Volume 13, Issue 8, August 1972
Index of content:

Studies in the C*‐Algebraic Theory of Nonequilibrium Statistical Mechanics: Dynamics of Open and of Mechanically Driven Systems
View Description Hide DescriptionWe construct a C*‐algebraic formulation of the dynamics of a pair of mutually interacting quantum‐mechanical systems S̃ and Ŝ, the former being finite and the latter infinite. Our basic assumptions are that: (i) Ŝ, when isolated, satisfies the Dubin‐Sewell dynamical axioms; (ii) the coupling between S̃ and Ŝ is energetically bounded and spatially localized; and (iii) the initial states of S̃ and Ŝ are mutually uncorrelated, with S̃ in an arbitrary normal state and Ŝ in a Gibbs state. Our formulation leads to a rigorous theory of (a) the dynamics of a finite open system, i.e., of a finite system (S̃) coupled to an ``infinite reservoir'' (Ŝ), and of (b) the dynamics of an infinite system (Ŝ), driven from equilibrium by a ``signal generator'' (S̃). As regards (a), we show that the state of S̃ always remains normal, and we derive a generalized master equation (in an appropriate Banach space) governing its temporal evolution. As regards (b), we show that the state of Ŝ always corresponds to a (time‐dependent) density matrix in the representation space of the algebra of observables for Ŝ, induced by the initial Gibbs state. By formulating the linear part (appropriately defined) of the response of Ŝ to S̃, we generalize the fluctuation‐dissipation theorem to infinite systems. Further, we show that the total effect of S̃ on Ŝ reduces to that of a ``classical'' time‐dependent external force in cases where the initial state of S̃ possesses certain coherence properties similar to those of the Glauber type.

Complex Time, Contour Independent Path Integrals, and Barrier Penetration
View Description Hide DescriptionBy developing an analogy between the Feynman path integral and contour integral representations of the special functions, we obtain WKB formulas for barrier penetration from a path integral. We first show that there exists for the path integral a notion of contour independence in the time parameter. We then select an appropriate contour to describe the physical situation of barrier penetration and obtain asymptotic formulas from the function space integral. The method is interpreted as a path integral derivation of the complex ray description of barrier penetration. In the last three sections we investigate several canonical problems of the theory of complex rays with these path integral techniques.

An O(2, 1) × O(3) Solution to a Generalized Quantum Mechanical Kepler Problem
View Description Hide DescriptionA differential equation directly related to the generalized Kepler equation of Infeld and Hull is solved in an O(2, 1) × O(3) group scheme. This equation contains as special cases the Schrödinger, Klein‐Gordon, and Dirac (two forms) hydrogen atoms. A generalized Pasternack and Sternheimer selection rule exists and some matrix elements can be evaluated group theoretically.

Solutions of a Class of Nonlinear Coupled Partial Differential Equations
View Description Hide DescriptionA class of nonlinear coupled partial differential equations are solved. These are generalizations of the general relativistic equations arising for a static distribution of massive charged particles.

On Multigroup Transport Theory with a Degenerate Transfer Kernel
View Description Hide DescriptionThe multigroup transport equation is studied in plane geometry assuming that the transfer kernel is representable in a degenerate form. The eigenvalue spectrum is analyzed and the associated eigensolutions are obtained in terms of generalized functions. Full‐range orthogonality relation is demonstrated. The full‐range completeness of the eigensolutions is established under rather general conditions. For the half‐range completeness to hold, it is additionally required that the scattering kernel be self‐adjoint and possesses reflection symmetry.

Radial Moments of Folding Integrals for Nonspherical Distributions
View Description Hide DescriptionSimple relations are exhibited between the radial moments of two nonspherical distributions when one is obtained by folding a scalar function into the other. Some applications are mentioned, and the generalization to nonscalar functions is indicated.

Variational Estimates and Generalized Perturbation Theory for the Ratios of Linear and Bilinear Functionals
View Description Hide DescriptionVariational functionals are presented which provide an estimate of ratios of linear and bilinear functionals of the solutions of the direct and adjoint equations (inhomogeneous and homogeneous) governing linear systems. These variational functionals are used as the basis for a generalized perturbation theory for estimating the effects of changes in system parameters upon these ratios of linear and bilinear functionals. The relation of the present theory to the variational theory of Pomraning and to the generalized perturbation theory of Usachev and Gandini is discussed. Potential applications of the theory to nuclear reactor physics are outlined.

Amplitude Dispersion and Stability of Nonlinear Weakly Dissipative Waves
View Description Hide DescriptionA method to solve nonlinear dissipative wave equations using ideas of Luke, Krylov, and Bogolyubov is presented. The method is compared to Whitham's theory. Dispersion relations for nonlinear dissipative waves, including amplitude dispersion, are discussed. Furthermore, stability problems of such waves are investigated.

Structure of Simplexes of Equilibrium States in Quantum Statistical Mechanics
View Description Hide DescriptionThe structure of Choquet simplexes of equilibrium states of infinite system in quantum statistical mechanics is investigated. It is shown that a facial simplex is a Bauer simplex whenever its extreme points form a physically equivalent class; however, the simplex of KMS states for a given inverse temperature is not a Bauer simplex if it is not a singleton.

Hypergeometric Structure of the Generalized Veneziano Amplitudes
View Description Hide DescriptionA multiple integral representation for a function of n(n − 1)/2 variables, which reduces to the (n + 3)‐point generalized Veneziano amplitude for unit values of the variables, is integrated out once to obtain a recurrence formula for the amplitude. The result of complete integration obtained through repeated use of the recurrence formula is shown to belong to a class of generalized hypergeometric functions of many variables which are similar to but are more complex than the Lauricella functions. It is shown also that the (n + 3)‐point amplitude for n ≥ 3 can be represented as a linear combination of an infinite number of _{ n+1} F_{n} [1] series with varying parameters.

Is the Maxwell Field Local?
View Description Hide DescriptionIs the classical Maxwell field truly local? This question is raised by several observations, among them the Aharonov‐Bohm effect; but the question cannot be answered without a systematic definition and characterization of local subalgebras of observables. This paper reformulates classical field theory in analogy to axiomatic quantum field theory and introduces a precise statement for local independence. (Synonyms for local independence are Einstein causality and principle of maximum signal velocity.) The formal answer of the analysis is: The free Maxwell field does not have local independence. This conclusion is critically discussed.

A Watson Sum for a Cubic Lattice
View Description Hide DescriptionThe origin‐origin value of the Green's function for a simple cubic lattice with axial anisotropy is evaluated exactly.

Multigroup Neutron Transport
View Description Hide DescriptionWe investigate the nature of the approximations involved in the multigroup treatment of the time‐dependent neutron transport equation by using the method of approximating sequences of Banach spaces. We prove that solutions of the multigroup system converge, in a suitable sense, to the corresponding solutions of the exact transport equation. Moreover, we indicate the order of magnitude of the rate of convergence.

Local and Covariant Quantization of Linearized Einstein's Equations
View Description Hide DescriptionAn analysis is given of all the possible quantizations of the linearized Einstein equations in terms of a weakly local and/or covariant potential h _{μν}(x). The discussion is done without making the apparently arbitrary choices which characterize the standard formulations, and special attention is paid in proving those general features which follow from basic principles and are therefore common to all local and/or covariant formulations. It is shown that the requirement of locality and/or covariance alone implies that the Einstein equations cannot hold as mean values on a dense set of states, and therefore unphysical states must be introduced in an essential way. Moreover, the requirement that the Einstein equations hold as mean values on the physical states forces the existence of states of negative norm in order to define h _{μν} as a local and/or covariant operator. Thus the characteristic features of Gupta's formulation are shown to be shared by any local and/or covariant theory. The arbitrary choices which occur in the representation of the field operator h _{μν}, in the definition of the metric operator and in the choice of the subsidiary condition which identifies the physical states, are shown to lead to only a one‐parameter family of theories. They can be classified according to the subsidiary condition,each q ≠ −¼ identifying a possible theory. This arbitrariness, which makes the literature on the subject rather confusing, is resolved by proving that all the theories are (isometrically) equivalent. Such formulations are discussed in the framework of axiomatic quantum field theory, with particular emphasis on their group theoretical contents. Finally, an extensive treatment of Gupta's formulation is given along the lines discussed by Wightman and Gärding for quantum electrodynamics.

On the Reduction of the Generalized RPA Eigenvalue Problem
View Description Hide DescriptionThe 2n‐dimensional eigenvalue problem, which arises when the random phase approximation (RPA) matrix is not real, is reduced to an n‐dimensional eigenvalue problem. Some properties of the reduced eigenvalue problem are studied. A numerical example is considered for illustrative purposes.

Conservation of Kinks and Spin in Nonlinear Quantum Field Theories
View Description Hide DescriptionCertain nonlinear field theories exhibit particlelike structures called ``kinks'' that have fermionlike properties. This paper uses a Feynman path integral approach to show that, in the quantum theory, the number of kinks and the spin associated with them are both conserved quantities.

Rigorous Statistical Mechanics for Nonuniform Systems
View Description Hide DescriptionThe thermodynamic limit of a classical system with many‐body interactions and under the influence of an external potential is investigated for the canonical ensemble. By scaling the external potential to a sequence of domains which are isotropic expansions of an initial domain confining the system, it is shown that the canonical free energy per particle has an infinite system limit. Moreover, by restricting consideration to internal interactions which have the property that separated groups of particles have negligible mutual attractive energy as the system becomes infinite, it is proven that the free energy per particle limit is precisely the free energy per particle obtained by minimizing the integral ∫[φρ + f(ρ, β)] with respect to all properly normalized functions ρ(r). φ is the external potential; f(ρ, β) is the free energy per unit volume for a uniform system of density ρ and inverse temperature β. The only technical complication is the above‐mentioned restriction on the allowed internal interactions. It is demonstrated that there exists a wide class of many‐body interactions which have the required separation properties. The simplest example is a two‐body interaction which includes a hard core.

Stability of Finite‐Sized Radiating Plasmas
View Description Hide DescriptionIt is shown that the onset of instabilities in a finite‐sized, optically thin, radiating plasma coincides with the loss of uniqueness of solutions to the governing nonlinear differential equations. Furthermore, conditions on the luminosity function are derived for the existence of metastable states of such plasmas with dimensions smaller than the critical maximum size derived from conventional normal mode stability analysis. The results are applied to a simple configuration modelling free‐free emission from a confined radiating, high‐temperature plasma.

Exactly Soluble Model of Interacting Electrons
View Description Hide DescriptionWe diagonalize a many‐fermion Hamiltonian consisting of terms quadratic as well as quartic in the field operators. A dual spectrum of eigenstates is an interesting result. We also derive a formula for obtaining the free energy at finite temperature.

Unbounded Solutions of Coupled Mode Equations
View Description Hide DescriptionA number of sufficient conditions for the existence of unbounded solutions of two and three coupled mode equations are obtained when some modes are linearly unstable and all initial amplitudes are arbitrarily small. The difficulty of obtaining sufficient conditions for boundedness of all solutions is discussed, and only two such conditions are obtained. In certain cases it is proved that the unbounded solutions are not more rapid than exponential, whereas they can be shown to be singular (``explosive'') in other cases.