Volume 21, Issue 12, December 1980
Index of content:

U(N) Integrals, 1/N, and the De Wit–’t Hooft anomalies
View Description Hide DescriptionFormulas for the evaluation of all U(N) integrals are derived. Tables display the results for integrands involving up to six U’s and six U^{°}’s. The complete pole structure of De Wit–’t Hooft anomalies is unveiled. The effects of 1/N ^{2} corrections and De Wit–’t Hooft anomalies on two‐dimensional U(N) lattice gauge theories in the strong coupling 1/Nexpansion is discussed.

Lie groups, spin equations, and the geometrical interpretation of solitons
View Description Hide DescriptionThe integrable evolution equations imbeddable in SU (2) are shown to have two gauge equivalent forms; the AKNS form, and a spin form for which the field is a three‐dimensional vector of unit length. These equations are the compatibility conditions for the existence of a bilocal Lie group in two distinct frames of reference. These frames are associated with moving bases on surfaces formed by the motion of the strings introduced by Lamb. Both forms of the evolution equation are derivable from a locality assumption for the generators of the bilocal Lie group. The assumption is sufficient to distinguish between integrable and nonintegrable systems imbedded in SU (2).

Generalized groups as global or local symmetries
View Description Hide DescriptionWe extend global particle symmetries from the traditional group framework to that of generalized groups. The nature of these latter are presented, and various invariants constructed for them. The problem of gauging generalized groups is discussed and a no–go theorem proved under reasonable conditions on the generalized group structure.

Group actions on principal bundles and invariance conditions for gauge fields
View Description Hide DescriptionInvariance conditions for gauge fields under smooth group actions are interpreted in terms of invariant connections on principal bundles. A classification of group actions on bundles as automorphisms projecting to an action on a base manifold with a sufficiently regular orbit structure is given in terms of group homorphisms and a generalization of Wang’s theorem classifying invariant connections is derived. Illustrative examples on compactified Minkowski space are given.

A matrix representation of the translation operator with respect to a basis set of exponentially declining functions
View Description Hide DescriptionThe matrix elements of the translation operator with respect to a complete orthonormal basis set of the Hilbert spaceL _{2}(R^{3}) are given in closed form as functions of the displacement vector. The basis functions are composed of an exponential, a Laguerre polynomial, and a regular solid spherical harmonic. With this formalism, a function which is defined with respect to a certain origin, can be ’’shifted’’, i.e., expressed in terms of given functions which are defined with respect to another origin. In this paper we also demonstrate the feasibility of this method by applying it to problems that are of special interest in the theory of the electronic structure of molecules and solids. We present new one‐center expansions for some exponential‐type functions (ETF’s), and a closed‐form expression for a multicenter integral over ETF’s is given and numerically tested.

Infinities of polynomial conserved densities for nonlinear evolution equations
View Description Hide DescriptionThe infinities of polynomial conserved densities for several nonlinear evolutiopn equations are investigated using Noether’s theorem, and are identified as energy or momentum densities of higher‐order enveloping equations. A recursive operator formula is derived for the densities.

An infinity of polynomial conserved densities for a class of nonlinear evolution equations
View Description Hide DescriptionNoether’s theorem is applied to the infinity of polynomial conserved densities possessed by a general class of nonlinear evolution equations. The densities are identified on the solution sets of higher‐order enveloping equations as canonical energy or momentum densities, and a new recursive formula is derived for these densities.

The nonabelian Toda lattice: Discrete analogue of the matrix Schrödinger spectral problem
View Description Hide DescriptionWe investigate the discrete analog of the matrix Schrödinger spectral problem and derive the simplest nonlinear differential‐difference equation associated to such problem solvable by the inverse spectral transform. We also display the one and two soliton solution for this equation and tersely discuss their main features.

An addition theorem for vector Helmholtz harmonics
View Description Hide DescriptionAn addition theorem for the vector solutions of Helmholtz equations under translation of the coordinate axes is proposed and its results compared with those of a previous addition theorem for Hansen’s M and N vectors. The resulting comparisons are also separated into their radial and transverse components.

Particle trajectories in 1/r fields
View Description Hide DescriptionThe trajectory of a particle subjected to an attractive 1/r force is discussed. The general mathematical solution is given. Various analytical results are derived including the representations for the trajectory function.

A concise and accurate solution for Poiseuille flow in a plane channel
View Description Hide DescriptionThe recently developed F _{ N } method of solving problems in particle transport theory is used to establish a concise and accurate solution for the flow of a rarefied gas between two parallel plates. The Bhatnagar, Gross, and Krook model is used, and numerical results are given for a wide range of the Knudsen number.

Operator methods for time‐dependent waves in random media with applications to the case of random particles
View Description Hide DescriptionThe random medium is represented by the operator, constructed from the characteristic functional of the medium, and this representation is shown to considerably facilitate the formulation of various equations of waves in random media, as well as obtaining the physical insight into the equations. A specific application is made to waves in the medium of random particles, and the equations obeyed by the characteristic functional of wave are derived with the aid of the effective medium method. Here, the optical condition is exhibited by the condition of an operator in space and time. Independent of this operator method, the general theory is extended, in an unperturbative way, for the equations of the second‐order coherence functions, being given in form of the Bethe–Salpeter equation, and the coherent potential equations are formulated for the basic matrices of two kinds appeared in the equations. The explicit expressions of these matrices are obtained, on utilizing the coherent potential approximation, and are shown to be exactly the same as those obtained by the effective medium method, in both cases of weak‐scattering limit and of random particles. Finally, on employing the appropriate Fourier representations in space and time, the theory is presented in a few different forms, one being particularly suited to derive the equations of multifrequency coherence functions.

A note on the Schrödinger equation for the x ^{2}+λx ^{2}/(1+g x ^{2}) potential
View Description Hide DescriptionThe energy levels and wave functions of the Schrödinger equation involving the potential x ^{2}+λx ^{2}/(1+g x ^{2}) are calculated by the variational method, for any range of λ and g, without having to resort to numerical quadrature. Using properly scaled (in λ and g) harmonic oscillator functions as a basis set, an easy to compute analytical expression of the current Hamiltonian matrix element is derived. Perturbative results are also given.

Exactly solvable eigenvalue problem with hypergeometric eigenfunctions
View Description Hide DescriptionA Schrödinger equation with a momentum dependent interaction leads to exact solutions ψ∈ L ^{2}(R^{3},d ^{3} x) with radial parts of the wave function which are hypergeometric functions and their appropriate analytic continuations. The normalization integrals are obtained in closed form.

K‐surfaces in the Schwarzschild space‐time and the construction of lattice cosmologies
View Description Hide DescriptionWe investigate spacelike spherically symmetric hypersurfaces of constant mean curvature K (which we call K‐surfaces) in spherically symmetric static spacetimes. We obtain the differential equation satisfied by these surfaces from a variational principle. The spacetime Killing vector leads to a first integral in the form of a conservation of energy for a particle moving in an effective potential. An embedding of the K‐surfaces’ intrinsic geometry in flat space likewise follows from an effective potential motion. We apply the formalism to the Schwarzschild solution, and display results of numerical integrations for a variety of K‐surfaces and their flat space embeddings. We use these to construct ’’lattice’’ cosmological models, and obtain a foliation of K‐surfaces of such models with large scale behavior of both the open and closed Friedmann type.

Gödel‐like cosmological solutions
View Description Hide DescriptionGödels cosmological solutions have been generalized by Novello and Reboucas [Astrophys. J. 225, 719–24 (1978)]. An attempt is made further to generalize their work. A class of solutions is obtained which are Gödel‐like in the sense of Novello and Reboucas, but which have singularities at both ends of time.

Static gravitational and Maxwell fields in the general scalar tensor theory
View Description Hide DescriptionThe expression for g _{00} as a function of the scalar field Ψ is obtained in the general scalar tensortheory of gravitation proposed by Nordtvedt and later discussed by Barker, assuming that there exists a functional relationship between them. Exact solutions for a plane symmetric static gravitational field are also obtained in this theory. Further the calculations are extended for the static electrovac with the assumption that here both g _{00} and the scalar field Ψ are functions of the electrostatic potential φ, and the results are different from those previously obtained in the corresponding situation of Brans–Dicke theory.

Dynamics in nonglobally hyperbolic, static space‐times
View Description Hide DescriptionOrdinary Cauchy evolution determines a solution of a partial differential equation only within the domain of dependence of the initial data surface. Hence, in a nonglobally hyperbolic space‐time, one does not have fully deterministic dynamics. We show here that for the case of a Klein–Gordon scalar field propagating in an arbitrary static space‐time, a physically sensible, fully deterministic dynamical evolution prescription can be given. If the cosmic censor hypothesis should be overthrown, a prescription of this sort could rescue deterministic physics.

The use of anticommuting variable integrals in statistical mechanics. I. The computation of partition functions
View Description Hide DescriptionIntegrals over anticommuting variables are use to rewrite partition functions as fermionic field theories. The method is used to solve the two‐dimensional Ising model, the planar close‐packed dimer problems, and the free‐fermion eight vertex model.

The use of anticommuting variable integrals in statistical mechanics. II. The computation of correlation functions
View Description Hide DescriptionBy using integrals over anticommuting variables all the correlation functions in the two‐dimensional Ising model and free‐fermion eight vertex model are computed. The method is quite general and applicable to other solvable systems.