Volume 13, Issue 1, January 1972
Index of content:

The Jost Function of a Momentum‐Dependent Potential
View Description Hide DescriptionIt is proved that the Jost function of a central, momentum‐dependent, S‐state interaction is equal to the ratio of the Fredholm determinant of the kernel of the integral equation for the outgoing scattering solution in the momentum space representation to that of the regular solution in the same representation. This proof of the expected result is more general than that given by Warke and Bhaduri for a non‐local interaction.

Green's Dyadics for Elastic and Electromagnetic Waves in a Random Medium
View Description Hide DescriptionA method for obtaining Green's dyadics of renormalized stochastic vector wave operators is presented schematically and is applied to propagation of elastic and electromagnetic waves in statistically homogeneous and isotropic media. Formula for effective propagation constants and renormalized Green's dyadics are derived and then computed explicitly for various special cases of elastic and electromagnetic waves with a special type of correlation function. Furthermore, parameters which affect the validity of such a renormalization approximation are also derived by estimating the second nonvanishing term of the renormalized series for average waves.

Weaker Versions of Zeeman's Conjectures on Topologies for Minkowski Space
View Description Hide DescriptionIn a paper [Topology 6, 161 (1967)] Zeeman conjectured that (a) the finest topology on Minkowski space that induces the one‐dimensional Euclidean topology on every timelike line and (b) the finest topology on Minkowski space that induces the three‐dimensional Euclidean topology on every spacelike hyperplane have the same group of homeomorphisms G which is generated by the inhomogeneous Lorentz group and the dilatations. This paper deals with two topologies on Minkowski space which are weaker than those in (a) and (b), respectively, and have the property that they induce the one‐dimensional and the three‐dimensional Euclidean topology on timelike lines and spacelike hyperplanes, respectively. It is then shown that both topologies have G as their homeomorphism group. Thus, what we have shown amounts to proving the weaker versions of Zeeman's conjectures.

Lattice Green's Function for the Orthorhombic Lattice in Terms of the Complete Elliptic Integral
View Description Hide DescriptionThe real and imaginary parts of the orthorhombic lattice Green's function at the origin are expressed as a sum of simple integrals of the complete elliptic integrals of the first kind. In order to give the expressions for all values of the variable from − ∞ to + ∞, use is made of the method of analytic continuation. The results of the numerical computations are shown by figures.

On Representations of the Conformal Group Which When Restricted to Its Poincaré or Weyl Subgroups Remain Irreducible
View Description Hide DescriptionUnitary irreducible representations of the conformal group which, when restricted to its Poincaré or Weyl (= Poincaré group extended by dilatations) subgroups, remain irreducible are found. In particular it is proved that the continuous spin representations (p _{μ}·p ^{μ} = 0) of the Poincaré group cannot be extended to the conformal group and that, on the other hand, a known extension in the discrete spin case is unique (up to a unitary equivalence). Similar results hold for Weyl group for which, in addition, extensions also exist in the case p _{μ}·p ^{μ} ≠ 0. Namely, each unitary irreducible representation of the Weyl group characterized by the sign of p _{μ}·p ^{μ} (≠0) and by invariants of the corresponding little group can be extended to a one‐parameter family of irreducible representations of the conformal group. Finally, it is shown that, besides the above mentioned extensions of unitary irreducible representations of the Weyl group, there are no others.

On an Application of the Theory of Invariants
View Description Hide DescriptionThe problem of finding highest‐weight polynomials in certain chains of subgroups of the unitary group is shown to be related to finding semi‐invariants of certain ground forms according to the theory of invariants developed by mathematicians long ago.

Radial Matrix Elements of the Radial‐Angular Factorized Hydrogen Atom
View Description Hide DescriptionThe hydrogen atom is factorized according to the scheme and the radial group O(2, 1) studied. It is shown that r^{k}D _{ n/(n+q)}, where D_{a} is a dilatation operator, is proportional to a tensor operator in this scheme, allowing a group theoretical study of the radial matrix elements of r^{k} , including an explanation of the Pasternack and Sternheimer selection rule.

Example Related to the Foundations of Quantum Theory
View Description Hide DescriptionWe construct a very simple example of a statistical system which satisfies almost all of the axioms used by various authors in the quantum logic approach to the foundations of quantum mechanics. Since the example is not quantum mechanical, it is seen that present arguments are a long way from characterising quantum mechanics in terms of a set of physically meaningful axioms.

Uniform Asymptotic Solution of the Radial Schrödinger Equation for an Electron Bound in a Central Field Potential
View Description Hide DescriptionAssuming Z (the nuclear charge) is large, it is shown, using a theory of McKelvey, how a uniform asymptotic solution of the radial Schrödinger equation for an electron bound in a central field potential and for a related problem may be obtained. Requiring the asymptotic solutions to be bounded leads to a determination of the energy expansion parameters.

On Canonical Commutation Relations and Infinite‐Dimensional Measures
View Description Hide DescriptionRecently, Araki has proved ray continuity for the measure‐space realization of the CCR's. In this paper, stronger continuity properties of this realization are derived, from which Araki's result follows as a corollary. It is shown that V(g) is continuous on complete metric subspaces of for any metric stronger than the weak topology. Surprisingly no further assumptions on the measure are needed. If is an F‐ or LF‐space, V(g) is continuous on the whole of . These results are equivalent to certain continuity properties of the Radon‐Nikodym derivative of the measure. Via these, it is then shown that every quasi‐invariant measure on a nuclear F‐ or LF‐space, such as S or , is a superposition of ergodic measures. In the derivation, their close connection with irreducible representations of the CCR's is exploited.

Application of the Theory of Orlicz Spaces to Statistical Mechanics. I. Integral Equations
View Description Hide DescriptionIn this paper it is suggested that there may exist a fundamental relationship between the variables of thermodynamics, the operators associated with certain nonlinear integral equations of statistical mechanics, and the properties of a class of convex functions, called N functions, investigated by Krasnosel'skii and Rutickii. In particular, it is pointed out that the most general theoretical framework within which all these problems can be studied is that provided by the theory of Orlicz spaces. In the first part of our study, presented here, it is shown that the existence of solutions to certain nonlinear integral equations, derived either from the BBKYG hierarchy or from the grand partition function using a variational approach, can be established with some generality. The relationship between our results and those obtained by Ruelle is discussed.

Wave Propagation in Certain One‐Dimensional Random Media
View Description Hide DescriptionIn this paper we investigate the stochastic ordinary differential equation with y(t) a random process. Two specific types of process y(t) are considered. Both of these arise from a bounded mapping y(t) = f(x(t)) of a countable state space Markov processx(t). Exact equations are derived for the statistical moments of u(t), and the behavior of the first two moments is discussed in the limit of small ε. A description of the layered media to which our results apply is given and a comparison of our exact results with certain perturbation methods is made.

Asymptotic Behavior of Vacuum Space‐Times
View Description Hide DescriptionNewman and Penrose have given conditions on the asymptotic form of the Weyl tensor in empty space‐time that are sufficient to insure that the space‐time is asymptotically flat at null infinity and has the peeling property. We give considerably weaker conditions and show them to be sufficient for asymptotic flatness. Under the weaker conditions the asymptotic behavior of the Weyl tensor is more general than the case where the peeling property holds. The asymptotic dependence on a suitably defined radial coordinate is given for the basis null tetrad, the spin coefficients, and the tetrad components of the Weyl tensor.

Non‐L ^{2} Solutions of Exactly Soluble Relativistic Wave Equations and the Lee Model
View Description Hide DescriptionIn nonrelativistic potential theory there can exist singularities of the S matrix which are associated with wavefunctions belonging to a non‐L ^{2} class. In this paper the non‐L ^{2} character of these singularities is shown to persist in several relativistic models: in (1) two schemes, including that of the Klein‐Gordon equation and the coupling of a classical relativistic field to an external exponential source, in (2) the pair theory, classical and quantized, with separable interactions, and in (3) the Lee model. For (2) and (3), poles of the ``proper'' Jost function f̄_(k, 0), i.e., the Fredholm determinant for the outgoing scattering state, correspond to non‐L ^{2} class solutions of the associated dynamical field equation in coordinate space. These non‐L ^{2}solutions will be called ``shadow'' fields. They are of special importance and bear the same relation as the ``shadow'' states in potential theory because they also are of dynamical origin and do not appear in the completeness or unitarity relations by virtue of their non‐L ^{2} status.

Electromagnetic Radiation in a Uniformly Moving, Homogeneous Medium
View Description Hide DescriptionA new method of treatingradiation problems in a uniformly moving, homogeneous medium is presented. A certain transformation technique in connection with the four‐dimensional Green's function method makes it possible to elaborate the Green's functions of the governing differential equations in the rest system of the medium, whereas the final integrals determining the field may be calculated in the rest system of the source.

Structure of the Wigner 9j Coefficients in the Bargmann Approach
View Description Hide DescriptionBargmann's treatment of the Clebsch‐Gordan (3j) and Racah (6j) coefficients is here extended to the case of Wigner 9j coefficients. The generating function for the 9j coefficient is computed by the analytic method. The result is compared to the Schwinger's expression derived with the algebraic (boson operator) method. The full symmetry of the Wigner 9j coefficients is manifest and transparent in the Bargmann's formalism. A new explicit expression for the Wigner 9j coefficient is derived as a sixfold sum which may be regarded as the analog of the Racah's formula for the Racah coefficient.

A Class of Matrix Ensembles
View Description Hide DescriptionA class of random matrix ensembles is defined, with the purpose of providing a realistic statistical description of the Hamiltonian of a complicated quantum‐mechanical system (such as a heavy nucleus) for which an approximate model Hamiltonian is known. An ensemble of the class is specified by the model Hamiltonian H _{0}, an observed eigenvalue distribution‐function r(E), and a parameter τ which may be considered to be a fictitious ``time.'' Each of H _{0}, r(E), and τ may be chosen independently. The ensemble consists of matrices M which are obtained from H _{0} by an invariant random Brownian‐motion process, lasting for a time τ and tending to pull the eigenvalues of M toward the distribution r(E). For small τ the ensemble allows only small perturbations of H _{0}. As τ → ∞, the ensemble tends to a stationary limit independent of H _{0} and depending on r(E) alone. The following quantitative results are obtained. (1) It is proved that the global eigenvalue distribution in the limit τ → ∞ becomes identical with the observed distribution r(E). (2) A nonlinear partial differential equation is obtained for the global eigenvalue distribution ρ(E, τ) as a function of E and τ. Solution of this equation will show how the distribution changes from the initial form specified by H _{0} at τ = 0 to the final form r(E) at τ = ∞. Approximate solution shows that deviations of ρ(E, τ) from r(E) extending over an interval containing meigenvalues will disappear exponentially as soon as τ is of the order of mD ^{2}, where D is the local mean level spacing. (3) Exact analytic expressions are obtained for the correlation functions representing the probabilities for finding neigenvalues at assigned positions (E _{1}, …, E_{n} ) in the ensemble in the limit τ → ∞, irrespective of the positions of the remaining (N‐n) eigenvalues. It is made plausible, but not proved, that these correlation functions tend to limits as N → ∞, which are universal functions independent of r(E). If proved, this statement would imply that the local statistical properties (spacing distributions, etc.) of eigenvalues in the ensemble become, when τ and N are both large, universal properties independent of the global eigenvalue distributions. In particular, the spacing distributions would be identical with those calculated for more special ensembles by Wigner, Gaudin, and Mehta.

Currents in Classical Field Theories
View Description Hide DescriptionThe currentvector field associated with a one‐parameter group of transformations of a classical field is defined in a coordinate‐free way.

Velocity‐Dominated Singularities in Irrotational Dust Cosmologies
View Description Hide DescriptionWe consider irrotational dust solutions of the Einstein equations. We define ``velocity‐dominated'' singularities of these solutions. We show that a velocity‐dominated singularity can be considered as a three‐dimensional manifold with an invariantly and uniquely defined inner metric tensor, extrinsic curvature tensor, and scalar bang time function. We compute this structure for a variety of known exact models. The structure of the singularity uniquely determines the solution in a certain class of spatially inhomogeneous models. We briefly discuss the b boundary (Schmidt boundary). In an appendix we generalize conformal transformations to ``stretch'' transformations and calculate the curvature form of a stretched metric.

Statistically Exact Kinematic Dynamo Action in a Box
View Description Hide DescriptionA set of statistically exact equations is set up to describe dynamo action brought about by velocity turbulence confined to a finite spatial domain (a ``box''). Using a variation of a limit‐theorem employed elsewhere by Kac, we demonstrate that in large enough boxes a large‐scale magnetic field is regenerated by such velocity turbulence.