Volume 52, Issue 7, July 2011
Index of content:
 ARTICLES


Quantum Mechanics (General and Nonrelativistic)

On the SO(10, 2) dynamical symmetry group of the MICZKepler problem in a ninedimensional space
View Description Hide DescriptionWe propose an effective algebraic method to investigate the dynamical symmetry of a 9dimensional MICZKepler problem by using the connection between this problem and a 16dimensional isotropic harmonic oscillator. The dynamical symmetry group of the considered problem is found as SO(10,2). Explicit forms of all group elements are given. We also obtain all group elements in the algebraic representation of annihilation and creation operators that are very useful for concrete calculations.

Pseudobosons arising from standard ladder operators
View Description Hide DescriptionPseudobosons in the form: , with are considered, the α's and β's being real coefficients which depend on real parameters s _{1}, …, s _{ n }. The eigenstates of the two number operators and their norm are explicitly obtained. Pseudobosons in Bagarello's sense are recovered: the states form two sets of biorthogonal bases of the full Hilbert space, but Riesz bases are obtained only in the ordinary bosonic case. Some examples of this setting are analyzed in detail.

The Casimir spectrum revisited
View Description Hide DescriptionWe examine the mathematical and physical significance of the spectral density σ(ω) introduced by Ford [Phys. Rev. D38, 528 (1988)]10.1103/PhysRevD.38.528, defining the contribution of each frequency to the renormalisedenergy density of a quantum field. Firstly, by considering a simple example, we argue that σ(ω) is well defined, in the sense of being regulator independent, despite an apparently regulator dependent definition. We then suggest that σ(ω) is a spectral distribution, rather than a function, which only produces physically meaningful results when integrated over a sufficiently large range of frequencies and with a high energy smooth enough regulator. Moreover, σ(ω) is seen to be simply the difference between the bare spectral density and the spectral density of the reference background. This interpretation yields a simple “rule of thumb” to writing down a (formal) expression for σ(ω) as shown in an explicit example. Finally, by considering an example in which the sign of the Casimir force varies, we show that the spectrum carries no manifest information about this sign; it can only be inferred by integrating σ(ω).

The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach
View Description Hide DescriptionThe quantum free particle on the sphere (κ > 0) and on the hyperbolic plane (κ < 0) is studied using a formalism that considers the curvature κ as a parameter. The first part is mainly concerned with the analysis of some geometric formalisms appropriate for the description of the dynamics on the spaces (, , ) and with the transition from the classical κdependent system to the quantum one using the quantization of the Noether momenta. The Schrödinger separability and the quantum superintegrability are also discussed. The second part is devoted to the resolution of the κdependent Schrödinger equation. First the characterization of the κdependent “curved” plane waves is analyzed and then the specific properties of the spherical case are studied with great detail. It is proved that if κ > 0 then a discrete spectrum is obtained. The wavefunctions, that are related with a κdependent family of orthogonal polynomials, are explicitly obtained.

Uncertainty relations for joint localizability and joint measurability in finitedimensional systems
View Description Hide DescriptionTwo quantities quantifying uncertainty relations are examined. Busch and Pearson [J. Math. Phys.48, 082103 (2007)] investigated the limitation on joint localizability and joint measurement of position and momentum by introducing overall width and error bar width. In this paper, we show a simple relationship between these quantities for finitedimensional systems. Our result indicates that if there is a bound on joint localizability, it is possible to obtain a similar bound on joint measurability. For finitedimensional systems, uncertainty relations for a pair of general projectionvalued measures are obtained as byproducts.

Localized spectral asymptotics for boundary value problems and correlation effects in the free Fermi gas in general domains
View Description Hide DescriptionWe rigorously derive explicit formulae for the pair correlation function of the ground state of the free Fermi gas in the thermodynamic limit for general geometries of the macroscopic regions occupied by the particles and arbitrary dimension. As a consequence we also establish the asymptotic validity of the local density approximation for the corresponding exchange energy. At constant density these formulae are universal and do not depend on the geometry of the underlying macroscopic domain. In order to identify the correlation effects in the thermodynamic limit, we prove a local Weyl law for the spectral asymptotics of the Laplacian for certain quantum observables which are themselves dependent on a small parameter under very general boundary conditions.

Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

Generalized selfduality equations of polynomial type in YangMills theories
View Description Hide DescriptionThe purpose of this paper is to generalize the selfduality equation by Tchrakian and Corrigan et al.Novel generalized selfduality equations on higherdimensional spaces are discussed. This class of equations includes the usual selfduality equation for fourdimensional spaces. Some of the generalized selfduality equations overdetermine configurations and the existence of solutions is not trivial. Several examples of solutions of the equations are demonstrated. As an application of the equations, it is proved that some of those solutions solve the equations of motion derived from rotationally invariant actions, which consist of singletrace terms and are second order in the time derivative.

On solutions to the “FaddeevNiemi” equations
View Description Hide DescriptionRecently it has been pointed out that the socalled FaddeevNiemi equations that describe the YangMillsequations of motion in terms of a decomposed gauge field, can have solutions that obey the standard YangMillsequations with a source term. Here we argue that the source term is covariantly constant. Furthermore, we find that there are solutions of the YangMillsequation with a covariantly constant source term that are not solutions to the FaddeevNiemi equations. We also present a general class of gauge field configurations that obey the FaddeevNiemi equation but do not solve the YangMillsequation. We propose that these configurations might have physical relevance in a strongly coupled phase, where spincharge separation takes place and the YangMillstheory cannot be described in terms of a Landau liquid of asymptotically free gluons.

Extensions of the Poincaré group
View Description Hide DescriptionWe construct an extension of the Poincaré group which involves a mixture of internal and spacetime supersymmetries. The resulting group is an extension of the superPoincaré group with infinitely many generators which carry internal and spacetime indices. It is a closed algebra since all Jacobi identities are satisfied and it has, therefore, explicit matrix representations. We investigate the massless case and construct the irreducible representations of the extended symmetry. They are divided into two sets, longitudinal and transversal representations. The transversal representations involve an infinite series of integer and halfinteger helicities. Finally, we suggest an extension of the conformal group along the same line.

General Relativity and Gravitation

4dimensional spinfoam model with quantum Lorentz group
View Description Hide DescriptionWe study the quantum group deformation of the Lorentzian EPRL spinfoam model. The construction uses the harmonic analysis on the quantum Lorentz group. We show that the quantum group spinfoam model so defined is free of the infrared divergence, thus gives a finite partition function on a fixed triangulation. We expect this quantum group spinfoam model is a spinfoam quantization of discrete gravity with a cosmological constant.

Dynamical Systems

Asymptotic behavior of twodimensional stochastic magnetohydrodynamics equations with additive noises
View Description Hide DescriptionThis paper is devoted to the investigation of the asymptotic behavior of solutions of the stochastic magnetohydrodynamics equations driven by random exterior forced terms both in the velocity and in the magnetic field. The nonlinear term is supposed to be timedependent and satisfies certain dissipative conditions. The existence of a random attractor for this random dynamical system is obtained.

Geometric shape of invariant manifolds for a class of stochastic partial differential equations^{a)}
View Description Hide DescriptionInvariant manifolds play an important role in the study of the qualitative dynamical behaviors for nonlinear stochastic partial differential equations. However, the geometric shape of these manifolds is largely unclear. The purpose of the present paper is to try to describe the geometric shape of invariant manifolds for a class of stochastic partial differential equations with multiplicative white noises. The local geometric shape of invariant manifolds is approximated, which holds with significant likelihood. Furthermore, the result is compared with that for the corresponding deterministic partial differential equations.

Classical Mechanics and Classical Fields

Nonrelativistic ChernSimons vortices on the torus
View Description Hide DescriptionA classification of all periodic selfdual static vortexsolutions of the JackiwPi model is given. Physically acceptable solutions of the Liouville equation are related to a class of functions, which we term Ωquasielliptic. This class includes, in particular, the elliptic functions and also contains a function previously investigated by Olesen. Some examples of solutions are studied numerically and we point out a peculiar phenomenon of lost vortex charge in the limit where the period lengths tend to infinity, that is, in the planar limit.

Statistical Physics

A microscopic twoband model for the electronhole asymmetry in highT _{ c } superconductors and reentering behavior
View Description Hide DescriptionTo our knowledge there is no rigorously analyzed microscopic model explaining the electronhole asymmetry of the critical temperature seen in highT _{ c }cuprate superconductors – at least no model not breaking artificially this symmetry. We present here a microscopic twobandmodel based on the structure of energetic levels of holes in CuO_{2} conducting layers of cuprates. In particular, our Hamiltonian does not contain ad hoc terms implying – explicitly – different masses for electrons and holes. We prove that two energetically nearlying interacting bands can explain the electronhole asymmetry. Indeed, we rigorously analyze the phase diagram of the model and show that the critical temperatures for fermion densities below halffilling can manifest a very different behavior as compared to the case of densities above halffilling. This fact results from the interband interaction and intraband Coulomb repulsion in interplay with thermal fluctuations between two energetic levels. So, if the energy difference between bands is too big (as compared to the energy scale defined by the critical temperatures of superconductivity) then the asymmetry disappears. Moreover, the critical temperature turns out to be a nonmonotonic function of the fermion density and the phase diagram of our model shows “superconducting domes” as in highT _{ c }cuprate superconductors. This explains why the maximal critical temperature is attained at donor densities away from the maximal one. Outside the superconducting phase and for fermion densities near halffilling the thermodynamics governed by our Hamiltonian corresponds, as in real highT _{ c }materials, to a Mottinsulating phase. The nature of the interband interaction can be electrostatic (screened Coulomb interaction), magnetic (for instance, some Heisenbergtype onesite spin–spin interaction), or a mixture of both. If the interband interaction is predominately magnetic then – additionally to the electronhole asymmetry – we observe a reentering behavior meaning that the superconducting phase can only occur in a finite interval of temperatures. This phenomenon is rather rare, but has also been observed in the socalled magnetic superconductors. The mathematical results here are direct consequences of [J.B. Bru and W. de Siqueira Pedra, Rev. Math. Phys.22, 233 (2010)] which is reviewed in the introduction.

Phase induced transport of a Brownian particle in a periodic potential in the presence of an external noise: A semiclassical treatment
View Description Hide DescriptionWe develop, invoking a suitable systemreservoir model, the Langevin equation with a statedependent dissipation associated with a quantum Brownian particle submerged in a heat bath that offers a statedependent friction to study the directed motion (by studying the phaseinduced current) in the presence of an external noise. We study the phase induced current when both system and bath are subjected to external modulation by the noise and thereby expose the system to two crosscorrelated noises. We also demonstrate the wellknown fact that two noises remain mutually correlated if they share a common origin. We study the effects of correlation on the current in a periodic potential and envisage that the steady state current increases with increase in the extent of correlation, implying that exercising control on the degree of correlation can enhance the current in a properly designed experiment. To establish our model, we analyze numerically the effect of the external noise on system and bath separately as well as on composition of both.

Methods of Mathematical Physics

Sparse onedimensional discrete Dirac operators II: Spectral properties
View Description Hide DescriptionWe study spectral properties of some discrete Dirac operators with nonzero potential only at some sparse and suitably randomly distributed positions. As observed in the corresponding Schrödinger operators, we determine the Hausdorff dimension of its spectral measure and identify a sharp spectral transition from point to singular continuous.

Poisson YangBaxter maps with binomial Lax matrices
View Description Hide DescriptionA construction of multidimensional parametric YangBaxter maps is presented. The corresponding Lax matrices are the symplectic leaves of firstdegree matrix polynomials equipped with the Sklyanin bracket. These maps are symplectic with respect to the reduced symplectic structure on these leaves and provide examples of integrable mappings. An interesting family of quadrirational symplectic YB maps on with 3 × 3 Lax matrices is also presented.

Integrable hierarchies related to the KuperCH spectral problem
View Description Hide DescriptionIn this paper, from a given KuperCH spectral problem, we propose two kinds of super integrable hierarchies. One is the KuperCH hierarchy, the other is the generalized KuperHarryDym hierarchy. Moreover, we construct their zero curvature representations and superbiHamiltonian structures.

Dirac method and symplectic submanifolds in the cotangent bundle of a factorizable Lie group
View Description Hide DescriptionWe study some symplectic submanifolds in the cotangent bundle of a factorizable Lie group defined by second class constraints. By applying the Dirac method, we study many issues of these spaces as fundamental Dirac brackets, symmetries, and collective dynamics. As the main application, we study integrable systems on these submanifolds as inherited from a system on the whole cotangent bundle, meeting in a natural way with the AdlerKostantSymes theory of integrability.

Classification of integrable twocomponent Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions
View Description Hide DescriptionHamiltonian systems of hydrodynamic type occur in a wide range of applications including fluid dynamics, the Whitham averaging procedure, and the theory of Frobenius manifolds. In 1 + 1 dimensions, the requirement of the integrability of such systems by the generalised hodograph transform implies that integrable Hamiltonians depend on a certain number of arbitrary functions of two variables. On the contrary, in 2 + 1 dimensions the requirement of the integrability by the method of hydrodynamic reductions, which is a natural analogue of the generalised hodograph transform in higher dimensions, leads to finitedimensional moduli spaces of integrable Hamiltonians. In this paper we classify integrable twocomponent Hamiltonian systems of hydrodynamic type for all existing classes of differentialgeometric Poisson brackets in 2D, establishing a parametrisation of integrable Hamiltonians via elliptic/hypergeometric functions. Our approach is based on the Godunovtype representation of Hamiltonian systems, and utilises a novel construction of Godunov's systems in terms of generalised hypergeometric functions.
