Index of content:
Volume 18, Issue 1, January 1977

Lie theory and the wave equation in space–time. I. The Lorentz group
View Description Hide DescriptionIn this article we begin a study of the relationship between separation of variables and the conformal symmetry group of the wave equation ψ_{ t t } −Δ_{3}ψ=0 in space–time. In this first article we make a detailed study of separation of variables for the Laplace operator on the one and two sheeted hyperboloids in Minkowski space. We then restrict ourselves to homogeneous solutions of the wave equation and the Lorentz subgroup SO(3,1) of the conformal group SO(4,2). We study the various separable bases by using the methods of integral geometry as developed by Gel’fand and Graev. In most cases we give the spectral analysis for these bases, and a number of new bases are developed in detail. Many of the special function identities derived appear to be new. This preliminary study is of importance when we subsequently study models of the Hilbert space structure for solutions of the wave equation and the Klein–Gordon equation ψ_{ t t }−Δ_{3}ψ=λψ.

Structural stability of the phase transition in Dicke‐like models
View Description Hide DescriptionThe free energy for a general class of Dicke models is computed and expressed simply as the minimum value of a potential function Φ = (E−T S)/N. The function E/N is the image of the Hamiltonian under the quantum–classical correspondence effected by the atomic and field coherent state representations, and the function S is the logarithm of an SU(2) multiplicity factor. The structural stability of the second order phase transition under changes in the functional form of the Hamiltonian is determined by searching for stability changeovers along the thermal critical branch of Φ. The necessary condition for the presence of a second order phase transition is completely determined by the canonical kernel of the Hamiltonian. The sufficient condition is that a first order phase transition not occur at higher temperature. The critical temperature for a second order phase transition is given by a gap equation of Hepp–Lieb type.

Metastability and the analytic continuation of eigenvalues
View Description Hide DescriptionA metastable analytic continuation of the Ising modelfree energy is conjectured to follow from certain analyticity properties of the eigenvalues of the transfer matrix. The resulting description of metastability is applicable to any system whose phase transition is associated with eigenvalue degeneracy. Motivation for the conjectures concerning the Ising model is provided by the study of eigenvalue continuation in several simpler systems.

Factored irreducible symmetry operators
View Description Hide DescriptionIrreducible symmetry operators (ISO’s) are defined and their properties are displayed. A method for the systematic construction of ISO’s in the form of products of interchangeable factors is described. The key to the method is a set of conditions for directly testing for the irreducibility of induced symmetry operators formulated in terms of the operators themselves. Detailed examples are presented for the tetrahedral group and double group. The tetrahedral double group provides an example in which the induction process in its simplest form fails, but which does yield to a modification of the induction techniques.

The Wegner approximation of the plane rotator model as a massless, free, lattice, Euclidean field
View Description Hide DescriptionThe approximate plane rotator model proposed by Wegner can be reinterpreted as a massless, free, lattice, Euclidean field model. Using this approach, we compute the two‐point correlation function and the magnetization for the spin model as functions of the covariance of the fields.

The positive energy conjecture and the cosmic censor hypothesis
View Description Hide DescriptionWe show that a positive energy argument of Geroch can be modified to rule out a possible class of counterexamples to the cosmic censor hypothesis proposed by Penrose.

Canonical parameters of the 6j coefficient
View Description Hide DescriptionA set of six canonical parameters is derived for the 6j coefficient in a way analogous to that used previously for the 3j coefficient. The procedure involves implicitly a twelve‐to‐one homomorphism which reduces the known 144‐element symmetry group to a 12‐element group. The symmetries are completely and elegantly described by only five of the six parameters. A further property of the canonical parameters is their complete independence.

Nonstationary multiple scattering
View Description Hide DescriptionThis paper is devoted to the abstract formulation of the general nonstationary multiple scattering problem in radiative transfer. We consider the linear operators on curved surfaces. Therefore, the problem is attacked from the unified and general point of view. Special geometric considered are spherical shells and slabs. Special cases lead to stationary, instantaneous, and time invariant cases. The extension from stationary to nonstationary involves the distribution theory and the concept of nonpredictive operators. Many well−known physical problems in astrophysics are solved in the unified way.

Geometry of superspaces with Bose and Fermi coordinates and applications to graded Lie bundles and supergravity
View Description Hide DescriptionThe geometry of superspaces with Bose‐ and Fermi‐type coordinates is presented from a coordinate independent point of view. Various geometrical quantities of conventional manifolds are generalized so as to be applicable to superspaces. It is shown that these generalizations can be basically arrived at algebraically by replacing, in the definitions of various geometrical quantities, the Lie derivative of the conventional manifolds with a generalized graded Lie bracket. Explicit expressions for connection coefficients, Riemann curvature tensor, etc., are derived. The general formalism is then applied to graded Lie bundles the relevance of which to supergravitytheories is demonstrated.

The Kähler structure of asymptotic twistor space
View Description Hide DescriptionAsymptotic twistor space T is a 4‐complex‐dimensional Kähler manifold (of signature ++−−) which can be constructed from an asymptotically flat space–time containing gravitational radiation. The properties of this Kähler structure are investigated, the Kähler metric being of a particular type, arising from a scalar Σ with special homogeneity properties. The components of the Kähler curvature K ^{αβ} _{γδ} are found explicitly in terms of the asymptotic Weyl curvature of the space–time. When gravitational radiation is present, K ^{αβ} _{γδ} ≠0, whereas for a stationary field K ^{αβ} _{γδ}=0. The ’’Ricci‐flat’’ condition K ^{αβ} _{αγ}=0 is found always to hold.

On the existence of localized solutions in nonlinear chiral theories
View Description Hide DescriptionWe study the existence of localized solutions to theories based on Weinberg’s nonlinear realization of chiral SU(2) ⊗SU(2). The analysis is done by using specific variations of the action integral and then checking the ensuing global conditions. The following cases are studied: (i) π fields only and without time dependence, (ii) π fields with simple time dependence, (iii) π fields coupled to gauge fields, (iv) the above with certain chiral symmetry breaking potentials. We find that only in certain special cases could there be localized solutions. In most cases the intrinsic nonlinearity of the system does not seem to be enough to guarantee their existence.

Subgroups of the Euclidean group and symmetry breaking in nonrelativistic quantum mechanics
View Description Hide DescriptionA systematic study of explicit symmetry breaking in the nonrelativistic quantum mechanics of a scalar and a spinor particle is presented. The free Schrödinger (or Pauli) equation is invariant under the Euclidean group E(3); an external field will break this symmetry to a lower one. We first find all continuous subgroups of E(3) and then for each subgroup construct the most general (within certain restrictions) external field that breaks the symmetry from E(3) to the corresponding subgroup. For a scalar particle the interaction term is assumed to be of the form V (r)+A(r)P, where P is the momentum, i.e., it involves an arbitrary scalar and vector potential. For a spinor particle it is of the form V (r)+A(r)P +B(r) σ+M _{ i k }(r) σ_{ i } P _{ k } (σ_{ i } are the Pauli matrices). A one‐to‐one correspondence between subgroups of E(3) and classes of ’’symmetry breaking potentials’’ is established. The remaining subgroup symmetry is then used to solve or at least simplify the obtained Schrödinger equation. The existence of a one‐dimensional invariance group (for a particle in a field) leads to the partial separation of variables and determines the functional dependence of the wavefunction on one variable. A two‐dimensional group implies the complete separation of variables and the functional dependence on two variables. A higher dimensional invariance group implies the separation of variables in one or more systems of coordinates and in some cases specifies the wavefunction completely.

Clebsch potentials in the theory of electromagnetic fields admitting electric and magnetic charge distributions
View Description Hide DescriptionAny skew‐symmetric tensor field on a four‐dimensional pseudo‐Riemannian space V _{4} admits a representation in terms of Clebsch potentials and their derivatives. Since the usual 4−potential representation of the electromagnetic field tensor of classical electrodynamics breaks down in the presence of magnetic charges, these Clebsch potentials are treated as the field variables of an invariant variational principle. The resulting Euler–Lagrange equations determine not only a useful representation of the electromagnetic field tensor (in the presence of magnetic charges), but also give rise to Maxwell‐type field equations. The associated Lagrange density defines a unique energy–momentum tensor entirely on the basis of invariance consideration. A generalized variational principle is postulated, whose Euler–Lagrange equations specify the behavior of both the electromagnetic field tensor and the metric tensor of V _{4}. These are generalized Einstein–Maxwell equations. For the case of a spherically symmetric line‐element and a static electromagnetic field an explicit solution of these equations is found which generalizes the well known Reissner–Nordström metric. In the course of the construction of this solution the magnetic charge of the central mass appears naturally as a constant of integration of the associated differential equations for the Clebsch potentials. The equations of motion of a test particle in an external electromagnetic field are also deduced from a variational principle; subject to fairly weak restrictions, the expected generalization of the classical Lorentz force emerges from this analysis.

On Wentzel’s proof of the quantization condition for a single‐well potential
View Description Hide DescriptionWentzel’s elegant method for deriving the quantal generalization of the Sommerfeld quantization condition, which has been criticized by other authors, is justified and put on an irrefutable basis. Clarifying comments on some questions related to the discussion in the present paper are also made.

Symmetry and exact dyon solutions for classical Yang–Mills field equations
View Description Hide DescriptionWe show that the generalized electromagnetic field tensor _{μν} and the magnetic and electric charges in non‐Abelian gauge theories have little to do with the Higgs scalars and/or the dynamics of the Lagrangian. They are consequences of a symmetry in the theory. We present several exact static dyon solutions to the nonlinear classical fieldequations in both massless and massive Yang–Mills theories, which possess both electric and magnetic charges. The implications of _{μν} are also discussed.

Maxwell and Wien processes as special cases of the generalized Feller diffusion process
View Description Hide DescriptionIt is shown that Maxwell and Wien type processes are special cases of the generalized Feller diffusion process. In particular, both are obtained for specific parameter values from the delta function initial condition solution of the generalized Feller equation. For specific values of the independent time variable, one obtains the well‐known distribution laws of Maxwell and Wien of statistical physics.

Nonexistence theorem for spherically asymmetric solutions of the Fermi–Thomas model of atoms and ions
View Description Hide DescriptionIt is proved that there are no spherically asymmetric solutions of the Fermi–Thomas equation ∇^{2}(u/r) = (u/r)^{3/2}ξ, where ξ is unity inside the surfaceu=0, and zero otherwise. This is proved by showing that the boundary conditions on u at the origin and on the surface at infinity necessarily lead to unphysical discontinuities in u at the surfaceu=0, except for the case where u is spherically symmetric. A simple, only approximately statistical model is presented, which may replace the Fermi–Thomas model for asymmetric systems.

A classical perturbation theory
View Description Hide DescriptionA compact formula is found for the perturbation expansion of a general one‐dimensional Hamiltonian system in classical mechanics. The technique is also applied to the mathematical problem of functional inversion.

Gleason measures on infinite tensor products of Hilbert spaces
View Description Hide DescriptionNowak [Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys. 22, 393–5 (1974)] has given an example of a consistent (in the sense of Kolmogorov) family of Gleason measures [A. M. Gleason, J. Math. Mech. 6, 885–94 (1957)] {m _{ n }} defined over ⊗^{ n } _{ i=1} H _{ i } which do not extend to a Gleason measure on ⊙^{∞} _{ i=1} ^{Φ} H _{ i } for a given construction of the infinite tensor product. In this paper we show: (1) In the example of Nowak it is not necessary to assume, as is done, that the H _{ i } are infinite dimensional. (2) That every consistent family developed from pure states, which is the type considered by Nowak, extends over the complete infinite tensor product of von Neumann [Compositio. Math. 6, 1–77 (1938)]. (3) Even if each H _{ i } is two‐dimensional and the complete infinite tensor product of von Neumann is used, it is possible to give a simple counterexample to the conjecture that every consistent family of Gleason measures extends by the use of nonpure states.

Exact results for second‐neighbor surface magnons in an fcc lattice
View Description Hide DescriptionAn exact dispersion equation f (Ω) =0 is obtained for (001) surface magnons in an fcc lattice with nearest and second nearest neighbor Heisenberg coupling. On the Brillouin zone boundary a closed expression is given for the frequency Ω, while a simple root search yields Ω for wave vectors inside the zone. These exact frequencies confirm the results obtained by Trullinger from a Gottlieb polynomial expansion; a discussion of the latter method is given.