Volume 18, Issue 10, October 1977
Index of content:

Scalar meson fields in a conformally flat space
View Description Hide DescriptionScalar meson fields are studied for the conformally flat metric e ^{+ψ}(d x ^{1} ^{ 2 }+d x ^{2} ^{ 2 }+d x ^{3} ^{ 2 } −d x ^{4} ^{ 2 }). Complete sets of solutions of the field equations have been obtained for massless mesons, and the field equations have been reduced to a single ordinary differential equation in ψ for massive mesons. In either case ψ is found to be a function of x ^{1} ^{ 2 }+x ^{2} ^{ 2 }+x ^{3} ^{ 2 }−x ^{4} ^{ 2 } or c _{1} x ^{1}+c _{2} x ^{2}+c _{3} x ^{3}+c _{4} x ^{4} (c _{1},c _{2},c _{3},c _{4} being constants), and φ the meson field is found to be constant or a function of ψ.

Invariants and wavefunctions for some time‐dependent harmonic oscillator‐type Hamiltonians
View Description Hide DescriptionRecently the author has shown that the Hamiltonian, H= (1/2) ω^{ T } A (t) ω+B (t)^{ T }ω+C (t), in which A (t) is a positive definite symmetric matrix and ω^{μ}=q _{ i }, μ=1,n, i=1,n, ω^{μ}=p _{ i }, μ=n+1,2n, i=1,n, may be transformed to the time‐independent Hamiltonian, H̄= (1/2) ω̄^{ T }ω̄, by a time‐dependent linear canonical transformation, ω̄=Sω+r. H̄ is an exact invariant of the motion described by H. A matrix invariant may also be constructed which provides a basis for the generators of the dynamical symmetry group SU(n) which may always be associated with H, usually as a noninvariance group. In this paper we examine, by way of example, an oscillator with source undergoing translation, the two‐dimensional anisotropicoscillator, general one‐ and two‐dimensional oscillators with Hamiltonians of homogeneous quadratic form and obtain explicit invariants and Schrödinger wavefunctions with the aid of the linear canonical transformations.

Ground state properties and lower bounds for energy levels of a particle in a uniform magnetic field and external potential
View Description Hide DescriptionThe Hamiltonian H (B), for a particle of mass μ and charge e in a uniform magnetic field of strength B in the z direction and an external axially symmetric potential V, is a direct sum of operators H (m,B) acting in the subspace of eigenvaluem of the z component of angular momentumL _{ z }. Let λ (B) [λ (m,B)] denote the smallest eigenvalue of H (B) [H (m,B)]. For V=−e ^{2}/r (r=‖x‖), the attractive Coulomb potential, we obtain lower bounds l (m,B), for the spectrum of H (m,B) such that l (0,B) ≳l (0,0) =λ (0,0) for B≳0, l (m,B) ≳l (0,B) for m≠0, and l (‖m‖,B)−l (−‖m‖,B) =e B‖m‖/μ, l (m′,B) ≳l (m,B) if m′<m?0. We show at least for an interval [0,B′≳0] of B that the ground state of H (B) is the lowest eigenvalue, λ (0,B), of H (0,B) and is an almost everywhere positive function. If V=A/r ^{2}+r ^{2} for B=0, λ (0) =λ (0,0) and the ground statewavefunction is an almost everywhere positive function with L _{ z }eigenvalue zero. However, for large A, we prove that for an interval of B away from B=0, the lowest eigenvalue, λ (−1,B), of H (−1,B) is below the lowest eigenvalue, λ (0,B), of H (0,B) and that the ground state of H (B) is not an almost everywhere positive function.

On Schwinger functionals—positive extensions, moment problems, and representations
View Description Hide DescriptionWe construct extensions for a class of Schwinger functions at noncoincident arguments to symmetric states on the Borchers algebra. Conditions are given for these states to be strongly positive. For strongly positive states the relation between uniqueness of the Euclidean measure,polynomial density, and self‐adjointness for the Euclidean field is examined.

On the evolution equations for Killing fields
View Description Hide DescriptionThe problem of finding necessary and sufficient conditions on the Cauchy data for Einstein equations which insure the existence of Killing fields in a neighborhood of an initial hypersurface has been considered recently by Berezdivin, Coll, and Moncrief. Nevertheless, it can be shown that the evolution equations obtained in all these cases are of nonstrictly hyperbolic type, and, thus, the Cauchy data must belong to a special class of functions. We prove here that, for the vacuum and Einstein–Maxwell space–times and in a coordinate independent way, one can always choose, as evolution equations for the Killing fields, a strictly hyperbolic system: The above theorems can be thus extended to a l l Cauchy data for which the Einstein evolution problem has been proved to be well set.

Geometric aspects of supergauge theory
View Description Hide DescriptionIt is shown that the Wess–Zumino supergauge algebra can be obtained from the theory of automorphisms of a complex structure on spacetime, and that this is related to a formal geometric scheme of quantization. The geometric interpretation requires h/ to be proportional to the square of the basic unit of length.

Correlations near phase transition in a simple fluid
View Description Hide DescriptionCorrelations in a system with a weak, long‐ranged attractive potential are studied. Using a natural small parameter, the asymptotic orders of all correlations are established. Explicit leading order solutions are obtained for all correlations when the system is away from phase change As it approaches transition, It is shown that the many‐body correlations, even though still small, develop very long range and that they are shown extremely slowly varying functions over space. As a results all the correlations contribut equally significantly to the computation of the pair correlation. In an asymptotic ragion near transition, a hierarchy is derived in which each corelation is expressed in terms of its immediate predecessor, its immediate successor and the pair correlation. The pair potential does not appear in these equations. Instead the correlations are expressed in terms of a Yukawa potential whose strength and range are related to the density and other parameters but not to the detailed form of the potential. In the leading approximation, the pair correlation is proportional to this Yukawa function.

Structure of the Azzarelli–Collas representation for the scattering amplitude and generalization to the Rice representation and the Euler–Pochhammer representation
View Description Hide DescriptionThe Azzarelli–Collas integral representation for the scattering amplitude is extended to the Rice representation. The latter representation in turn is viewed as an example of the Euler–Pochhammer representation.

The quantum mechanical representations of the anisotropic harmonic oscillator group
View Description Hide DescriptionA group G associated with the n‐dimensional anisotropic harmonic oscillator is constructed : G is essentially a group generated by the position and momentum observables, the identity operator, and the Hamiltonian of the system. All the quantum mechanical irreducible representations of G are evaluated, using Mackey’s theory of induced representations.

On the accidental degeneracy of the anisotropic harmonic oscillator. I
View Description Hide DescriptionA group G associated with the n‐dimensional anisotropic harmonic oscillator is shown to be embedded in a semidirect product L of the Weyl group N and the symplectic group Sp(2n,R). A particular induced representation of the group L, when restricted to G, is proved to be unitarily equivalent to ⊕_{ s } d _{ω,s } U _{ G } ^{ v,−(sgnv)} ^{ s)} where d _{ω,s } is the degeneracy of the energy level E _{ω,s } of the n‐dimensional anisotropic harmonic oscillator with frequencies (ω_{1}, ω_{2},...,ω_{ n }) =ω, U _{ G } ^{ v,−(sgnv)} ^{ s } is an irreducible representation of G, and s may be regarded as indexing all distinct energy levels of the system.

A noninvariance group for the n‐dimensional isotropic harmonic oscillator
View Description Hide DescriptionA semidirect product of the Weyl group N, with the special unitary groupSU(n), is proved to be a possible noninvariance group for the n‐dimensional isotropic harmonic oscillator. In order to obtain this result, a method is developed for finding the representation W of SU(n) which intertwines the representations U _{ N } ^{ v } and (A+i B) U _{ N } ^{ v } of N [where A+i BεSU(n)]; it is also shown that W=⊕^{∞} _{ a=0} W _{ a }, where W _{ a } is an irreducible representation of SU(n), of dimension equal to the degeneracy of the (a+1)th energy level of the n‐dimensional isotropic harmonic oscillator.

On the accidental degeneracy of the n‐dimensional anisotropic harmonic oscillator. II
View Description Hide DescriptionIn an earlier paper, a group, G, associated with the n‐dimensional anisotropic harmonic oscillator was shown to be embedded in a semidirect product, L, of the Weyl group N and the symplectic group Sp(2n,R). A particular representation R ^{ v } of L, when restricted to G, was proved to be unitarily equivalent to ⊕_{ s } d _{ω,s } U _{ G } ^{ v,−(sgnv)} ^{ s)}, where d _{ω,s } is the degeneracy of the energy level E _{ω,s } of the n‐dimensional anisotropic harmonic oscillator with frequencies (ω_{1},ω_{2},...,ω_{ n }) =ω, U _{ G } ^{ v,−(sgnv)} ^{ s } is an irreducible representation of G and s may be regarded as indexing all distinct energy levels of the system. In the present paper, the representation R ^{ v } of L is shown to be unitarily equivalent to the representation U _{ N } ^{ v } W⊕? of L, where U _{ N } ^{ v } is an irreducible representation of N, W is the projective representation of Sp(2n,R) which intertwines the representations U _{ N } ^{ v } and S U _{ N } ^{ v } of N [where S ε Sp(2n,R)], and ? is the complex conjugate of W. This alternative form for the representation R ^{ v } of L enables it to be decomposed, into two irreducible representations.

A finite quantum electrodynamical base for many photon collective phenomena
View Description Hide DescriptionWe analyze the interaction of many ’’nonoverlapping’’ atoms with the common radiation field of transverse photons over a continuum of modes. Typical effects of quantum optics arising from this interaction require extensions to very high powers in the coupling constant e and diverge therefore within a systematic perturbation approach. Since renormalization to such high orders is not feasible in practice, we formulate for the purpose of quantum optics ’’a new peace treaty between QED and its infinities.’’ This is a compromise between the necessity of including effects of very high powers in e and the trivial demands for finite results. We show that a unitary time evolution operator exists if, in essence, only any finite number of levels of each atom is treated as ’’existent.’’ We convince ourselves that the limits of the concept of ’’nonoverlapping atoms’’ are reached long before any infinities can enter. The resulting theories meet still reasonable requirements with respect to the dualism of light, Dicke principle, and causality. They are formulated so that some practical demands of quantum optics (homogeneous and inhomogeneous line broadening, Doppler effects) can be met easily if desired.

A multiple‐scales space–time analysis of a randomly perturbed one‐dimensional wave equation
View Description Hide DescriptionAn initial value problem for one‐dimensional wave propagation is considered; the medium is assumed to be randomly perturbed as a function of both space and time. The stochastic perturbation theory of Papanicolaou and Keller [SIAM J. Appl. Math. 21, 287 (1971)] is applied directly in the space–time regime to derive transport equations for the first and second moments of the solution. These equations are solved in special cases.

The interaction function and lattice duals
View Description Hide DescriptionAn interaction function is defined for lattice models in statistical mechanics. A correlation function expansion is derived, giving a direct proof of the duality relations for correlation functions.

Extensions of Lie‐graded algebras
View Description Hide DescriptionFor Lie‐graded algebras which are generalizations of Lie algebras with respect to a graduation, used recently in physics for the classification of elementary particles, and extension G of F by T, i.e., a short exact sequence T≳→G→≳F can be described by a Lie‐graded composition on T⊕F, which is formulated in terms of a pair of mappings ∂:F→derT and Δ:F×F→T. The congruence of two extensions of F by T, i. e., the equivalence of the corresponding short exact sequences, is related to an equivalence relation on the set Z ^{2}(F,T) of such 2‐cocycles (∂,Δ) such that there exists a bijection between the set of congruence classes of extensions of F by T and the set H ^{2}(F,T) of classes in Z ^{2}(F,T). This generalizes Lie algebraical results which are also known for groups. Examples for two special cases are given: the semidirect sums with Δ trivial and the almost direct sums with ∂ trivial. Both generalize the concepts of tangent and cotangent algebras of Lie algebras and their central extensions with R, the latter being used in the Bargmann theory of ray representations of these semidirect sums.

Invariant integrals in nonmetric supersymmetry spaces
View Description Hide DescriptionDefinitions are given for the exterior product, infinitesimal extension of a cell, Levi‐Civita symbol, and generalized Kronecker delta in order to identify invariant integrals in spaces with both Fermi and Bose dimensions. Stokes’ and Green’s theorems for such spaces are constructed as a preliminary to defining a generalized action principle in supersymmetry spaces not necessarily equipped with a metric or a connection.

The equivalence of a one‐dimensional turbulence problem and the one‐dimensional Coulomb gas
View Description Hide DescriptionWe show that Burger’s equation subject to initial conditions which are governed by a canonical (Gaussian) distribution in the kinetic energy can be related to the properties of a one‐dimensional Coulomb gas in a certain limit. Some consequences of this are worked out.

Neutron transport in plane geometry with general anisotropic, energy‐dependent scattering
View Description Hide DescriptionWe consider the neutron transportequation in plane geometry with a general energy‐dependent anisotropicscattering kernel. We construct the solution of the subcritical half‐space albedo problem as a contour integral around the positive half of the spectrum of a reduced transport operator K. The integrand involves the boundary data and two operators which provide the Wiener–Hopf factorization of a third operator contained in (λI−K)^{−1}. Bounds are obtained for the location of the spectrum of K in the complex plane. We also obtain representations of the solutions of the Milne problem and of the full‐space and half‐space problems with sources. Various simplifications of the general theory, which occur for particular scattering models, are discussed as an illustration of the results.

A Green’s function for a linear equation associated with solitons
View Description Hide DescriptionA linear equation associated with nonlinear waveequations which support solitons is analyzed. A complete set of solutions of this linear equation is described through the techniques of scattering theory. This set is used to construct an explicit representation of a Green’s function for perturbation theory. The cases of the nonlinear Schrödinger and sine‐Gordon equations are discussed in some detail.