Volume 17, Issue 3, March 1976
Index of content:

Transition probability spaces
View Description Hide DescriptionThe set of pure quantum states is described as an abstract space with a geometry determined by transition probabilities. We describe all possible structures for three‐dimensional transition probability spaces with less than ten states, as well as some even larger spaces of a certain symmetric type. It is shown that the orthoclosed subspaces of a transition probability space form an atomistic orthomodular poset.

Einstein–Maxwell metrics admitting a dual interpretation
View Description Hide DescriptionConditions are given under which the metric part of a solution of the source‐free Einstein–Maxwell equations may be interpreted as the metric part of a solution with sources. Examples are given of space–times which admit this dual interpretation and also of space–times admitting one interpretation only.

An effective‐potential approach to stationary scattering theory for long‐range potentials
View Description Hide DescriptionIt is shown that previously derived integral equations for two‐body scattering with long‐range potentials (equations which replace the Lippmann‐Schwinger equations) can be reduced to a form which is solvable by iterative methods. The method is applicable to potentials V (r) which behave asymptotically as r ^{−α}, 1/2<α?1, and in particular to Coulomb‐like potentials.

On the duality condition for quantum fields
View Description Hide DescriptionA general quantum field theory is considered in which the fields are assumed to be operator‐valued tempered distributions. The system of fields may include any number of boson fields and fermion fields. A theorem which relates certain complex Lorentz transformations to the T C P transformation is stated and proved. With reference to this theorem, duality conditions are considered, and it is shown that such conditions hold under various physically reasonable assumptions about the fields. Extensions of the algebras of field operators are discussed with reference to the duality conditions. Local internal symmetries are discussed, and it is shown that these commute with the Poincaré group and with the T C P transformation.

Gauge‐covariant differentiation and Green’s functions for the Yang–Mills field
View Description Hide DescriptionA gauge‐independent definition of differentiation is given for non‐Abelian gauge fields in terms of parallel translation. This is achieved by a suitable definition of time‐ordered operator products. Equal time commutation relations are used to derive the differential equations for the related Green’s functions. The Green’s functions are discussed for general linear gauges. In comparison with electrodynamics the Green’s functions have the well‐known ghost‐loop terms.

Lie theory and separation of variables. 9. Orthogonal R‐separable coordinate systems for the wave equation ψ_{ t t }−Δ_{2}ψ=0
View Description Hide DescriptionA list of orthogonal coordinate systems which permit R‐separation of the wave equation ψ_{ t t }−Δ_{2}ψ=0 is presented. All such coordinate systems whose coordinate curves are cyclides or their degenerate forms are given. In each case the coordinates and separation equations are computed. The two basis operators associated with each coordinate system are also presented as symmetric second order operators in the enveloping algebra of the conformal group O(3,2).

Lie theory and separation of variables. 10. Nonorthogonal R‐separable solutions of the wave equation ∂_{ t t }ψ=Δ_{2}ψ
View Description Hide DescriptionWe classify and discuss the possible nonorthogonal coordinate systems which lead to R‐separable solutions of the wave equation. Each system is associated with a pair of commuting operators in the symmetry algebra so(3,2) of this equation, one operator first order and the other second order. Several systems appear here for the first time.

Lie theory and separation of variables. 11. The EPD equation
View Description Hide DescriptionWe show that the Euler–Poisson–Darboux equation {∂_{ t t }−∂_{ r r }−[(2m+1)/r]∂_{ r }}Θ=0 separates in exactly nine coordinate systems corresponding to nine orbits of symmetric second‐order operators in the enveloping algebra of SL(2,R), the symmetry group of this equation. We employ techniques developed in earlier papers from this series and use the representation theory of SL(2,R) to derive special function identities relating the separated solutions. We also show that the complex EPD equation separates in exactly five coordinate systems corresponding to five orbits of symmetric second‐order operators in the enveloping algebra of SL(2,C).

On a phase interchange relationship for composite materials
View Description Hide DescriptionA theorem exists relating the transverse conductivity of a fiber reinforced material in a determinate manner to the conductivity of the composite with the phase properties interchanged. It is shown that no such theorem can exist in the three‐dimensional case, e.g., for a statistically isotropic composite material. However, an inequality is established relating the two effective conductivities.

On the conductivity of fiber reinforced materials
View Description Hide DescriptionA two‐phase material in which the phase boundaries are cylindrical surfaces is considered. A technique exists for finding upper and lower bounds on the effective thermal conductivity (or electrical conductivity,permittivity, or magnetic permeability) of the composite in the direction perpendicular to the generators of the phase boundaries in terms of two different three point correlation functions. It is shown how a phase interchange theorem can be introduced into these bounds enabling us to express them in terms of a single geometrical constant of phase geometry. We determine what range of values of this factor is realizable for real phase geometries, and we show that the bounds thus obtained span exactly all realizable effective conductivities for such composites. Finally, we show that the bounds as expressed here enable us to use a knowledge of the effective conductivity of a composite for one ratio of constituent conductivities to narrow the bounds for some other ratio.

Long‐wavelength normal mode vibrations of infinite, ionic crystal lattices. II
View Description Hide DescriptionIn an earlier paper [J. Math. Phys. 16, 1156 (1975)] we presented a mathematical theory of the long‐wavelength normal mode vibrations of infinite crystal lattices whose particles interact with Coulomb forces. (Retardation was neglected.) The paper showed how the eigenvalues and eigenvectors of the complete long‐wavelength dynamical matrix are related to the eigenvalues and eigenvectors of the dynamical matrix obtained by neglecting the contribution of the macroscopic electric field. Rules were obtained for determining whether or not the various branches of the dispersion relations for a lattice approach definite frequencies in the long‐wavelength limit. The paper was restricted to the rigid ion approximation. In this paper we show that the above treatment can be easily extended to include lattices with polarizable and deformable atoms.

Spectrum generating algebras and Lie groups in classical mechanics
View Description Hide DescriptionWe give a general framework for a geometric foundation of time dependent classical mechanics. The theory is based on the concept of evolution space which is phase space extended by time. Lie algebras of constants of motion which may possess explicit time dependence are constructed, and general conditions for getting global Lie group actions from infinitesimal actions are derived. In a natural way these groups map solutions of the Hamiltonian equations of motion onto one another and act on the orbit space via symplectic transformations. The theory is applied to the nonrelativistic free particle, the harmonic and damped oscillator, nonstationary quadratic systems, and to the motion of a particle in constant electromagnetic fields.

Dynamical quantization of the Kepler manifold
View Description Hide DescriptionThe dynamical quantization of the ’’Kepler manifold’’ in any number of degrees of freedom is constructed. The Kepler manifold is the phase space of the regularized Kepler motion and is shown to be an SO(n,2) ‐homogeneous symplectic manifold, corresponding to an extremely singular orbit in the co‐adjoint representation; the quantization is obtained by ’’approximating’’ this orbit by more regular ones, which are equivalent to homogeneous bounded domains. The most relevant result is that the usual quantum‐mechanical ’’hydrogen atom’’ model is recovered in the particular representation introduced by Fock in 1935 [SO(n) ‐homogeneous integral equation in momentum space].

On the equality of S operators corresponding to unitarily equivalent Hamiltonians in single channel scattering
View Description Hide DescriptionSome years ago, conditions in order that unitarily equivalent Hamiltonians H and W*H W yield the same S operator were investigated by Ekstein and by Coester and his collaborators in the framework of nonrelativistic time‐dependent single‐channel quantum scattering theory. This subject has turned out to be of considerable importance for nuclear physics, since it constitutes the foundation of a widely used method for constructing phase‐equivalent nucleon–nucleon potentials. In the present paper we derive a rigorous and concise necessary and sufficient condition for a pair of unitarily equivalent Hamiltonians, governing the dynamics of a pair of nonrelativistic particles in the center‐of‐mass system, to yield the same S operator. Our theory, based on a time‐dependent approach, applies to very general types of short‐range potentials, with or without hard cores, and to an extensive class of long‐range potentials. Our necessary and sufficient condition simplifies when certain strong limits W _{±}, related to W, are unitary and when W _{+}=W _{−}. Requirements sufficient for each of these properties to hold are determined. Various examples of operators W such that H and W*H W have the same S operator are discussed.

Bounds on the admittance for KMS states
View Description Hide DescriptionUpper and lower bounds are proved for the static admittance of observables in a KMS state on a von Neumann algebra. As an application some exact results for the transverse Ising model are derived.

Conformal algebra in superspace and supergauge theory
View Description Hide DescriptionThe conformal algebra in a superspace with orthosymplectic metric is found to be an orthosymplectic algebra with two extra Bose dimensions. It is argued that all supergauge fields are also Nambu–Goldstone fields. In supergauge theory the nature of the gauged internal symmetry is found to be severely restricted: It must be O(2N), where N is the number of fundamental Dirac fermions or alternatively one‐eightth of the number of Fermi dimensions of superspace.

A theorem on Ruch’s principle of nonequilibrium statistics
View Description Hide DescriptionIt is proved that whenever two probability distributions, on a finite set, p and p′ are given such that p′ has a bigger mixing character than p, it is possible to find a unitary matrix U, such that p′_{ i } =Σ_{ j }‖U _{ i j }‖^{2} p _{ j }. This theorem ensures that Ruch’s principle of increasing mixing character provides the strongest assertion to be made upon the diagonal of the density matrix of some quantum mechanical system at time t≳0, without one’s knowing the Hamiltonian operator but given that the density matrix was initially diagonal with a known but arbitrary diagonal.

Matrix elements of multiple electron exchange between spin functions of maximum multiplicity
View Description Hide DescriptionA general formula for such matrix elements is obtained, in terms of a sum of terms each proportional to the square root of a product of eight binomial coefficients.

Multipole expansion of the density of states about a crystal cell
View Description Hide DescriptionWe construct the expansion of a Bloch wave with energy E into a complete set of multipole waves around a ’’center’’ of a crystal as an analog of the expansion in spherical waves in free space. Crystal point group symmetry is used to classify the set. The density of states in a cell is then analyzed into multipole components whose magnitude depends on the cell’s distance from the ’’center.’’

Duality transformation in a many‐component spin model
View Description Hide DescriptionIt is shown that the duality transformation relates a spin model to its dual whose Boltzmann factors are the eigenvalues of the matrix formed by the Boltzmann factors of the original spin model. The duality relation valid for finite lattices is obtained, and applications are given.