Volume 43, Issue 2, February 2002
Index of content:
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Covariant localizations in the torus and the phase observables
View Description Hide DescriptionWe describe all the localization observables of a quantum particle in a onedimensional box in terms of sequences of unit vectors in a Hilbert space. An alternative representation in terms of positive semidefinite complex matrices is furnished and the commutative localizations are singled out. As a consequence, we also get a vector sequence characterization of the covariant phase observables.

Global time asymmetry as a consequence of a wave packets theorem
View Description Hide DescriptionWhen any wave packet in the Liouvillian representation of the density matrices becomes a Hardy class function from below. This fact, in the global frame of the Reichenbach diagram, is used to explain the observed global time asymmetry of the universe.

Generalized coherent and intelligent states for exact solvable quantum systems
View Description Hide DescriptionThe socalled Gazeau–Klauder and Perelomov coherent states are introduced for an arbitrary quantum system. We give also the general framework to construct the generalized intelligent states which minimize the Robertson–Schrödinger uncertainty relation. As illustration, the Pöschl–Teller potentials of trigonometric type will be chosen. We show the advantage of the analytical representations of Gazeau–Klauder and Perelomov coherent states in obtaining the generalized intelligent states in analytical way.

Semiclassical properties and chaos degree for the quantum Baker’s map
View Description Hide DescriptionWe study the chaotic behavior and the quantumclassical correspondence for the Baker’s map. Correspondence between quantum and classical expectation values is investigated and it is numerically shown that it is lost at the logarithmic timescale. The quantum chaos degree is computed and it is demonstrated that it describes the chaotic features of the model. The correspondence between classical and quantum chaos degrees is considered.

Symplectic areas, quantization, and dynamics in electromagnetic fields
View Description Hide DescriptionA gauge invariant quantization in a closed integral form is developed over a linear phase space endowed with an inhomogeneous Faraday electromagnetic tensor. An analog of the Groenewold product formula (corresponding to Weyl ordering) is obtained via a membrane magnetic area, and extended to the product of symbols. The problem of ordering in quantization is related to different configurations of membranes: A choice of configuration determines a phase factor that fixes the ordering and controls a symplectic groupoid structure on the secondary phase space. A gauge invariant solution of the quantum evolution problem for a charged particle in an electromagnetic field is represented in an exact continual form and in the semiclassical approximation via the area of dynamical membranes.

Ensemble fluctuations and the origin of quantum probabilistic rule
View Description Hide DescriptionWe demonstrate that the origin of the socalled quantum probabilistic rule (which differs from the classical Bayes’ formula by the presence of factor) might be explained in the framework of ensemble fluctuations which are induced by preparation procedures. In particular, quantum rule for probabilities (with nontrivial factor) could be simulated for macroscopic physical systems via preparation procedures producing ensemble fluctuations of a special form. We discuss preparation and measurement procedures which may produce probabilistic rules which are neither classical nor quantum; in particular, hyperbolic “quantum theory.”

Global minimizer for the GinzburgLandau functional of an inhomogeneous superconductor
View Description Hide DescriptionIn this paper, we prove that the global minimizer of the GinzburgLandau functional of an inhomogeneous superconductor in an external magnetic field below the first critical field is the vortexless solution. Here, is defined as the value of the applied field, for which the minimal energy among vortexless configurations is equal to the minimal energy among singlevortex configurations.

A spectral quadruple for de Sitter space
View Description Hide DescriptionA set of data supposed to give possible axioms for spacetimes with a sufficient number of isometries in spectral geometry is given. These data are shown to be sufficient to obtain dimensional de Sitter spacetime. The data rely at the moment somewhat on the guidance given by a required symmetry, in part to allow explicit calculations in a specific model. The framework applies also to the noncommutative case. Finite spectral triples are discussed as an example.

Baxter TQ equation for shape invariant potentials. The finitegap potentials case
View Description Hide DescriptionThe Darboux transformation applied recurrently on a Schrödinger operator generates what is called a dressing chain, or from a different point of view, a set of supersymmetric shape invariant potentials. The finitegap potential theory is a special case of the chain. For the finitegap case, the equations of the chain can be expressed as a time evolution of a Hamiltonian system. We apply Sklyanin’s method of separation of variables to the chain. We show that the classical equation of the separation of variables is the Baxter TQ relation after quantization.

Regularizing divergences in the von Neumann entropy
View Description Hide DescriptionWe study the decoherence process of a harmonic oscillator in a dissipative environment by considering the von Neumann entropy. Derivatives of the von Neumann entropy around the initial time exhibit divergences when the system is initially in a pure state. A regularization procedure based on the zeta function technique is considered in order to extract information about decoherence.

Noncommutative cohomological field theory and GMS soliton
View Description Hide DescriptionWe show that it is possible to construct a quantum field theory that is invariant under the translation of the noncommutative parameter This is realized in a noncommutative cohomological field theory. As an example, a noncommutative cohomological scalar field theory is constructed, and its partition function is calculated. The partition function is the Euler number of Gopakumar, Minwalla, and Strominger (GMS) soliton space.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Closure of orbits and dynamical symmetry of screened Coulomb potential and isotropic harmonic oscillator
View Description Hide DescriptionIt is shown that for any central potential there exist a series of conserved aphelion and perihelion vectors However, if and only if is a pure or screened Coulomb potential, and L constitute an algebra in the subspace spanned by the degenerate states with a given energy eigenvalue at the aphelia and perihelia For a pure Coulomb potential, is reduced to the Pauli–Runge–Lenz (PRL) vector R and for a screened Coulomb potential is reduced to the extended PRL vector While always holds, holds only at the aphelia and perihelia. Moreover, the space spanning the algebra for a screened Coulomb potential is smaller than that for a pure Coulomb potential. The relation of closed orbits for a screened Coulomb potential with that for a pure Coulomb potential is clarified. The ratio of the radial frequency and angular frequency for a pure Coulomb potential irrespective of the angular momentum and energy For a screened Coulomb potential κ is determined by the angular momentum and when κ is any rational number (κ<1), the orbit is closed. The situation for a pure or screened isotropic harmonic oscillator is similar.

 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


On the Kolmogorovlike generalization of Tsallis entropy, correlation entropies and multifractal analysis
View Description Hide DescriptionThe generalization à la Kolmogorov of Tsallis entropy, introduced by the authors in a previous work [J. Math. Phys. 37, 4480 (1996)], is revisited. Invariance properties are pointed out under weaker conditions than before. This result leads us to wonder if Tsallis entropy at the Kolmogorov abstraction level brings new information with respect to the generalization that Kolmogorov did of Shannon entropy. The negative answer motivates us to look for other generalizations of Tsallis entropy in order to avoid the lack of new information. Correlation entropies seem to be good candidates for this purpose. The relationship of this kind of entropy with the multifractal analysis is studied with the help of the thermodynamic formalism. We also outline its usefulness to generalize properties of Tsallis entropy.

Short distance asymptotics of Ising correlations
View Description Hide DescriptionWe prove that the short distance asymptotics for the even Ising model scaling functions from below is given by the Luther–Peschel formula. Generalizations to the odd scaling functions and holonomic fields are given.

 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


Classical Hamiltonian systems with coalgebra symmetry and their integrable deformations
View Description Hide DescriptionSeveral families of classical integrable systems with two degrees of freedom are derived from phasespace realizations of Poisson coalgebras. As a remarkable fact, the existence of the dimensional integrable generalization of all these systems is always ensured (by construction) due to their underlying dynamical coalgebra symmetry. By following the same approach, different integrable deformations for such systems are obtained from the deformed analogues of The wellknown JordanSchwinger realization is also proven to be related to a (noncoassociative) coalgebra structure on and the dimensional integrable Hamiltonian generated by such JordanSchwinger representation is obtained. Finally, the relation between complete integrability and the properties of the initial phasespace realization is elucidated through two more examples based on the HeisenbergWeyl and Poisson coalgebras.

Superintegrability in a twodimensional space of nonconstant curvature
View Description Hide DescriptionA Hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion. This property has been extensively studied in the case of twodimensional spaces of constant (possibly zero) curvature when all the independent integrals are either quadratic or linear in the canonical momenta. In this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions. This is done by examining in detail one of the spaces of revolution found by G. Koenigs. We determine that there are essentially three distinct potentials which when added to the free Hamiltonian of this space have this type of superintegrability. Separation of variables for the associated Hamilton–Jacobi and Schrödinger equations is discussed. The classical and quantum quadratic algebras associated with each of these potentials are determined.

Dark soliton generation for the intermediate nonlinear Schrödinger equation
View Description Hide DescriptionThe generation of dark solitons due to a small change of the initial data is studied within the framework of the intermediate nonlinear Schrödinger (INLS) equation. In particular, we analyze the spectral problem associated with the INLS equation when the potential consists of a small perturbation imposed on a constant background. We derive a criterion for the perturbation to generate a pair of new discrete eigenvalues as well as their explicit expressions in terms of the perturbation. In addition, we demonstrate that the eigenvalues appear without a threshold on the magnitude of the perturbation. We also consider both the shallow and deepwater limits of various results obtained for the INLS spectral problem. In the former case, the limiting procedure can be performed smoothly whereas in the latter case, the spectral equation exhibits a new feature of bound states, showing that the eigenvalue is exponentially small compared with the magnitude of the perturbation.

Remarks on several 2+1 dimensional lattices
View Description Hide DescriptionIn this paper, five 2+1 dimensional lattices considered by several authors are revisited again. First of all we will show that two lattices proposed by Blaszak and Szum [J. Math. Phys. 42, 225 (2001)] become the socalled differentialdifference KP equation due to Date, Jimbo, and Miwa [J. Phys. Soc. Jpn. 51, 4116 (1982); 51, 4125 (1982); 52, 388 (1983); 52, 761 (1983); 52, 766 (1983)] by simple variable transformations, while another lattice found by Blaszak and Szum can be viewed as a higherdimensional generalization of a lattice given by Wu and Hu [J. Phys. A 32, 1515 (1999)]. Some integrable properties on these three lattices are derived. Second, it is shown that a 2+1 dimensional Todalike lattice studied by Cao, Geng, and Wu [J. Phys. A 32, 8059 (1999)] can be transformed into the bilinear equation given by Hu, Clarkson, and Bullough [J. Phys. A 30, L669 (1997)]. For this bilinear version we also present some rational solutions and Lie symmetries. Finally, a lattice due to Levi, Ragnisco, and Shabat [Can. J. Phys. 72, 439 (1994)] is transformed into coupled bilinear equations. It is shown that these coupled bilinear equations do not have twosoliton solutions. This further confirms that the lattice under consideration is not completely integrable.

 RELATIVITY AND GRAVITATION


Euclidean scalar Green function in a higher dimensional global monopole space–time
View Description Hide DescriptionWe construct the explicit Euclidean scalar Green function associated with a massless field in a higher dimensional global monopole space–time, i.e., a space–time with which presents a solid angle deficit. Our result is expressed in terms of an infinite sum of products of Legendre functions with Gegenbauer polynomials. Although this Green function cannot be expressed in a closed form, for the specific case where the solid angle deficit is very small, it is possible to develop the sum and obtain the Green function in a more workable expression. Having this expression it is possible to calculate the vacuum expectation value of some relevant operators. As an application of this formalism, we calculate the renormalized vacuum expectation value of the square of the scalar field, and the energymomentum tensor, for the global monopole space–time with spatial dimensions and

Integrable cases of gravitating static isothermal fluid spheres
View Description Hide DescriptionIt is shown that different approaches toward the solution of the Einstein equations for a static spherically symmetric perfect fluid with a γlaw equation of state lead to an Abel differential equation of the second kind. Its only integrable cases at present are flat space–time, de Sitter solution and its Buchdahl transform, Einstein static universe, and the Klein–Tolman solution.
