Volume 18, Issue 11, November 1977
Index of content:

Free particle systems and their dynamics in de Sitter space
View Description Hide DescriptionA first step in determining all the global projective unitary representations describing free particle systems in imaginary Lobachevsky space is made. Essentially we determine explicitly the global form of any representation describing a free particle of spin j on the group generated by rotations and translations of space–time at time t and time translations. We also discuss whether or not positional observables should be preserved under physical equivalence and determine the effects this has on the representation theory of free particle systems.

A tachyon dust universe
View Description Hide DescriptionIn the present paper some investigations have been made on the model suggested by Ray for a tachyon dust universe and the results obtained have been compared with the results in the flat Friedmann universe filled with ordinary dust (here called bradyon dust) moving slower than light, by various scientists, on the ground that the role played by time for ordinary matter is played by spatial coordinates for tachyons. The effect of the cosmological constant (Λ) on the expanding tachyon universe also has been discussed here. The tetrad technique has been used as a mathematical tool for handling the problems of gravitational field equations and perturbation of momentum flux.

K matrix for the Woods–Saxon potential
View Description Hide DescriptionThe s‐wave part of the off‐shell K matrix elements for the Woods–Saxon potential has been obtained in terms of elementary transcendental functions by using the differential equation approach to off‐shell scattering.

Stability, equilibrium and KMS for an infinite classical system
View Description Hide DescriptionThe stability condition as a property which characterizes the thermodynamic equilibrium is studied from an abstract point of view. Furthermore, an application of the main result to the case of an infinite classical harmonic system is given.

On notions of Markov property
View Description Hide DescriptionWe exhibit the logical connection between two mathematically and physically interesting notions of Markov property due to Nelson [J. Funct. Anal. 12, 97 (1973)] and to Wong [Ann. Math. Stat. 40, 1625 (1969)], respectively, in the case of Gaussian generalized stochastic fields.

Evolution of a stable profile for a class of nonlinear diffusion equations with fixed boundaries
View Description Hide DescriptionA class of quasilinear parabolic equations with fixed boundaries arising in studies of cross‐field diffusion in toroidal multipole plasmas is presented. It is well known that these equations have separable solutions which decay in time. Surprisingly, both octupole and numerical experiments show, in particular cases, that the separable solution evolves from an arbitrary initial distribution of particles. The evolution and stability properties of these solutions are demonstrated in this paper. When the coefficients of the equations are independent of the spatial variable, infinitesimal perturbations decay as the fourth power (or higher) of the separable solution time dependence; the separable solution is therefore stable. When the initial particle distribution has no nulls except at the boundaries, an approximate analysis shows that large perturbations decay exponentially causing the rapid evolution of the separable solution. The analysis allows the asymptotic behavior of the system to be predicted approximately from knowledge of the initial particle distribution.

Global structure of the ’’Kantowski–Sachs’’ cosmological models
View Description Hide DescriptionA discussion is given of the ’’Kantowski–Sachs’’ cosmological models; these are defined locally as admitting a four‐parameter continuous isometry group which acts on spacelike hypersurfaces, and which possesses a three‐parameter subgroup whose orbits are 2‐surfaces of constant curvature (i.e., the models possess spherical symmetry, combined with a translational symmetry, and can thus be regarded as nonempty analogs of part of the extended Schwarzschild manifold). It is shown that all general relativistic models in which the matter content is a perfect fluid satisfying reasonable energy conditions are geodesically incomplete, both to the past and to the future, and that at each resulting singularity the fluid energy density is infinite. In the case where the fluid obeys a barotropic equation of state (which includes all known exact perfect fluid solutions) the field equations are shown to decouple to form a plane autonomous subsystem. This subsystem is examined using qualitative (Poincaré–Bendixson) theory, and phase–plane diagrams are drawn depicting the behavior of the fluid’s energy density and shear anisotropy in the course of the models’ evolution. Further diagrams depict the conformal structure of these universes, and a table summarizes the asymptotic properties of all physically relevant variables.

A necessary condition for the validity of Huygens’ principle on a curved space–time
View Description Hide DescriptionIt is proved that a necessary condition for the validity of Huygens’ principle on a curved space–time V ^{4} is that V ^{4} be an Einstein space. In connection with this result some remarks about the strong and weak formulations of Mach’s principle are also pointed out.

Perturbation solution of the Percus–Yevick equation for the square‐mound potential
View Description Hide DescriptionThe perturbation solution of the Percus–Yevick equation, based on the known solution for hard‐sphere interactions, is found for the square‐mound potential. The correlation functions are expanded in powers of the parameter α, related to the height of the mound, i.e., describing the deviation of the interactions from the hard‐sphere ones. An algorithm for the calculation of subsequent coefficients is given. Numerical calculations show, however, that the series converges slowly and thus a few terms approximate the whole series with sufficient accuracy for small (but still finite) values of α, i.e., for almost hard‐sphere interactions, only. For mounds high enough the system behaves very similarly to the hard‐sphere system, but it quickly loses its ’’hard’’ characteristics as the mound decreases. This result, in the light of the success of the recent perturbation theory of liquids, seems to suggest that, whereas fairly significant changes of the potential in the outside of the hard core may be treated as small perturbations, even a small change inside the hard core very strongly perturbs the properties of the system.

Functional equations for extended hadrons
View Description Hide DescriptionWe consider the problem of solution of some functional equations occurring in the theory of extended hadrons. By means of stochastic methods solutions of these equations are obtained in the form of a contractive (Markov) semigroup in Hilbert space. Analytic continuation to a unitary group implementing time evolution is performed. The problem of unitary implementability of Lorentz and gauge symmetries, essential for physical interpretation, remains unsolved.

Quantum counting processes
View Description Hide DescriptionIt is proposed that counting experiments in quantum physics should be analyzed in terms of point processes (QPP) defined in the framework of quantum probability theory. A coincidence approach is developed for a class QPP called the regular QPP. A counting formula is derived which determines completely the counting statistics of a regular QPP by means of a pair of ’’generators.’’

The gravitational influence of a beam of light of variable flux
View Description Hide DescriptionAn exact solution is obtained for the Einstein field equations of a columnated, time varying beam of light. The beam is circular in cross section, infinite in path length, and is considered in the geometrical limit. The beam is described in a retarded time coordinate system. The flux density is dependent on the radial coordinate and on the retarded time. The solution is sufficiently general so as to describe a single pulse of light traveling through a vacuum. It also allows the description of acceleration fields which propagate in the direction of the beam at the speed of light. Geodesics are considered in order to test the interpretation of the solutions and the stability of the time varying beam.

Spaces of positive and negative frequency solutions of field equations in curved space–times. I. The Klein–Gordon equation in stationary space–times
View Description Hide DescriptionIn stationary space–times V _{ n }×R with compact space‐section manifold without boundary V _{ n }, the Klein–Gordon equation is solved by the one‐parameter group of unitary operators generated by the energy operator i ^{−1} T ^{−1} in the Sobolev spaces H ^{ l }(V _{ n }) ×H ^{ l−1}(V _{ n }). The canonical symplectic and complex structures of the associated dynamical system are calculated. The existence and the uniqueness of the Lichnerowicz kernel are established. The Hilbert spaces of positive and negative frequency‐part solutions defined by means of this kernel are constructed.

A note on the range of the applicability of the Ornstein–Zernike theory in the van der Waals model
View Description Hide DescriptionWe show that the condition determining the range of the applicability of the Ornstein–Zernike theory obtained previously by Hemmer for the van der Waals model is too strong. The weaker condition obtained by us is in agreement with the existing results for different physical systems, and it requires the temperature to satisfy the following inequality: (T−T _{ c })/T _{ c }≫ (γd)^{6}, where γd denotes the ratio of the short‐range part of the potential to the range of the long‐range part and T _{ c } is the critical temperature predicted in the van der Waals model.

Generators of infinite direct products of unitary groups
View Description Hide DescriptionUnder suitable conditions, an infinite direct product ⊗_{ n } U _{ n }(t) of continuous unitary one‐parameter groups U _{ n }(t) is again a continuous unitary one‐parameter group. This question is discussed here in terms of the generators A _{ n } of U _{ n }(t). It is shown that the generator A of ⊗_{ n } U _{ n }(t) has a total set of product vectors in its domain of definition. As examples, the particle number, energy–momentum, and angular momentum operators in non‐Fock direct product representations of free fields are investigated. The spectra of these operators are determined.

Sequences of transformations for particle‐field Hamiltonians
View Description Hide DescriptionThe nonrelativistic Hamiltonian for a particle interacting with a scalar field is studied by a method that involves a product of unitary transforms. We use intermediate coupling transforms for finite particle momenta to treat the interaction with oscillators with a range of wave vectors. Elimination of the oscillators in the range leads to a new Hamiltonian of the identical operator structure but with a modified interaction and mass. We show that with any finite number of divisions the self energy is always lower than the intermediate coupling result. In the limit of infinitely many divisions differential equations are derived for the interaction and effective mass as a function of the smallest wave vector of the oscillators that have been removed.

Magnetic monopoles in SU(4) gauge theories
View Description Hide DescriptionAll spherically symmetric magnetic poles are explictly constructed in SU(4). We show the relevance of the concept of the little group which transforms spherically symmetric solutions among themselves. Unlike the SU(2) and SU(3) situation this little group is not always Abelian in SU(4). On the other hand, it is shown that while the total strength of the pole is always quantized its projection along the Higgs field is sometimes arbitrary. This is also in contrast with the SU(2) and SU(3) cases.

Stationary gravitational fields of a charged perfect fluid
View Description Hide DescriptionEinstein’s field equations which include electromagnetism are investigated when the metric admits a timelike Killing motion and the source is a charged perfect fluid under isometric motion. It is shown that the pressure must necessarily be a function of the electrostatic and gravitational potentials. A class of solutions is found under the following simplifying assumptions: (i) The pressure is a constant, (ii) the Lorentz force vanishes, and (iii) the magnetic and twist potentials are functionally related. In this class the ratio of σ/(ρ+3p) is a constant and this resembles an equilibrium condition. Finally a four‐parameter group (maximal) is supplied which can generate new solutions of this class.

Lee model with 3V particles
View Description Hide DescriptionThe 3V‐particle Lee model is proposed. The NΘ scattering amplitude, the VΘ scattering amplitude and the nΘΘ production amplitude are evaluated. The integral equation in the VΘ sector is solved. The usual deductive method cannot be applied here to solve the integral equation of the VΘ sector. A nondeductive method is applied to solve the integral equation. The solution obtained correctly reduces to the solution of the VΘ integral equation of the ordinary Lee model, whenever any two bare interaction constants in the Hamiltonian are switched off. The two resonances that appear in the NΘ sector, again appear in the VΘ sector at the same energies.

Fermi–Bose and internal symmetries with universal Clifford algebras
View Description Hide DescriptionSupersymmetry in four‐dimensional space–time is approached from the theory of universal Clifford algebras. The representation produced results in a class of new algebras characterized by an index n which indicates the natural appearance of su(2n) ×u(1) as a subalgebra interacting only with the spinor (or odd) parts of the surrounding Fermi–Bose algebra. Finally, a reformulation of the Dirac equation in this formalism is presented which is not plagued with the empirically untenable problem of a continuous range of eigenvalues.