Volume 23, Issue 7, July 1982
Index of content:

Determination of basis for the irreducible representations of the unitary group for U( p+q)↓U( p)⊗U( q)
View Description Hide DescriptionA direct procedure is outlined for determining the basis spanning finite dimensional irreducible representations of U( p+q) adapted to the subgroup U( p)⊗U( q). Using a tableau based analysis, it is shown that the realization of the semimaximal states follows readily from a knowledge of the matrix elements of the generators E _{ i+1} of U( p), U( q)⊆U( p+q).

Branching rules for the supergroup SU(N/M) from those of SU(N+M)
View Description Hide DescriptionThe decomposition of representations of supergroups into representations of subgroups is needed in practical applications. In this paper we set up and exploit a fruitful one‐to‐one correspondence between the Lie group branching SU (N+M)⊇SU(N)⊗SU(M)⊗U(1) and the supergroup branchings SU(N/M)⊇SU(N)⊗SU(M)⊗U(1) and SU(N _{1}+N _{2}/M _{1}+M _{2})⊇SU(N _{1}/M _{1}) ⊗SU(N _{2}/M _{2})⊗U(1). A simple and useful prescription is discovered for obtaining the SU(N/M) branching rules from those of SU(N+M) for any representation. A large class of examples, sufficient for many physical applications we can foresee, are explicitly worked out and tabulated.

Spectral theory for the periodic sine‐Gordon equation: A concrete viewpoint
View Description Hide DescriptionA summary of the spectral theory for quasiperiodic sine‐ and sinh‐Gordon equations is given. Analogies with whole‐line solitons and scattering theory motivates the discussion. The relation between the ingredients in the inverse spectral solution of the periodic sine‐Gordon equation and physical characteristics of sine‐Gordon waves is emphasized. The explicit topics covered are summarized in the table of contents in the Introduction.

New sum rule for products of Bessel functions with application to plasma physics
View Description Hide DescriptionIn our investigations of the linear theory of the stability of relativistic beam‐plasma systems immersed in a magnetic field we have been led to consider sum rules for an infinite series of products of Bessel functions of the form J^{∞} _{ n = −∞}(n ^{ j } J ^{2} _{ n })/ (n+μ). In this work we report on the sum of this series treated as a special case of a more general infinite series. We also mention the extension of the results beyond the range of the parameters for which formulae are explicitly given and indicate how intermediate results obtained may be useful in their own right. Finally, an additional application of our result is indicated.

Some algebraic, geometric, and system‐theoretic properties of the Special Functions of mathematical physics
View Description Hide DescriptionIt is known that many of the Special Functions of mathematical physics appear as matrix elements of Lie group representations. This paper is concerned with a beginning attack on the converse problem, i.e., finding conditions that a given function be a matrix element. The methods used are based on a combination of ideas from system theory,functional analysis, Lie theory, differential algebra, and linear ordinary differential equationtheory. A key idea is to attach a s y m b o l as an element of a commutative algebra. In favorable cases, this symbol defines a Riemann surface, and a meromorphic differential form on that surface. The topological and analytical invariants attached to this form play a key role in system theory. The Lie algebras of the groups appear as linear differential operators on this Riemann surface. Finally, it is shown how the Picard‐Vessiot‐Infeld‐Hull theory of factorization of linear differential operators leads to realization of many Special Functions as matrix representations of group representations.

New similarity solutions for the Ernst equations with electromagnetic fields
View Description Hide DescriptionA new class of exact similarity solutions is found for the Ernst equations with electromagnetic fields. The original coupled nonlinear partial differential equations are reduced to a system of coupled nonlinear ordinary differential equations. The reduced system is solvable in a manner identical to previous similarity solutions found by Kaliappan and Lakshmanan. These solutions may be considered the extension of the Curzon solution (static, uncharged) to the stationary, charged solution.

Quantization as a consequence of the symmetry group: An approach to geometric quantization
View Description Hide DescriptionA method is proposed to obtain the dynamics of a system which only makes use of the group law. It incorporates many features of the traditional geometric quantization program as well as the possibility of obtaining the classical dynamics: The classical or quantum character of the theory is related to the choice of the group, avoiding thus the need of quantizing preexisting classical systems and providing a group connection between the quantum and classical systems, i.e., the classical limit. The method is applied to the free‐particle dynamics and the harmonic oscillator.

Joint distributions, quantum correlations, and commuting observables
View Description Hide DescriptionWe provide necessary and sufficient conditions for several observables to have a joint distribution. When applied to the bivalent observables of a quantum correlation experiment, we show that these conditions are equivalent to the Bell inequalities, and also to the existence of deterministic hidden variables. We connect the no‐hidden‐variables theorem of Kochen and Specker to these conditions for joint distributions. We conclude with a new theorem linking joint distributions and commuting observables, and show how violations of the Bell inequalities correspond to violations of commutativity, as in the theorem.

Theory of nonbijective canonical transformations in mechanics: Application to the Coulomb problem
View Description Hide DescriptionIt is shown that the study of nonbijective transformation requires a fiber bundle formulation of mechanics. The conditions upon which nonbijective canonical point transformations can be defined are given. Then, as an example, we apply that theory to the study of the Coulomb problem in two and three dimensions. The Hopf fibration leads to an inverse harmonic oscillator problem. Since the completion of this work, a paper by G. H. Ringwood and J. T. Devreese has been published in J. Math. Phys. 21, 1390 (1980), dealing with the same problem. Their work is based on the construction of propagators in quotient spaces. The identity between the propagator prescriptions and nonlinear canonical transformations is not automatically fulfilled. Therefore it seems that the reliability of their results is not due to their method, which in general is not correct, but to an underlying property of the transformation used, namely the Kustaanheimo‐Stiefel map (see our results).

The quartic anharmonic oscillator in stochastic electrodynamics
View Description Hide DescriptionThe case of a slightly anharmonic oscillator (with a βx ^{4} perturbing potential) is examined in the framework of stochastic electrodynamics (SED) in full detail. We obtain the stationary probability density and the mean energy, which differs from the quantum result at order β^{2}. Using Kubo’s linear response theory we obtain the absorption curve: the maximum absorption frequencies do not coincide with the quantum transition frequencies. From the calculation of the emission energy we show that the ’’radiation balance’’ is n o t exactly satisfied as soon as β≠0, a property which disagrees with the quantum results. Finally, we discuss the consequences of this lack of radiation balance concerning Kirchhoff’s law.

Continued fraction theory of the rotating harmonic oscillator
View Description Hide DescriptionWe study the rotating‐vibrating system, consisting of the rotating harmonic oscillator, using the analytic theory of continued fractions. We prove that there is a convergent continued fraction representation of the Green’s function which is analytic in the complex coupling constant plane, except for a cut along the negative real axis. The perturbation series for the Green’s function is unambiguously defined by the continued fraction but diverges on account of an essential singularity at the origin. An infinite but incomplete set of exact solutions for certain specific valeus of the coupling follows from the representation of the Green’s function as a continued fraction. Finally, we use Worpitzky’s theorem in continued fraction theory to show that in the strong coupling limit α→0^{+} (α being the inverse of the coupling parameter), there exists a lower bound to all energy eigenvalues for a given value of l, the orbital angular momentum.

An E_{6}⊗U(1) invariant quantum mechanics for a Jordan pair
View Description Hide DescriptionQuantum mechanical spaces associated with geometries based on exceptional groups are of interest as models for internal (quark) symmetries. Using the concept of a Jordan pair, two copies of complex 3×3 octonionic Jordan algebras (M^{8} _{3}) are shown to define a quantum mechanics over the complex octonionic plane having E_{6}⊗U(1) as automorphism group. The unusual features of this new quantal structure (neither a projective geometry, nor a lattice) are discussed.

Linear invariants of a time‐dependent quantal oscillator
View Description Hide DescriptionExplicitly time‐dependent invariants, linear in x and p, are found to simplify the solution of the Schrödinger equation with oscillator‐type Hamiltonian. As an example, exact dynamics is obtained of the time‐dependent quantal oscillator, with damping and subject to external force. The invariant and hence the solution of the Schrödinger equation involve only the amplitude of the classical damped oscillator. The identity of our solution with a recently obtained result is illustrated. A digressive remark on quadratic invariants and the Dirac operator formalism is added.

HH spaces with an algebraically degenerate right side
View Description Hide DescriptionWe find the most general form of the key function for an HH space whose curvature on the right side is algebraically degenerate.

The investigation of some self‐similar solutions of Einstein’s equations
View Description Hide DescriptionNew application of self‐similar generalizations of homogeneous cosmological models is shown in the problem of nonstationary nonspherically‐symmetric accretion of self‐gravitating gas on the center. For self‐similar Bianchi types II and III, solutions of the system of Einstein equations are reduced to some dynamical system of small order. A number of exact solutions in empty space and for the stiff equation of state of matter is found.

Electrovac generalization of Neugebauer’s N = 2 solution of the Einstein vacuum field equations
View Description Hide DescriptionWe show that the N = 2 Neugebauer solution of the Einstein vacuum field equations is easily reproduced by employing two successive Kinnersley–Chitre (K–C) transformations of a type considered earlier by I. Hauser. Furthermore, by employing two successive K–C transformations of a type considered recently by C. Cosgrove we are able to produce a new electrovac generalization of the N = 2 Neugebauer solution. In principle, an analogous approach could be employed for the explicit construction of electrovac generalizations of Neugebauer solutions corresponding to higher values of N.

New exact static solutions to Einstein’s equations for spherically symmetric perfect fluid distributions
View Description Hide DescriptionNew exact solutions to Einstein’s equations are given which are spherically symmetric and static with perfect fluid distributions satisfying a linear equation of state p = nρ and n∈(0,1]. Heintzmann’s generating method is then used to build up a family of new solutions for each value of n.

Static gravitational fields in a general class of scalar–tensor theories
View Description Hide DescriptionThe static field equations are investigated within the framework of a general class of scalar‐tensor theories of gravitation proposed by Nordtvedt. In the Brans‐Dicke and Barker theories, it is shown that in vacuum g _{00} is functionally related to the scaler field. Two families of exact solutions are also obtained for a static spherically symmetric gravitational field in Barker theory.

Path integrals and stationary phase approximation for quantum dynamical semigroups. Quadratic systems
View Description Hide DescriptionThe solution of the Markovian master equation for the quantum open system of n degrees of freedom is formally written in terms of a path integral and the stationary phase approximation is discussed. The exactly soluble models with generators quadratic in position and momentum operators are investigated and the explicit expressions for the space–time propagators for one‐dimensional systems are derived.

On inverse problems for plane‐parallel media with nonuniform surface illumination
View Description Hide DescriptionElementary considerations are used to solve the inverse problem in linear transport theory for the case of variable illumination over the surface of a plane‐parallel layer. The developed formalism yields as a special case the inverse solution for the classical searchlight problem.