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Dec 1980

Volume 21, Issue 12, pp. 2695-2852

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U(N) Integrals, 1/N, and the De Wit–’t Hooft anomalies

Stuart Samuel

J. Math. Phys. 21, 2695 (1980); http://dx.doi.org/10.1063/1.524386 (9 pages) | Cited 36 times

Online Publication Date: 21 July 2008

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Formulas 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/N2 corrections and De Wit–’t Hooft anomalies on two‐dimensional U(N) lattice gauge theories in the strong coupling 1/N expansion is discussed.
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02.20.-a Group theory

Lie groups, spin equations, and the geometrical interpretation of solitons

George Reiter

J. Math. Phys. 21, 2704 (1980); http://dx.doi.org/10.1063/1.524387 (11 pages) | Cited 10 times

Online Publication Date: 21 July 2008

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The 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).
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02.20.Qs General properties, structure, and representation of Lie groups
02.30.Jr Partial differential equations

Generalized groups as global or local symmetries

J. G. Taylor

J. Math. Phys. 21, 2715 (1980); http://dx.doi.org/10.1063/1.524388 (4 pages)

Online Publication Date: 21 July 2008

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We 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.
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02.20.Qs General properties, structure, and representation of Lie groups
11.30.-j Symmetry and conservation laws

Group actions on principal bundles and invariance conditions for gauge fields

J. Harnad, S. Shnider, and Luc Vinet

J. Math. Phys. 21, 2719 (1980); http://dx.doi.org/10.1063/1.524389 (6 pages) | Cited 75 times

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Invariance 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.
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02.20.Sv Lie algebras of Lie groups

A matrix representation of the translation operator with respect to a basis set of exponentially declining functions

Eckhard Filter and E. Otto Steinborn

J. Math. Phys. 21, 2725 (1980); http://dx.doi.org/10.1063/1.524390 (12 pages) | Cited 64 times

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The matrix elements of the translation operator with respect to a complete orthonormal basis set of the Hilbert space L2(R3) 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.
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02.30.-f Function theory, analysis
31.15.-p Calculations and mathematical techniques in atomic and molecular physics

Infinities of polynomial conserved densities for nonlinear evolution equations

Mark J. McGuinness

J. Math. Phys. 21, 2737 (1980); http://dx.doi.org/10.1063/1.524391 (6 pages) | Cited 1 time

Online Publication Date: 21 July 2008

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The 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.
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02.30.Jr Partial differential equations

An infinity of polynomial conserved densities for a class of nonlinear evolution equations

Mark J. McGuinness

J. Math. Phys. 21, 2743 (1980); http://dx.doi.org/10.1063/1.524392 (6 pages) | Cited 1 time

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Noether’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.
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02.30.Jr Partial differential equations

The nonabelian Toda lattice: Discrete analogue of the matrix Schrödinger spectral problem

M. Bruschi, S. V. Manakov, O. Ragnisco, and D. Levi

J. Math. Phys. 21, 2749 (1980); http://dx.doi.org/10.1063/1.524393 (5 pages) | Cited 31 times

Online Publication Date: 21 July 2008

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We 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.
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02.30.Ks Delay and functional equations

An addition theorem for vector Helmholtz harmonics

F. Borghese, P. Denti, G. Toscano, and O. I. Sindoni

J. Math. Phys. 21, 2754 (1980); http://dx.doi.org/10.1063/1.524394 (2 pages) | Cited 28 times

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An 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.
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02.30.Px Abstract harmonic analysis

Particle trajectories in 1/r fields

M. Arnow

J. Math. Phys. 21, 2756 (1980); http://dx.doi.org/10.1063/1.524395 (4 pages)

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The 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.
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45.05.+x General theory of classical mechanics of discrete systems

A concise and accurate solution for Poiseuille flow in a plane channel

C. E. Siewert, R. D. M. Garcia, and P. Grandjean

J. Math. Phys. 21, 2760 (1980); http://dx.doi.org/10.1063/1.524396 (4 pages) | Cited 14 times

Online Publication Date: 21 July 2008

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The recently developed FN 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.
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47.10.-g General theory in fluid dynamics
47.15.-x Laminar flows
47.45.-n Rarefied gas dynamics
47.60.-i Flow phenomena in quasi-one-dimensional systems

Operator methods for time‐dependent waves in random media with applications to the case of random particles

K. Furutsu

J. Math. Phys. 21, 2764 (1980); http://dx.doi.org/10.1063/1.524397 (16 pages) | Cited 5 times

Online Publication Date: 21 July 2008

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The 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.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation
02.50.Ey Stochastic processes

A note on the Schrödinger equation for the x2+λx2/(1+gx2) potential

N. Bessis and G. Bessis

J. Math. Phys. 21, 2780 (1980); http://dx.doi.org/10.1063/1.524398 (6 pages) | Cited 61 times

Online Publication Date: 21 July 2008

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The energy levels and wave functions of the Schrödinger equation involving the potential x2+λx2/(1+gx2) 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.
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03.65.Ge Solutions of wave equations: bound states

Exactly solvable eigenvalue problem with hypergeometric eigenfunctions

Michael J. King and Fritz Rohrlich

J. Math. Phys. 21, 2786 (1980); http://dx.doi.org/10.1063/1.524399 (3 pages) | Cited 3 times

Online Publication Date: 21 July 2008

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A Schrödinger equation with a momentum dependent interaction leads to exact solutions ψ∊ L2(R3,d3x) with radial parts of the wave function which are hypergeometric functions and their appropriate analytic continuations. The normalization integrals are obtained in closed form.
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03.65.Ge Solutions of wave equations: bound states

K‐surfaces in the Schwarzschild space‐time and the construction of lattice cosmologies

Dieter R. Brill, John M. Cavallo, and James A. Isenberg

J. Math. Phys. 21, 2789 (1980); http://dx.doi.org/10.1063/1.524400 (8 pages) | Cited 30 times

Online Publication Date: 21 July 2008

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We 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.
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04.20.Cv Fundamental problems and general formalism
02.40.-k Geometry, differential geometry, and topology

Gödel‐like cosmological solutions

Dipankar Ray

J. Math. Phys. 21, 2797 (1980); http://dx.doi.org/10.1063/1.524401 (2 pages) | Cited 25 times

Online Publication Date: 21 July 2008

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Gödels 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.
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04.20.Jb Exact solutions

Static gravitational and Maxwell fields in the general scalar tensor theory

A. Banerjee and S. B. Duttachoudhury

J. Math. Phys. 21, 2799 (1980); http://dx.doi.org/10.1063/1.524402 (3 pages) | Cited 9 times

Online Publication Date: 21 July 2008

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The expression for g00 as a function of the scalar field Ψ is obtained in the general scalar tensor theory 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 g00 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.
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04.20.Jb Exact solutions
04.50.-h Higher-dimensional gravity and other theories of gravity

Dynamics in nonglobally hyperbolic, static space‐times

Robert M. Wald

J. Math. Phys. 21, 2802 (1980); http://dx.doi.org/10.1063/1.524403 (4 pages) | Cited 45 times

Online Publication Date: 21 July 2008

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Ordinary 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.
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04.20.Jb Exact solutions
95.30.Sf Relativity and gravitation

The use of anticommuting variable integrals in statistical mechanics. I. The computation of partition functions

Stuart Samuel

J. Math. Phys. 21, 2806 (1980); http://dx.doi.org/10.1063/1.524404 (9 pages) | Cited 65 times

Online Publication Date: 21 July 2008

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Integrals 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.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

The use of anticommuting variable integrals in statistical mechanics. II. The computation of correlation functions

Stuart Samuel

J. Math. Phys. 21, 2815 (1980); http://dx.doi.org/10.1063/1.524405 (5 pages) | Cited 14 times

Online Publication Date: 21 July 2008

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By 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.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

The use of anticommuting variable integrals in statistical mechanics. III. Unsolved models

Stuart Samuel

J. Math. Phys. 21, 2820 (1980); http://dx.doi.org/10.1063/1.524406 (14 pages) | Cited 10 times

Online Publication Date: 21 July 2008

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The Ising model in three dimensions is fermionized by using integrals over anticommuting variables. The result is generalized to the Ising model in arbitrary dimensions and in a magnetic field. Approximation methods are developed to attack unsolved statistical mechanics models. Perturbation theory and the Hartree approximation are applied to the unsolved monomer‐dimer problems. The result is a numerical solution to this unsolved class of problems. Anticommuting variables appear to be a powerful approach to unsolved problems.
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05.20.-y Classical statistical mechanics
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

Geometrical reinterpretation of Faddeev–Popov ghost particles and BRS transformations

Jean Thierry‐Mieg

J. Math. Phys. 21, 2834 (1980); http://dx.doi.org/10.1063/1.524385 (5 pages) | Cited 64 times

Online Publication Date: 21 July 2008

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A classical geometrical interpretation of the ghosts fields is presented. BRS rules follow from the Cartan‐Maurer fibration theorem. The statistics of ghosts are explained and the effective quantum Lagrangian is derived without factorizing the volume of the gauge group. Topologically nontrivial ghost configurations are defined.
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11.15.-q Gauge field theories
11.10.Ef Lagrangian and Hamiltonian approach
11.10.Gh Renormalization

Gauge symmetry and its breakdown: The example of a BCS superconductor

S. K. Bose

J. Math. Phys. 21, 2839 (1980); http://dx.doi.org/10.1063/1.524407 (3 pages) | Cited 1 time

Online Publication Date: 21 July 2008

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The mathematical structure of an infinitely extended BCS super‐conductor is re‐examined in the light of the theory of bundle representations. The role of the homotopy group in the BCS model is clarified. The precise characterization of the constant gauge transformation in terms of a principal fiber bundle (with discrete fiber and group) is pointed out.
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11.30.Qc Spontaneous and radiative symmetry breaking
74.20.Fg BCS theory and its development

On the proper vectors of real third‐order matrices

Vijay K. Stokes

J. Math. Phys. 21, 2842 (1980); http://dx.doi.org/10.1063/1.524725 (2 pages) | Cited 2 times

Online Publication Date: 21 July 2008

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The result that a pseudovector can be associated with a real third order skew‐symmetric matrix has been used for establishing some properties of the proper vectors of real third order matrices. It turns out that the pseudovector associated with the skew‐symmetric part of such matrices characterizes some interesting properties of proper vectors, such as the question of their orthogonality.
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83.10.Ff Continuum mechanics

Bloch electrons in a magnetic field: Reduction to one dimension

G. H. Wannier

J. Math. Phys. 21, 2844 (1980); http://dx.doi.org/10.1063/1.524384 (3 pages) | Cited 9 times

Online Publication Date: 21 July 2008

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A reduction to one dimension of the above problem, found by Schellnhuber and Obermair for a special model lattice, is shown to be valid for all lattices without restriction. As was the case in their problem, the field must be rational. If the rational number is the reciprocal of an integer a single equation results. This condition is well adapted to the study of fields of practically attainable magnitude. If the rational number is of the form q/p a system of q coupled equations is obtained.
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71.10.-w Theories and models of many-electron systems
75.90.+w Other topics in magnetic properties and materials (restricted to new topics in section 75)
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