Volume 46, Issue 8, August 2005
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Finitetime stochastic reduction models
View Description Hide DescriptionA new energybased stochastic extension of the Schrödinger equation for which the wave function collapses after the passage of a finite amount of time is proposed. An exact closedform solution to the dynamical equation, valid for all finitedimensional quantum systems, is presented and used to verify explicitly that reduction of the state vector to an energy eigenstate occurs. A timechange technique is introduced to construct a “clock” variable that relates the asymptotic and the finitetime collapse models by means of a nonlinear transformation.

Nocloning of nonorthogonal states does not require inner product preserving
View Description Hide DescriptionThe nocloning theorem says there is no quantum copy machine which can copy any onequbit state. Inner product preserving was always used to prove the nocloning of nonorthogonal states. In this paper we show that the nocloning of nonorthogonal states does not require inner product preserving and discuss the minimal properties which a linear operator possesses to copy two different states at the same device. In this paper, we obtain the following necessary and sufficient condition. For any two different states and , assume that a linear operator can copy them, that is, and . Then the two states are orthogonal if and only if and are unit length states. Thus we only need linearity and that and are unit length states to prove the nocloning of nonorthogonal states. It implies that inner product preserving is not necessary for the nocloning of nonorthogonal states.

Quantum channels and representation theory
View Description Hide DescriptionIn the study of dimensional quantum channels , an assumption which includes many interesting examples, and which has a natural physical interpretation, is that the corresponding Kraus operators form a representation of a Lie algebra. Physically, this is a symmetry algebra for the interaction Hamiltonian. This paper begins a systematic study of channels defined by representations; the famous WernerHolevo channel is one element of this infinite class. We show that the channel derived from the defining representation of is a depolarizing channel for all , but for most other representations this is not the case. Since the standard Bloch sphere only exists for the qubit representation of , we develop a consistent generalization of Bloch’s technique. By representing the density matrix as a polynomial in Lie algebra generators, we determine a class of positive semidefinite matrices which represent quantum states for various channels defined by finitedimensional representations of semisimple Lie algebras. We also give a general method for finding positive semidefinite matrices using Lie algebraic trace identities. This includes an analysis of channels based on the exceptional Lie algebra and the Clifford algebra.

On approximately symmetric informationally complete positive operatorvalued measures and related systems of quantum states
View Description Hide DescriptionWe address the problem of constructing positive operatorvalued measures (POVMs) in finite dimension consisting of operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SICPOVMs) for which the inner products are perfectly uniform. However, SICPOVMs are notoriously hard to construct and, despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SICPOVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases. Moreover, we construct vector systems in which are almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on results of analytic number theory.

Geometry of the Schrödinger equation and stochastic mass transportation
View Description Hide DescriptionThe Schrödinger equation is deduced from a geometric principle. Lagrange, HamiltonJacobi, and Hamilton formalisms are defined almost analogously to the deterministic case, which can be identified as geometric optic. The form of these formalisms is identical to the deterministic formalisms. Furthermore, it will be shown how the deterministic case is “superficial” to the stochastic one if Planck’s constant is very small. An elementary proof of Heisenberg’s uncertainty relation finishes the paper.

PTinvariant periodic potentials with a finite number of band gaps
View Description Hide DescriptionWe obtain the band edge eigenstates and the midband states for the complex, generalized associated Lamé potentials , where , and there are four parameters , , , . By construction, this potential is PTinvariant since it is unchanged by the combined parity and time reversal transformations. This work is a substantial generalization of previous work with the associated Lamé potentials and their corresponding PTinvariant counterparts , both of which involving just two parameters ,. We show that for many integer values of ,,,, the PTinvariant potentials are periodic problems with a finite number of band gaps. Further, using supersymmetry, we construct several additional, complex, PTinvariant, periodic potentials with a finite number of band gaps. We also point out the intimate connection between the above generalized associated Lamé potential problem and Heun’s differential equation.

Bounds on localizable information via semidefinite programming
View Description Hide DescriptionWe investigate the socalled localizable information of bipartite states and a parallel notion of information deficit. Localizable information is defined as the amount of information that can be concentrated by means of classical communication and local operations where only maximally mixed states can be added for free. The information deficit is defined as difference between total information contents of the state and localizable information. We consider a larger class of operations, the socalled PPT operations, which in addition preserve maximally mixed state (PPTPMM operations). We formulate the related optimization problem as a semidefinite program with suitable constraints. We then provide bound for fidelity of transition of a given state into product pure state on Hilbert space of dimension . This allows to obtain a general upper bound for localizable information (and also for information deficit). We calculated the bounds exactly for Werner states and isotropic states in any dimension. Surprisingly it turns out that related bounds for information deficit are equal to relative entropy of entanglement (in the case of Werner states, regularized one). We compare the upper bounds with lower bounds based on simple protocol of localization of information.

A constructive algorithm for the Cartan decomposition of
View Description Hide DescriptionWe present an explicit numerical method to obtain the CartanKhanejaGlaser decomposition of a general element in terms of its “Cartan” and “nonCartan” components. This effectively factors in terms of group elements that belong in with , a procedure that can be iterated down to . We show that every step reduces to solving the zeros of a matrix polynomial, obtained by truncation of the BakerCampbellHausdorff formula, numerically. All computational tasks involved are straightforward and the overall truncation errors are well under control.

Clean positive operator valued measures
View Description Hide DescriptionIn quantum mechanics the statistics of the outcomes of a measuring apparatus is described by a positive operator valued measure (POVM). A quantum channel transforms POVMs into POVMs, generally irreversibly, thus losing some of the information retrieved from the measurement. This poses the problem of which POVMs are “undisturbed,” i.e., they are not irreversibly connected to another POVM. We will call such POVMs clean. In a sense, the clean POVMs would be “perfect,” since they would not have any additional “extrinsical” noise. Quite unexpectedly, it turns out that such a “cleanness” property is largely unrelated to the convex structure of POVMs, and there are clean POVMs that are not extremal and vice versa. In this article we solve the cleannes classification problem for number of outcomes ( dimension of the Hilbert space), and we provide a set of either necessary or sufficient conditions for , along with an iff condition for the case of informationally complete POVMs for .

The potential generates real eigenvalues only, under symmetric rapid decay boundary conditions
View Description Hide DescriptionWe consider the nonHermitian eigenvalue problems , under every rapid decay boundary condition that is symmetric with respect to the imaginary axis in the complex plane. We prove that the eigenvalues are all real and positive.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Monopoleantimonopole and vortex rings
View Description Hide DescriptionThe SU(2) YangMillsHiggs theory supports the existence of monopoles, antimonopoles, and vortex rings. In this paper, we would like to present new exact static antimonopolemonopoleantimonopole (AMA) configurations. The net magnetic charge of these configurations is always , while the net magnetic charge at the origin is always for all positive integer values of the solution’s parameter . However, when increases beyond 1, vortex rings appear coexisting with these AMA configurations. The number of vortex rings increases proportionally with the value of . They are located in space where the Higgs field vanishes along rings. We also show that a singlepoint singularity in the Higgs field does not necessarily correspond to a structureless 1monopole at the origin but to a zerosize monopoleantimonopolemonopole (MAM) structure when the solution’s parameter is odd. This monopole is the WuYangtype monopole and it possesses the Dirac string potential in the Abelian gauge. These exact solutions are a different kind of Bogomol’nyiPrasadSommerfield (BPS) solutions as they satisfy the firstorder Bogomol’nyi equation but possess infinite energy due to a point singularity at the origin of the coordinate axes. They are all axially symmetrical about the axis.

Extending the PicardFuchs system of local mirror symmetry
View Description Hide DescriptionWe propose an extended set of differential operators for local mirror symmetry. If is CalabiYau such that , then we show that our operators fully describe mirror symmetry. In the process, a conjecture for intersection theory for such is uncovered. We also find operators on several examples of type through similar techniques. In addition, open string PicardFuchs systems are considered.

Superselection in the presence of constraints
View Description Hide DescriptionSuperselection and constraints occur together in many gauge theories, and here we begin a study of such systems. Our main focus will be to analyze compatibility questions between constraining and superselection, and we will develop an example modelled on QED in which our framework is realized. We proceed from a generalization of DoplicherRoberts superselection theory to the case of the nontrivial center, and a set of Dirac quantum constraints and find conditions under which the superselection structures will survive constraining in some form. This involves an analysis of the restriction and factorization of superselection structures.

Operator product expansions as a consequence of phase space properties
View Description Hide DescriptionThe paper presents a modelindependent, nonperturbative proof of operator product expansions in quantum field theory. As an input, a recently proposed phase space condition is used that allows a precise description of point field structures. Based on the product expansions, we also define and analyze normal products (in the sense of Zimmermann).

Stability of LandauGinzburg branes
View Description Hide DescriptionWe evaluate the ideas of stability at the LandauGinzburg (LG) point in moduli space of compact CalabiYau manifolds, using matrix factorizations to model the topological brane category. The standard requirement of unitarity at the IR fixed point is argued to lead to a notion of “stability” for matrix factorizations of quasihomogeneous LG potentials. The brane on the quintic at the LandauGinzburg point is not obviously unstable. Aiming to relate stability to a moduli space problem, we then study the action of the gauge group of similarity transformations on matrix factorizations. We define a naive moment maplike flow on the gauge orbits and use it to study boundary flows in several examples. Gauge transformations of nonzero degree play an interesting role for braneantibrane annihilation. We also give a careful exposition of the grading of the LandauGinzburg category of branes, and prove an index theorem for matrix factorizations.

 GENERAL RELATIVITY AND GRAVITATION


Homogeneous cosmologies from the quasiMaxwell formalism
View Description Hide DescriptionWe show how to use the quasiMaxwell formalism to obtain solutions of Einstein’s field equations corresponding to homogeneous cosmologies—namely Einstein’s universe, Gödel’s universe, and the OzsvathFarnsworthKerr class I solutions—written in frames for which the associated observers are stationary.

The recollapse problem of closed Friedmann–Robertson–Walker models in higherorder gravity theories
View Description Hide DescriptionWe study the closed universe recollapse conjecture for positively curved Friedmann–Robertson–Walker models with a perfect fluid matter source and a scalar field which arises in the conformal frame of the theory. By including ordinary matter, we extend the analysis of a previous work. We analyze the structure of the resulted fourdimensional dynamical system with the methods of the center manifoldtheory and the normal form theory. It is shown that an initially expanding closed FRW universe, starting close to the Minkowski spacetime, cannot avoid recollapse. We discuss the posibility that potentials with a positive minimum may prevent the recollapse of closed universes.

 DYNAMICAL SYSTEMS


Instabilities of multiphase wave trains in coupled nonlinear Schrödinger equations: A bisymplectic framework
View Description Hide DescriptionHamiltonian systems, with bisymplectic structure, are known to model a wide range of interesting phenomena occurring in optics, oceanography, biochemistry, geology, and materials science. Examples of such systems are nonlinear Schrödinger (NLS) equations and KleinGordon (KG) equations. The paper focuses on a general class of the former and presents a linear stability theory for the interaction of a basic class of periodic traveling wavesolutions, which exploits the geometric structure of the system. A criterion for linear instability is derived. Additionally, for the qualitatively tractable cases, criteria for linear instability are given explicitly in terms of: the amplitudes of the modes; the parameters of the system that characterize the medium as well as the interaction between component modes; and, when the solutions of the system are both timeand space dependent, the wave numbers. An extension to the coupled NLS equations case study is introduced, namely the consideration of a related class of coupled KG equations, which has the potential to lead to further development for the underlying bisymplectic systems theory.

Supersymmetric nonlocal gas equation
View Description Hide DescriptionIn this paper we study systematically the question of supersymmetrization of the nonlocal gas equation. We obtain both the and the supersymmetric generalizations of the system which are integrable. We show that both the systems are biHamiltonian. While the supersymmetrization allows the hierarchy of equations to be extended to negative orders (local equations), we argue that this is not the case for the supersymmetrization. In the bosonic limit, however, the system of equations lead to a new coupled integrable system of equations.

 FLUIDS


Existence theorems for twofluid magnetohydrodynamics
View Description Hide DescriptionThe description of a plasma as composed by two types of fluids, formed by ions and electrons, is more complete than the classical onefluid magnetohydrodynamics(MHD) model and it has proved necessary to explain the phenomena of fast magnetic reconnection. We prove a finitetime theorem of existence and uniqueness of solutions for this system for either Dirichlet or periodic boundary conditions in dimension three. It turns out that the regularity estimates for the magnetic field are finer than the MHD ones.
