Volume 48, Issue 3, March 2007
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Transport properties of quasifree fermions
View Description Hide DescriptionUsing the scattering approach to the construction of nonequilibrium steady states proposed by Ruelle [J. Stat. Phys.98, 57–75 (2000)], we study the transport properties of systems of independent electrons. We show that LandauerBüttiker and GreenKubo formulas hold under very general conditions.

Exact path integral treatment of a diatomic molecule potential
View Description Hide DescriptionA rigorous evaluation of the path integral for Green’s function associated with a fourparameter potential for a diatomic molecule is presented. A closed form of Green’s function is obtained for different shapes of this potential. When the deformation parameter is or , it is found that the quantization conditions are transcendental equations that require a numerical solution. For and , the energy spectrum and the normalized wave functions of the bound states are derived. Particular cases of this potential which appear in the literature are also briefly discussed.

Coupling in the singular limit of thin quantum waveguides
View Description Hide DescriptionWe analyze the problem of approximating a smooth quantum waveguide with a quantum graph. We consider a planar curve with compactly supported curvature and a strip of constant width around the curve. We rescale the curvature and the width in such a way that the strip can be approximated by a singular limit curve, consisting of one vertex and two infinite, straight edges, i.e., a broken line. We discuss the convergence of the Laplacian, with Dirichlet boundary conditions on the strip, in a suitable sense and we obtain two possible limits: the Laplacian on the line with Dirichlet boundary conditions in the origin and a nontrivial family of point perturbations of the Laplacian on the line. The first case generically occurs and corresponds to the decoupling of the two components of the limit curve, while in the second case a coupling takes place. We present also two families of curves which give rise to coupling.

Semiclassical coherentstate propagator for many particles
View Description Hide DescriptionWe obtain the semiclassical coherentstate propagator for a manyparticle system with an arbitrary Hamiltonian.

MICZKepler problems in all dimensions
View Description Hide DescriptionThe Kepler problem is a physical problem about two bodies which attract each other by a force proportional to the inverse square of the distance. The MICZKepler problems are its natural cousins and have been previously generalized from dimension 3 to dimension 5. In this paper, we construct and analyze the (quantum) MICZKepler problems in all dimensions higher than 2.

Quantum probabilities for timeextended alternatives
View Description Hide DescriptionWe study the probability assignment for the outcomes of timeextended measurements. We construct the class operator that incorporates the information about a generic timesmeared quantity. These class operators are employed for the construction of positiveoperatorvalued measures for the timeaveraged quantities. The scheme highlights the distinction between velocity and momentum in quantum theory. Propositions about velocity and momentum are represented by different class operators, hence they define different probability measures. We provide some examples, we study the classical limit, and we construct probabilities for generalized timeextended phase space variables.

deformations and vector coherent states of the JaynesCummings model in the rotating wave approximation
View Description Hide DescriptionClasses of deformations of the JaynesCummings model in the rotating wave approximation are considered. Diagonalization of the Hamiltonian is performed exactly, leading to useful spectral decompositions of a series of relevant operators. The latter include ladder operators acting between adjacent energy eigenstates within two separate infinite discrete towers, except for a singleton state. These ladder operators allow for the construction of deformed vector coherent states. Using arithmetics, explicit and exact solutions to the associated moment problem are displayed, providing new classes of coherent states for such models. Finally, in the limit of decoupled spin sectors, our analysis translates into deformations of the supersymmetric harmonic oscillator, such that the two supersymmetric sectors get intertwined through the action of the ladder operators as well as in the associated coherent states.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


On the mathematical structure and hidden symmetries of the BornInfeld field equations
View Description Hide DescriptionThe mathematical structure of the BornInfeld field equations was analyzed from the point of view of the symmetries. To this end, the field equations were written in the most compact form by means of quaternionic operators constructed according to all the symmetries of the theory, including the extension to a noncommutative structure. The quaternionic structure of the phase space was explicitly derived and described from the Hamiltonian point of view, and the analogy between the BornInfeld theory and the Maxwell (linear) electrodynamics in curved spacetime was explicitly shown. Our results agree with the observation of Gibbons and Rasheed [Nucl. Phys. B454, 185 (1995); Phys. Lett. B365, 46 (1996)] that there exists a discrete symmetry in the structure of the field equations that is unique in the case of the BornInfeld nonlinear electrodynamics.

Conservation laws in Skyrmetype models
View Description Hide DescriptionThe zero curvature representation of Zakharov and Shabat [V. E. Zakharov and A. B. Shabat, Soviet Phys. JETP34, 62 (1972)] has been generalized recently to higher dimensions and has been used to construct nonlinear field theories which are integrable or contain integrable submodels. The Skyrme model, for instance, contains an integrable subsector with infinitely many conserved currents, and the simplest Skyrmion with baryon number 1 belongs to this subsector. Here we use a related method, based on the geometry of target space, to construct a whole class of theories which are integrable or contain integrable subsectors (where integrability means the existence of infinitely many conservation laws). These models have threedimensional target space, like the Skyrme model, and their infinitely many conserved currents turn out to be Noether currents of the volumepreserving diffeomorphisms on target space. Specifically for the Skyrme model, we find both weak and strong integrability conditions, where the conserved currents form a subset of the algebra of volumepreserving diffeomorphisms in both cases, but this subset is a subalgebra only for the weak integrable submodel.

KazhdanLusztigdual quantum group for logarithimic extensions of Virasoro minimal models
View Description Hide DescriptionWe derive and study a quantum group that is KazhdanLusztig dual to the algebra of the logarithmic conformal field theorymodel. The algebra is generated by two currents and of dimension and the energymomentum tensor. The two currents generate a vertexoperator ideal with the property that the quotient is the vertexoperator algebra of the Virasoro minimal model. The number of irreducible representations is the same as the number of irreducible representations on which acts nontrivially. We find the center of and show that the modular group representation on it is equivalent to the modular group representation on the characters and “pseudocharacters.” The factorization of the ribbon element leads to a factorization of the modular group representation on the center. We also find the Grothendieck ring, which is presumably the “logarithmic” fusion of the model.

 GENERAL RELATIVITY AND GRAVITATION


On recovering continuum topology from a causal set
View Description Hide DescriptionAn important question that discrete approaches to quantum gravity must address is how continuum features of spacetime can be recovered from the discrete substructure. Here, we examine this question within the causal set approach to quantum gravity, where the substructure replacing the spacetime continuum is a locally finite partial order. A new topology on causal sets using “thickened antichains” is constructed. This topology is then used to recover the homology of a globally hyperbolic spacetime from a causal set which faithfully embeds into it at sufficiently high sprinkling density. This implies a discretecontinuum correspondence which lends support to the fundamental conjecture or “Hauptvermutung” of causal set theory.

 DYNAMICAL SYSTEMS


On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator
View Description Hide DescriptionUsing the modified PrelleSinger approach, we point out that explicit time independent first integrals can be identified for the damped linear harmonic oscillator in different parameter regimes. Using these constants of motion, an appropriate Lagrangian and Hamiltonian formalism is developed and the resultant canonical equations are shown to lead to the standard dynamical description. Suitable canonical transformations to standard Hamiltonian forms are also obtained. It is also shown that a possible quantum mechanical description can be developed either in the coordinate or momentum representations using the Hamiltonian forms.

compact uniform attractors for a nonautonomous incompressible nonNewtonian fluid with locally uniformly integrable external forces in distribution space
View Description Hide DescriptionWe consider the longtime behavior of solutions for a twodimensional nonautonomous incompressible nonNewtonian fluid with external forces in distribution space. When the external force is locally uniformly integrable (see Definition 3.1) in , we prove the existence of compact uniform attractor in space and reveal its structure for the families of processes corresponding to the fluid. Moreover, if is properly small, we establish the unique existence of bounded asymptotically stable solutions and give two interesting corollaries.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Global actionangle coordinates for completely integrable systems with noncompact invariant submanifolds
View Description Hide DescriptionThe obstruction to the existence of global actionangle coordinates of Abelian and noncommutative (nonAbelian) completely integrable systems with compact invariant submanifolds has been studied. We extend this analysis to the case of noncompact invariant submanifolds.

EulerPoincaré reduction for discrete field theories
View Description Hide DescriptionIn this note, we develop a theory of EulerPoincaré reduction for discrete Lagrangian field theories. We introduce the concept of EulerPoincaré equations for discrete field theories, as well as a natural extension of the MoserVeselov scheme, and show that both are equivalent. The resulting discrete field equations are interpreted in terms of discrete differential geometry. An application to the theory of discrete harmonic mappings is also briefly discussed.

Chetaev versus vakonomic prescriptions in constrained field theories with parametrized variational calculus
View Description Hide DescriptionStarting from a characterization of admissible Chetaev and vakonomic variations in a field theory with constraints we show how the so called parametrized variational calculus can help to derive the vakonomic and the nonholonomic field equations. We present an example in field theory where the nonholonomic method proved to be unphysical.

 STATISTICAL PHYSICS


Duality and exact correlations for a model of heat conduction
View Description Hide DescriptionWe study a model of heat conduction with stochastic diffusion of energy. We obtain a dual particle process which describes the evolution of all the correlation functions. An exact expression for the covariance of the energy exhibits longrange correlations in the presence of a current. We discuss the formal connection of this model with the simple symmetric exclusion process.

Independent electron model for open quantum systems: LandauerBüttiker formula and strict positivity of the entropy production
View Description Hide DescriptionA general argument leading from the formula for currents through an open noninteracting mesoscopic system given by the theory of nonequilibrium steady states to the LandauerBüttiker formula is pointed out. Time reversal symmetry is not assumed. As a consequence it is shown that, as far as the system has a nontrivial scattering theory and the reservoirs have different temperatures and/or chemical potentials, the entropy production is strictly positive.

Symmetries in coherence theory of partially polarized light
View Description Hide DescriptionThe symmetry properties of the second order statistical characteristics of partially polarized and partially coherent light that appear when one multiplies the electric fields by nonsingular Jones matrices at two different points in the spacetime domain are analyzed. Since the application of such deterministic linear transformations does not introduce random fluctuations between the two fields, they do not modify their intrinsic coherenceproperties. For a better understanding of coherenceproperties, it is thus interesting to analyze the symmetry properties introduced by these transformations. It is first shown that the invariant subsets generated by the action of deterministic nonsingular Jones matrices applied to each electric field correspond to constant intrinsic degrees of coherence. Then, the subgroup of transformations that keep invariant the second order statistical properties is determined and it is demonstrated that four kinds of different symmetry groups can appear. It is shown that the recently introduced intrinsic degrees of coherence allow one to characterize the kind of symmetry group of the analyzed light, while such a characterization cannot be obtained with a unique scalar degree of coherence. A practical consequence of this result is that two degrees of coherence are necessary to characterize the kind of symmetry of the second order statistical properties of partially coherent and partially polarized light.

 METHODS OF MATHEMATICAL PHYSICS


Systematic investigation of twodimensional static array sums
View Description Hide DescriptionWe discuss general properties of doubly periodic sums over the square lattice, linking phased, Blochtype sums in the direct lattice with displaced sums in the reciprocal lattice using the Poisson summation formula. We discuss cardinal points, where the sums reduce to a single product of Dirichlet functions, and exhibit all cardinal points for the square lattice. We introduce a new method for evaluating lattice sums and illustrate this by solving low order systems of sums of integer order 2,3,4,5. For the last case, the analytic expressions for the sums involve complementary functions, or alternatively functions with complex characters.
