Volume 40, Issue 3, March 1999
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Analytical solution of the relativistic Coulomb problem with a hard core interaction for a onedimensional spinless Salpeter equation
View Description Hide DescriptionIn this paper, we construct an analytical solution of the onedimensional spinless Salpeter equation with a Coulomb potential supplemented by a hard core interaction, which keeps the particle in the x positive region.

A new scaling property of the Casimir energy for a piecewise uniform string
View Description Hide DescriptionAn unexpected and very accurate scaling invariance of the Casimir energy of the piecewise uniform relativistic string is pointed out. The string consists of pieces of equal length, of alternating type I/type II material, endowed with different tensions and mass densities but adjusted such that the velocity of transverse sound equals c. If denotes the Casimir energy as a function of the tension ratio it turns out that the ratio which lies between zero and one, will be practically independent of N for integers Physical implications of this scaling invariance are discussed. Finite temperature theory is also considered.

DyonSkyrmion lumps
View Description Hide DescriptionWe make a numerical study of the classical solutions of the combined system consisting of the Georgi–Glashow model and the SO(3) gauged Skyrme model. Both monopoleSkyrmion and dyonSkyrmion solutions are found. A new bifurcation is shown to occur in the gauged Skyrmion solution sector.

Measurevalued solution to the strongly degenerate compressible Heisenberg chain equations
View Description Hide DescriptionIn this paper we are concerned with the existence of solutions to the compressible Heisenberg chain equations. By the vanishing viscosity method we prove that this system admits at least one measurevalued solution.

Rectangular well as perturbation
View Description Hide DescriptionWe discuss a finite rectangular well of a depth as a perturbation for the infinite one with λ as a perturbation parameter. In particular, we consider a behavior of energy levels in the well as functions of complex λ. It is found that all the levels of the same parity are defined on infinitely sheeted Riemann surfaces whose topological structures are described in detail. These structures differ considerably from those found in models investigated earlier. It is shown that perturbation series for all the levels converge what is in a contrast with the known results of Bender and Wu. The last property is shown to hold also for the infinite rectangular well with the Dirac delta barrier as a perturbation considered earlier by Ushveridze.

A braided interpretation of fractional supersymmetry in higher dimensions
View Description Hide DescriptionA many variable calculus is introduced using the formalism of braided covector algebras. Its properties are discussed in detail and related to fractional supersymmetry when certain of its deformation parameters are roots of unity. The special cases of twodimensional supersymmetry and fractional supersymmetry are developed in detail.

On spectral properties of Harperlike models
View Description Hide DescriptionWe study spectral properties of Harperlike models by algebraic and combinatorial methods and derive sufficient conditions for the existence of spectral gaps with qualitative estimates. For this class the Chambers relation holds and we obtain an analytic expression for the representation dependent part. Models corresponding to the rectangular and triangular lattice are studied. In the second case we show that one class of spectral gaps is open for magnetic fields with “rational magnetic flux per unit cell.” A quantitative estimate for the gap widths is given for the anisotropic case and for “irrational magnetic flux” fulfilling some Liouville condition the spectrum is a Cantor set.

Weak coupling limit and removing an ultraviolet cutoff for a Hamiltonian of particles interacting with a quantized scalar field
View Description Hide DescriptionAn interaction system consisting of particles and a quantized scalar field is considered. The Hamiltonian of the system is defined as a selfadjoint operator in a Hilbert space. An ultraviolet cutoff is imposed on the Hamiltonian. A renormalized Hamiltonian is defined by subtracting a renormalization term from the Hamiltonian. Our aim in this paper is to remove the ultraviolet cutoff and take the weak coupling limit simultaneously for the renormalized Hamiltonian. By using a functional integral that contains a vectorvalued stochastic integral, a Schrödinger Hamiltonian with a manybody Coulomb potential (resp., Yukawa potential) is derived, if the mass of the quantized scalar field is zero (resp., positive).

Uncertainty principle for proper time and mass
View Description Hide DescriptionWe review Bohr’s reasoning in the Bohr–Einstein debate on the photon box experiment. The essential point of his reasoning leads us to an uncertainty relation between the proper time and the rest mass of the clock. It is shown that this uncertainty relation can be derived if only we take the fundamental point of view that the proper time should be included as a dynamic variable in the Lagrangian describing the system of the clock. Some problems and some positive aspects of our approach are then discussed.

Path integral for the relativistic threedimensional Aharonov–Bohm–Coulomb system
View Description Hide DescriptionThe path integral for the relativistic threedimensional spinless Aharonov–Bohm–Coulomb system is solved, and the energy spectra are extracted from the resulting amplitude.

Metric symmetries and spin asymmetries of Ricciflat Riemannian manifolds
View Description Hide DescriptionThe Calabi–Yau and Joyce manifolds used in string and Mtheory compactifications have no continuous groups of isometries, but they often have nontrivial discrete (actually finite) isometry groups. Discrete isometries of nonsimply connected Riemannian manifolds do not necessarily map spin structures into themselves, however; thus, inconsistencies are possible if a spin connection is used to construct the gauge vacuum. We consider this problem in detail and show how it may be avoided.

Nonextensive Bose–Einstein condensation model
View Description Hide DescriptionThe imperfect Boson gas supplemented with a gentle repulsive interaction is completely solved. In particular, it is proved that it has nonextensive Bose–Einstein condensation, i.e., there is condensation without macroscopic occupation of the ground state level.

Extended supersymmetries for the Schrödinger–Pauli equation
View Description Hide DescriptionIt is argued that extended, reducible, and generalized supersymmetry (SUSY) are common in many systems of standard nonrelativistic quantum mechanics. For example, it is proved that a wellstudied quantum mechanical system of a spin particle interacting with constant and homogeneous magnetic field admits the SUSY and has the internal symmetry so(3,3). Then an approach of energy spectra of a SUSY nature is presented and developed. It is applied to a wide class of systems described by the Schrödinger–Pauli equation admitting and SUSY. Some of these supersymmetries have a very peculiar property—their supercharges are realized without usual fermionic variables. It is shown that for them, the usual extension to SUSY is no longer guaranteed.

Determination of Wigner distribution function for the ddimensional Coulomb problem
View Description Hide DescriptionIn this work we present a theoretical study of the ddimensional Coulomb problem in quantum phase space. A coordinate transformation in hyperspherical space is used that maps the ddimensional Coulomb problem into the Ddimensional harmonic oscillator and the Wigner distribution function for the ddimensional Coulomb problem is then obtained. This exactly soluble model can shed some light on finitesize features of Wigner’s distribution, which will be a vital experience for various dynamic problems.

Dynamical semigroups for interacting quantum and classical systems
View Description Hide DescriptionA mathematical framework for the completely positive semigroup coupling between classical and quantum systems is proposed. The coupling ensures a flow of information from the quantum system to the classical one and the influence of the classics on the dynamics of the quantum system in a dissipative way. The classical evolution on average is modified by the expectation value of some quantum operator. Examples of a classical particle moving along a geodesic line in a curved space interacting with the quantum system, and the coupling of a two state quantum system to all pure states, are discussed.

Arithmetic properties of spectra produced by Farey hierarchies of approximants
View Description Hide DescriptionWe discuss the consequences of the hierarchical nature of series of approximants of aperiodic crystals on their diffraction patterns and spectra of elementary excitations. We show how a linear form defined on can be used to order Bragg reflections in diffraction patterns according to their amplitudes, and gaps in spectra of elementary excitations according to their widths, for all the structures in the hierarchy. Bragg peaks amplitudes and gap widths are projective functions on recursively defined on 2D Farey sets (generalization of Farey series).

On a trace formula of the Buslaev–Faddeev type for a longrange potential
View Description Hide DescriptionWe propose an approach to obtaining new trace formulas of the Gel’fand–Levitan–Buslaev–Faddeev type, valid for Hilbert–Schmidt perturbations. In this way we obtain a new trace formula for Schrödinger operators on the halfline with longrange potentials.

Toward quantum mathematics. I. From quantum set theory to universal quantum mechanics
View Description Hide DescriptionWe develop the old idea of von Neumann of a set theory with an internal quantum logic in a modern categorical guise [i.e., taking the objects of the category H of (pre)Hilbert spaces and linear maps as the sets of the basic level]. We will see that in this way it is possible to clarify the relationship between categorification and quantization and besides this to understand that in some sense a categorificational approach to quantization is a discretized version of the one taken by noncommutative geometry. The tower of higher categorifications will appear as the analog of the von Neumann hierarchy of classical set theory (where by classical set theory, we will understand the usual Zermelo–Fraenkel system). Finally, we make a suggestion of how to understand all the different categorifications as different realizations of one and the same abstract structure by viewing quantum mechanics as universal in the sense of category theory. This gives the possibility to view extended topological quantum field theories purely as involving an abstract notion of quantum mechanics plus representation theory without the need to enlarge the class of kinematic structures of quantum systems on each step of categorification. In a future part of the work we will apply the language developed here to deal especially with the question of a categorification of the manifold notion.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


On the anisotropic Manev problem
View Description Hide DescriptionWe consider the Manev potential, given by the sum between the inverse and the inverse square of the distance, in an anisotropic space, i.e., such that the force acts differently in each direction. Using McGehee coordinates, we blow up the collision singularity,paste a collision manifold to the phase space, study the flow on and near the collision manifold, and find a positivemeasure set of collision orbits. Besides frontal homothetic, frontal nonhomothetic, and spiraling collisions and ejections, we put into the evidence the surprising class of oscillatory collision and ejection orbits. Using the infinity manifold, we further tackle capture and escape solutions in the zeroenergy case. By finding the connection orbits between equilibria and/or cycles at impact and at infinity, we describe a large class of capturecollision and ejectionescape solutions.

Nonholonomic constraints in timedependent mechanics
View Description Hide DescriptionThe constraint reaction force of ideal nonholonomic constraints in timedependent mechanics on a configuration bundle is obtained. Using the vertical extension of Hamiltonian formalism to the vertical tangent bundle of the Hamiltonian of a nonholonomic constrained system is constructed. The present setting is more general than the one usually considered in the literature on nonholonomic mechanics.
