Volume 16, Issue 3, March 1975
Index of content:

Tensor operators and twisted group algebras
View Description Hide DescriptionTwo generalizations of Frame’s theory of the conjugating representation of a finite group G are explored and applied to the problem of forming tensor operators out of a group algebra. In each case the group G acts on the group algebraA (Ḡ), where Ḡ either contains G or covers G as a central extension. By construction, an affirmative answer is given to the question, raised by de Vries and van Zanten, as to whether or not Ḡ may be found such that A (Ḡ) carries all the irreducible representations of G. Several examples are given, with the groups chosen from among the crystallographic point groups.

Vector potentials and physical optics
View Description Hide DescriptionThe general problem considered is to obtain solutions w to the vector equation v = curl(w), where v is a given divergence−free vector field with singularities. Two methods are discussed: A special method, which applies when v is of the type which occurs in the Kirchhoff theory of diffraction, and a general method, which applies to any divergence−free vector field whatever. As an example the general method is used to obtain the Maggi−Rubinowicz representations of the Kirchhoff−Helmholtz (double) integral as a line integral. The singularities of the solutions w are known to produce important optical effects, and the nature of these singularities is largely determined by topological properties of the domain on which v is regular.

Useful extremum principle for the variational calculation of matrix elements. II
View Description Hide DescriptionRecent work [Phys. Rev. A 9, 108 (1974)] on variational principles for diagonal bound state matrix elements of arbitrary Hermitian operators is extended. In particular, it is shown that the previously derived minimum principle for the trial auxiliary function appearing in such variational principles can be constructed using a modified Hamiltonian possessing not heretofore recognized positive definite properties. Thus there is at least one alternative to the particular modified Hamiltonian on which the results of Phys. Rev. A 9, 108 (1974) originally were based.

Perturbative−variational approximations to the spectral properties of semibounded Hilbert space operators, based on the moment problem with finite or diverging moments. Application to quantum mechanical systems
View Description Hide DescriptionWe introduce in a systematic way the properly defined arctangent of the mean value of the resolvent of a Hilbert space operator. We consider more precisely the case of semibounded self−adjoint operators H, in the region of the discrete spectrum. The arctangent of the Padé approximations to the mean value of the resolvent are constructed out of the moments. They are shown to provide converging monotonic increasing sequences of lower bounds to the arctangent of the resolvent. Consequently bounds with the same properties are derived for the discrete eigenvalues of H and, most remarkable, t h e o r d e r i n g of the corresponding poles of the approximations reproduces the ordering of the exact poles. The Padé method is shown to provide a way to fully exploit the content of the Rayleigh−Ritz variational method, by providing a simple mechanical procedure to build up the variational subspaces: It defines a powerful both perturbative and variational approximation to semibounded operators. The difficulties of the Ritz method in the degenerate case are overcome by the fact that all bounds in the Padé method are s t r i c t bounds. In a second part, we consider the important case in which the moments are given by diverging algorithms. By properly regularizing them, we show that the Padé−Ritz variational principle generalizes to produce absolute maxima of the arctangent of the Padé approximations in the regulator and that these maxima form monotonic converging sequences of lower bounds. In the last part, we discuss an application to quantum mechanical systems for which the perturbative variational method is applied to the e n e r g y, allowing us to treat the case of strong coupling. As a consequence it appears possible to solve (approximately) the R−dimensional anharmonic oscillator in a purely a l g e b r a i c way.

Static cylindrically symmetric solutions of the Einstein−Maxwell equations
View Description Hide DescriptionA general class of solutions of Einstein−Maxwell equations with static cylindrical symmetry is obtained. The equations are derived using canonical methods and the fields are shown to satisfy a certain Painlevé differential equation of the third transcendental type. A particular algebraic solution is studied in detail, and is found to have a certain mass and current on the axis and a helical magnetic field around it.

A spinor field theory on a seven−dimensional homogeneous space of the Poincaré group
View Description Hide DescriptionA field theory of half−integer spin particles is constructed on a seven−dimensional homogeneous space of the Poincaré group. The mass spectrum consists of nonparallel linear trajectories. The field theory has no spacelike or lightlike solutions. Electromagnetic form factors and structure functions of the theory are discussed.

Boundaries of spacetimes
View Description Hide DescriptionA review of the Schmidt technique and the Sachs technique for assigning boundary points to spacetimes is given. A modification of the Sachs process which makes it obviously identical with Schmidt’s is suggested. Some simple examples are discussed.

Spacetime symmetries and linearization stability of the Einstein equations. I
View Description Hide DescriptionWe consider the Marsden−Fischer conditions for linearization stability applied to vacuum spacetimes with compact Cauchy hypersurfaces. We show that if a vacuum spacetimeS admits a Killing vector field, then the Marsden−Fischer criterion fails to be satisfied at any Cauchy surface for S. We also show that if the Marsden−Fischer criterion fails to hold on an initial surface, then there is a Cauchy development of this intial data which admits one or more Killing vectors. The number of independent Killing fields present is shown to equal the dimension of the kernel of the linear map defined by Marsden and Fischer.

Lie theory and separation of variables. 6. The equation i U _{ t } + Δ_{2} U = 0
View Description Hide DescriptionThis paper constitutes a detailed study of the nine−parameter symmetry group of the time−dependent free particle Schrödinger equation in two space dimensions. It is shown that this equation separates in exactly 26 coordinate systems and that each system corresponds to an orbit consisting of a commuting pair of first− and second−order symmetry operators. The study yields a unified treatment of the (attractive and repulsive) harmonic oscillator, linear potential and free particle Hamiltonians in a time−dependent formalism. Use of representation theory for the symmetry group permits simple derivations of addition and expansion theorems relating various solutions of the Schrödinger equation, many of which are new.

Lie theory and separation of variables. 7. The harmonic oscillator in elliptic coordinates and Ince polynomials
View Description Hide DescriptionAs a continuation of Paper 6 we study the separable basis eigenfunctions and their relationships for the harmonic oscillator Hamiltonian in two space variables with special emphasis on products of Ince polynomials, the eigenfunctions obtained when one separates variables in elliptic coordinates. The overlaps connecting this basis to the polar and Cartesian coordinate bases are obtained by computing in a simpler Bargmann Hilbert space model of the problem. We also show that Ince polynomials are intimately connected with the representation theory of S U (2), the group responsible for the eigenvalue degeneracy of the oscillator Hamiltonian.

Stationary electrovacuum spacetimes with bifurcate horizons
View Description Hide DescriptionGeneral features of all stationary electrovacuum solutions of Einstein−Maxwell equations which contain regular bifurcate horizons are studied. A certain set of invariant quantities is found in whose values the full information about the solutions is recorded. The quantities have a simple physical meaning and generalize directly the ’’local invariants’’ defined for the axially symmetric case in the previous paper [J. Math. Phys. 15, 1554 (1974)]. A necessary condition that the solutions represent a neighborhood of a black hole in an asymptotically flat spacetime is given. The condition has the form of an inequality which places an upper bound on the magnitudes of the gravimagnetic, electric, and magnetic fields at the horizon. In the case of axial symmetry, the inequality reduces to that derived in the previous paper.

Static electrovacuum spacetimes with bifurcate horizons
View Description Hide DescriptionWe prove a necessary and sufficient condition for a stationary electrovacuum spacetime with a bifurcate horizon to be static. The condition is expressed by means of the invariant quantities introduced in the preceding paper.

Algebraic identities among U (n) infinitesimal generators
View Description Hide DescriptionSome algebraic identities among infinitesimal generators of the n−dimensional unitary group U (n) have been found. They satisfy a simple quadratic equation for degenerate representations. A generalization of Holstein−Primakoff boson realization for the U (n) group is also given.

Wave propagation with stochastically coupled propagating and evanescent modes
View Description Hide DescriptionThe problem of electromagnetic wave propagation in a randomly perturbed waveguide is analyzed in the forward scattering approximation. Both propagating and evanescent modes are taken into account. Coupled power equations are derived in the asymptotic limit of long guide length and small perturbations. More generally, a diffusion equation is derived which governs the evolution of functions of the process in this same asymptotic limit. The resulting dynamical equations characterize the evolution of the propagating modes; the evanescent modes affect this propagation through their modification of the parameters in these equations. However, in the presence of evanescent modes, the forward scattering approximation leads to nonconservative coupled power equations; energy is apparently exchanged with the neglected backward waves through their coupling to the evanescent modes.

Some remarks on the evolution of a Schrödinger particle in an attractive 1/r ^{2} potential
View Description Hide DescriptionComparing the different solutions of Case and Nelson for the evolution operators of a Schrödinger particle in the potential V (r) = −1/r ^{2}, we show that Nelson’s nonunitary solution is a simple average, over a physical parameter related to a boundary condition at the singularity, of Case’s family of solutions.

Time−dependent dynamical symmetries, associated constants of motion, and symmetry deformations of the Hamiltonian in classical particle systems
View Description Hide DescriptionA study is made of time−dependent dynamical symmetry mappings of Hamilton’s equations for classical particle systems. The conditions that an infinitesimal mapping (δx ^{ A }, δt) ≡ (ξ^{ A }(x,t) δa, ξ^{0}(x,t) δa), A = 1, ..., 2n, be a symmetry mapping are expressed in terms of a ’’symmetry vector’’ Z ^{ A }(x,t) ≡ ξ^{ A } − ξ^{0} η^{ A B }∂_{ B } H (x,t) (where η^{ A B } defines the symplectic matrix of phase space). These conditions imply that ξ^{0} is arbitrary. It is shown that the symmetry deformation of a constant of motion M (x,t) will also be a (’’derived’’) constant of motion (time−dependent related integral theorem). It follows for the case H = H (x) that every time−dependent symmetry deformation of H (x) is a constant of motion, and it is shown conversely that every constant of motion M (x,t) can be expressed as a symmetry deformation of the Hamiltonian, that is, there exists a symmetry vector Z ^{ A }(x,t) such that M = Z ^{ A }∂_{ A } H. It is found that if Z ^{ A }(≠0) is a symmetry vector, then M (x,t) Z ^{ A } will be a (scaled) symmetry vector if and only if M is a constant of motion. The existence of groups of symmetry vectors is considered, and it is shown that a complete set of r symmetry vectors Z ^{ A } _{α}, α = 1, ..., r, determines an r−parameter continuous group of symmetries. A special class of symmetry vectors Z ^{ A } _{(P)} (x,t) ≡ η^{ A B } (∂_{ B } M − N∂_{ B } H), (’’extended Poisson vectors’’), where M (x,t), N (x,t) are constants of motion is defined and conditions that such vectors determine a symmetry group are obtained. Poisson vectors are also used to show that the related integral theorem mentioned above may be considered as a generalization of Poisson’s theorem on constants of motion. Dependency relations between derived constants of motion with respect to vectors of a symmetry group are obtained.

Observables, operators, and complex numbers in the Dirac theory
View Description Hide DescriptionThe geometrical formulation of the Diractheory with spacetimealgebra is shown to be equivalent to the usual matrix formalism. Imaginary numbers in the Diractheory are shown to be related to the spin tensor. The relation of observables to operators and the wavefunction is analyzed in detail and compared with some purportedly general principles of quantum mechanics. An exact formulation of Larmor and Thomas precessions in the Diractheory is given for the first time. Finally, some basic relations among local observables in the nonrelativistic limit are determined.

Consistency in the formulation of the Dirac, Pauli, and Schrödinger theories
View Description Hide DescriptionProperties of observables in the Pauli and Schrödinger theories and first order relativistic approximations to them are d e r i v e d from the Dirac theory. They are found to be i n c o n s i s t e n t with customary interpretations in many respects. For example, failure to identify the ’’Darwin term’’ as the s−state spin−orbit energy in conventional treatments of the hydrogen atom is traced to a failure to distinguish between charge and momentum flow in the theory. Consistency with the Dirac theory is shown to imply that the Schrödinger equation describes not a spinless particle as universally assumed, but a particle in a spin eigenstate. The bearing of spin on the interpretation of the Schrödinger theory is discussed. Conservation laws of the Dirac theory are formulated in terms of relative variables, and used to derive virial theorems and the corresponding conservation laws in the Pauli−Schrödinger theory.

On Chapman−Kolmogorov equations for neutron population in a multiplying assembly
View Description Hide DescriptionWe prove that the Chapman−Kolmogorov system for neutron population in a multiplying assembly admits a unique positive and norm invariant solution belonging to the Banach space of summable sequences. We then show that the standard kinetic equation can be deduced in a rigorous way from such a system of a countably infinite number of partial differential equations. Finally, we indicate how the Chapman−Kolmogorov initial−value problem can be approximated by a problem in a finite−dimensional space.

Wave propagation in media undergoing uniform linear acceleration
View Description Hide DescriptionWave propagation in a homogenous material undergoing uniform linear acceleration is considered. The covariant constitutive equations for a holohedral, nondispersive, dielectric material derived by Lianis are applied. Expressions are found for initially plane waves propagating in the direction of acceleratedmotion for the case where the material functions are constant. Expressions are also found for the radiation reflected and refracted at the surface of a half−space undergoing uniform linear acceleration. A detailed energy balance across this interface is also presented.