Volume 52, Issue 7, July 2011
Index of content:
- Quantum Mechanics (General and Nonrelativistic)
52(2011); http://dx.doi.org/10.1063/1.3606515View Description Hide Description
We propose an effective algebraic method to investigate the dynamical symmetry of a 9-dimensional MICZ-Kepler problem by using the connection between this problem and a 16-dimensional 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.
52(2011); http://dx.doi.org/10.1063/1.3606580View Description Hide Description
Pseudo-bosons 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. Pseudo-bosons 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.
52(2011); http://dx.doi.org/10.1063/1.3614003View Description Hide Description
We 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).
52(2011); http://dx.doi.org/10.1063/1.3610674View Description Hide Description
The 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 κ-dependent 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 κ-dependent Schrödinger equation. First the characterization of the κ-dependent “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 κ-dependent family of orthogonal polynomials, are explicitly obtained.
Uncertainty relations for joint localizability and joint measurability in finite-dimensional systems52(2011); http://dx.doi.org/10.1063/1.3614503View Description Hide Description
Two 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 finite-dimensional 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 finite-dimensional systems, uncertainty relations for a pair of general projection-valued measures are obtained as by-products.
Localized spectral asymptotics for boundary value problems and correlation effects in the free Fermi gas in general domains52(2011); http://dx.doi.org/10.1063/1.3610167View Description Hide Description
We 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)
52(2011); http://dx.doi.org/10.1063/1.3603815View Description Hide Description
The purpose of this paper is to generalize the self-duality equation by Tchrakian and Corrigan et al.Novel generalized self-duality equations on higher-dimensional spaces are discussed. This class of equations includes the usual self-duality equation for four-dimensional spaces. Some of the generalized self-duality equations over-determine 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 single-trace terms and are second order in the time derivative.
52(2011); http://dx.doi.org/10.1063/1.3603993View Description Hide Description
Recently it has been pointed out that the so-called Faddeev-Niemi equations that describe the Yang-Millsequations of motion in terms of a decomposed gauge field, can have solutions that obey the standard Yang-Millsequations with a source term. Here we argue that the source term is covariantly constant. Furthermore, we find that there are solutions of the Yang-Millsequation with a covariantly constant source term that are not solutions to the Faddeev-Niemi equations. We also present a general class of gauge field configurations that obey the Faddeev-Niemi equation but do not solve the Yang-Millsequation. We propose that these configurations might have physical relevance in a strongly coupled phase, where spin-charge separation takes place and the Yang-Millstheory cannot be described in terms of a Landau liquid of asymptotically free gluons.
52(2011); http://dx.doi.org/10.1063/1.3607971View Description Hide Description
We construct an extension of the Poincaré group which involves a mixture of internal and space-time supersymmetries. The resulting group is an extension of the superPoincaré group with infinitely many generators which carry internal and space-time 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 half-integer helicities. Finally, we suggest an extension of the conformal group along the same line.
- General Relativity and Gravitation
52(2011); http://dx.doi.org/10.1063/1.3606592View Description Hide Description
We study the quantum group deformation of the Lorentzian EPRL spin-foam model. The construction uses the harmonic analysis on the quantum Lorentz group. We show that the quantum group spin-foam model so defined is free of the infra-red divergence, thus gives a finite partition function on a fixed triangulation. We expect this quantum group spin-foam model is a spin-foam quantization of discrete gravity with a cosmological constant.
- Dynamical Systems
Asymptotic behavior of two-dimensional stochastic magneto-hydrodynamics equations with additive noises52(2011); http://dx.doi.org/10.1063/1.3614884View Description Hide Description
This paper is devoted to the investigation of the asymptotic behavior of solutions of the stochastic magneto-hydrodynamics equations driven by random exterior forced terms both in the velocity and in the magnetic field. The nonlinear term is supposed to be time-dependent and satisfies certain dissipative conditions. The existence of a random attractor for this random dynamical system is obtained.
52(2011); http://dx.doi.org/10.1063/1.3614777View Description Hide Description
Invariant 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
52(2011); http://dx.doi.org/10.1063/1.3610643View Description Hide Description
A classification of all periodic self-dual static vortexsolutions of the Jackiw-Pi model is given. Physically acceptable solutions of the Liouville equation are related to a class of functions, which we term Ω-quasi-elliptic. 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 two-band model for the electron-hole asymmetry in high-T c superconductors and reentering behavior52(2011); http://dx.doi.org/10.1063/1.3600202View Description Hide Description
To our knowledge there is no rigorously analyzed microscopic model explaining the electron-hole asymmetry of the critical temperature seen in high-T c cuprate superconductors – at least no model not breaking artificially this symmetry. We present here a microscopic two-bandmodel based on the structure of energetic levels of holes in CuO2 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 near-lying interacting bands can explain the electron-hole asymmetry. Indeed, we rigorously analyze the phase diagram of the model and show that the critical temperatures for fermion densities below half-filling can manifest a very different behavior as compared to the case of densities above half-filling. This fact results from the inter-band interaction and intra-band 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 non-monotonic function of the fermion density and the phase diagram of our model shows “superconducting domes” as in high-T 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 half-filling the thermodynamics governed by our Hamiltonian corresponds, as in real high-T c materials, to a Mott-insulating phase. The nature of the inter-band interaction can be electrostatic (screened Coulomb interaction), magnetic (for instance, some Heisenberg-type one-site spin–spin interaction), or a mixture of both. If the inter-band interaction is predominately magnetic then – additionally to the electron-hole 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 so-called 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 treatment52(2011); http://dx.doi.org/10.1063/1.3614776View Description Hide Description
We develop, invoking a suitable system-reservoir model, the Langevin equation with a state-dependent dissipation associated with a quantum Brownian particle submerged in a heat bath that offers a state-dependent friction to study the directed motion (by studying the phase-induced 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 cross-correlated noises. We also demonstrate the well-known 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
52(2011); http://dx.doi.org/10.1063/1.3600536View Description Hide Description
We 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.
52(2011); http://dx.doi.org/10.1063/1.3601520View Description Hide Description
A construction of multidimensional parametric Yang-Baxter maps is presented. The corresponding Lax matrices are the symplectic leaves of first-degree 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.
52(2011); http://dx.doi.org/10.1063/1.3603817View Description Hide Description
In this paper, from a given Kuper-CH spectral problem, we propose two kinds of super integrable hierarchies. One is the Kuper-CH hierarchy, the other is the generalized Kuper-Harry-Dym hierarchy. Moreover, we construct their zero curvature representations and super-bi-Hamiltonian structures.
52(2011); http://dx.doi.org/10.1063/1.3603427View Description Hide Description
We 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 Adler-Kostant-Symes theory of integrability.
Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions52(2011); http://dx.doi.org/10.1063/1.3602081View Description Hide Description
Hamiltonian 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 finite-dimensional moduli spaces of integrable Hamiltonians. In this paper we classify integrable two-component Hamiltonian systems of hydrodynamic type for all existing classes of differential-geometric Poisson brackets in 2D, establishing a parametrisation of integrable Hamiltonians via elliptic/hypergeometric functions. Our approach is based on the Godunov-type representation of Hamiltonian systems, and utilises a novel construction of Godunov's systems in terms of generalised hypergeometric functions.