Index of content:
Volume 19, Issue 2, February 1978

Note on the Bertotti–Robinson electromagnetic universe
View Description Hide DescriptionIt is shown in a concise manner using Debever’s vectorial formalism that the Bertotti–Robinson solution is the most general conformally flat solution of the source free Einstein–Maxwell equations for nonnull electromagnetic fields.

Correlation functions and higher order coherence in inelastically scattered quantum radiation
View Description Hide DescriptionThe concept of coherence is analyzed in a system of interacting radiation and matter. Using the projection technique, the reduced field density operator is found, and with it the first and higher order field correlation functions are computed. It is proved that the inelastically scattered field is, for the most part, incoherent in any order at any time interval after collision with an atomic ensemble, except some specific time intervals and positions, where the order of coherence is determined by the atomic correlations considered.

The reduced field density operator for an inelastically scattered quantum radiation
View Description Hide DescriptionThe projection technique used for setting a non‐Markoffian differential equation for the reduced field density operator is discussed, and a detailed solution of the equation is found.

A consequence of the invariance of Biot’s variational principle in thermal conduction
View Description Hide DescriptionThe invariance of Biot’s variational principle in thermal conduction suggests that this variational principle can be formulated in terms of a self‐similar variational principle. This work presents such a derivation and applies it to a thermodynamics example.

A kinetic formulation of the three‐dimensional quantum mechanical harmonic oscillator under a random perturbation
View Description Hide DescriptionThe behavior of a three‐dimensional, nonrelativistic, quantum mechanical harmonic oscillator is investigated under the influence of three distinct types of randomly fluctuating potential fields. Specifically, kinetic (or transport) equations are derived for the corresponding stochastic Wigner equation (the exact equation of evolution of the phase‐space Wigner distribution density function) and the stochastic Liouville equation (correspondence limit approximation) using two closely related statistical techniques, the first‐order smoothing and the long‐time Markovian approximations. Several physically important averaged observables are calculated in special cases. In the absence of a deterministic inhomogeneous potential field (randomly perturbed, freely propagating particle), the results reduce to those reported previously by Besieris and Tappert.

Lorentz transformations as space–time reflections. II. Timelike reflections
View Description Hide DescriptionContinuing previous work, some features of the covariant factorization of Lorentz transformations, into complementary space–time reflections, are further discussed in terms of timelike reflections. Several properties of timelike reflections are shown, which bear interesting relations with the Lorentztensor performing active isometric transformations between two inertial observers, while their geometric meaning is also briefly examined. Some product rules for timelike reflections appear, as a background enabling the discussion of the group multiplication laws for Lorentz isometry tensors. This brings into the fore the realization of the restricted Lorentz group attained in the present formalism. The instances of two multiplication laws of the Lorentztensors are examined. Next, the same problem (i.e., factorization of an ordinary rotation into two complementary reflections) is readily solved for the Euclidean 3‐space. Both formalisms (Euclidean and Minkowskian) are essentially the same. Indeed, factorization of an isometric transformation into two complementary reflections is a general property of flat Riemannian geometry. A few concluding remarks are presented.

Strings and gauge invariance
View Description Hide DescriptionWithin the context of the nonrelativistic theory of dyons, we study a number of interrelated issues concerning the quantum formulation of magnetic charge. We begin by solving the two‐body Schrödinger equation for an arbitrarily oriented singularity line (string) in terms of the known solutions with the string on the z axis. Charge quantization conditions emerge by requiring that the wavefunctions be single valued. The general solutions express the necessary gauge covariance of the wavefunctions. These results provide a basis for the reconsideration of the phase factor in the dyon–dyon scattering amplitude. Finally, the connection between the formulations in terms of vector potentials and in terms of intrinsic spin is investigated. This approach leads to a rederivation of the gauge transformation properties of the theory.

Subcriticality and supercriticality of energy dependent neutron transport in slab geometry
View Description Hide DescriptionThe energy dependent transport system in an anisotropic medium in slab geometry subjecting possible internal sourceq and incoming fluxes ψ_{0}, ψ_{1} is discussed. It has been shown in an earlier paper that under certain conditions on the average number of secondary neutrons per collision c, the scattering cross section σ, and the optical slab length 2a, this system has a unique nonnegative solution for all inputs q, ψ_{0}, ψ_{1}. The aim of this paper is to establish analogous conditions on c, σ, a so that the system has no nonnegative solution when there is either internal source or incoming fluxes (or both), and it only has the trivial solution when neither internal source nor incoming fluxes are present in the system. This conclusion together with the earlier results yield explicit conditions for insuring the supercriticality and the subcriticality of the energy dependent system and therefore lead to analytical upper and lower bounds for the critical value c* in terms of σ and a.

Interaction and stability of localized solutions in a classical nonlinear scalar field theory
View Description Hide DescriptionThe interaction between the localized solutions of a nonlinear scalar field theory are studied. We also study the stability of the above solutions under certain time‐dependent perturbations.

On equivalence of parabolic and hyperbolic super‐Hamiltonians
View Description Hide DescriptionThree types of super‐Hamiltonians occur in generally covariant field theories: linear in the momenta (hypersurface kinematics), parabolic in the momenta (parametrized field theories on a given Riemannian background), and hyperbolic in the momenta (geometrodynamics). Three simple models are discussed in which the linear or parabolic super‐Hamiltonian can be cast, essentially by a canonical transformation, into an equivalent hyperbolic form: (1) The scalar field propagating on a (1+1) ‐dimensional flat Minkowskian background, (2) hypersurface kinematics on a (1+n) ‐dimensional flat Minkowskian background, and (3) geometrodynamics of a (1+2) ‐dimensional vacuum spacetime. The implications for constraint quantization are mentioned.

Random linear systems and temporal homogeneity
View Description Hide DescriptionThe class of random linear systems having stochastic Green’s functions whose moments are invariant under arbitrary uniform translations of all time variables is defined and investigated. It is pointed out that this class is very broad, including, for example, virtually all treatments of wave propagation through a random medium. Proceeding by analogy with quantum field theory the quantities G and M, related to the first and second moments of the stochastic Green’s function, are defined. Various properties of G and M (which in quantum field theory correspond respectively to a propagator and an elastic scattering amplitude) are discussed, and it is shown that they may be conveniently used to describe the principal physical effects induced by transmission through a randomly fluctuatingsystem. Specific examples are given in which these quantities are explicitly calculated and used to illustrate the general results.

Lower bounds on the total cross section and the slope parameter for some measurable sequences of s→∞
View Description Hide DescriptionThe lower bounds on the total cross section and the slope parameter are obtained on the basis of the analyticity and polynomial upper boundedness of the scattering amplitude and the unitarity of the S matrix: σ_{tot}⩽C s ^{−6} (log s)^{−2}, B⩽C s ^{−5}(log s)^{−4} for some measurablesequences of s→∞. These bounds hold for any t in 0?t<4m ^{2} _{π}. It is unnecessary in order to obtain our bounds that the scattering amplitude has the crossing even property. If we assume this property, we can suppress the logarithmic factors of our bounds. Also we obtain our lower bounds for any sequence of s→∞, if we take the average scattering amplitude.

Isotropy subgroups of SO(3) and Higgs potentials
View Description Hide DescriptionA method is given for determining the isotropy subgroups of an arbitrary, irreducible representation of SO(3). These subgroups are explicitly worked out for low‐dimensional representations. As an application of these results we contruct the most general, renormalizable, SO(3) invariant Higgs potentials for these representations, determine the local minima of the potentials and discuss patterns of spontaneous symmetry breaking.

An alternative approach to the normal frequencies of a randomly disordered linear chain
View Description Hide DescriptionAn attempt is made to gain some insight into the statistical character of the normal frequencies of a long chain of harmonic oscillators with randomly disordered masses. Instead of being concerned with the limit L→∞ of a chain whose random characteristics do not depend on its length L, the paper deals with the so‐called diffusion limit where the ’’size’’ of the randomness in each of the masses tends to 0 like L ^{−1/2}.

Explicit results for the quantum‐mechanical energy states basic to a finite square‐well potential
View Description Hide DescriptionThe theory of complex variables is used to establish explicit expressions for the discrete energy states relevant to a square‐well potential.

Symmetry conditions and non‐Abelian gauge fields
View Description Hide DescriptionClassical gauge fields are envisaged in the context of fibre bundle theory. General symmetry conditions are found which lead to an Abelian holonomy group. This, in turn, has important consequences on the solutions of the gauge fieldequations: Symmetric solutions have unphysical properties. Non‐Abelian holonomy groups are thus needed.

Some static and nonstatic solutions of Brans–Dicke theory of gravitation
View Description Hide DescriptionStatic and nonstatic vacuum solutions of Brans–Dicke field equations are derived. For this purpose, a new and convenient technique is proposed. Results are applied to some known solutions.

Quadratic Hamiltonians, quadratic invariants and the symmetry group SU(n)
View Description Hide DescriptionWe show that any 2n‐dimensional quadratic Hamiltonian may be transformed by a (usually time‐dependent) linear canonical transformation into any other 2n‐dimensional quadratic Hamiltonian, in particular that of the isotropic harmonic oscillator. This latter Hamiltonian possesses the symmetry group SU(n) and n ^{2}−1 linearly independent quadratic invariants which provide a basis for the generators of the group. Every other quadratic Hamiltonian is shown to have a quadratic invariant possessing SU(n) symmetry. The free particle structure is given explicitly. The anisotropicoscillator is shown not to possess SU(3) symmetry based on quadratic invariants. However, its wavefunctions and energy levels may be obtained directly from those of the isotropic oscillator whether the frequencies are commensurable or not.

Splitting and representation groups for Polish groups
View Description Hide DescriptionIt is shown that all continuous unitary/antiunitary projective representations of a Polish group G, with the same subgroup of elements represented into the projective unitary groups, arise from continuous unitary/antiunitary (ordinary) representations of a topological group (called a splitting group) obtained from an extension of G by an Abelian topological group. Moreover, there exists always a splitting group which is ’’minimal’’ in some well‐defined sense (a representation group). A sufficient (resp. a necessary and sufficient) condition for the existence of a Polish (resp. of a second countable locally compact) representation group is given.

Inequalities and uncertainty principles
View Description Hide DescriptionSobolev inequalities give lower bounds for quantum mechanical Hamiltonians. These inequalities are derived from commutator inequalities related to the Heisenberg uncertainty principle.