariant second-order wave equation, free of subsidiary conditions, is deduced from the familiar linear relativistic wave equation for a free particle of arbitrary spin by use of the representation-invariant Lie algebra of O(4, 1). The correspondence principle is used to interpret the physical content of this generalized equation, which explicitly admits zitterbewegung and implies an inverse spin dependence for the rest energy. Without further assumption this generalized second-order equation is equivalent to the Klein-Gordon equation for the particular Lie algebras of the Dirac and Duffin-Kemmer rings. For higher spins the imposition of a subsidiary condition, understood via the Bargmann-Wigner analysis, extends the equivalence with the Klein-Gordon equation and explicitly displays the above mass spectrum.
Ensemble of Random Matrices with a Random Bias
Authors: Favro L. D. , McDonald J. F.
Published: February 2004

The ensemble exp [−γ Tr (H − H0)2], where the eigenvalues λn of H0 are given by some distribution f(λ), is investigated. In particular, the limits of large and small γ for the orthogonal case are considered. Formal expressions are obtained for various distributions of the eigenvalues in the two limits. The approximation developed in the large-γ limit is also applied to the thermodynamics of an incompletely specified system. Further, it is shown that this approximation is easily extended to include the unitary and symplectic ensembles.
Special Functions and the Complex Euclidean Group in 3-Space. III
Authors: Miller Willard
Published: February 2004

This paper is the third in a series analyzing identities for special functions which can be derived from a study of the local representations of the Euclidean group in 3-space. Here identities are derived which relate Gegenbauer polynomials, Whittaker functions, Jacobi polynomials, and Bessel functions. Among the results are generalizations of the addition theorems for solid-spherical harmonics and a group-theoretic interpretation of the Maxwell theory of poles.
Natural Boundary and Initial Conditions from a Modification of Hamilton's Principle
Authors: Tiersten H. F.
Published: February 2004

In a Hamiltonian variational formulation of a field theory, certain boundary conditions arise naturally whereas others arise as constraints on the admissible variations, and no initial conditions arise naturally. When the Hamiltonian formulation is used to obtain approximate solutions to boundary-value problems, the approximating functions need not satisfy the natural conditions but must satisfy the constraint conditions. Although, in many instances, it is desirable for the approximating functions to satisfy the constraint—and even the natural—conditions, in other instances it is imperative that the approximating functions do not satisfy certain constraint conditions. A procedure is introduced for transforming Hamilton's principle so that the initial conditions and all conditions at boundaries and internal surfaces of discontinuity arise naturally and no constraint conditions are required. The transformation is effected by modifying the principle slightly, using Lagrange multipliers in the classical manner, and adding an appropriate initial-value term to the Lagrangian. A particularly useful approximation technique is applied to a problem with an internal surface of discontinuity, and it is shown that the transformed principle can be used whereas the usual form of Hamilton's principle cannot. It is noted that the transformed principle has an important advantage over the method of least squares.
Quantum-Ergodic Problem
Authors: Sposito Garrison
Published: February 2004

The Gibbs approach to quantum dynamics is justified for nonrelativistic systems. It is shown that infinite-time and microcanonical ensemble averages may be equated wherever the former exists and whenever a maximal set of constants of the motion can be determined, provided that the ensemble average is over states whose images under the maximal set are virtually stationary. The relevance of the demonstration for quantum-statistical mechanics is discussed in the light of the ensemble formalism and the classical ergodic problem.
Reduction of the Bethe-Salpeter Equation for a Zero-Mass System
Authors: Keam R. F.
Published: February 2004

The ladder-approximation Bethe-Salpeter equation for a bound spin-½ fermion-antifermion system of zero total mass is reduced in the general case to coupled radial equations for general radial potential functions. Symmetries of the radial equations are also demonstrated.
Elastic-Wave Propagation and the Factorization of Matrix-Valued Functions of Several Complex Variables
Authors: Kraut Edgar A.
Published: February 2004

The solution of systems of equations of the Wiener-Hopf type leads directly to the problem of factoring matrix-valued functions of one or more complex variables. The matrix-factorization problem is solved here for matrices that can be diagonalized by a similarity transformation and also have eigenvalues whose logs are analytic and quadratically integrable in a set of strips. As an application of the technique, Green's tensor for the elastic wave equation is factored.
Note on Zak's Method for Constructing Representations of Space Groups
Authors: Klauder L. T. , Gay J. G.
Published: February 2004

Zak [J. Math. Phys. 1, 165 (1960)] has given a method for constructing the irreducible representations of space groups which involves inducing the representations of the full group from those of an invariant subgroup. When a representation of the subgroup is self-conjugate, Zak's prescription for the induction is subject to a restriction which makes it inapplicable in some practical applications of the method. This paper presents a general prescription for carrying out the induction from self-conjugate representations.
Diffraction Scattering and Vector-Meson Resonances
Authors: Atkinson D. , Contogouris A. P.
Published: February 2004

It is shown that, under reasonable assumptions about inelasticity and asymptotic behavior, the usual diffraction picture combined with the N/D approach to π-π scattering leads to a singular integral equation. The authors' formalism is then used to show that, in conformity with the nearby-singularities philosophy, a constant left-hand discontinuity is by itself incapable of producing resonances in the GeV region. Next, a model for the creation of vector resonances, which combines a long-range force (defined by exchange of a cutoff vector meson) plus a short-range force (compatible with diffraction requirements), is introduced. The effect of the short-range force on the self-consistent (bootstrap) solutions is investigated in an approximate scheme. For a self-consistent solution with the correct ρ-meson mass, which is found to exist, inclusion of the short-range force is shown to decrease the self-consistent width by a factor of 2, which is nevertheless still greater than the experimental value.
Threshold-Behavior Requirements for P-Wave N/D Equations
Authors: Cornille H. , Rubinstein G.
Published: February 2004

For nonrelativistic P-wave N/D equations we consider the two different cases depending on whether the solutions of the subtracted equations, assuring automatically the right threshold-behavior conditions, satisfy the unsubtracted equations or not. In both cases we investigate the meaning of the corresponding interactions.
Curvature and the Petrov Canonical Forms
Authors: Thorpe John A.
Published: February 2004

The Petrov classification for the curvature tensor of an Einstein space M4 is related to the critical-point theory of the sectional-curvature function σ, regarded as a function on the manifold of nondegenerate tangent 2-planes at each point of the space. It is shown that the Petrov type is determined by the number of critical points. Furthermore, all the invariants in the canonical form can be computed from a knowledge of the critical value and the Hessian quadratic form of σ at any single critical point.
High-Frequency Scattering by a Transparent Sphere. II. Theory of the Rainbow and the Glory
Authors: Nussenzveig H. M.
Published: February 2004

The treatment, initiated in Paper I [J. Math. Phys. 10, 82 (1969)], of the high-frequency scattering of a scalar plane wave by a transparent sphere is continued. The main results here are an improved theory of the rainbow and a theory of the glory. The modified Watson transformation is applied to the third term of the Debye expansion of the scattering amplitude in terms of multiple reflections. Only the range 1<N<√2, where N is the refractive index, is considered. In the geometrical-optic approximation, this term is associated with rays transmitted after one internal reflection, and there are three angular regions, corresponding to one ray, two rays, or no ray (shadow) passing through each direction. Together with transition regions, this leads to six different angular domains. In the 1-ray and 2-ray regions, geometrical-optic terms are dominant. Correction terms corresponding to the 2nd-order WKB approximation are also evaluated. In the 0-ray region, the amplitude is dominated by complex rays and surface waves. The 1-ray/2-ray transition is a Fock-type region. The rainbow appears in the 2-ray/0-ray transition region. The extension of the method of steepest descents due to Chester, Friedman and Ursell is applied. The result is a uniform asymptotic expansion for the scattering amplitude. It reduces to Airy's theory in the lowest-order approximation, but its domain of validity is considerably greater, both with regard to size parameter and to angles. The glory is an example of strong ``Regge-pole dominance'' of the near-backward scattering amplitude. Van de Hulst's conjecture that surface waves are responsible for the glory is confirmed. However, besides surface waves taking two shortcuts through the sphere, higher-order terms in the Debye expansion must also be taken into account. By considering also the effect of higher-order surface-wave contributions, all the features observed in the glory (apart from the polarization) are explained. Resonance effects associated with nearly-closed paths of diffracted rays lead to large, rapid, quasiperiodic intensity fluctuations. The same effects are responsible for the ripple in the total cross-section. Similar fluctuations appear in any direction, but their amplitude increases with the scattering angle, becoming a maximum near the backward direction, where they are dominant. They can also be interpreted as a collective effect due to many nearly-resonant partial waves in the edge domain. The dominant surface-wave contributions can also be summed to all orders for N < 1, leading to a renormalization of the propagation constants of surface waves.
Instability of Transverse Waves in a Relativistic Plasma
Authors: Lerche I.
Published: February 2004

Linearized equations are set up to describe disturbances in an infinite, spatially uniform, relativistic plasma without an ambient magnetic field. It is shown that, as well as the usual electrostatic waves, there also exists a class of electromagnetic waves. The two sets of waves are coupled in general, but can still be classified as mainly longitudinal or mainly transverse. Under the assumption that the system is stable against the longitudinal disturbances it is shown that the relativistic plasma will be unstable to the transverse waves unless it is virtually isotropic.
Some Asymptotic Behavior of Stieltjes Transforms
Authors: Zimering Shimshon
Published: February 2004

We consider an integrable function g(t) which behaves as ζt−α when t tends to infinity (ζ is a finite number) for 0<α<1, and show that its Stieltjes transform has the same behavior when |z| approaches infinity and provided that z is in the sector |arg z|<π−δ, where π>δ>0. (Theorem 1). In addition, we study the cases of α equal to zero, one, and larger than one (Theorems 2–4). Our results contain those of L. Lanz and G. M. Prosperi [Nuovo Cimento 33, 201 (1964)] and those of W. S. Woolcock [J. Math. Phys. 8, 1270 (1967)]. They are proved in a direct manner, using a theorem of D. V. Widder [The Laplace Transform, (Princeton University Press, Princeton, N.J., 1959), fifth printing, p. 329] and the regularity of the integral transforms that arise.
Unitary Representations of SL(2, C) in an E(2) Basis
Authors: Chang Shau-Jin , O'Raifeartaigh L.
Published: February 2004

Starting from the functional representation of Gel'fand and Naimark, the unitary irreducible representations of SL(2, C) are described in a basis of the subgroup E(2)⊗D, where E(2)⊗D is the subgroup of all 2 × 2 matrices of the form (α0γδ), αδ=1. Physically, this is the subgroup into which SL(2, C) degenerates at infinite momentum and may be thought of as the 2-dimensional Euclidean group together with its dilations. Advantages to using the E(2)⊗D basis are: (1) It is convenient to calculate form factors; (2) the generators of E(2)⊗D are represented either multiplicatively or by first-order differential operators and are independent of the values of the SL(2, C) Casimir operators; (3) the principal and supplementary series of SL(2, C) are treated on the same footing and, in particular, have the same inner product; and (4) the transformation coefficients to the usual angular-momentum basis are related to Bessel functions. The E(2)⊗D is used to compute explicitly the finite matrix elements of an arbitrary Lorentz transformation and to investigate the structure of vector operators in unitary representation of SL(2, C).