Volume 47, Issue 2, February 2006
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Remarkable identities related to the (quantum) elliptic CalogeroSutherland model
View Description Hide DescriptionWe present remarkable functional identities related to the elliptic CalogeroSutherland (eCS) system. We derive them from a second quantization of the eCS model within a quantum field theorymodel of anyons on a circle and at finite temperature. The identities involve two eCS Hamiltonians with arbitrary and, in general, different particle numbers and , and a particular function of variables arising as anyoncorrelation function of particles and antiparticles. In addition to identities obtained from anyons with the same statistics parameter , we also obtain “dual” relations involving “mixed” correlation functions of anyons with two different statistics parameters and . We also give alternative, elementary proofs of these identities by direct computations.

Optimal estimation of quantum observables
View Description Hide DescriptionWe consider the problem of estimating the ensemble average of an observable on an ensemble of equally prepared identical quantum systems. We show that, among all kinds of measurements performed jointly on the copies, the optimal unbiased estimation is achieved by the usual procedure that consists in performing independent measurements of the observable on each system and averaging the measurement outcomes.

Regge trajectories of the Coulomb potential in the space of constant negative curvature
View Description Hide DescriptionAnalytic properties of the scattering amplitude for Coulomb potential on the background of the space of constant negative curvature are studied. Special attention is given to the comparison of the Regge trajectories for curved and flat spaces. We show that there exist considerably differences in the behavior of the Regge trajectories in these spaces.

Quantum and Fisher information from the Husimi and related distributions
View Description Hide DescriptionThe two principal/immediate influences—which we seek to interrelate here—upon the undertaking of this study are papers of Życzkowski and Słomczyński [J. Phys. A34, 6689 (2001)] and of Petz and Sudár [J. Math. Phys.37, 2262 (1996)]. In the former work, a metric (the Monge one, specifically) over generalized Husimi distributions was employed to define a distance between two arbitrary density matrices. In the PetzSudár work (completing a program of Chentsov), the quantum analog of the (classically unique) Fisher information (monotone) metric of a probability simplex was extended to define an uncountable infinitude of Riemannian (also monotone) metrics on the set of positive definite density matrices. We pose here the questions of what is the specific/unique Fisher information metric for the (classically defined) Husimi distributions and how does it relate to the infinitude of (quantum) metrics over the density matrices of Petz and Sudár? We find a highly proximate (small relative entropy) relationship between the probability distribution (the quantum Jeffreys’ prior) that yields quantum universal data compression, and that which (following Clarke and Barron) gives its classical counterpart. We also investigate the Fisher information metrics corresponding to the escort Husimi, positive and certain Gaussian probability distributions, as well as, in some sense, the discrete Wigner pseudoprobability. The comparative noninformativity of prior probability distributions—recently studied by Srednicki [Phys. Rev. A71, 052107 (2005)]—formed by normalizing the volume elements of the various information metrics, is also discussed in our context.

Rigorous braket formalism and wave function operator for one particle quantum mechanics
View Description Hide DescriptionFollowing previous works dedicated to the mathematical meaning of the “braket” formalism [I. M. Gel’fand and G. E. Shilov, Generalized Functions (Academic, New York, 1964), Vol. I; J. P. Antoine, J. Math. Phys.10, 53 (1969); Yu. M. Berezanskii, Expansions of Selfadjoint Operators (American Mathematical Society, Providence, RI 1968); E. Prugovečki, J. Math. Phys.14, 1410 (1973); J. P. Antoine and A. Grossmann, J. Funct. Anal.23, 369 (1976); 23, 379 (1976)], we develop a new rigorous mathematical approach, based on an operator representation of bras and kets. This leads to a formalism very similar to second quantization. Welldefined operators associated with local observables can be exhibited, intimately related to previous works of E. Prugovečki [Stochastic Quantum Mechanics and Quantum SpaceTime (Reidel, Dordrecht, 1986)].

Largeorder behavior of the perturbation energies for the hydrogen atom in magnetic field
View Description Hide DescriptionLargeorder behavior for the perturbation energies of the hydrogen atom in magnetic field is derived. By means of the dispersion relations, the largeorder behavior of the series is determined by calculating the lifetime of the quasistationary states in an imaginary magnetic field. This problem is treated by means of the modified multidimensional WKB method. The asymptotic formula for the perturbation energies derived by Avron is generalized to the states with an arbitrary degeneracy. The first order correction to the resulting formula is also found. Thus, the multidimensional WKB method is for the first time explicitly carried out beyond the leading approximation. The analytical results are verified numerically and an excellent agreement between the two is found. The connection between our and conventional semiclassical approximation is also briefly discussed.

Mathematical analysis of a Bohr atom model
View Description Hide DescriptionBohr proposed in 1913 a model for atoms and molecules by synthesizing Planck’s quantum hypothesis with classical mechanics. When the atom number is small, his model provides good accuracy for the groundstate energy. When is large, his model is not as accurate in comparison with the experimental data but still provides a good trend agreeing with the experimental values of the groundstate energy of atoms. The main objective of this paper is to provide a rigorous mathematical analysis for the Bohr atom model. We have established the following: (1) An existence proof of the global minimizer of the groundstate energy through scaling. (2) A careful study of the critical points of the energy function. Such critical points include both the stable steadystate electron configurations as well as unstable saddletype configurations. (3) Coplanarity of certain electron configurations. Numerical examples and graphics are also illustrated.

New families of finite coherent orthoalgebras without bivaluations
View Description Hide DescriptionIn the present paper we study the following problem: how to construct a coherent orthoalgebra which has only a finite number of elements, but at the same time does not admit a bivaluation (i.e., a morphism with a codomain being an orthoalgebra with just two elements). This problem is important in the perspective of BellKochenSpecker theory, since one can associate such an orthoalgebra to every saturated noncolorable finite configuration of projective lines. The first result obtained in this paper provides a general method for constructing finite orthoalgebras. This method is then applied to obtain a new infinite family of finite coherent orthoalgebras that do not admit bivaluations. The corresponding proof is combinatorial and yields a description of the groups of symmetries for these orthoalgebras.

Upper and lower bounds for an eigenvalue associated with a positive eigenvector
View Description Hide DescriptionWhen an eigenvector of a semibounded operator is positive, we show that a remarkably simple argument allows to obtain upper and lower bounds for its associated eigenvalue. This theorem is a substantial generalization of Bartatype inequalities and can be applied to nonnecessarily purely quadratic Hamiltonians. An application for a magnetic Hamiltonian is given and the case of a discrete Schrödinger operator is also discussed. It is shown how this approach leads to some explicit bounds on the groundstate energy of a system made of an arbitrary number of attractive Coulombian particles.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Covariant canonical formalism for fourdimensional BF theory
View Description Hide DescriptionThe covariant canonical formalism for fourdimensional BF theory is performed. The aim of the paper is to understand in the context of the covariant canonical formalism both the reducibility that some first class constraints have in Dirac’s canonical analysis and also the role that topological terms play. The analysis includes also the cases when both a cosmological constant and the second Chern character are added to the pure BF action. In the case of the BF theory supplemented with the second Chern character, the presymplectic 3form is different to the one of the BF theory in spite of the fact both theories have the same equations of motion while on the space of solutions they both agree to each other. Moreover, the analysis of the degenerate directions shows some differences between diffeomorphisms and internal gauge symmetries.

Instability of coherent states of a real scalar field
View Description Hide DescriptionWe investigate stability of both localized timeperiodic coherent states (pulsons) and uniformly distributed coherent states (oscillating condensate) of a real scalar field satisfying the KleinGordon equation with a logarithmic nonlinearity. The linear analysis of timedependent parts of perturbations leads to the Hill equation with a singular coefficient. To evaluate the characteristic exponent we extend the LindemannStieltjes method, usually applied to the Mathieu and Lamé equations, to the case that the periodic coefficient in the general Hill equation is an unbounded function of time. As a result, we derive the formula for the characteristic exponent and calculate the stabilityinstability chart. Then we analyze the spatial structure of the perturbations. Using these results we show that the pulsons of any amplitudes, remaining welllocalized objects, lose their coherence with time. This means that, strictly speaking, all pulsons of the model considered are unstable. Nevertheless, for the nodeless pulsons the rate of the coherence breaking in narrow ranges of amplitudes is found to be very small, so that such pulsons can be longlived. Further, we use the obtained stabilityinstability chart to examine the AffleckDinetype condensate. We conclude the oscillating condensate can decay into an ensemble of the nodeless pulsons.

Integrability from an Abelian subgroup of the diffeomorphisms group
View Description Hide DescriptionIt has been known for some time that for a large class of nonlinear field theories in Minkowski space with twodimensional target space the complex eikonal equation defines integrable submodels with infinitely many conservation laws. These conservation laws are related to the areapreserving diffeomorphisms on target space. Here we demonstrate that for all these theories there exists, in fact, a weaker integrability condition which again defines submodels with infinitely many conservation laws. These conservation laws will be related to an Abelian subgroup of the group of areapreserving diffeomorphisms. As this weaker integrability condition is much easier to fulfill, it should be useful in the study of those nonlinear field theories.
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 GENERAL RELATIVITY AND GRAVITATION


Global visibility of naked singularities
View Description Hide DescriptionGlobal visibility of naked singularities is analyzed here for a class of spherically symmetric spacetimes, extending previous studies—limited to inhomogeneous dust cloud collapse—to more physical valid situations in which pressures are nonvanishing. Existence of nonradial geodesics escaping from the singularity is shown, and the observability of the singularity from faraway observers is discussed.

Killing vectors in asymptotically flat spacetimes. II. Asymptotically translational Killing vectors and the rigid positive energy theorem in higher dimensions
View Description Hide DescriptionWe show that the borderline cases in the proof of the positive energy theorem for initial data sets, on spin manifolds, in dimensions , are only possible for initial data arising from embeddings in Minkowski spacetime.

A systematic derivation of the Riemannian BarrettCrane intertwiner
View Description Hide DescriptionThe BarrettCrane intertwiner for the Riemannian general relativity is systematically derived by solving the quantum BarrettCrane constraints corresponding to a tetrahedron (except for the nondegeneracy condition). It was shown by Reisenberger that the BarrettCrane intertwiner is the unique solution. The systematic derivation can be considered as an alternative proof of the uniqueness. The new element in the derivation is the rigorous imposition of the crosssimplicity constraint.
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 DYNAMICAL SYSTEMS


Analysis of a particle antiparticle description of a soliton cellular automaton
View Description Hide DescriptionWe present a derivation of a formula that gives dynamics of an integrable cellular automation associated with crystal bases. This automaton is related to type affine Lie algebra and contains usual boxball systems as a special case. The dynamics is described by means of such objects as carriers, particles, and antiparticles. We derive it from an analysis of a recently obtained formula of the combinatorial (an intertwiner between tensor products of crystals) that was found in a study of geometric crystals.

Positive Lyapunov exponents for continuous quasiperiodic Schrödinger equations
View Description Hide DescriptionWe prove that the continuous onedimensional Schrödinger equation with an analytic quasiperiodic potential has positive Lyapunov exponents in the bottom of the spectrum for large couplings.

Novel solvable variants of the goldfish manybody model
View Description Hide DescriptionA recent technique to identify solvablemanybody problems in twodimensional space yields, via a new twist, new manybody problems of “goldfish” type. Some of these models are isochronous, namely their genericsolutions are completely periodic with a fixed period (independent of the initial data). The investigation of the behavior of some of these isochronous systems in the vicinity of their equilibrium configurations yields some amusing diophantine relations.

Formal and analytical integrability of the Bianchi IX system
View Description Hide DescriptionIn this paper we provide a complete description of the first integrals of the classical Bianchi IX system that can be described by a general class of formal power series. As a corollary we also obtain a complete description of some of its analytic first integrals in a neighborhood of the origin. In particular, we prove that the system is not completely integrable by analytic first integrals.

Discrete dynamical systems embedded in Cantor sets
View Description Hide DescriptionWhile the notion of chaos is well established for dynamical systems on manifolds, it is not so for dynamical systems over discrete spaces with variables, as binary neural networks and cellular automata. The main difficulty is the choice of a suitable topology to study the limit . By embedding the discrete phase space into a Cantor set we provided a natural setting to define topological entropy and Lyapunov exponents through the concept of error profile. We made explicit calculations both numerical and analytic for wellknown discrete dynamical models.
