Volume 52, Issue 8, August 2011
Index of content:
 ARTICLES


Quantum Mechanics (General and Nonrelativistic)

Phase operators, phase states and vector phase states for SU _{3} and SU _{2, 1}
View Description Hide DescriptionThis paper focuses on phase operators, phase states, and vector phase states for the sl _{3}Lie algebra. We introduce a oneparameter generalized oscillatoralgebra which provides a unified scheme for dealing with su _{3} (for κ < 0), su _{2, 1} (for κ > 0), and h _{4}⊗h _{4} (for κ = 0) symmetries. Finite and infinitedimensional representations of are constructed for κ < 0 and κ ⩾ 0, respectively. Phase operators associated with are defined and temporally stable phase states (as well as vector phase states) are constructed as eigenstates of these operators. Finally, we discuss a relation between quantized phase states and a quadratic discrete Fourier transform and show how to use these states for constructing mutually unbiased bases.

Levinson's theorem for graphs
View Description Hide DescriptionWe prove an analog of Levinson's theorem for scattering on a weighted (m + 1)vertex graph with a semiinfinite path attached to one of its vertices. In particular, we show that the number of bound states in such a scattering problem is equal to m minus half the winding number of the phase of the reflection coefficient (where each socalled halfbound state is counted as half a bound state).

Special features of the relation between Fisher information and Schrödinger eigenvalue equation
View Description Hide DescriptionIt is well known that a suggestive relation exists that links Schrödinger's equation (SE) to the informationoptimizing principle based on Fisher's information measure. The connection entails the existence of a Legendre transform structure underlying the SE. Here, we show that appeal to this structure leads to a first order differential equation for the SE's eigenvalues that, in certain cases, can be used to obtain the eigenvalues without explicitly solving the SE. Complying with the above mentioned equation constitutes a necessary condition to be satisfied by an energy eigenvalue. We show that the general solution is unique.

Completeness of †categories and the complex numbers
View Description Hide DescriptionThe complex numbers are an important part of quantum theory, but are difficult to motivate from a theoretical perspective. We describe a simple formal framework for theories of physics, and show that if a theory of physics presented in this manner satisfies certain completeness properties, then it necessarily includes the complex numbers as a mathematical ingredient. Central to our approach are the techniques of category theory, and we introduce a new categorytheoretical tool, called the †limit, which governs the way in which systems can be combined to form larger systems. These †limits can be used to characterize the properties of the †functor on the category of finitedimensional Hilbert spaces, and so can be used as an equivalent definition of the inner product. One of our main results is that in a nontrivial monoidal †category with finite †limits and a simple tensor unit, the semiring of scalars embeds into an involutive field of characteristic 0 and orderable fixed field.

Generalized coherent states and their statistical characteristics in powerlaw potentials
View Description Hide DescriptionGeneralized coherent states based on GazeauKlauder formalism are developed for onedimensional powerlaw potentials and their quantum statistical characteristics, together with generalized Heisenberg algebracoherent states, are reported. We show that these states exhibit superPoissonian, Poissonian, or subPoissonian distributions as a function of the powerlaw exponent. The analytical results are supported by numerical calculations. In addition, we explain possible sources of errors in numerical analysis.

Exponential operators and the algebraic description of quantum confined systems
View Description Hide DescriptionWe study the relations and transformations produced by exponential operators, the argument of which are dependent on the basic algebraic elements of supersymmetric and shapeinvariant potential systems, and obtain explicit expressions. We apply our results to selfsimilar potential systems and to a set of translational shapeinvariant systems, including the Morse, PöschlTeller, Scarf, and RosenMorse potentials, and obtain closedform expressions. We show that our results reproduce those obtained for the harmonic oscillator in the appropriate limits.

Quantum Information and Computation

Hilbert's projective metric in quantum information theory
View Description Hide DescriptionWe introduce and apply Hilbert's projective metric in the context of quantum informationtheory. The metric is induced by convex cones such as the sets of positive, separable or positive partial transpose operators. It provides bounds on measures for statistical distinguishability of quantum states and on the decrease of entanglement under protocols involving local quantum operations and classical communication or under other conepreserving operations. The results are formulated in terms of general cones and base norms and lead to contractivity bounds for quantum channels, for instance, improving Ruskai's tracenorm contraction inequality. A new duality between distinguishability measures and base norms is provided. For two given pairs of quantum states we show that the contraction of Hilbert's projective metric is necessary and sufficient for the existence of a probabilistic quantum operation that maps one pair onto the other. Inequalities between Hilbert's projective metric and the Chernoff bound, the fidelity and various norms are proven.

Extremal quantum protocols
View Description Hide DescriptionGeneralized quantum instruments correspond to measurements where the input and output are either states or more generally quantum circuits. These measurements describe any quantum protocol including games, communications, and algorithms. The set of generalized quantum instruments with a given input and output structure is a convex set. Here, we investigate the extremal points of this set for the case of finite dimensional quantum systems and generalized instruments with finitely many outcomes. We derive algebraic necessary and sufficient conditions for extremality.

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

Existence and properties of radial solutions in the selfdual ChernSimons O(3) sigma model
View Description Hide DescriptionIn this paper, we study the selfdual equations arising from the ChernSimons gauged O(3) sigma model with symmetric potential. We prove the existence of radially symmetric solutions of the reduced elliptic equation having topological and nontopological boundary conditions.

Topics in cubic special geometry
View Description Hide DescriptionWe reconsider the subleading quantum perturbative corrections to cubic special Kähler geometries. Imposing the invariance under axionshifts, all such corrections (but the imaginary constant one) can be introduced or removed through suitable, lower unitriangular symplectic transformations and dubbed PecceiQuinn (PQ) transformations. Since PQ transformations do not belong to the d = 4 Uduality group G _{4}, in symmetric cases they generally have a nontrivial action on the unique quartic invariant polynomial of the charge representation of G _{4}. This leads to interesting phenomena in relation to theory of extremal black hole attractors; namely, the possibility to make transitions between different charge orbits of , with corresponding change of the supersymmetry properties of the supported attractor solutions. Furthermore, a suitable action of PQ transformations can also set to zero, or vice versa it can generate a nonvanishing : this corresponds to transitions between “large” and “small” charge orbits, which we classify in some detail within the “special coordinates” symplectic frame. Finally, after a brief account of the action of PQ transformations on the recently established correspondence between Cayley's hyperdeterminant and elliptic curves, we derive an equivalent, alternative expression of , with relevant application to black hole entropy.

One real function instead of the Dirac spinor function
View Description Hide DescriptionThree out of four complex components of the Dirac spinor can be algebraically eliminated from the Dirac equation (if some linear combination of electromagnetic fields does not vanish), yielding a partial differential equation of the fourth order for the remaining complex component. This equation is generally equivalent to the Dirac equation. Furthermore, following Schrödinger [Nature (London), 169, 538 (1952)], the remaining component can be made real by a gauge transform, thus extending to the Dirac field the Schrödinger conclusion that charged fields do not necessarily require complex representation. One of the two resulting real equations for the real function describes current conservation and can be obtained from the Maxwell equations in spinor electrodynamics (the DiracMaxwell electrodynamics). As the Dirac equation is one of the most fundamental equations, these results both belong in textbooks and can be used for development of new efficient methods and algorithms of quantum chemistry.

On generalized SethiVafaWitten formulas
View Description Hide DescriptionWe present a formula for computing proper pushforwards of classes in the Chow ring of a projective bundle under the projection , for B being a nonsingular compact complex algebraic variety of any dimension. Our formula readily produces generalizations of formulas derived by Sethi, Vafa, and Witten to compute the Euler characteristic of elliptically fibered CalabiYau fourfolds used for Ftheory compactifications of string vacua. The utility of such a formula is illustrated through applications, such as the ability to compute the Chern numbers of any nonsingular complete intersection in such a projective bundle in terms of the Chern class of a line bundle on B.

General Relativity and Gravitation

Classical gauge theory in Riem
View Description Hide DescriptionIn the geometrodynamical setting of general relativity in Lagrangian form, the objects of study are the Riemannian metrics (and their time derivatives) over a given 3manifold M. It is our aim in this paper to study some geometrical aspects of the space Riem(M) of all metrics over M. For instance, the Hamiltonian constraints by themselves do not generate a group, and thus its action on Riem(M) cannot be viewed in a geometrical gauge setting. It is possible to do so for the momentum constraints however. Furthermore, in view of the recent results representing GR as a dual theory, invariant under foliation preserving 3–diffeomorphisms and 3D conformal transformations, but not under refoliations, we are justified in considering the gauge structure pertaining only to the groups of diffeomorphisms of M, and , of conformal diffeomorphisms on M. For these infinitedimensional symmetry groups, has a natural principal fiber bundle structure, which renders the gravitational field amenable to the full range of gaugetheoretic treatment. The aim of the paper is to use the geometrical structure present in the configuration space of general relativity to build gauge connection forms. The interpretation of the gauge connection form for the 3diffeomorphism group is that it yields parallel translation of coordinates. For the conformal group, it yields parallel translation of scale. We focus on the concept of a gauge connection forms for these structures and construct explicit formulae for supermetricinduced gauge connections. To apply the formalism, we compute general properties for a specific connection bearing strong resemblance to the one naturally induced by the deWitt supermetric, showing it has desirable relationalist properties.

Dynamical Systems

On the periodic orbits and the integrability of the regularized Hill lunar problem
View Description Hide DescriptionThe classical Hill's problem is a simplified version of the restricted threebody problem where the distance of the two massive bodies (say, primary for the largest one and secondary for the smallest one) is made infinity through the use of Hill's variables. The LeviCivita regularization takes the Hamiltonian of the Hill lunar problem into the form of two uncoupled harmonic oscillators perturbed by the Coriolis force and the Sun action, polynomials of degree 4 and 6, respectively. In this paper, we study periodic orbits of the planar Hill problem using the averaging theory. Moreover, we provide information about the C ^{1} integrability or nonintegrability of the regularized Hill lunar problem.

Modulated amplitude waves with nonzero phases in BoseEinstein condensates
View Description Hide DescriptionIn this paper we give a frame for application of the averaging method to BoseEinstein condensates(BECs) and obtain an abstract result upon the dynamics of BECs. Using the averaging method, we determine the location where the modulated amplitude waves (periodic or quasiperiodic) exist and obtain that all these modulated amplitude waves (periodic or quasiperiodic) form a foliation by varying the integration constant continuously. Compared with the previous work, modulated amplitude waves studied in this paper have nontrivial phases and this makes the problem become more difficult, since it involves some singularities.

On the recursion operators for the Gerdjikov, Mikhailov, and Valchev system
View Description Hide DescriptionWe consider the Recursion Operator approach to the soliton equations related to an auxiliary linear system introduced recently by Gerdjikov, Mikhailov, and Valchev (GMV system). We discuss the recursion operators obtained for the GMV system and show that they are particular cases of the generating operators related to a ZakharovShabat type linear system on the algebra in pole gauge.

Wave attraction in resonant counterpropagating wave systems
View Description Hide DescriptionWave attraction is a general phenomenon that was first established in the context of the attraction of the polarization between two counterpropagating waves in optical fibers. This phenomenon has been observed experimentally, and its properties were studied through numerical simulations. The relevant models are Hamiltonian hyperbolic systems of partial differential equations, with timedependent boundary conditions on a finite interval. The underlying mechanism can be traced back to the existence of singular tori in the corresponding stationary equations. In this article, we analyze in detail the simplest example in this family of models. We show that most of the phenomena of the wave attraction process are already present in a linear model with resonant interaction. We establish the existence and regularity of the solutions and analyze the relaxation towards a stationary solution that features the wave attraction properties.

Classical Mechanics and Classical Fields

On the Greenfunctions of the classical offshell electrodynamics under the manifestly covariant relativistic dynamics of Stueckelberg
View Description Hide DescriptionIn previous papers derivations of the Green function have been given for 5D offshell electrodynamics in the framework of the manifestly covariant relativistic dynamics of Stueckelberg (with invariant evolution parameter τ). In this paper, we reconcile these derivations resulting in different explicit forms, and relate our results to the conventional fundamental solutions of linear 5D wave equations published in the mathematical literature. We give physical arguments for the choice of the Green function retarded in the fifth variable τ.

Fluids

L ^{ p }bounds for quasigeostrophic equations via functional analysis
View Description Hide DescriptionWe give a proof of L ^{ p }bounds for the quasigeostrophic equation and other nonlocal equations. The proof uses mainly tools from functional analysis, notably the product formulas (also known as “operator splitting methods”) and the BochnerPollard subordination identities, hence it could be applicable to other equations.

Statistical Physics

Exact propagator for a FokkerPlanck equation, first passage time distribution, and anomalous diffusion
View Description Hide DescriptionWe obtain an exact form for the propagator of the FokkerPlanck equation ∂_{ t }ρ = −∂_{ x }(F(x, t)ρ), with in presence of the external force . Using the results found here, we also investigate the mean square displacement, survival probability, and first passage time distribution. In addition, we discuss the connection of these results with anomalous diffusion phenomena.
