Volume 40, Issue 1, January 1999
Index of content:
 QUANTUM PHYSICS; PARTICLES AND FIELDS


On the BornInfeld electron: Spin effects
View Description Hide DescriptionWe start from a natural generalization of the Born–Infeld Lagrangian which involves two constants h,k and two parameters and show, by investigating static solutions of a first order approximation with respect to a small parameter that and that h is directly proportional to Planck’s constant. It seems reasonable to interpret as the spin of the electron and the angle as its orientation. Thus we obtain solutions that appear to reflect the influence of the states of spin on the electromagnetic field.

Thermalization of quantum states
View Description Hide DescriptionAn exact stochastic model for the thermalization of quantum states is proposed. The model has various physically appealing properties. The dynamics are characterized by an underlying Schrödinger evolution, together with a nonlinear term driving the system toward an asymptotic equilibrium state and a stochastic term reflecting fluctuations. There are two free parameters, one of which can be identified with the heat bath temperature, while the other determines the characteristic time scale for thermalization. Exact expressions are derived for the evolutionary dynamics of the system energy, the system entropy, and the associated density operator.

Existence and nonexistence in Chern–Simons–Higgs theory with a constant electric charge density
View Description Hide DescriptionIn this paper we are devoted to proving the existence and nonexistence of selfdual equations arising in Chern–Simons–Higgs theory with a constant electric charge density. There are three kinds of boundary conditions that admit solitonic structures. It is shown that there exist solutions in two cases of them. In the other case, we prove that there is a critical electric charge density with negative value such that above the value there exists a solution and below it we have no solution. We also study asymptotic behaviors for solutions as the electric charge density goes to zero. It is found that they converge to solutions of a topological Chern–Simons system without constant electric charge density.

On the spectral theory of dispersive Nbody Hamiltonians
View Description Hide DescriptionIn this work we describe a general class of dispersive Nbody Hamiltonians for which we prove the Hunziker, van Winter, and Zislin (HWZ) theorem and a Mourre estimate outside a closed and countable set of energies called thresholds. As a consequence of the Mourre estimate we prove a strong form of the limiting absorption principle, which implies the absence of a singular continuous spectrum and gives criteria of local smoothness.

Gauge transformations in quantum mechanics and the unification of nonlinear Schrödinger equations
View Description Hide DescriptionBeginning with ordinary quantum mechanics for spinless particles, together with the hypothesis that all experimental measurements consist of positional measurements at different times, we characterize directly a class of nonlinear quantum theories physically equivalent to linear quantum mechanics through nonlinear gauge transformations. We show that under two physically motivated assumptions, these transformations are uniquely determined: they are exactly the group of timedependent, nonlinear gauge transformations introduced previously for a family of nonlinear Schrödinger equations. The general equation in this family, including terms considered by Kostin, by BialynickiBirula and Mycielski, and by Doebner and Goldin, with timedependent coefficients, can be obtained from the linear Schrödinger equation through gauge transformation and a subsequent process we call gauge generalization. We thus unify, on fundamental grounds, a rather diverse set of nonlinear time evolutions in quantum mechanics.

A rigorous real time Feynman path integral
View Description Hide DescriptionUsing improper Riemann integrals, we will formulate a rigorous version of the realtime, timesliced Feynman path integral for the transition probability amplitude. We will do this for nonvector potential Hamiltonians with potential which has, at most, a finite number of discontinuities and singularities. We will also provide a Nonstandard Analysis version of our formulation.

Secondorder mixtures in relativistic Schrödinger theory
View Description Hide DescriptionIn relativistic Schrödinger theory the mixtures and pure states can be treated from a unified point of view such that a pure state merely emerges as a special case of a mixture. Here the concept of mixture is of purely local nature and therefore the mixture character (degree of order) can change over space and time. Although the general dynamics does not forbid the transitions from mixtures to pure states (and vice versa), the considered models do admit these transitions only in an asymptotic sense. The general concepts and results are demonstrated by considering the fourcomponent Dirac theory for spinning matter over the Robertson–Walker universes. A detailed study is made for a specific subclass of secondorder mixtures sharing many of their properties with the pure states (i.e., wave functions).

oscillator and Kepler problem: The case of nontrivial constraints
View Description Hide DescriptionThe topologically nontrivial correspondence between the Kepler problem and the singular oscillator is considered. It is shown that both “isospinor” and “isovector” particles on the instanton background in can be described in terms of an singular oscillator. The energy spectrum is calculated for an arbitrary “isospin” case.

Isolated versus nonisolated periodic orbits in variants of the twodimensional square and circular billiards
View Description Hide DescriptionSquare and circular infinite wells are among the simplest twodimensional potentials which can completely solved in both classical and quantum mechanics. Using the methods of periodic orbit theory, we study several variants of these planar billiard systems which admit both singular isolated and continuous classes of nonisolated periodic orbits. (In this context, isolated orbits are defined as those which are not members of a continuous family of paths whose orbits are all of the same length.) Examples include (i) various “folded” versions of the standard infinite wells (i.e., potentials whose geometrical shapes or “footprints” can be obtained by repeated folding of the basic square and circular shapes) and (ii) a square well with an infinitestrength repulsive function “core,” which is a special case of a Sinai billiard. In each variant case considered, new isolated orbits are introduced and their connections to the changes in the quantum mechanical energy spectrum are explored. Finally, we also speculate about the connections between the periodic orbit structure of supersymmetric partner potentials, using the twodimensional square well and it superpartner potential as a specific example.

Chaotic observables for a free quantum particle
View Description Hide DescriptionThis paper is devoted to the time evolution of observables in the quantum mechanics of a single particle without interaction. It is assumed that wave functions belong to a certain set that is dense in The paper applies to observables represented by positive selfadjoint operators A on with the property that maps into Quadratic forms with form domains are used to generate a topology for operators defining a topological space The space X provides the framework to define sensitive dependence on initial conditions, topological transitivity, and existence of a dense set of periodic points, the three aspects of chaos in Devaney’s definition of chaos for maps on metric spaces. It is shown that every neighborhood of every operator A in X contains operators that establish sensitive dependence on initial conditions, and similarly for the other aspects of chaos. Hence, the time evolution of operators in the Heisenberg picture is chaotic in the sense of this paper.

Nonlocal potentials, isolated states, and Levinson’s theorem
View Description Hide DescriptionA compact expression for the calculation of phase shifts is derived for a potential which is the sum of local and nonlocal parts. Nonlocal potentials can support positive energy bound states, that is, states embedded in the continuous energy spectrum. These states, sometimes referred to as “isolated” states, are not associated with any poles of the S matrix. Some controversy exists in the literature on how such bound states are included in Levinson’s theorem; it is found that the phase shift should be taken continuous at the energy of the bound state rather than taken to have a discontinuity of π. For simplicity, the analysis is restricted to the radial s wave Schrödinger equation and separable nonlocal potentials.

An estimate of the ground state energy of the fractional quantum Hall effect
View Description Hide DescriptionSuppose that there are N electrons in a disk of radius R with a perpendicular magnetic field. We give an estimate for the ground state energy of such a system in the case

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Rigorous application of the stochastic functional method to planewave scattering from a random cylindrical surface
View Description Hide DescriptionThe stochastic functional method is applied to planewave scattering from a random cylindrical surface, whereupon the Dirichlet boundary condition is rigorously imposed. Analytical results, accurate to second and fourth order in surface roughness, are obtained for the coefficients of the Wiener–Hermite expansion of the secondary scattered wave field. The validity of approximate solutions is numerically investigated by means of the boundary condition criterion and of the energy consistency criterion. The former, which is introduced herein, states that any approximate solution should be in conformity with the boundary condition, whereas the latter pertains to the energy conservation law. The numerical investigation indicates that the rigorous application of the stochastic functional method yields more accurate results in terms of both criteria than did previous treatments of the problem under consideration. Moreover, it is suggested that applicability limits should be set through the mean boundary condition criterion instead of the energy consistency criterion; the latter may lead to underestimating deficiencies of the approximate solution under test.

Lattice electromagnetic theory from a topological viewpoint
View Description Hide DescriptionThe language of differential forms and topological concepts are applied to study classical electromagnetictheory on a lattice. It is shown that differential forms and their discrete counterparts (cochains) provide a natural bridge between the continuum and the lattice versions of the theory, allowing for a natural factorization of the field equations into topological field equations (i.e., invariant under homeomorphisms) and metric field equations. The various potential sources of inconsistency in the discretization process are identified, distinguished, and discussed. A rationale for a consistent extension of the lattice theory to more general situations, such as to irregular lattices, is considered.

 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


Pseudoorthogonal groups and integrable dynamical systems in two dimensions
View Description Hide DescriptionIntegrable systems in low dimensions, constructed through the symmetry reduction method, are studied using phase portrait and variable separation techniques. In particular, invariant quantities and explicit periodic solutions are determined. Widely applied models in Physics are shown to appear as particular cases of the method.

Symmetries of Hamiltonian systems with two degrees of freedom
View Description Hide DescriptionWe classify the Lie point symmetry groups for an autonomous Hamiltonian system with two degrees of freedom. With the exception of the harmonic oscillator or a free particle where the dimension is 15, we obtain all dimensions between 1 and 7. For each system in the classification we examine integrability.

Superintegrability of the Calogero–Moser system: Constants of motion, master symmetries, and timedependent symmetries
View Description Hide DescriptionThe classical dimensional Calogero–Moser system is a maximally superintegrable system endowed with a rich variety of symmetries and constants of motion. In the first part of the article some properties related with the existence of several families of constants of motion are analyzed. In the second part, the master symmetries and the timedependent symmetries of this system are studied.

The study of dromion interactions of dimensional integrable systems
View Description Hide DescriptionStarting from a twoline soliton solution of an integrable dimensional system in bilinear form, one can find a dromion solution that is localized in all directions for a suitable potential. The interaction between two dromions is studied in detail through the method of figure analysis for a dimensional modified Korteweg–deVries (KdV) system, sine–Gordon system and Sawada–Kotera system. Except for a phase shift, there are no changes in the shape and velocity of the dromions after interactions for these models. The interactions of dromions for these models are not only elastic (there is no exchange of energy) but also irrotational (there is no exchange of angular momentum).

Spin and wave function as attributes of ideal fluid
View Description Hide DescriptionAn ideal fluid whose internal energy depends on density, density gradient, and entropy is considered. Dynamic equations are integrated, and a description in terms of hydrodynamic (Clebsch) potentials occurs. All essential information on the fluid flow (including initial and boundary conditions) appears to be carried by the dynamic equations for hydrodynamic potentials. Information on initial values of the fluid flow is carried by arbitrary integration functions. Initial and boundary conditions for potentials contain only nonessential information concerning the fluid particle labeling. It is shown that the description in terms of ncomponent complex wave function is a kind of such description in terms of hydrodynamic potentials. Spin determined by the irreducible number of the wave function components appears to be an attribute of the fluid flow. Classification of fluid flows by the spin appears to be connected with invariant subspaces of the relabeling group.

The Stäckel systems and algebraic curves
View Description Hide DescriptionWe show how the Abel–Jacobi map provides all the principal properties of an ample family of integrable mechanical systems associated to hyperelliptic curves. We prove that derivative of the Abel–Jacobi map is just the Stäckel matrix, which determines northogonal curvilinear coordinate systems in a flat space. The Lax pairs, rmatrix algebras and explicit form of the flat coordinates are constructed. An application of the Weierstrass reduction theory allows us to construct several flat coordinate systems on a common hyperelliptic curve and to connect among themselves different integrable systems on a single phase space.
