Volume 22, Issue 5, May 1981
Index of content:

Matrices, differential operators, and polynomials
View Description Hide DescriptionIt is noted that the matrix Z, of order n, defined in terms of the n arbitrary numbers x _{ j } by the formula Z _{ j k } = δ_{ j k } J′_{ l = 1,η} (x _{ j }−x _{ l })^{−1} +(1−δ_{ j k })(x _{ j }−x _{ k })^{−1}, may be considered (in an appropriate framework) to correspond to the differential operator d/d x. There follow prescriptions to construct explicit matrices of (arbitrary) order n in terms of n (or more) arbitrary numbers or of the n roots of given polynomials, matrices whose eigenvalues (and eigenvectors) are given, fully or in part, by very simple formulas. Novel representations of the classical polynomials (Hermite, Laguerre, Lagrange, Gegenbauer, Jacobi) are also obtained, such as the formula for Hermite polynomialsH _{ n }(x) = 2^{ n } det[xI−H(φ)], where I is the unit matrix (of order n) and the matrix H(φ), of order n, is defined by H _{ j k }(φ) = (2n)^{−1/2}(δ_{ j k }(n−1)[exp(2iϑ_{ j }) + 1/2 exp(−2iϑ_{ j })] +(1−δ_{ j k }) {−exp(2iϑ_{ j }) +2i sin(ϑ_{ j }−ϑ_{ k })]^{−1} exp[−i(ϑ_{ j }+ϑ _{ k })]}), with ϑ_{ j } = φ+πj/n, φ arbitrary.

Explicit evaluation of the representation functions of IU(n)
View Description Hide DescriptionAll representation functions of IU(n) have been found in explicitly closed form. They are obtained through the contraction of U(n+1) or U(n,1). These expressions are closely connected with the generalized beta functions of Gel’fand and Graev.

Super Lie groups: global topology and local structure
View Description Hide DescriptionA general mathematical framework for the super Lie groups of supersymmetric theories is presented. The definition of super Lie group is given in terms of supermanifolds, and two theorems (analogous to theorems in classical Lie grouptheory) are proved. The relationship of the super Lie groups defined here to the formal groups of Berezin and Kac and the graded Lie groups of Kostant is analyzed.

The Lorentz group in the oscillator realization. III. The group SO(3,1)
View Description Hide DescriptionEmploying the boson operators of Barut and Böhm, we study the oscillator realization of the Lie algebra of the Lorentz group SO(3,1) in the coordinate representation. The construction yields a direct sum of the principal series of representations ( j _{0},ρ) belonging to the integral or half‐integral class. The decomposition of the representation space into the eigenspacesL ^{2} _{ j 0ρ} of irreducible representations leads to a two‐variable second order realization of the SO(3,1) algebra acting on f _{ j 0 ρ} ∈L ^{2} _{ j 0ρ}. The construction is shown to be highly symmetric. While the elements of the SO(2,1) subalgebra are invariant under the pseudorotation group SO(2,2), those of the full SO(3,1) algebra are invariant under the SO(2)×SO(1,1) subgroup of SO(2,2). We use this intrinsic symmetry in the construction to identify the generalized SO(2,1)⊆SO(3,1) eigenbases with the SO(2,2) harmonics in an SO(2)×SO(1,1) basis, and thereby achieve a significant unification among results which would normally appear disconnected.

Partial‐range completeness and existence of solutions to two‐way diffusion equations
View Description Hide DescriptionSeparating variables to solve a two‐way diffusion equation leads to a nonstandard eigenvalue problem in one variable. It is shown that the eigenfunctions having negative eigenvalues are complete on the part of the domain where initial conditions are imposed, while those with positive eigenvalues are complete where final conditions are imposed. The corresponding exponentially growing and exponentially decaying solutions may be used to expand arbitrary solutions on semi‐infinite intervals in the ’’time’’ variable. A natural iterative procedure for obtaining solutions on finite intervals is shown to converge. In some cases a linearly growing solution must also be taken into account.

Bäcklund transformations for the Ernst equation
View Description Hide DescriptionPresented is a systematic approach to the transformation theories for the Ernst equation from the viewpoint of the Bäcklund transformation. It is explicitly shown that the method of Clairin gives a simple derivation of various transformations such as transformations found by Ehlers, Neugebauer, and Harrison.

Inverse scattering connections
View Description Hide DescriptionWe establish and study a transformation which connects the Schroedinger, the Klein‐Gordon, and the Dirac operators. This provides an equivalence between their associated direct and inverse spectral transforms.

Symmetries and the Dirac equation
View Description Hide DescriptionA new class of symmetries are given for the Dirac equation without external fields. We consider the two cases of massive and massless particles.

Some spectral properties in algebras of unbounded operators
View Description Hide DescriptionSome aspects of spectral theory in algebras of unbounded operators are studied. After having pointed out the pathologies of the spectral behavior of these operators we give a sufficient condition in order that a self‐adjoint operator admit a spectral decomposition with spectral measure with values in the same algebra. Some examples illustrating the developed ideas are given.

On Euler characteristics of compact Einstein 4‐manifolds of metric signature (++−−)
View Description Hide DescriptionIn the present paper, the Euler characteristic is studied for a compact oriented Einstein 4‐manifold of signature (++−−). For such a manifold, there are three types of normal forms of the curvature tensor. It is shown that for each type the Euler characteristic is nonnegative and even. Several new inequalities are obtained concerning the Euler characteristic and the volume of the manifold. The second Betti number is even, and if it is zero, then the first Betti number also vanishes. The arguments developed here are based upon the famous work of Chern about the Gauss–Bonnet formula for a pseudo‐Riemannian manifold.

On the substructure of the classical observables x ^{α}, p _{α}, and J ^{αβ}
View Description Hide DescriptionThe position, momentum, and angular momentum (spin + orbital) of a classical massive magnetic dipole particle are constructed from certain pairs of O(3,3) spinors and a scalar σ. These spinor pairs (and also σ) are endowed with a translation transformation law (which is fundamentally different from that of twistors), and are given the name hyperspinors. An action of the covering group of the Poincaré group is defined on hyperspinors and σ. Equations of motion for these hyperspinors and σ are proposed, special cases of which lead to the Lorentz force law for the momentum and the BMT (Bargmann, Michel, and Telegdi) equation for the Pauli–Lubanski pseudovector. A generalization to include SU(N) internal degrees of freedom in this model is suggested.

Geometrical spacetime perturbation theory: Regular higher order structures
View Description Hide DescriptionA coordinate‐independent formulation of spacetime perturbation‐theory is extended beyond the first‐order. Higher‐order analogs of the second fundamental form (first metric variation) and corresponding higher‐order projection identities lead to higher‐order perturbation equations for the spacetime metric fields coupled to spacetime. A deformation‐geometrical vertex functional or Frechet derivative is introduced and used to express deformation‐covariant perturbation equations in terms of action functionals. The deformation‐covariant vertex functionals of the Einstein–Hilbert action functional are computed to fourth order (sufficient for third‐order perturbation equations).

Note on Finslerian relativity
View Description Hide DescriptionFinslerian structure of spacetime is investigated. For a special type of generalized Finsler metric the explicit expression of Cartan‐like metrical connection is derived and it is shown that it resembles the usual one. Causal problems in Finsler‐type spacetime are discussed and, based on the arguments, the Einstein‐type equations for the Finslerian quantities are derived by using the lifting of a Finsler metric to a tangent bundle. It is shown that a solution of the proposed equations can also be obtained from the complex structure of the tangent bundle. One‐form type metrics are used to discuss the geometrical interpretation of isosymmetry. A simple way of obtaining a metrical connection for a general generalized Finsler metric is given.

Existence of localized solutions for certain model field theories
View Description Hide DescriptionWe study the existence of static solutions of certain modelequations. In particular we consider coupled systems of equations and allow the coefficients to be variable, even singular.

Quantum mechanics and stochastic control theory
View Description Hide DescriptionA time‐symmetric stochastic control theory is proposed as one of the representatives of quantum mechanics. The main idea is based on Nelson’s probability theoretical approach to quantum mechanics. His approach is reformulated as a time‐symmetric stochastic control problem. Several different control constraints equivalent to Nelson’s are obtained. One of them has a close connection to the Lagrangian formalism of classical mechanics. This suggests to us the use of stochastic calculus of variations. Within the realm of this time‐symmetric stochastic control theory it is shown why Schrödinger’s original variational method of quantization was successful. Several advantageous points of the stochastic control theoretical approach to quantum mechanics, including the analysis of the classical limit, are also discussed.

A multidimensional extension of the combinatorics function technique. I. Linear and homogeneous partial difference equations
View Description Hide DescriptionThis paper is aimed at series solutions of physical phenomena that are described by linear homogeneous differential equations, like the Schrödinger differential equation. Series solutions to such equations, when they exist, lead to multidimensional, linear, and homogeneous recurrence relations among the expansion coefficients. The physical constraints imposed on the solutions of an ordinary differential equation (in the case of the Schrödinger equation that would be on the wavefunctions) then lead to a set of ’’initial values’’ on the expansion coefficients. The consistency of the initial values with the recurrence relation or partial difference equation(PDE), is one of the major problems in such cases. Until now, there was no systematic way of obtaining the solution of a PDE in terms of the initial values, and no systematic technique dealing with the consistency check. In this paper, we have been able to solve both of these problems by a natural extension of the combinatorics function technique developed by Antippa and Phares for one‐dimensional linear recurrence relations.

A multidimensional extension of the combinatorics function technique. II. Linear and inhomogeneous partial difference equations
View Description Hide DescriptionRecently the solution of multidimensional, linear, and homogeneous recurrence relations, or partial difference equations(PDE), was obtained via a multidimensional extension of the combinatorics function technique, developed by Antippa and Phares. Combinatorics functions of the first and second kind are representations of ’’restricted’’ paths connecting two points in an n‐dimensional space. These functions are shown to give the solution of the most general linear and inhomogeneous PDE. The consistency of the PDE with the initial value conditions is also discussed. Applications of the method are given elsewhere.

Schrödinger operators with l ^{ l/2} _{ w }(R^{ l }) ‐potentials
View Description Hide DescriptionWe consider the Schrödinger operators −Δ+V with V in the weak L ^{ q } class, with q = 1/2 the underlying dimension, which is the borderline for a definition to be possible. We concentrate first on optimal bounds on how large the weak norm of V can be and then on spectral properties on L ^{ p } spaces.

Variable‐phase approach to off‐shell scattering by nonlocal potentials
View Description Hide DescriptionThe basic ideas of the variable‐phase approach are used to develop first‐order differential equations for the quasiphase parameters which describe the half‐off‐shell scattering amplitudes for a nonlocal potential. The on‐shell equation of Calogero and Sobel’s equation for a local potential are obtained as special cases.

Bound on the diffraction peak for the spin 0–spin 1/2 case
View Description Hide DescriptionWe apply a recently developed method by the author to the problem of the diffraction peak bound for the spin 0–spin 1/2 particle scattering with the spin taken fully into account. The variational technique is used with the constraints, the total cross section, elastic cross section, and the unitarity of partial waves.