Volume 20, Issue 11, November 1979
Index of content:

A new method of matrix transformation. I. Matrix diagonalizations via involutional transformations
View Description Hide DescriptionIt is shown that two matrices A and B of order n×n which satisfy a monic quadratic equation with two roots λ_{1} and λ_{2} are connected by A T _{ A B }=T _{ A B } B where T _{ A B }=A+B−(λ_{1}+λ_{2}) I with I being the n×n unit matrix (Theorem 1). The condition for T _{ A B } to be involutional is that the anticommutator of ?=A−(1/2)(λ_{1}+λ_{2}) I and ?=B−(1/2)(λ_{1}+λ_{2}) is a c number (Theorem 2). A 2m×2m matrix Q ^{(2m)} is introduced as a typical form of a matrix which can be diagonalized by an involutional transformation. These theorems are further extended through the matrix representation of the group of the general homogeneous linear transformations, GL(n). IUH (involutional, unitary, and Hermitian) matrices are introduced and discussed. The involutional transformations are shown to play a fundamental role in the transformations of Dirac’s Hamiltonian and of the field Hamiltonians which are quadratic in particle creation and annihilation operators in solid state physics.

A new method of matrix transformations. II. General theory of matrix diagonalizations via reduced characteristic equations and its application to angular momentum coupling
View Description Hide DescriptionA general formalism is given to construct a transformation matrix which connects two matrices A and B of order n×n satisfying any given polynomialequation of degree r, p ^{(r)}(x) =0; r?n. The transformation matrix T _{ A B } is explicitly given by a polynomial of degree (r−1) in A and B based on p ^{(r)}(x). A special case where B is a diagonal matrix Λ equivalent to A leads to the general theory of matrix diagonalizations with the transformation matrix T _{ AΛ}, which can be made nonsingular with a proper choice of Λ. In another special case where B is a constant matrix with the constant being a simple root λ_{ν} of p ^{(r)}(x), the transformation matrix T _{ A B } reduces to the idempotent matrix P _{ν} belonging to the eigenvalue λ_{ν} of A. Based on the relation which exists between T _{ AΛ} and P _{ν}, one can construct a transformation matrix U which is more effective than T _{ AΛ} and becomes unitary when A is Hermitian. Illustrative examples of the formalism are given for the problem of angular momentum coupling.

Representation groups for semiunitary projective representations of finite groups
View Description Hide DescriptionGiven a finite group G and a subgroup K of index 1 or 2, a method is developed for finding all the finite groups ? (up to equivalence) such that any semiunitary projective representation of G can be lifted to a semiunitary representation of ?. The method is used in simple but interesting groups in physics.

Micu‐type invariants of simple Lie algebras
View Description Hide DescriptionAn algorithm is presented for computing the eigenvalues of Micu‐type invariants for Lie algebrasA _{ r }, B _{ r }, C _{ r }, and D _{ r }. Inasmuch as the definitions of these invariants is recursive and not in terms of a closed formula, only a recursion relation is obtained for computation of their eigenvalues, which is easily programmable on a computer. Some of the formulas developed here are valid in arbitrary complex semisimple Lie algebras and may be useful in other contexts.

Quantum dynamics of Hamiltonians perturbed by pulses
View Description Hide DescriptionThere has been a recent controversy between Gzyl [J. Math Phys. 18, 1327 (1977)] and Blume [J. Math. Phys. 19, 2004 (1978)] concerning the correct way to integrate the equationi∂ψ/∂t=H _{0}ψ+δ (t) Vψ where δ is the Dirac delta function. In this paper we: (1) Suggest some physical reasons for rejecting Gzyl’s procedure and (2) Extend the limiting procedure suggested the Blume to unbounded Hamiltonians.

On the asymptotic behavior of the derivatives of Airy functions
View Description Hide DescriptionWe present a noniterative functional solution to the three‐term recursion relation satisfied by the higher order derivatives of the Airy function. This solution allows one to obtain the asymptotic behaviors of these derivatives for large argument and also for large order.

Schwinger equations
View Description Hide DescriptionA compact description of the Schwinger equation is given and its different transformations and projections are considered.

An application of the maximum principle to the study of essential self‐adjointness of Dirac operators. I
View Description Hide DescriptionWe formulate a theorem saying that the Dirac operator corresponding to a one‐electron atomic ion is essentially self‐adjoint on the usual domain, provided that the nuclear charge Z is less than 118. Furthermore, for such nuclear charges the domains of the closure of the free particle and total Dirac operators are equal. In the present part I of this paper we prove this theorem for the part of the operator over each of the usual reducing subspaces.

Cyclic pursuit in a plane
View Description Hide DescriptionThe motion of an arbitrary set of points (or bugs) in a plane chasing one another in cyclic pursuit is studied. It is shown that for regular center‐symmetric configurations, analytic solutions are easily obtained by going to an appropriate rotating frame of reference. A few cases of nonsymmetric configurations are discussed. In particular, it is shown that for three bugs in a triangular configuration, the center of the rotating coordinate system relative to which the bugs have no tangential velocity is the point of collapse and coincides with one of the two Brocard points of the triangle. For the case when all the bugs have the same speed, a theorem is proved that whenever a premature (i.e., nonmutual) capture occurs, the collision must be head on. This theorem is then applied to the case of three and four‐bug configurations to show that these systems collapse to a point, i.e., the capture is mutual. Some aspects of these results are generalized to the case of the n‐bug systems.

Stability and boundedness results of stochastic differential equations relative to a generalized norm
View Description Hide DescriptionWe consider results previously obtained in the context of a generalized norm. In particular, we develop a general comparison principle which permits us to give sufficient conditions for conditional stability and boundedness in the mean relative to a generalized norm. Finally, we provide a decomposition technique together with additional comments which demonstrate the applicability of our results.

Some applications of time‐dependent canonical transformations to nonlinear nonconservative classical systems
View Description Hide DescriptionThe recently proposed time‐dependent canonical transformation method of Leach for the quadratic Hamiltonian has been extended to deal with nonlinear nonconservative classical systems. It is observed that the linear time dependent canonical transformations are not adequate to remove the time dependence of any arbitrary time‐dependent nonlinear Hamiltonian. Alternatively, we propose that such a Hamiltonian may be transformed to a quadratic form by means of successive nonlinear canonical transformations. It is also shown that the canonical method is useful to obtain solutions for differential equations governing certain dissipative classical systems.

Invariant *‐product quantization of the one‐dimensional Kepler problem
View Description Hide DescriptionIt is shown that, with proper interpretation, phase space trajectories for the one‐dimensional Kepler problem retain their significance after quantization. The classical and quantum theories are both entirely conventional; only the Wigner correspondence (Weyl quantization rule) is not. The usual quantum theory may be obtained by so(2,1) ‐invariant *‐product quantization (generalized Moyal mechanics). [The analog in three dimensions is so(4,2) invariant *‐quantization; it is believed that this should encounter no new difficulties of principle.] New results concerning invariant *‐products on polynomials, in the case of so(2,1), are presented. The Kepler problem is the first known example of a nonanalytic *‐representation. Piecewise analytic *‐representations are defined and are shown to provide a general framework for invariant quantization on singular orbits of semisimple groups. When piecewise analytic *‐representations are allowed, then the specification of an invariant *‐product on polynomials is no longer sufficient to determine a unique quantum theory.

Explicit normalization of bound‐state wave functions and the calculation of decay widths
View Description Hide DescriptionWe present a general method for normalizing nontrivial bound‐state wave functions. In particular, we consider explicit and formally complete asymptotic expansions of the solutions of the wave equation for an arbitrary quark confining power potential. The regular solution is identified and shown to be continuable to an eigensolution around a local minimum of the associated potential. The solutions are then normalized and used to derive explicit series expressions for certain decay rates.

Quantum theory of infinite component fields
View Description Hide DescriptionThe quantum theory of the infinite component SO(4,2) fields is formulated as a model for relativistic composite objects. We discuss three classes (timelike, lightlike, and spacelike) of physical solutions to a general class of infinite component wave equations. These solutions provide a definite physical interpretation to infinite component wave equations and are obtained by reducing SO(4,2) with respect to its orthogonal, pseudo‐orthogonal, and Euclidean subgroups. The analytic continuations among these solutions are established. In the nonrelativistic limit the timelike physical states exactly reduce to the Schrödinger solution for the hydrogen atom—the simplest composite object. The wave equations for the three classes are studied in two different realizations. In one case the equations describe a three‐dimensional internal Kepler motion with a discrete and a continuous energy spectrum and in the other case the equations describe a four‐dimensional internal oscillatory motion with attractive as well as repulsive potentials. It is found that the Kustaanheimo–Steifel transformation of classical mechanics exactly relates these two internal motions also in the quantum case. Thus a completely relativistic theory of composite systems is established for which the internal dynamics is the generalization of the nonrelativistic two‐body dynamics.

Borel multipliers for the Bondi–Metzner–Sachs group
View Description Hide DescriptionIt is shown that there exist only two cohomology classes of Borel multipliers for the Bondi–Metzner–Sachs group B when the subgroup of ’’supertranslations’’ is the additive group of a separable real Hilbert space. This implies that all Borel multipliers for the universal covering group ? of B are coboundaries and so every continuous unitary projective representation of B can be obtained from a continuous unitary (ordinary) representation of ?.

Deformations of gauge groups. Gravitation
View Description Hide DescriptionThis is a review, and an attempt at completion, of the ’’Gupta program,’’ the ultimate goal of which is e i t h e r to show that Einstein’s theory of gravitation is the only self‐consistent field theory of interacting, massless, spin‐2 particles in flat space o r to discover interesting alternatives. It is useful to notice that the gauge group of general relativity is a deformation (in a mathematically precise sense) of the gauge group associated with the massless, spin‐2 free field. The uniqueness of Einstein’s theory depends on the stability of its gauge group with respect to a class of differentiable deformations. A generalized Gupta program for massless fields of arbitrary spins is proposed.

Stationary charged C‐metric
View Description Hide DescriptionThe physical properties of the stationary charged C‐metric are fully investigated. Tetrad components of curvature and Maxwelltensors are studied in a Bondi–Sachs coordinate system to show that this solution represents a uniformly accelerating and rotating charged particle with magnetic monopole and NUT parameter. The physical quantities—news function, mass loss, mass, charge, and multipole moments—are calculated. The different mass loss definitions are compared and it is shown that the definition given by Bondi is the more appropriate one. It is also shown that the magnetic monopole in the presence of rotation affects the electric charge.

Generalized static electromagnetic fields in Brans–Dicke theory
View Description Hide DescriptionA general class of static, axially symmetric solutions of the Brans–Dicke–Maxwell equations is obtained under the assumption −r ^{2}⋅g _{ r r }=g _{ t t }⋅g _{φφ}⋅g _{ z z } These solutions have been subjected to conformal transformation and they are found to be new solutions of the static, axially symmetric Einstein–Maxwell–scalar fields. We have also developed a more general set of solutions under the unit transformations given by Morganstern.

Dissipative operators for infinite classical systems and equilibrium
View Description Hide DescriptionDissipative differential operators for infinite classical systems are characterized and used to obtain correlation inequalities for equilibrium states.

One particle reduced density matrix of impenetrable bosons in one dimension at zero temperature
View Description Hide DescriptionWe compute exactly the one particle reduced density matrix ρ (r) of a system of impenetrable bosons in one dimension at zero temperature. We do this by relating ρ (r) to a certain double scaling limit of the transverse correlation function of the one‐dimensional spin 1/2 X–Ymodel. We study the asymptotic behavior of ρ (r) for large r. This expansion contains oscillatory terms which arise due to the intrinsic quantum mechanical nature of the problem. We use these results to discuss the analytic structure of the momentum density function n (k).