Volume 10, Issue 10, October 1969
Index of content:

Cluster Expansion in a Bose System
View Description Hide DescriptionWhen a trial function for Bose systems contains both one‐particle and two‐particle functions, the cluster expansion is no longer simply related to the classical expansion of Mayer and Mayer. But by observing that the permutations in one‐particle functions can be represented exactly as in the high‐temperature expansion of the spin‐½ XY model, the cluster expansion is developed analogously to the classical case.

Formal Solutions of Inverse Scattering Problems
View Description Hide DescriptionFormal solutions of inverse scattering problems for scattering from a potential, a variable index of refraction, and a soft boundary are developed using a method devised by Jost and Kohn.

Two‐Variable Expansion of the Scattering Amplitude: An Application of Appell's Generalized Hypergeometric Functions
View Description Hide DescriptionAppell's polynomials in two variables orthogonal in a triangle are described and some of their properties and those of related generalized hypergeometric functions are given. An application to the expansion of the scattering amplitude is suggested, the equal‐mass case being discussed in some detail. A simple crossing matrix is derived. Difficulties introduced by inequality of the particle masses are explained. A Neumann formula is presented which permits an analytic continuation in the parameters to be made of the expansion coefficients for parts of the amplitude: This is in analogy with the Froissart‐Gribov continuation. A conjectured analog of the Sommerfeld‐Watson transformation then suggests the existence of fixed cuts in the partial‐wave scattering amplitude.

On Continuous Eigenvalues in Neutron Thermalization
View Description Hide DescriptionThe continuous eigenvalue spectra of the linearized Boltzmann operator describing the energy distribution of neutrons in an infinite Einstein crystal are studied. This operator consists of two terms: a multiplication operator and an integral operator with δ‐function‐type singular kernel. The eigenvalue problem is transformed into the solution of an inhomogeneous integral equation by applying Case's method on the one‐velocity transport equation. The existence of the solution of the integral equation is examined by the Neumann‐series expansion. It is found that for sufficiently low temperature the range of numerical values of the multiplication operator forms the continuous eigenvalue spectra of the Boltzmann operator and the corresponding eigenfunctions are of δ‐function type.

Solutions of the Einstein and Einstein‐Maxwell Equations
View Description Hide DescriptionAlgebraically degenerate solutions of the Einstein and Einstein‐Maxwell equations are studied. Explicit solutions are obtained which contain two arbitrary functions of a complex variable, one function being associated with the gravitational field and the other mainly with the electromagnetic field.

Particle‐Hole Matrix: Its Connection with the Symmetries and Collective Features of the Ground State
View Description Hide DescriptionA detailed study is made of the properties of the ``particle‐hole matrix'',where g> is a many‐fermion ground state and ρ_{ ab } is the 1‐body density matrix. It is shown that the zero eigenvalues of the particle‐hole matrix are intimately and simply related to the one‐body symmetries of the ground state. It is also shown that the large eigenvalues of G_{abcd} are closely related to the collective features of the ground state.

Diffusion in Nonlinear Multiplicative Media
View Description Hide DescriptionThe time‐dependent behavior of the nonlinear distributions defined by the diffusion equation with several nonlinear source terms is studied. The nonlinear diffusion equation is solved by an eigenfunction‐expansion method, which is in principle independent of geometry or number of dimensions. The qualitative time behavior of the distributions and their steady states can be ascertained from a simple analysis of the fundamental mode approximation only. Explicit solutions are presented in one‐ and two‐dimensional geometries.

Observables for Massive Relativistic Particles of Arbitrary Spin
View Description Hide DescriptionThe expressions for the dynamical operators corresponding to a Bargmann‐Wigner particle of mass κ and arbitrary spin s are given, making use of the Foldy‐Wouthuysen and Chakrabarti transformations for the Bargmann‐Wigner equations given by Pursey, and Azcárraga and Boya, respectively. The operators selecting the positive and negative energy states are also given in the two representations. The operators for arbitrary spin are studied and a condition for an operator be constant of motion is given. Finally, the Newton‐Wigner position operator and those obtained in the Foldy‐Wouthuysen and Chakrabarti generalized representations are compared.

Properties of Generalizations to Padé Approximants
View Description Hide DescriptionThe Padé‐approximant method has recently been generalized in such a way that convergence of the new approximants can be proved for a larger class of functions than the series of Stieltjes and conditions can be obtained for these approximants to form sequences of converging upper and lower bounds. In this paper properties of these approximants to a class of functions, which correspond to a series of Stieltjes with nonzero radius of convergence, are considered. It is shown that under certain conditions they form only sequences of lower bounds to the exact function, but one can then define new approximants, which give upper bounds. Conditions are also obtained under which an upper bound can be put on the error of an approximant, using only the information necessary to calculate this approximant.

One More Technique for the Dimer Problem
View Description Hide DescriptionThe problem of counting the dimer coverings of a square lattice is recast as a counting of coverings by oriented closed loops. Thus the answer is expressed as the value of a suitable permanent. This permanent is transformed to a determinant, which on evaluation recovers the familiar result.

Plane‐Symmetric Gauge Field
View Description Hide DescriptionPlane‐symmetric solutions to the gauge field equations are considered on the classical level. The conditions of plane symmetry are gauge‐invariantly defined in terms of the internal holonomy group. It is shown that, if analytic gauge fields satisfy the plane‐symmetry conditions and if there exists at least one source‐free region in event space, then the internal holonomy group is Abelian and a gauge exists in which the gauge field satisfies Maxwell's equations.

Representation Theory of SO(4, 1) and E(3, 1): An Explicit Spinor Calculus
View Description Hide DescriptionThe explicit spinor basis states for the irreducible unitary representations of the spinor covering group of the de Sitter group SO(4, 1) are obtained by analytic continuation from those of the compact group Sp(4). Some features of the contraction of these basis states to those of the Poincaré group are discussed; the explicit asymptotic form of the basis state of a continuous representation of SO(4, 1) in the limit of the contraction to a basis state of a finite‐mass discrete‐spin representation of E(3, 1) is investigated.

Remarks on Lie‐Algebra‐Invariant Equations. I
View Description Hide DescriptionWe construct first‐order‐invariant equations with respect to the inhomogeneous Euclidean algebraIE(n) (isomorphic to the n‐dimensional Galilei algebra) by embedding its Euclidean subalgebra E(n) in O(n + 1, 1). This embedding allows us to utilize Gel'fand's method [I. Gel'fand et al., Representations of the Rotation and Lorentz Groups and Their Applications (Pergamon Press, Inc., New York, 1963)] adapted to the construction of invariant equations on semisimple algebras to nonsemisimple ones also.

Remarks on Lie‐Algebra‐Invariant Equations. II
View Description Hide DescriptionWe deal with a generalization of Gel'fand method which provides us with a simplified and constructive derivation of some known and new results on the construction and general properties of tensor and nth‐order Lie‐algebra‐invariant equations.

Finite Transformations and Basis States of SU(n)
View Description Hide DescriptionA convenient parameterization for finite transformations of SU(n) is developed which explicitly exhibits the special unitary subgroups. This is also used to parameterize the defining space. Higher‐dimensional representations are discussed. The question of which representations carry the trivial representation of SU(n − 1) is considered, as well as the parameterization of these states. Application is made to the transformations and basis states of SU(3).

Diffraction by a Slit or Strip
View Description Hide DescriptionThe problem of diffraction by a slit plane (or an infinite strip) is solved exactly for the case of a two‐dimensional source distribution, the Green's function being given explicitly in terms of the fundamental solution of the wave operator. Also treated are the limiting case of plane waves and the asymptotic behavior with respect to large wavenumber.

Soluble Model of Condensation
View Description Hide DescriptionA soluble model of gas‐liquid condensation is constructed by a quantum‐mechanical extension of van Kampen's classical analysis. The Hamiltonian of a system of identical particles is separated into a ``core'' and a ``tail'' Hamiltonian. Following van Kampen, we simulate the actual core Hamiltonian by an effective Hamiltonian with the same free energy. The tail Hamiltonian is replaced by its diagonal part in a single‐particle basis chosen self‐consistently so that the thermodynamics is exactly soluble. The resultant free energy agrees with that of Lieb's quantum‐mechanical extension of the van der Waals‐Maxwell theory, without the necessity of taking the van der Waals limit of a suitably infinite‐ranged and infinitely‐weak attractive tail. However, the exchange contributions omitted from the model become negligible only in that limit.

N‐Representability Theorem for Reduced Density Matrices
View Description Hide DescriptionA reduced density matrix representing a statistical mixture is shown to provide no definite information concerning its N‐representability. A theorem about simultaneous N‐representability and non‐N‐representability of a reduced density matrix is formulated and proved.

Quantum Harmonic Oscillator with Time‐Dependent Frequency
View Description Hide DescriptionThe temporal evolution of the state vector relative to a harmonic oscillator with time‐dependent frequency is examined. The Schrödinger equation is solved by choosing the instantaneous eigenstates of the Hamiltonian as the basis, thus getting an infinite set of coupled linear differential equations. This formulation is particularly suitable for studying the cases in which the Hamiltonian undergoes a very slow or a sudden variation from an initial constant value into a final one. A rigorous proof of adiabatic invariance to all orders in the slowness parameter is given for the transition probabilities. An application to the evolution of an initially coherent state is made.

Alternative Dynamics for Classical Relativistic Particles
View Description Hide DescriptionEquations of motion for interacting classical relativistic particles (different from Van Dam‐Wigner equations) are derived from an action principle. In the classical limit they reduce to Newton's equations with an interparticle potential depending on distance only.