Volume 48, Issue 7, July 2007
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Simplifying additivity problems using direct sum constructions
View Description Hide DescriptionWe study the additivity problems for the classical capacity of quantum channels, the minimal output entropy, and its convex closure. We show for each of them that additivity for arbitrary pairs of channels holds if and only if it holds for arbitrary equal pairs, which in turn can be taken to be unital. In a similar sense, weak additivity is shown to imply strong additivity for any convex entanglement monotone. The implications are obtained by considering direct sums of channels (or states) for which we show how to obtain several information theoretic quantities from their values on the summands. This provides a simple and general tool for lifting additivity results.

Algebraic structure of the Feynman propagator and a new correspondence for canonical transformations
View Description Hide DescriptionWe investigate the algebraic structure of the Feynman propagator with a general timedependent quadratic Hamiltonian system. Using the Liealgebraic technique, we obtain a normalordered form of the timeevolution operator, and then the propagator is easily derived by a simple “integration within ordered product” technique. It is found that this propagator contains a classical generatingfunction, which demonstrates a new correspondence between classical and quantum mechanics.

Type structure and chiral breaking in the standard model
View Description Hide DescriptionIn modern theories of computation, a significant role is played by the notions of type and typing, and many benefits accrue from careful attention to the ebb and flow of such structures through the course of a computation. We argue here that types are similarly inherent in quantum theory and that benefits may accrue from their careful handling—something that has apparently not been done in physics proper heretofore. In particular, we investigate the type structure of the interactionLagrangians of the standard model and deduce the chiral symmetry breaking of the weak interaction as a simple and immediate consequence of the careful maintenance of this structure through the course of a possible computation of the appropriate term.

New construction for a QMA complete threelocal Hamiltonian
View Description Hide DescriptionWe present a new way of encoding a quantum computation into a threelocal Hamiltonian. Our construction is novel in that it does not include any terms that induce legalillegal clock transitions. Therefore, the weights of the terms in the Hamiltonian do not scale with the size of the problem as in previous constructions. This improves the construction by Kempe and Regev [Quantum Inf. Comput.3, 258–264 (2003);eprint quantph∕0302079] who were the first to prove that threelocal Hamiltonian is complete for the complexity class QMA, the quantum analog of NP. Quantum SAT, a restricted version of the local Hamiltonian problem using only projector terms, was introduced by Bravyi (eprint quantph∕0602108) as an analog of the classical SAT problem. Bravyi proved that quantum 4SAT is complete for the class QMA with onesided error and that quantum 2SAT is in P. We give an encoding of a quantum circuit into a quantum 4SAT Hamiltonian using only threelocal terms. As an intermediate step to this threelocal construction, we show that quantum 3SAT for particles with dimensions (a qutrit and two qubits) is complete. The complexity of quantum 3SAT with qubits remains an open question.

Some solutions to the space fractional Schrödinger equation using momentum representation method
View Description Hide DescriptionThe space fractional Schrödinger equation with linear potential, deltafunction potential, and Coulomb potential is studied under momentum representation using Fourier transformation. By use of Mellin transform and its inverse transform, we obtain the energy levels and wave functions expressed in function for a particle in linear potential field. The wave function expressed also by the function and the unique energy level of the bound state for the particle of even parity state in deltafunction potential well, which is proved to have no action on the particle of odd parity state, is also obtained. The integral form of the wave functions for a particle in Coulomb potential field is shown and the corresponding energy levels which have been discussed in Laskin’s paper [Phys. Rev. E66, 056108 (2002)] are proved to satisfy an equality of infinite limit of the function. All of these results contain the ones of the standard quantum mechanics as their special cases.

Phase space representations and perturbation theory for continuoustime histories
View Description Hide DescriptionWe consider two technical developments of the formalism of continuoustime histories. First, we provide an explicit description of histories of the simple harmonic oscillator on the classical history phase space, comparing and contrasting the Q, P, and Wigner representations; we conclude that a representation based on coherent states is the most appropriate. Second, we demonstrate a generic method for implementing a perturbative approach for interacting theories in the history formalism, using the quartic anharmonic oscillator. We make use of the identification of the closedtimepath generating functional with the decoherence functional to develop a perturbative expansion for the latter up to second order in the coupling constant. We consider both configuration space and phase space histories.

Quantumlike representation of extensive form games: Probabilistic aspects
View Description Hide DescriptionWe consider an application of the mathematical formalism of quantum mechanics outside physics, namely, to game theory. We present a simple game between macroscopic players, say, Alice and Bob (or in a more complex form—Alice, Bob, and Cecilia), which can be represented in the quantumlike (QL) way—by using a complex probability amplitude (game’s “wave function”) and noncommutative operators. The crucial point is that games under consideration are socalled extensive form games. Here the order of actions of players is important; such a game can be represented by the tree of actions. The QL probabilistic behavior of players is a consequence of incomplete information (which is available to, e.g., Bob) about the previous action of Alice. In general one could not construct a classical probability space underlying a QL game. This can happen even in a QL game with two players. In a QL game with three players Bell’s inequality can be violated. The most natural probabilistic description is given by the socalled contextual probability theory completed by the frequency definition of probability.

Quantum superintegrable systems with quadratic integrals on a two dimensional manifold
View Description Hide DescriptionThere are two classes of quantum integrable systems on a manifold with quadratic integrals, the Liouville and the Lie integrable systems, as it happens in the classical case. The quantum Liouville quadratic integrable systems are defined on a Liouville manifold and the Schrödinger equation can be solved by separation of variables in one coordinate system. The Lie integrable systems are defined on a Lie manifold and are not generally separable ones but can be solved. Therefore, there are superintegrable systems with two quadratic integrals of motion not necessarily separable in two coordinate systems. The quantum analogs of the two dimensional superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion. There are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones. Analytic formulas for the involved integrals are calculated in all the cases. All the known quantum superintegrable systems are classified as special cases of these six general classes. The coefficients of the associative algebra of the general cases are calculated. These coefficients are the same as the coefficients of the classical case multiplied by plus quantum corrections of orders and .

Uncertainty principle and quantum Fisher information. II.
View Description Hide DescriptionHeisenberg and Schrödinger uncertainty principles give lower bounds for the product of variances if the observables are not compatible, namely, if the commutator is not zero. In this paper, we prove an uncertainty principle in Schrödinger form where the bound for the product of variances depends on the area spanned by the commutators and with respect to an arbitrary quantum version of the Fisher information.

Weighted complex projective 2designs from bases: Optimal state determination by orthogonal measurements
View Description Hide DescriptionWe introduce the problem of constructing weighted complex projective 2designs from the union of a family of orthonormal bases. If the weight remains constant across elements of the same basis, then such designs can be interpreted as generalizations of complete sets of mutually unbiased bases, being equivalent whenever the design is composed of bases in dimension . We show that, for the purpose of quantum state determination, these designs specify an optimal collection of orthogonal measurements. Using highly nonlinear functions on Abelian groups, we construct explicit examples from orthonormal bases whenever is a prime power, covering dimensions , 10, and 12, for example, where no complete sets of mutually unbiased bases have thus far been found.

Spectral decomposition of Bell operators for multiqubit systems
View Description Hide DescriptionThe spectral decomposition of multipartite Bell operators for two dichotomic observables per site, as introduced by Werner and Wolf [Phys. Rev. A64, 032112 (2001)], is done. Implications on the characterization of Bell inequalities as criteria of entanglement are discussed.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


A mathematical formalism for the Kondo effect in WessZuminoWitten branes
View Description Hide DescriptionIn the paper, we adapt our previous formalism for a mathematical treatment of branes to include processes, specifically the Kondo flow for WessZuminoWitten (WZW) branes. In this framework, we give the precise mathematical definitions and formulate a mathematical conjecture relating WZW branes to nonequivariant twisted theory in the case of the group . We also discuss regularization of the Kondo flow, thereby giving a first step toward proving our conjecture.

Structure of higher spin gauge interactions
View Description Hide DescriptionIn a previous paper, higher spin gauge field theory was formulated in an abstract way, essentially only keeping enough machinery to discuss gauge invariance of an action. The approach could be thought of as providing an interface (or syntax) toward an implementation (or semantics) yet to be constructed. The structure then revealed turned out to be that of a strongly homotopy Lie algebra. In the present paper, the framework will be connected to more conventional field theoretic concepts. The Fock complex vertex operator implementation of the interactions in the BRSTBV formulation of the theory will be elaborated. The relation between the vertex order expansion and homological perturbation theory will be clarified. A formal nonobstruction argument is reviewed. The syntactically derived shLie algebrastructure is semantically mapped to the Fock complex implementation, and it is shown that the equations governing the higher order vertices are reproduced. Global symmetries and subsidiary conditions are discussed and as a result the tracelessness constraints are discarded. Thus, all equations needed to compute the vertices to any order are collected. The framework is general enough to encompass all possible interaction terms. Finally, the abstract framework itself will be strengthened by showing that it can be naturally phrased in terms of category theory.
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 GENERAL RELATIVITY AND GRAVITATION


Warped product spaces and geodesic motion in the neighborhood of branes
View Description Hide DescriptionWe study the classical geodesic motions of nonzero rest mass test particles and photons in fivedimensional warped product spaces. We show that it is possible to obtain a general picture of these motions using the natural decoupling that occurs in such spaces between the motions in the fifth dimension and the motion in the hypersurfaces. This splitting allows the use of phase space analysis in order to investigate the possible confinement of particles and photons to hypersurfaces in fivedimensional warped product spaces. Using such an analysis, we find a novel form of quasiconfinement which is oscillatory and neutrally stable. We also find that this class of warped product spaces locally satisfy the symmetry by default. The importance of such a confinement is that it is purely due to the classical gravitational effects, without requiring the presence of branetype confinement mechanisms.
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 DYNAMICAL SYSTEMS


Analyticity of the SinaiRuelleBowen measure for a class of simple Anosov flows
View Description Hide DescriptionWe consider perturbations of the Hamiltonian flow associated with the geodesic flow on a surface with constant negative curvature. We prove that, under a small perturbation, not necessarily of Hamiltonian character, the SinaiRuelleBowen measure associated with the flow exists and is analytic in the strength of the perturbation. An explicit example of “thermostated” dissipative dynamics is considered.

Polytropic gas dynamics revisited
View Description Hide DescriptionWe investigate the Hamiltonian as well as Lax formulations of the polytropic gas dynamics which can be characterized by the polytropic exponent . We show that when with , the resonant phenomena occur in biHamiltonian formulation and logarithmictype conserved charges emerge naturally. We provide a logarithmic Lax representation for conserved charges and the whole hierarchy flows, which coincides with the biHamiltonian formulation constructed from a twodimensional Frobenius manifold associated with the polytropic gas dynamics.

Stochastic embedding of dynamical systems
View Description Hide DescriptionMost physical systems are modeled by an ordinary or a partial differential equation, like the body problem in celestial mechanics. In some cases, for example, when studying the long term behavior of the solar system or for complex systems, there exist elements which can influence the dynamics of the system which are not well modeled or even known. One way to take these problems into account consists of looking at the dynamics of the system on a larger class of objects that are eventually stochastic. In this paper, we develop a theory for the stochastic embedding of ordinary differential equations. We apply this method to Lagrangiansystems. In this particular case, we extend many results of classical mechanics, namely, the least action principle, the EulerLagrange equations, and Noether’s theorem. We also obtain a Hamiltonian formulation for our stochastic Lagrangiansystems. Many applications are discussed at the end of the paper.
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 STATISTICAL PHYSICS


Irreversibility implies the occurrence of nonmonotonic power spectra
View Description Hide DescriptionBoth continuoustime and discretetime systems driven by exponential ergodic stationary Markov processes are investigated. We prove that in both cases irreversibility implies the occurrence of nonmonotonic power spectra of some observables. This result combined with the work of Jiang and Zhang [J. Math. Phys.44, 4681–4689 (Year: 2003)] tells us that in the continuoustime case, reversibility can be characterized by monotonicity in all power spectra, while this is not true in the discretetime case.

Deformed multivariable FokkerPlanck equations
View Description Hide DescriptionIn this paper new multivariable deformed FokkerPlanck (FP) equations are presented. These deformed FP equations are associated with the Ruijsenaars–Schneider–van Diejen (RSvD)type systems in the same way that the usual onevariable FP equation is associated with the oneparticle Schrödinger equation. As the RSvD systems are the “discrete” counterparts of the celebrated exactly solvable manybody CalogeroSutherlandMoser systems, the deformed FP equations presented here can be considered as discrete deformations of the ordinary multivariable FP equations.

Remarks on the stability of the frictionless VlasovPoissonFokkerPlanck system
View Description Hide DescriptionWe present the uniform stability estimate of classical solutions to the frictionless VlasovPoissonFokkerPlanck system, when initial data are small and decay algebraically fast enough in phase space. The hypoelliptic structure of the system plays a key role in the stability analysis in low dimensions two and three, which are missing in the VlasovPoisson system [M. Chae and S.Y. Ha, SIAM J. Math. Anal.37, 1704–1731 (2006)].
