Volume 20, Issue 9, September 1979
Index of content:

A double angle sum formula for Gegenbauer polynomials
View Description Hide DescriptionA formula is given by which a Gegenbauer polynomial whose argument is the cosine of twice an angle can be equated with a finite, alternating series of products of Gegenbauer polynomials whose arguments are all cosines of the angle.

On the applicability of the variation of action method to some one‐field solitons
View Description Hide DescriptionThe applicability of the widely used method of variation of action is discussed by investigating the transverse stability of the kink solution of the nonlinear Schrödinger equation and the bell solution of the Kadomtsev–Petviashvili equation. An exact calculation shows that neither instability nor stability can be predicted correctly by the variation of action method. The reasons for that defect are discussed and the correct stability regions are presented.

When does a projective system of state operators have a projective limit?
View Description Hide DescriptionIn analogy to Kolmogorov’s classical extension theorem, we establish necessary and sufficient conditions under which a family {W _{(t 1,⋅⋅⋅,t n )}} of state operators, defined on the finite tensor products H_{ t } _{1}⊗⋅⋅⋅⊗H_{ t } _{ n } of some family {H_{ t }∈T} of complex Hilbert spaces, extends to a state operator on the infinite tensor product ⊗_{ t∈T }H_{ t }.

Birman–Schwinger bounds for the Laplacian with point interactions
View Description Hide DescriptionBirman–Schwinger bounds on the eigenvalues of −Δ are described, where −Δ is the self adjoint operator obtained from the three‐dimensional Laplacian by imposing boundary conditions at N distinct points in R^{3}.

On rotating plane‐fronted waves and their Poincaré‐invariant differential geometry
View Description Hide DescriptionAfter reviewing the rotating plane‐fronted wave type solutions of the scalar wave equation and Maxwell’s vacuum equations in flat space (we point out that there are Yang–Mill analogs as well), we study their Poincaré‐invariant geometry by discussing their characteristic differential invariants and a noninertial curvilinear coordinate system canonically associated with them. In one of the appendices we treat the shearfree and the nondiverging null hypersurfaces in c o m p l e x Minkowski space, in another one we derive the Yang–Mills version of Robinson’s theorem on null electromagnetic fields.

The role of the Onsager–Machlup Lagrangian in the theory of stationary diffusion process
View Description Hide DescriptionThe Onsager–Machlup Lagrangian is shown to have a direct relevance to a cost function for a stochastic control problem. It is found that any stationary diffusion process can be regarded as a solution to the stochastic control problem, that is, it is controlled optimally by the Onsager–Machlup Lagrangian. A deterministic limit of the stationary diffusion process is also obtained as a solution to an ordinary (nonrandom) control problem which is equivalent to the usual variational problem with respect to the Onsager–Machlup Lagrangian.

A model for the stochastic origins of Schrödinger’s equation
View Description Hide DescriptionA model for the motion of a charged particle in the vacuum is presented which, although purely classical in concept, yields Schrödinger’s equation as a solution. It suggests that the origins of the peculiar and nonclassical features of quantum mechanics are actually inherent in a statistical description of the radiative reactive force.

Exact solution of a time‐dependent quantal harmonic oscillator with damping and a perturbative force
View Description Hide DescriptionThe problem of a quantal harmonic oscillator with damping and a time‐dependent frequency acted on by a time‐dependent perturbative force is exactly solved. The wavefunctions are found in Schrödinger representation using the theory of explicitly time‐dependent invariants and also by an expansion of the Feynman propagator. The propagator is obtained in exactly closed form by an explicit path integration of the classical Lagrangian. It is found that the wavefunctions and the propagator depend only on the solution of classical damped oscillator through a single function ρ (t). The function ρ (t) itself may be obtained as a solution of a second order nonlinear differential equation under the appropriate set of initial conditions.

Explicit solution of the wave equation for arbitrary power potentials with application to charmonium spectroscopy
View Description Hide DescriptionWe present an explicit and almost complete series solution of the Schrödinger equation for an arbitrary quark‐confining power potential with or without a weak Coulomb component or other corrections. In particular, we derive two pairs of high‐energy asymptotic expansions of the bound‐state eigenfunctions together with a corresponding expansion of the eigenvalue determined by the secular equation. We also obtain a pair of uniformly convergent expansions and discuss other types of solutions. Various properties of the solutions and eigenvalues are examined including the scattering problem of the cutoff potential and the behavior of Regge trajectories. Finally, the relevance of these investigations to the spectroscopy of heavy quark composites is discussed. In particular, we derive approximate expressions for leptonic decay rates. Examples are given to demonstrate the usefulness of these results for theoretical discussion and as alternatives for numerical integration techniques. A subsequent paper will deal with the normalization of the bound‐state wavefunctions and the corresponding derivation of explicit series expressions for certain decay rates.

Solution of a quantum mechanical eigenvalue problem with long range potentials
View Description Hide DescriptionWavefunctions and eigenvalues for the Schrödinger equation on the half‐line x⩾0 are examined in the presence of a potential v _{0} x ^{−2}+v _{1} x ^{−1}+v _{2} x ^{−1/2}. With a special choice for the constant v _{0} the wave equation can be solved in terms of parabolic cylinder functions. In this case the spectrum is determined by an implicit equation that arises from the boundary condition that must be imposed at x=0. Depending on v _{1} and v _{2}, the spectrum can contain and infinite number of discrete values, a finite number, or none. It is pointed out that continuous variations in v _{1} or v _{2} can convert negative energy bound states into positive energy responances, or v i c e v e r s a, and the threshold behavior has been investigated.

A scale limit of φ^{2} in φ^{4} Euclidean field theory. I
View Description Hide DescriptionIt is shown that a scale limit (appropriately defined) of the Wick square of the free Euclidean field in d<4 dimensions with mass m _{0} exists and is a random field with values independent at every point. When m _{0}→0 a stable distribution is obtained. The same limit is then calculated in φ^{4} cutoff field theory. After taking the scale limit the regularization can be removed. The limit field is again independent at every point but with different density and mean different from zero. The interpretation of the results in lattice approximation is given. The problem of restoring the correlations between different points is considered as a perturbation around an independent‐value field.

Comments on certain divergence‐free tensor densities in a 3‐space
View Description Hide DescriptionIt is well known that a necessary and sufficient condition for the conformal flatness of a three‐dimensional pseudo‐Riemannian manifold can be expressed in terms of the vanishing of a third‐order tensor density concomitant of the metric which has contravariant valence 2. This was first discovered by Cotton in 1899. It is shown that Cotton’s tensor density is not the Euler–Lagrange expression corresponding to a scalar density built from one metric tensor. This tensor density is shown to be uniquely characterized by its conformal properties coupled with the demand that it be differentiable for arbitrary metrics.

Plane symmetric static fields in Brans–Dicke theory of gravitation
View Description Hide DescriptionAn explicit form of the relationship between g _{00} and the Brans–Dicke scalar potential φ in the interior of a perfect fluid with an equation of statep=ερ as well as in the matter free space, is obtained assuming functional dependence of g _{00} on φ. Plane symmetric static perfect fluids in Brans–Dicke theory of gravitation are discussed. Explicit solutions are also obtained for a fluid with ε=1/3, that is, for a disordered radiation and some of its properties studied.

Neutrino, Maxwell, and scalar fields in the cylindrical magnetic or plasm universe
View Description Hide DescriptionIn this paper we discuss the solutions of the c‐number quantum‐mechanical equations of neutrino,Maxwell, and scalar fields in the background metric of a static, cylindrically symmetric magnetic or plasm universe. The magnetic universe consists of a parallel magnetic field distribution held together by the gravitational field of its own energy and stress. It is an exact solution of the Einstein–Maxwell equations for the gravitational field associated with a stationary source‐free cylindrically symmetric magnetic field. The solution does not depend specifically upon the assumption that the seat of the energy‐stress distribution is an electromagnetic field. A ’’plasm of index 2’’ would have the same metric. The main instrument of the present perturbation theory, valid in all algebraically special spacetimes, is the Cohen–Kegeles (CK) scalar wave equation for test fields of helicity h (or s). This equation, written in the updated Geroch–Held–Penrose (GHP) formalism, is made explicit for the chosen null tetrad by determining the spin coefficients and the operators of that formalism. Exact solutions of the CK wave equation are obtained for the case of zero orbital angular momentum. For the case of nonzero orbital angular momentum the JWKB approximation is used to obtain the quadratures for the orbits in the magnetic universe. It is shown that the neutrino orbits precess about the magnetic (or plasm) universe axis in a manner different from orbits of particles with helicity zero and helicity one. The expression for the neutrino precessional frequency (due to nonvanishing helicity) is given explicitly. The photon precessional frequency is twice that for neutrinos. Finally, the CK procedure is compared with that of Teukolsky for finding the neutrino and Maxwell fields in type D spacetimes.

Expansions of the affinity, metric and geodesic equations in Fermi normal coordinates about a geodesic
View Description Hide DescriptionFermi normal coordinates about a geodesic form a natural coordinate system for the nonrotating geodesic (freely falling) observer. Expansions of the affinity, metric, and geodesic equations in these coordinates in powers of proper distance normal to the geodesic are calculated here to third order, fourth order, and third order, respectively. An iteration scheme for calculation to higher orders is also given. For generality, we compute the affinity and the geodesic equations in an arbitrary affine manifold, and compute the metric in a Riemannian manifold with arbitrary signature.

Euclidean and Minkowski space formulations of linearized gravitational potential in various gauges
View Description Hide DescriptionWe show that there exists a unitary map connecting linearized theories of gravitational potential in vacuum, formulated in various covariant gauges and noncovariant radiation gauge. The free Euclidean gravitational potentials in covariant gauges satisfy the Markov property of Nelson, but are nonreflexive. For the noncovariant radiation gauge, the corresponding Euclidean field is reflexive but it only satisfies the Markov property with respect to special half spaces. The Feynman–Kac–Nelson formula is established for the Euclidean gravitational potential in radiation gauge.

The classification of all H spaces admitting a Killing vector
View Description Hide DescriptionWe show that all H spaces (self‐dual solutions of the complex Einstein vacuum equations) that admit (at least) one Killing vector may be gauged in such a way as to be divided into only five types, characterized by the type of equation which determines their potential function. In four of these types we show that this knowledge is sufficient to reduce the requirement of being an H space to a linear equation whose solutions are well known. The fifth case is reduced considerably and a large class of special solutions is given.

The nondiverging and nontwisting type D electrovac solutions with λ
View Description Hide DescriptionAssuming the type D of the metric and the alignment of the EM field along the double D–P directions all solutions of Einstein–Maxwell equations free of complex expansion in the presence of cosmological constant are studied. All solutions of this type are found equivalent to Carter’s branch [ (−)] in a previous paper and are derivable from the general 7‐parametric D’s by contractions. Various special cases are examined, and at least a 4‐parameter group of symmetries of these solutions is exhibited and studied.

Bures distance and relative entropy
View Description Hide DescriptionWe have previously constructed an entropy functional which characterizes statistical inference from partial measurement by maximum relative entropy. Here we discuss the mathematical properties of this concept in greater detail and establish its relation to the Bures distance and the Uhlmann transition probability.

Singular potentials and analytic regularization in classical Yang–Mills equations
View Description Hide DescriptionThe class of instaton solutions with ’’extension’’ parameter λ^{2} positive is extended to λ^{2} negative. The nature of the singular sphere of radius ‖λ‖ is analyzed in the light of the analytical regularization method. This leads to well defined solutions of the Yang–Mills equations. Some of them are sourceless (’’±i o’’ and ’’V p’’), others correspond to currents concentrated on the sphere of singularity (’’+’’ and ’’−’’). Although the equations are nonlinear, the ’’V p’’ solutions turn out to be the real part of the ’’±i o’’ solutions. The ansatz of the t’Hooft for the superposition of instantons is used to sum the contributions corresponding to λ^{2} with positive and negative signs. A subsequent limiting process allows then the construction of solutions of the ’’multipole’’ type. The general situation of potentials having a denominator D, with a corresponding surface of singularity at D=0, is also considered in the same light.