Volume 40, Issue 9, September 1999
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Point and line boundaries in scalar Casimir theory
View Description Hide DescriptionA simple imagecharge construction enables one to insert point or line boundaries (or planar or hyperplanar boundaries when sufficiently many spatial dimensions are available) at or into the central point, line, plane etc., of a great range of spatial backgrounds in quantum field theory which have appropriate symmetry. This nontrivial construction (which provides among other things the exact vacuum stress tensor of the quantum field if can be computed for the original background prior to point, line,…, insertion) works if all directions perpendicular to the inserted object are symmetric under In other respects the symmetric spatial background can be quite arbitrary. While the inserted object experiences (by symmetry) no net Casimir force from the background, it does exert Casimir forces throughout this background which were originally not present. In addition to general theory, detailed examples are given (which include exact field ’s and exact Casimir force densities) for arbitrary spatial dimension. First: point and line boundaries in otherwise empty space; then a planar boundary with a semiinfinite line extending from one side; finally, parallel planar boundaries with a point boundary halfway between them. Only scalar quantum fields are analyzed here; however the extension to the electromagnetic Casimir effect is discussed qualitatively.

Partially solvable quantum manybody problems in Ddimensional space
View Description Hide DescriptionA simple technique employed almost three decades ago to manufacture partially solvable quantum manybody problems is revisited. [A quantum problem is “partially solvable” if (only) some of its eigenvalues and eigenfunctions can be exhibited]. The models thereby generated are characterized by Hamiltonians of normal form, i.e., standard kinetic plus momentumindependent potential energy; in most cases the latter features threebody, in addition to twobody and onebody, interactions. The setting refers to Ddimensional space; the examples focus on and and include generalizations of, and additional results on, cases recently discussed in the literature, as well as new models.

An estimation theoretical characterization of coherent states
View Description Hide DescriptionWe introduce a class of quantum pure state models called the coherent models. A coherent model is an evendimensional manifold of pure states whose tangent space is characterized by a symplectic structure. In a rigorous framework of noncommutative statistics, it is shown that a coherent model inherits and expands the original spirit of the minimum uncertainty property of coherent states.

Exact solutions of Dirac equation for neutrinos in presence of external fields
View Description Hide DescriptionExact solutions of the Dirac equation in a chiral representation for neutrinos in the presence of external stresses are investigated in terms of special functions, using the algebraic method of the separation of variables in Cartesian, cylindrical, and spherical coordinates.

Exactly solvable models of sphere interactions in nonrelativistic quantum mechanics
View Description Hide DescriptionWe introduce and perform a systematic study of a new exactly solvable model of sphere interactions in quantum mechanics : the interaction, formally given by We also consider the cases of a plus a Coulomb interaction and finitely many sphere interactions with support on concentric spheres. For all these models, we provide basic properties and discuss the stationary scattering theory. We also briefly discuss the sphere interaction of the second type.

Degeneracy and parasupersymmetry of Dirac Hamiltonian in space–time
View Description Hide DescriptionThe quantum mechanics of a spin particle on a locally spatial constant curvature part of a space–time in the presence of a constant magnetic field of a magnetic monopole has been investigated. It has been shown that these twodimensional Hamiltonians have the degeneracy group of and parasupersymmetry of arbitrary order or shape invariance. Using this symmetry we have obtained its spectrum algebraically. The Dirac’s quantization condition has been obtained from the representation theory. Also, it is shown that the presence of angular deficit suppresses both the degeneracy and shape invariance.

On a purported local extension of the quantum formalism
View Description Hide DescriptionIt is widely believed that Bell has proved there can be no consistent local extension of the quantum formalism. Against this, Angelidis has presented a hidden variable theory which, he claims, makes precisely the same predictions as the quantum formalism and which also satisfies locality. In this note, we argue that Angelidis’ theory does not live up to its inventor’s claims.

On one generalization for the projection matrices method in the matrix factorization problem
View Description Hide DescriptionA boundary value problem for a 2×2 Smatrix is solved by the method of projection matrices. The Smatrix is assumed to be diagnoalized by orthogonal matrix where matrix elements of are polynomials. The equation has roots of any order. The example is considered for the roots with order 2.

Classical mechanics and geometric quantization on an infinite dimensional disc and Grassmannian
View Description Hide DescriptionWe discuss the classical mechanics on the Grassmannian and the disc modeled on the ideal We apply methods of geometric quantization to these systems. Their relation to a flat symplectic space is also discussed.

Gauge theory of w symmetry
View Description Hide DescriptionThe algebra is a higherspin extension of the Virasoro algebra. In this paper, we construct the gauge theory of w symmetry in terms of its representations.
 Top

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Method of handling the divergences in the radiation theory of sources that move faster than their waves
View Description Hide DescriptionThe nonintegrable singularities that arise when the retarded potentials associated with supersonically or superluminally moving sources are differentiated are closely related to those encountered in the context of the Cauchy problem for hyperbolic equations over odddimensional spacetimes. The purpose of this paper is to point out that the field components are given in these cases by Hadamard’s finite parts of the resulting divergent integrals, as in the case of the Cauchy problem, and to show that the procedure familiar from the subluminal regime—which is used by Hannay—is not applicable when there are source elements that approach the observer with the wave speed and zero acceleration at the retarded time.

Nonequilibrium dynamics of infinite particle systems with infinite range interactions
View Description Hide DescriptionWe discuss the existence and uniqueness of nonequilibrium dynamics of infinitely many particles interacting via superstable pair interactions in one and two dimensions. The interaction is allowed to be of infinite range and singular at the origin. Under suitable regularity conditions on the interaction potential, we show that if the potential decreases polynomially as the distance between interacting two particles increases, then the tempered solution to the system of Hamiltonian equations exists. Moreover, if the potential satisfies further that either it has a subexponential decreasing rate or it is everywhere twotimes continuously differentiable, then we show that the tempered solution is unique. The results extend those of Dobrushin and Fritz obtained for finite range interactions.
 Top

 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


Backscattering and inverse problem in random media
View Description Hide DescriptionWith use of the invariant imbedding method and Markov process approximation, the statistical characteristics of the backscattering field from the impulse point source in a nonstationary multidimensional random medium are considered. The equations for the characteristic function and the density of the probability in the functional space of the backscattering are obtained. The equations for the statistical moments of the field are yielded and then solutions are presented in the matrix form. The additional averaging of the fast field variations is used for the simplification of the statistical equations. The procedures for the investigation of the direct and inverse statistical problem are proposed. The role of the phenomenological transfer theory is discussed from the statistical point of view.
 Top

 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


Solution of a discrete inverse scattering problem and of the Cauchy problem of a class of discrete evolution equations
View Description Hide DescriptionThe Cauchy problem of a class of nonlinear evolution equations is solved by finding explicit solutions of a discrete inverse scattering problem that are not restricted to the pure soliton case and implementing appropriate time evolution of the scattering data. This yields operatorvalued functions, which are shown to solve a hierarchy of operator evolution equations by applying methods similar to those in Marchenko’s work. In addition the relation to canonical Lax constructions is investigated. Using methods introduced by Aden and Carl and Schiebold, one obtains scalar solutions to corresponding scalar equations, sometimes representable by determinants on operator ideals.

Generalized Gel’fand–Levitan integral equation for two block Ablowitz–Kaup–Newell–Segur systems
View Description Hide DescriptionWe derive a generalized Gel’fand–Levitan integral equation for two block Ablowitz–Kaup–Newell–Segur systems. This is possible if we suppose that the matrix coupling coefficients are invertible and come from simple zeroes of the determinant of the diagonal blocks of the scattering matrix.

Irreversible weak limits of classical dynamical systems
View Description Hide DescriptionA general discussion is given of weak limits of classical dynamical systems depending on a parameter. The resulting maps are shown to be invertible if and only if they define a group of measure preserving point transformations. The irreversible case automatically leads to positive bistochastic maps and is characterized in terms of convergence properties of the corresponding automorphisms of the observable algebra. Necessary and sufficient conditions are given for the limit to define a timeindependent Markov process. Two models are discussed, for a particle in a periodic potential, and for a particle interacting with fixed configurations of external obstacles.

A coupled AKNS–Kaup–Newell soliton hierarchy
View Description Hide DescriptionA coupled AKNS–Kaup–Newell hierarchy of systems of soliton equations is proposed in terms of hereditary symmetry operators resulted from Hamiltonian pairs. Zero curvature representations and triHamiltonian structures are established for all coupled AKNS–Kaup–Newell systems in the hierarchy. Therefore all systems have infinitely many commuting symmetries and conservation laws. Two reductions of the systems lead to the AKNS hierarchy and the Kaup–Newell hierarchy, and thus those two soliton hierarchies also possess triHamiltonian structures.

Instability and chaos in spatially homogeneous field theories
View Description Hide DescriptionSpatially homogeneous field theories are studied in the framework of dynamical system theory. In particular, we consider a model of inflationary cosmology and a Yang–Mills–Higgs system. We discuss also the role of quantum chaos and its application to field theories.

Dimension of the global attractor for damped semilinear wave equations with critical exponent
View Description Hide DescriptionWe obtain an estimate of the upper bound of the Hausdorff dimension of the global attractor for damped semilinear wave equations with a critical exponent. The obtained Hausdorff dimension decreases as the damping grows for large damping.

Families of quasibiHamiltonian systems and separability
View Description Hide DescriptionIt is shown how to construct an infinite number of families of quasibiHamiltonian (QBH) systems by means of the constrained flows of soliton equations. Three explicit QBH structures are presented for the first three families of the constrained flows. The Nijenhuis coordinates defined by the Nijenhuis tensor for the corresponding families of QBH systems are proved to be exactly the same as the separated variables introduced by mean of the Lax matrices for the constrained flows.
