Index of content:
Volume 5, Issue 4, April 1964

Remarks Concerning Reciprocity in Quantum Mechanics
View Description Hide DescriptionA new derivation of the reciprocity theorem is given. The general invariance property of the Hamiltonian which leads to symmetry of the Green's function for a quantum mechanical system is exhibited. It is found that reciprocity does not necessarily imply Hermiticity of the Hamiltonian, so that the ``complex optical model potential,'' for example, satisfies the reciprocity relations. The concept of reciprocity is then generalized to include a somewhat wider class of symmetry properties. Some properties of antiunitary transformations are discussed.

Multiplicative Symmetries in Axiomatic Quantum Field Theory
View Description Hide DescriptionThe group of all multiplicative symmetries defined for a finite number of interacting fields is studied in detail. Various theorems are proved connecting the existence of multiplicative symmetries with properties of the Wightman distributions of the fields.

Relativistic Particle Dynamics and the S Matrix
View Description Hide DescriptionDirect‐interaction theories are examined from the viewpoint of relativistic scattering theory and the associated concept of ``asymptotic covariance.'' It is pointed out that with any two‐particle Hamiltonian which has no bound states there can be associated a variety of representations of the Lie algebra of the inhomogeneous Lorentz group (IHLG), although the S matrix is in general not covariant. It is shown that the requirement of asymptotic covariance ensures both the covariance of the S matrix and the existence of a unique representation of the IHLG to be associated with the relativistic two‐particle system. The connection between the Lie algebra, the covariant form of the S matrix, and the uniqueness of K, the generator of pure Lorentz transformations, is thereby clarified. The extension of these considerations to include bound states is made. The form of H given by Bakamjian and Thomas is shown to satisfy asymptotic covariance and, moreover, to be the most general form of interest from the viewpoint of relativistic scattering theory, thereby including as a special case a form of H suggested by Sudarshan. It is also proved that relativistic Hamiltonians of this type do not admit the usual notion of a coupling constant.

Field Operators for Bosons with Impenetrable Cores. II. Equations of Motion and General Operator Formalism
View Description Hide DescriptionContinuing the investigation started in part I, we derive the equations of motion in the Fock representation and a nonrelativistic field theory for bosons interacting as hard spheres.

Causality Implies the Lorentz Group
View Description Hide DescriptionCausality is represented by a partial ordering on Minkowski space, and the group of all automorphisms that preserve this partial ordering is shown to be generated by the inhomogeneous Lorentz group and dilatations.

New Methods for Reduction of Group Representations Using an Extension of Schur's Lemma
View Description Hide DescriptionThe generators of the group or group algebra are used in an analog of the Lie‐Cartan method, which can be applied to finite or infinite groups. This gives a mean for reduction of an arbitrary group representation, using only the matrix representatives of the generators. It is a set of algorithms using pivotal condensation and can easily be coded as a digital computer program. Connections with Lie‐Cartan theory are suggested, the reduction of the symmetric group discussed, and methods for the reduction of representations of the n × n unitary, orthogonal, and proper orthogonal groups suggested.

Representations for the Nonrelativistic Coulomb Green's Function
View Description Hide DescriptionDerivation of the representations for the nonrelativistic Coulomb Green's function is discussed. It is shown that the recently published representations, both in integral and closed forms, can be obtained from an expression for the difference of the diverging and converging wave solutions of the Green's function which is given in terms of the wavefunctions summed by Gordon.

Theory of Vibrational Structure in Optical Spectra of Impurities in Solids. II. Multiplets
View Description Hide DescriptionThe results of a previous paper describing vibrational structure in the spectra of singlet‐to‐singlet optical transitions of impurities in solids are extended to a general class of multiplet‐to‐multiplet transitions. The transitions considered are those between degenerate or nondegenerate multiplets which are not interconnected by the interaction coupling the impurities to the lattice phonons. The phonon field couples impurity states within each multiplet but does not couple states in different multiplets. The results apply to systems which display strong Jahn‐Teller distortions as well as to those which do not. As a specific application of our results we consider the singlet‐to‐doublet transitions of simple three‐level systems having C _{3} and C _{3v } symmetry.

Exact Conditions for the Preservation of a Canonical Distribution in Markovian Relaxation Processes
View Description Hide DescriptionNecessary and sufficient conditions have been determined for the exact preservation of a canonical distribution characterized by a time‐dependent temperature (canonical invariance) in Markovian relaxation processes governed by a master equation. These conditions, while physically realizable, are quite restrictive so that canonical invariance is the exception rather than the rule. For processes with a continuous energy variable, canonical invariance requires that the integral master equation is exactly equivalent to a Fokker‐Planck equation with linear transition moments of a special form. For processes with a discrete energy variable, canonical invariance requires, in addition to a special form of the level degeneracy, equal spacing of the energy levels and transitions between nearest‐neighbor levels only. Physically, these conditions imply that canonical invariance is maintained only for weak interactions of a special type between the relaxing subsystem and the reservoir. It is also shown that canonical invariance is a sufficient condition for the exponential relaxation of the mean energy. A number of systems (hard‐sphere Rayleigh gas, Brownian motion, harmonic oscillators,nuclear spins) are discussed in the framework of the above theory. Conditions for approximate canonical invariance valid up to a certain order in the energy are also developed and then applied to nuclear spins in a magnetic field.

Elastic, Electromagnetic, and Other Waves in a Random Medium
View Description Hide DescriptionPropagation of any type of wave in a random medium is analyzed on the assumption that the medium differs slightly from a homogeneous medium. An equation satisfied by the average wave is deduced which is correct through terms of order ε^{2}, where ε measures the deviation of the medium from homogeneity. From this equation, the propagation constant of the medium is determined. The general formulation applies to any type of linear differential or integral equation with random coefficients. It is applied to time‐harmonic waves satisfying the reduced wave equation, to the equations of elasticity and to Maxwell's equations. The propagation constant for the average or coherent wave is complex even for a nondissipative medium, because the coherent wave is continually scattered by the inhomogeneities and converted into the incoherent wave. The propagation velocity of the average wave is also diminished by the inhomogeneities. This propagation constant depends upon certain trigonometric integrals of the auto‐ and cross‐correlation functions of the coefficients in the original equations, i.e., of the various coefficients characterizing the medium. To illustrate the results, media with particular random variations are considered and the propagation constants are determined for them.

A Theorem on the Conductivity of a Composite Medium
View Description Hide DescriptionA composite medium consisting of a rectangular lattice of identical parallel cylinders of arbitrary cross section is considered. The cylinders have conductivity σ_{2} and are imbedded in a medium of conductivity σ_{1}. Simple properties of the conductivity tensor of the composite medium are deduced from the theory of harmonic functions.

Scattering of Electromagnetic Waves by a Composite Cylinder
View Description Hide DescriptionThe scattering of an obliquely incident electromagnetic wave by a composite cylinder has been obtained using fundamental electromagnetic principles. The general result has been reduced to simpler forms for certain special cases.

Application of Operational Methods to the Analysis of Uniform Plasmas
View Description Hide DescriptionVarious aspects of plasma theory are derived by the application of operational methods to the equation of electron motion. Discussed herein are the complex tensordielectric constant, the tensorpermittivity and conductivity, plasma transient behavior, and plasma oscillations.

Expansions of Integrals of Bessel Functions of Large Order
View Description Hide DescriptionSeries expansions in powers of 1/v of integrals over x of the product of the Bessel function J _{ v }(vx/y), [J _{ v }(vx/y)]^{2}, or J _{ v }(vx/y) J _{ v+1}(vx/y) with an arbitrary function of a restricted class are developed.