Volume 47, Issue 7, July 2006
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


A Finslerian version of ‘t Hooft deterministic quantum models
View Description Hide DescriptionUsing the Finsler structure living in the phase space associated to the tangent bundle of the configuration manifold, deterministic models at the Planck scale are obtained. The Hamiltonian functions are constructed directly from the geometric data and some assumptions concerning time inversion symmetry. The existence of a maximal acceleration and speed is proved for Finslerian deterministic models. We investigate the spontaneous symmetry breaking of the orthogonal symmetry of the Hamiltonian of a deterministic system. This symmetry break implies the nonvalidity of the argument used to obtain Bell’s inequalities for spin states. It is introduced and motivated in the context of Randers spaces, an example of a simple ’t Hooft model with interactions.

Quantum graphs as holonomic constraints
View Description Hide DescriptionWe consider the dynamics on a quantum graph as the limit of the dynamics generated by a oneparticle Hamiltonian in with a potential having a deep strict minimum on the graph, when the width of the well shrinks to zero. For a generic graph we prove convergence outside the vertices to the free dynamics on the edges. For a simple model of a graph with two edges and one vertex, we prove convergence of the dynamics to the one generated by the Laplacian with Dirichlet boundary conditions in the vertex.

Differential realization of pseudoHermiticity: A quantum mechanical analog of Einstein’s field equation
View Description Hide DescriptionFor a given pseudoHermitian Hamiltonian of the standard form: , we reduce the problem of finding the most general (pseudo)metric operator satisfying to the solution of a differential equation. If the configuration space is , this is a KleinGordon equation with a nonconstant mass term. We obtain a general series solution of this equation that involves a pair of arbitrary functions. These characterize the arbitrariness in the choice of . We apply our general results to calculate for the symmetric square well, an imaginary scattering potential, and a class of imaginary deltafunction potentials. For the first two systems, our method reproduces the known results in a straightforward and extremely efficient manner. For all these systems we obtain the most general up to secondorder terms in the coupling constants.

Phase space quantization and the operator moment problem
View Description Hide DescriptionWe consider questions related to a quantization scheme in which a classical variable on a phase space is associated with a (preferably unique) semispectral measure, such that the moment operators of are required to be of the form , with a suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in the Hilbert space of the quantum system. In particular, we investigate the situation where the map is implemented by the operator integral with respect to some fixed positive operator measure. The phase space is first taken to be an abstract measurable space, then a locally compact unimodular group, and finally , where we determine explicitly the relevant operators for certain variables , in the case where the quantization map is implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.

Representation of state property systems
View Description Hide DescriptionA “state property system” is the mathematical structure which models an arbitrary physical system by means of its set of states, its set of properties, and a relation of “actuality of a certain property for a certain state.” We work out a new axiomatization for standard quantum mechanics, starting with the basic notion of state property system, and making use of a generalization of the standard quantum mechanical notion of “superposition” for state property systems.

On the ground state energy for a magnetic Schrödinger operator and the effect of the DeGennes boundary condition
View Description Hide DescriptionMotivated by the GinzburgLandau theory of superconductivity, we estimate in the semiclassical limit the ground state energy of a magnetic Schrödinger operator with De Gennes boundary condition and we study the localization of the ground states. We exhibit cases when the De Gennes boundary condition has strong effects on this localization.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Scalar gauge theory on a cylinder
View Description Hide DescriptionIn this work, we study the complex bosonic matter coupled to the gauge fields on a cylinder, which has a circular space part. We extend the ideas used for fermions on a cylinder in [S. G. Rajeev and Guruswamy, Mod. Phys. Lett. A7, 3783 (1992)] to this case. Here, the normal ordering rule for bosonic bilinears is found. In the large limit, we can reformulate the whole problem in terms of a set of basic mesonic operators. The full equations of motion satisfied by these mesonic variables are too complicated. An approximation method, which is a kind of linearization, is suggested for this problem.

Laurent series expansion of a class of massive scalar oneloop integrals up to in terms of multiple polylogarithms
View Description Hide DescriptionIn a recent paper we have presented results for a set of massive scalar oneloop master integrals needed in the NNLO parton model description of the hadroproduction of heavy flavors. The oneloop integrals were evaluated in dimension and the results were presented in terms of a Laurent series expansion up to . We found that some of the coefficients contain a new class of functions which we termed the functions. The functions are defined in terms of onedimensional integrals involving products of logarithm and dilogarithm functions. In this paper we derive a complete set of algebraic relations that allow one to convert the functions of our previous approach to a sum of classical and multiple polylogarithms. Using these results we are now able to present the coefficients of the oneloop master integrals in terms of classical and multiple polylogarithms.
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 GENERAL RELATIVITY AND GRAVITATION


A Michelson interferometer in the field of a plane gravitational wave
View Description Hide DescriptionWe treat the problem of a Michelson interferometer in the field of a plane gravitational wave in the framework of general relativity. The arms of the interferometer are regarded as the world lines of the light beams, whose motion is determined by the HamiltonJacobi equation for a massless particle. In the case of a weak monochromatic wave we find that the formula for the delay of a light beam agrees with the result obtained by solving the linearized coupled EinsteinMaxwell equations. We also calculate this delay in the next (quadratic) approximation.

The orbital precession around oblate spheroids
View Description Hide DescriptionAn exact series will be given for the gravitational potential generated by an oblate gravitating source. To this end the corresponding EpsteinHubbell type elliptic integral is evaluated. The procedure is based on the Legendre polynomial expansion method and on combinatorial techniques. The result is of interest for gravitational models based on the linearity of the gravitational potential. The series approximation for such potentials is of use for the analysis of orbital motions around a nonspherical source. It can be considered advantageous that the analysis is purely algebraic. Numerical approximations are not required. As an important example, the expression for the orbital precession will be derived for an object orbiting around an oblate homogeneous spheroid.

The path topology and the causal completion
View Description Hide DescriptionIt is shown that the path topology of Hawking, King, and McCarthy can be extended to the causal completion of a globally hyperbolic Lorentzian manifold. The suggested topology is defined only in terms of chronological structures and is finer than the extended Alexandrov topology. It is also shown that a homeomorphism induces a conformal isomorphism and a homeomorphism in the extended Alexandrov topology.

Gravitational AharonovBohm effect due to weak fields
View Description Hide DescriptionWe study the behavior of relativistic quantum particles in the spacetimes generated by a rotating massive body and a moving mass current, in the weak field approximation. We solve the Dirac equation in these gravitational fields and calculate the currents associated with the particles. It is shown that these solutions and the currents depend on the angular momentum and on the velocity of the sources, in the cases of a massive rotating body and a moving mass current, respectively. These effects may be looked upon as a gravitational analog of the AharonovBohm effect.
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 DYNAMICAL SYSTEMS


PoincaréBirkhoff periodic orbits for mechanical Hamiltonian systems on
View Description Hide DescriptionHere, a version of the Arnol’d conjecture, first studied by Conley and Zehnder, giving a generalization of the PoincaréBirkhoff last geometricaltheorem, is proved inside Viterbo’s framework of the generating functions quadratic at infinity. We give brief overviews of some tools that are often utilized in symplectic topology.

Quasiperiodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems
View Description Hide DescriptionWe consider a class of ordinary differential equations describing onedimensional analytic systems with a quasiperiodic forcing term and in the presence of damping. In the limit of large damping, under some generic nondegeneracy condition on the force, there are quasiperiodic solutions which have the same frequency vector as the forcing term. We prove that such solutions are Borel summable at the origin when the frequency vector is either any onedimensional number or a twodimensional vector such that the ratio of its components is an irrational number of constant type. In the first case the proof given simplifies that provided in a previous work of ours. We also show that in any dimension , for the existence of a quasiperiodic solution with the same frequency vector as the forcing term, the standard Diophantine condition can be weakened into the Bryuno condition. In all cases, under a suitable positivity condition, the quasiperiodic solution is proved to describe a local attractor.

Longtime dynamics of variable coefficient modified Kortewegde Vries solitary waves^{a)}
View Description Hide DescriptionWe study the longtime behavior of solutions to the Kortewegde Vriestype equation, with initial conditions close to a stable, solitary wave. The coefficient is a bounded and slowly varying function, and is a nonlinearity. For a restricted class of nonlinearities, we prove that for long time intervals, such solutions have the form of the solitary wave, whose center and scale evolve according to a certain dynamical law involving the function, plus an small fluctuation. The result is stronger than those previously obtained for general nonlinearities .

Eigenvalue problems and their application to the wavelet method of chaotic control
View Description Hide DescriptionControlling chaos via wavelet transform was recently proposed by Wei, Zhan, and Lai [Phys. Rev. Lett.89, 284103 (2002)]. It was reported there that by modifying a tiny fraction of the wavelet subspace of a coupling matrix, the transverse stability of the synchronous manifold of a coupled chaotic system could be dramatically enhanced. The stability of chaotic synchronization is actually controlled by the second largest eigenvalue of the (wavelet) transformed coupling matrix for each and . Here is a mixed boundary constant and is a scalar factor. In particular, (respectively, 0) gives the nearest neighbor coupling with periodic (respectively, Neumann) boundary conditions. The first, rigorous work to understand the eigenvalues of was provided by Shieh et al. [J. Math. Phys. (to be published)]. The purpose of this paper is twofold. First, we apply a different approach to obtain the explicit formulas for the eigenvalues of and . This, in turn, yields some new information concerning . Second, we shed some light on the question whether the wavelet method works for general coupling schemes. In particular, we show that the wavelet method is also good for the nearest neighbor coupling with Neumann boundary conditions.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Diffraction of light by a planar aperture in a metallic screen
View Description Hide DescriptionWe present a complete derivation of the formula of Smythe [Phys. Rev.72, 1066 (1947)] giving the electromagnetic field diffracted by an aperture created in a perfectly conducting plane surface. The reasoning, valid for any excitating field and any hole shape, makes use only of the free scalar Green function for the Helmoltz equation without any reference to a Green dyadic formalism. We compare our proof with the one previously given by Jackson and connect our reasoning to the general Huygens Fresnel theorem.

Explicit actions for electromagnetism with two gauge fields with only one electric and one magnetic physical field
View Description Hide DescriptionWe extend the work of Mello et al. based on Cabbibo and Ferrari concerning the description of electromagnetism with two gauge fields from a variational principle, i.e., an action. We provide a systematic independent derivation of the allowed actions that have only one magnetic and one electric physical field and are invariant under the discrete symmetries and . We conclude that neither the Lagrangian, nor the Hamiltonian, are invariant under the electromagnetic duality rotations. This agrees with the weakstrong coupling mixing characteristic of the duality due to the Dirac quantization condition providing a natural way to differentiate dual theories related by the duality rotations (the energy is not invariant). Also, the standard electromagnetic duality rotations considered in this work violate both and by inducing Hopf terms (theta terms) for each sector and a mixed Maxwell term. The canonical structure of the theory is briefly addressed and the magnetic gauge sector is interpreted as a ghost sector.
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 STATISTICAL PHYSICS


Generalized OrnsteinUhlenbeck processes
View Description Hide DescriptionWe solve a physically significant extension of a classic problem in the theory of diffusion, namely the OrnsteinUhlenbeck process [Ornstein and Uhlenbeck, Phys. Rev.36, 823 (1930)]. Our generalized OrnsteinUhlenbeck systems include a force which depends upon the position of the particle, as well as upon time. They exhibit anomalous diffusion at short times, and nonMaxwellian velocity distributions in equilibrium. Two approaches are used. Some statistics are obtained from a closedform expression for the propagator of the FokkerPlanck equation for the case where the particle is initially at rest. In the general case we use spectral decomposition of a FokkerPlanck equation, employing nonlinear creation and annihilation operators to generate the spectrum which consists of two staggered ladders.

Heat equilibrium distribution in a turbulent flow
View Description Hide DescriptionWe consider a shear flow of a scale invariant Gaussian random velocity field that does not depend on the coordinates in the direction of the flow. We investigate a heat advection coming from a Gaussian random homogeneous source. We discuss a relaxation at large time of a temperature distribution determined by the forced advectiondiffusion equation. We represent the temperature correlation functions by means of the FeynmanKac formula. Jensen inequalities are applied for lower and upper bounds on the correlation functions. We show that at finite time there is no velocity dependence of long range temperature correlations (low momentum asymptotics) in the direction of the flow but the equilibrium heat distribution has large distance correlations (low momentum behavior) with an index depending on the scaling index of the random flow and of the index of the random forcing. If the velocity has correlations growing with the distance (a turbulent flow), then the large distance correlations depend in a crucial way on the scaling index of the turbulent flow. In such a case the correlations increase in the direction of the flow and decrease in the direction perpendicular to the flow, making the stream of heat more coherent.
