Volume 45, Issue 5, May 2004
Index of content:
 DYNAMICAL SYSTEMS


Supersymmetric exact sequence, heat kernel and super Korteweg–de Vries hierarchy
View Description Hide DescriptionWe introduce the free supersymmetric derivation ring and prove the existence of an exact sequence of supersymmetric rings and linear transformations. We apply necessary and sufficient conditions arising from this exact supersymmetric sequence to obtain the essential relations between conserved quantities, gradients and the super Korteweg–de Vries (KdV) hierarchy. We combine this algebraic approach with an analytic analysis of the super heat operator. We obtain the explicit expression for the Green’s function of the super heat operator in terms of a series expansion and discuss its properties. The expansion is convergent under the assumption of bounded bosonic and fermionic potentials. We show that the asymptotic expansion when of the Green’s function for the superheat operator evaluated over its diagonal generates all the members of the super KdV hierarchy.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


The Günther’s formalism in classical field theory: momentum map and reduction
View Description Hide DescriptionThe polysymplectic formalism in local field theory, developed by Günther [J. Diff. Geom. 25, 23 (1987)], is revised. A new approach and new results on momentum maps and reduction are given.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Two new types of quantum states generated via higher powers of Bogoliubov’s transformation
View Description Hide DescriptionBy applying higher powers of Bogoliubov’s transformation operator to the even and odd coherent states we construct mathematically two new types of quantum states. Various special functions and mathematical formulations are used to obtain the matrix elements for the states in the coherent state representation. The nonclassical nature of these states is found to be related to their quantum coherence in phase space. The interference effects can also be regarded as a consequence of the oscillatory behavior of the special functions in the structure of the states.

 FLUIDS


Interior structural bifurcation and separation of 2D incompressible flows
View Description Hide DescriptionWe study transitions in the topological structure of a family of divergencefree vector fields near an interior point. It is shown that structural bifurcation occurs at if has an isolated degenerate singular point with zero index and nonzero Jacobian at and with nonzero acceleration in the direction normal to the (unique) eigenspace of the Jacobian. This result is carried out by analyzing the orbit structure of near such an isolated degenerate interior singular point of Applications to typical interior separation phenomena in twodimensional fluid flows are addressed as well.

 METHODS OF MATHEMATICAL PHYSICS


On the exterior structure of graphs
View Description Hide DescriptionAfter a detailed ab initio description of the exterior structure of graphs as handled by Connes and Kreimer in their work on renormalization (illustrated by the example of the model in six dimensions) we spell out in detail their study of the Lie algebra of infinitesimal characters and of the group of characters of the Hopf algebra of Feynman graphs.

 STATISTICAL PHYSICS


Geodesic distances on density matrices
View Description Hide DescriptionWe find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Locality and orthomodular structure of compound systems
View Description Hide DescriptionA Plattice is defined as a σcomplete, orthomodular atomic lattice L which is formed by the set of propositions of a physical system. A composition of physical systems in the framework of Plattices is considered and some notions of locality are given. It is shown that the following statements about compound systems are equivalent. (a) All atoms of a compound system are reducible to those of its subsystems. (b) All pure states of a compound system are separable into those of its subsystems. (c) A compound system has statistical property independence. (d) At least one of the subsystems is classical. (e) Belltype inequalities hold.

 STATISTICAL PHYSICS


Anisotropic scattering kernel: Generalized and modified Maxwell boundary conditions
View Description Hide DescriptionThis article presents a model of a scattering kernel of boundary conditions for the Boltzmann equation. The proposed scattering kernel is based on an anisotropic accommodation argument. Three parameters equal to the momentum accommodation coefficients are shown as characterizing the influence of each direction. First the new scattering kernel is derived from a phenomenological criticism of the first form of the scattering kernel proposed by Maxwell; then the same result is established from an analytic approach based on the spectral nature of the linear integral operator associated to the scattering kernel problem. As a result, the model provides a correct form of scattering kernel to handle the influence of each direction in particle collisions with the wall. Finally independent accommodation of each internal mode is added to extend the model to the case of polyatomic gases.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


On the discrete spectrum of nonselfadjoint Schrödinger differential equation with an operator coefficient
View Description Hide DescriptionWe study the discrete part of spectrum of a singular nonselfadjoint secondorder differential equation on a semiaxis with an operator coefficient. Its boundedness is proved. The result is applied to the Schrödinger boundary value problem with a complex potential in an angular domain.

 STATISTICAL PHYSICS


On diffusion dynamics for continuous systems with singular superstable interaction
View Description Hide DescriptionWe consider the time evolution of states for continuous infinite particle systems which corresponds to nonequilibrium diffusion dynamics. For initial states which are perturbations of the equilibrium we obtain a bound for finite volume nonequilibrium correlation functions and their continuity in time uniformly in volume for any finite time interval. This gives the possibility to construct the time evolution of correlation functions and corresponding states in the thermodynamic limit.

 GENERAL RELATIVITY AND GRAVITATION


Generating Gowdy cosmological models
View Description Hide DescriptionUsing the analogy with stationary axisymmetric solutions, we present a method to generate new analytic cosmological solutions of Einstein’s equation belonging to the class of Gowdy cosmological models. We show that the solutions can be generated from their data at the initial singularity and present the formal general solution for arbitrary initial data. We exemplify the method by constructing the Kantowski–Sachs cosmological model and a generalization of it that corresponds to an unpolarized Gowdy model.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


On the extra phase correction to the semiclassical spin coherentstate propagator
View Description Hide DescriptionThe problem of an origin of the Solari–Kochetov extraphase contribution to the naive semiclassical form of a generalized phasespace propagator is addressed with the special reference to the su(2) spin case which is the most important in applications. While the extraphase correction to a flat phasespace propagator can straightforwardly be shown to appear as a difference between the principal and the Weyl symbols of a Hamiltonian in the nexttoleading order expansion in the semiclassical parameter, the same statement for the semiclassical spin coherentstate propagator holds provided the Holstein–Primakoff representation of the su(2) algebra generators is employed.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Green functions of the Dirac equation with magneticsolenoid field
View Description Hide DescriptionVarious Green functions of the Dirac equation with a magneticsolenoid field (the superposition of the Aharonov–Bohm field and a collinear uniform magnetic field) are constructed and studied. The problem is considered in and dimensions for the natural extension of the Dirac operator (the extension obtained from the solenoid regularization). Representations of the Green functions as proper time integrals are derived. The nonrelativistic limit is considered. For the sake of completeness the Green functions of the Klein–Gordon particles are constructed as well.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


The role of the irreducible representations of the Poincaré group in solving Maxwell’s equations
View Description Hide DescriptionMaxwell’sequations are universally solved when the charge and current densities are given, without reference to the transformation properties of the fields or charge density–current density four vector. However these transformation properties are generally held to be essential physical properties of the fields and the charge density–current density fourvector. They are required if the consequences of relativity are to hold. The way the fields and fourvectors transform constitutes a representation of the Poincaré group which can be reduced to the irreducible representations of that group first given by Wigner. The use of the irreducible representations corresponds to the expansion of the fields and currents into modes which have the simplest possible transformation properties. These modes can be identified as wave functions of particles of spin 1 and mass We compare the solutions of Maxwell’sequations utilizing the “usual” timedependent Green’s function and the method introduced in this paper. The solutions are identical, if we assume that the fields created by the timedependent sources have the same initial values for the fields. Among the new results, we demonstrate the mechanism by which one can transform transverse fields for which the charge density is zero to fields that are partially longitudinal and for which the charge density source is not zero under Poincaré transformations of the space–time coordinates. We give a concrete example of a transverse current density source and one for a longitudinal current density source and show that sources lead to a mass spectrum of the photons. Longitudinal current densities always lead to noncausal (as noncausal is commonly understood) solutions for the fields.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Group averaging in the oscillator representation of
View Description Hide DescriptionWe investigate refined algebraic quantization with group averaging in a finitedimensional constrained Hamiltonian system that provides a simplified model of general relativity. The classical theory has gauge group and a distinguished observable algebra. The gauge group of the quantum theory is the double cover of and its representation on the auxiliary Hilbert space is isomorphic to the oscillator representation. When and (mod 2), we obtain a physical Hilbert space with a nontrivial representation of the quantum observable algebra. For the system provides the first example known to us where group averaging converges to an indefinite sesquilinear form.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


On the generalized function calculus for infrared and ultraviolet singular quantum fields
View Description Hide DescriptionNew theorems on properties of the generalized functions defined on Gelfand–Shilov’s spaces are established. These functional classes are universal for the operator realization of quantum field theories whose infrared or/and ultraviolet behavior is more singular than that of the standard Wightman quantum field theories (QFT’s). The leading role in these applications is played by the notion of a carrier cone of analytic functional which generalizes and replaces the notion of support of distribution. An explicit representation for the generalized functions with a given carrier cone is obtained. It is proved that the restrictions of functionals defined on to the spaces with smaller subscripts have the same carrier cones. The precise characterization of the relation between the carrier cones of multilinear forms with respect to their arguments and the carrier cones of their associated generalized functions is given. Applications of the obtained results to indefinite metric QFT and to nonlocal models are discussed.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Integrable and superintegrable quantum systems in a magnetic field
View Description Hide DescriptionIntegrable quantum mechanical systems with magnetic fields are constructed in twodimensional Euclidean space. The integral of motion is assumed to be a first or second order Hermitian operator. Contrary to the case of purely scalar potentials, quadratic integrability does not imply the separation of variables in the Schrödinger equation. Moreover, quantum and classical integrable systems do not necessarily coincide: the Hamiltonian can depend on the Planck constant ℏ in a nontrivial manner.

 STATISTICAL PHYSICS


A statistical complexity measure with nonextensive entropy and quasimultiplicativity
View Description Hide DescriptionThe properties of a statistical complexity measure that are characterized by nonextensivity in entropy have been investigated, which is of socalled disequilibrium type. Considering the composition law for two systems with different nonextensivities (quasimultiplicativity), a nontrivial relation between the nonextensive parameters and the fluctuating bit number in information theory has been mentioned. To see the time evolution of the nonextensive complexity measure, we examine systems having a lognormal distribution, the underlying dynamics for which is known to obey a random multiplicative process in the presence of a boundary constraint.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Associated Bessel functions and the discrete approximation of the freeparticle time evolution operator in cylindrical coordinates
View Description Hide DescriptionA central finite difference approximation for the radial contribution to the Laplacian is considered in a threedimensional cylindrical coordinate system A freeparticle Schrödinger time evolution operator is constructed by exponentiation, Denoting the central finite difference approximation of by the matrix with is shown to be similar to a particular unitary representation of the group of motions on Euclidean threespace that has been described by Vilenkin and Klimyk. The matrix elements of generalize the Bessel function and provide an approximation of the leading term in the radial contribution to the evolution operator.

 GENERAL RELATIVITY AND GRAVITATION


Curvature singularity of the distributional Bañados, Teitelboim, and Zanelli black hole geometry
View Description Hide DescriptionFor the nonrotating Bañados, Teitelboim, and Zanelli black hole, the distributional curvature tensor field is found. It is shown to have singular parts proportional to a δdistribution with support at the origin. This singularity is related, through Einstein field equations, to a point source. Coordinate invariance and independence on the choice of differentiable structure of the results are addressed.
