Volume 56, Issue 3, March 2015
Index of content:

We consider the motion of test particles in the regular black hole spacetime given by AyónBeato and García [Phys. Rev. Lett. 80, 5056 (1998)]. The complete set of orbits for neutral and weakly charged test particles is discussed, including for neutral particles the extreme and overextreme metric. We also derive the analytical solutions for the equation of motion of neutral test particles in a parametric form and consider a postSchwarzschild expansion of the periastron shift to second order in the charge.
 ARTICLES

 Partial Differential Equations

On the steadystate solutions of a nonlinear photonic lattice model
View Description Hide DescriptionIn this paper, we consider the steadystate solutions of the following equation related with nonlinear photonic lattice model , , where u, v are realvalue function defined on R/(τ 1 Z) × R/(τ 2 Z). The existence and nonexistence of nonconstant semitrivial (with only one component zero) solutions are considered.

Wellposedness for the fractional FokkerPlanck equations
View Description Hide DescriptionIn this paper, we study the fractional FokkerPlanck equation and obtain the existence and uniqueness of weak L^{p} solutions (1 ⩽ p ⩽ ∞) under the assumptions that the coefficients are only in Sobolev spaces. Moreover, to L ^{∞}solutions, we gain the wellposedness for BV coefficients. Besides, the nonnegative weak L^{p} solutions and renormalized solutions are derived. After then, we achieve the stability for stationary solutions.
 Representation Theory and Algebraic Methods

ℓoscillators from secondorder invariant PDEs of the centrally extended conformal Galilei algebras
View Description Hide DescriptionWe construct, for any given , the secondorder, linear partial differential equations (PDEs) which are invariant under the centrally extended conformal Galilei algebra. At the given ℓ, two invariant equations in one time and space coordinates are obtained. The first equation possesses a continuum spectrum and generalizes the free Schrödinger equation (recovered for ) in 1 + 1 dimension. The second equation (the “ℓoscillator”) possesses a discrete, positive spectrum. It generalizes the 1 + 1dimensional harmonic oscillator (recovered for ). The spectrum of the ℓoscillator, derived from a specific osp(12ℓ + 1) h.w.r., is explicitly presented. The two sets of invariant PDEs are determined by imposing (representationdependent) onshell invariant conditions both for degree 1 operators (those with continuum spectrum) and for degree 0 operators (those with discrete spectrum). The onshell condition is better understood by enlarging the conformal Galilei algebras with the addition of certain secondorder differential operators. Two compatible structures (the algebra/superalgebra duality) are defined for the enlarged set of operators.
 Quantum Mechanics

On the nosignaling approach to quantum nonlocality
View Description Hide DescriptionThe nosignaling approach to nonlocality deals with separable and inseparable multiparty correlations in the same set of probability states without conflicting causality. The set of halfspaces describing the polytope of nosignaling probability states that are admitted by the most general class of Bell scenarios is formulated in full detail. An algorithm for determining the skeleton that solves the nosignaling description is developed upon a new strategy that is partially pivoting and partially incremental. The algorithm is formulated rigorously and its implementation is shown to be effective to deal with the highly degenerate nosignaling descriptions. Several applications of the algorithm as a tool for the study of quantum nonlocality are mentioned. Applied to a large set of bipartite Bell scenarios, we found that the corresponding nosignaling polytopes have a striking high degeneracy that grows up exponentially with the size of the Bell scenario.

Temporal evolution of instantaneous phonons in timedependent harmonic oscillators
View Description Hide DescriptionWe study a timedependent harmonic oscillator based on the dynamics of instantaneous phonons, which have obvious physical meaning and direct experimental relevance. We find simple analytic solutions for an important class of evolution and identify two parameterchangingrate regimes with qualitatively different oscillator behaviors. We show that rapid adiabatic processes are possible if the frequency and the mass of the oscillator change in opposite directions. The state vector in the Schrödinger picture is handily achieved by use of the eigenstates of the instantaneous phonon operators that are analytically known for arbitrary frequency and mass values.

Symmetry and conservation laws in semiclassical wave packet dynamics
View Description Hide DescriptionWe formulate symmetries in semiclassical Gaussian wave packet dynamics and find the corresponding conserved quantities, particularly the semiclassical angular momentum, via Noether’s theorem. We consider two slightly different formulations of Gaussian wave packet dynamics; one is based on earlier works of Heller and Hagedorn and the other based on the symplecticgeometric approach by Lubich and others. In either case, we reveal the symplectic and Hamiltonian nature of the dynamics and formulate natural symmetry group actions in the setting to derive the corresponding conserved quantities (momentum maps). The semiclassical angular momentum inherits the essential properties of the classical angular momentum as well as naturally corresponds to the quantum picture.

Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations
View Description Hide DescriptionFor a number of nonlocal nonlinear equations such as nonlocal, nonlinear Schrödinger equation (NLSE), nonlocal AblowitzLadik (AL), nonlocal, saturable discrete NLSE (DNLSE), coupled nonlocal NLSE, coupled nonlocal AL, and coupled nonlocal, saturable DNLSE, we obtain periodic solutions in terms of Jacobi elliptic functions as well as the corresponding hyperbolic soliton solutions. Remarkably, in all the six cases, we find that unlike the corresponding local cases, all the nonlocal models simultaneously admit both the bright and the dark soliton solutions. Further, in all the six cases, not only the elliptic functions dn(x, m) and cn(x, m) with modulus m but also their linear superposition is shown to be an exact solution. Finally, we show that the coupled nonlocal NLSE not only admits solutions in terms of Lamé polynomials of order 1 but also admits solutions in terms of Lamé polynomials of order 2, even though they are not the solution of the uncoupled nonlocal problem. We also remark on the possible integrability in certain cases.
 Quantum Information and Computation

Contextinvariant quasi hidden variable (qHV) modelling of all joint von Neumann measurements for an arbitrary Hilbert space
View Description Hide DescriptionWe prove the existence for each Hilbert space of the two new quasi hidden variable (qHV) models, statistically noncontextual and contextinvariant, reproducing all the von Neumann joint probabilities via nonnegative values of realvalued measures and all the quantum product expectations—via the qHV (classicallike) average of the product of the corresponding random variables. In a contextinvariant model, a quantum observable X can be represented by a variety of random variables satisfying the functional condition required in quantum foundations but each of these random variables equivalently models X under all joint von Neumann measurements, regardless of their contexts. The proved existence of this model negates the general opinion that, in terms of random variables, the Hilbert space description of all the joint von Neumann measurements for can be reproduced only contextually. The existence of a statistically noncontextual qHV model, in particular, implies that every Npartite quantum state admits a local quasi hidden variable model introduced in Loubenets [J. Math. Phys. 53, 022201 (2012)]. The new results of the present paper point also to the generality of the quasiclassical probability model proposed in Loubenets [J. Phys. A: Math. Theor. 45, 185306 (2012)].

Universal quantum computation with metaplectic anyons
View Description Hide DescriptionWe show that braidings of the metaplectic anyons X ϵ in SO (3) 2 = SU (2)4 with their total charge equal to the metaplectic mode Y supplemented with projective measurements of the total charge of two metaplectic anyons are universal for quantum computation. We conjecture that similar universal anyonic computing models can be constructed for all metaplectic anyon systems SO(p)2 for any odd prime p ≥ 5. In order to prove universality, we find new conceptually appealing universal gate sets for qutrits and qupits.
 Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

From simplicial ChernSimons theory to the shadow invariant I
View Description Hide DescriptionThis is the first of a series of papers in which we introduce and study a rigorous “simplicial” realization of the nonAbelian ChernSimons path integral for manifolds M of the form M = Σ × S ^{1} and arbitrary simply connected compact structure groups G. More precisely, we will introduce, for general links L in M, a rigorous simplicial version WLO rig (L) of the corresponding Wilson loop observable WLO(L) in the socalled “torus gauge” by Blau and Thompson [Nucl. Phys. B 408(2), 345–390 (1993)]. For a simple class of links L, we then evaluate WLO rig (L) explicitly in a nonperturbative way, finding agreement with Turaev’s shadow invariant .

From simplicial ChernSimons theory to the shadow invariant II
View Description Hide DescriptionThis is the second of a series of papers in which we introduce and study a rigorous “simplicial” realization of the nonAbelian ChernSimons path integral for manifolds M of the form M = Σ × S ^{1} and arbitrary simply connected compact structure groups G. More precisely, we introduce, for general links L in M, a rigorous simplicial version WLO rig (L) of the corresponding Wilson loop observable WLO(L) in the socalled “torus gauge” by Blau and Thompson [Nucl. Phys. B 408(2), 345–390 (1993)]. For a simple class of links L, we then evaluate WLO rig (L) explicitly in a nonperturbative way, finding agreement with Turaev’s shadow invariant .

On the concept of Bell’s local causality in local classical and quantum theory
View Description Hide DescriptionThe aim of this paper is to implement Bell’s notion of local causality into a framework, called local physical theory. This framework, based on the axioms of algebraic field theory, is broad enough to integrate both probabilistic and spatiotemporal concepts and also classical and quantum theories. Bell’s original idea of local causality will arise as the classical case of our definition. Classifying local physical theories by whether they obey local primitive causality, a property rendering the dynamics of the theory causal, we then investigate what is needed for a local physical theory to be locally causal. Finally, comparing local causality with the common cause principles and relating both to the Bell inequalities we find a nice parallelism: Bell inequalities cannot be derived neither from local causality nor from a common cause unless the local physical theory is classical or the common cause is commuting, respectively.
 General Relativity and Gravitation

Motion of test particles in a regular black hole space–time
View Description Hide DescriptionWe consider the motion of test particles in the regular black hole spacetime given by AyónBeato and García [Phys. Rev. Lett. 80, 5056 (1998)]. The complete set of orbits for neutral and weakly charged test particles is discussed, including for neutral particles the extreme and overextreme metric. We also derive the analytical solutions for the equation of motion of neutral test particles in a parametric form and consider a postSchwarzschild expansion of the periastron shift to second order in the charge.
 Dynamical Systems

The SharmaParthasarathy stochastic twobody problem
View Description Hide DescriptionWe study the SharmaParthasarathy stochastic twobody problem introduced by Sharma and Parthasarathy in [“Dynamics of a stochastically perturbed twobody problem,” Proc. R. Soc. A 463, 9791003 (2007)]. In particular, we focus on the preservation of some fundamental features of the classical twobody problem like the Hamiltonian structure and first integrals in the stochastic case. Numerical simulations are performed which illustrate the dynamical behaviour of the osculating elements as the semimajor axis, the eccentricity, and the pericenter. We also derive a stochastic version of Gauss’s equations in the planar case.

Invariant tori for a derivative nonlinear Schrödinger equation with quasiperiodic forcing
View Description Hide DescriptionThis paper is concerned with a one dimensional derivative nonlinear Schrödinger equation with quasiperiodic forcing under periodic boundary conditions , where g(βt) is real analytic and quasiperiodic on t with frequency vector β = (β 1, β 2, …, βm ). f is real analytic in some neighborhood of the origin in ℂ, f(0) = 0 and f′(0) ≠ 0. We show that the above equation admits Cantor families of smooth quasiperiodic solutions of small amplitude. The proof is based on an abstract infinite dimensional KolmogorovArnoldMoser theorem for unbounded perturbation vector fields and partial Birkhoff normal form.
 Classical Mechanics and Classical Fields

Existence of a lower bound for the distance between point masses of relative equilibria for generalised quasihomogeneous nbody problems and the curved nbody problem
View Description Hide DescriptionWe prove that if for relative equilibrium solutions of a generalisation of quasihomogeneous nbody problems the masses and rotation are given, then the minimum distance between the point masses of such a relative equilibrium has a universal lower bound that is not equal to zero. We furthermore prove that the set of such relative equilibria is compact and prove related results for nbody problems in spaces of constant Gaussian curvature.
 Statistical Physics

Positive contraction mappings for classical and quantum Schrödinger systems
View Description Hide DescriptionThe classical Schrödinger bridge seeks the most likely probability law for a diffusion process, in path space, that matches marginals at two end points in time; the likelihood is quantified by the relative entropy between the sought law and a prior. Jamison proved that the new law is obtained through a multiplicative functional transformation of the prior. This transformation is characterised by an automorphism on the space of endpoints probability measures, which has been studied by Fortet, Beurling, and others. A similar question can be raised for processes evolving in a discrete time and space as well as for processes defined over noncommutative probability spaces. The present paper builds on earlier work by Pavon and Ticozzi and begins by establishing solutions to Schrödinger systems for Markov chains. Our approach is based on the Hilbert metric and shows that the solution to the Schrödinger bridge is provided by the fixed point of a contractive map. We approach, in a similar manner, the steering of a quantum system across a quantum channel. We are able to establish existence of quantum transitions that are multiplicative functional transformations of a given Kraus map for the cases where the marginals are either uniform or pure states. As in the Markov chain case, and for uniform density matrices, the solution of the quantum bridge can be constructed from the fixed point of a certain contractive map. For arbitrary marginal densities, extensive numerical simulations indicate that iteration of a similar map leads to fixed points from which we can construct a quantum bridge. For this general case, however, a proof of convergence remains elusive.
 Methods of Mathematical Physics

Semiclassical states on Lie algebras
View Description Hide DescriptionThe effective technique for analyzing representationindependent features of quantum systems based on the semiclassical approximation (developed elsewhere) has been successfully used in the context of the canonical (Weyl) algebra of the basic quantum observables. Here, we perform the important step of extending this effective technique to the quantization of a more general class of finitedimensional Lie algebras. The case of a Lie algebra with a single central element (the Casimir element) is treated in detail by considering semiclassical states on the corresponding universal enveloping algebra. Restriction to an irreducible representation is performed by “effectively” fixing the Casimir condition, following the methods previously used for constrained quantum systems. We explicitly determine the conditions under which this restriction can be consistently performed alongside the semiclassical truncation.

A new recurrence formula for generic exceptional orthogonal polynomials
View Description Hide DescriptionA new recurrence relation for exceptional orthogonal polynomials is proposed, which holds for types 1, 2, and 3. To provide concrete examples, the recurrence relations are then given for Xj Hermite, Laguerre, and Jacobi polynomials in the j = 1, 2 cases.

Matrix Painlevé systems
View Description Hide DescriptionWe study a class of the isomonodromic deformation equations, which we call matrix Painlevé systems. They can be written in Hamiltonian form. The Hamiltonian is written as the trace of the Hamiltonian of the sixth Painlevé equation with canonical variables replaced by matrices. We also derive the matrix version of the fifth, fourth, third, and second Painlevé equation from the sixth matrix Painlevé system by the degeneration.