Volume 48, Issue 8, August 2007
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


On the decay rate of the false vacuum
View Description Hide DescriptionThe finite size theory of metastability in a quartic potential is developed by the semiclassical path integral method. In the quantum regime, the relation between temperature and classical particle energy is found in terms of the first complete elliptic integral. At the sphaleron energy, the criterion which defines the extension of the quantum regime is recovered. Within the latter, the temperature effects on the fluctuation spectrum are evaluated by the functional determinant method and computation. The eigenvalue which causes metastability is determined as a function of size/temperature by solving a Lamè equation. The ground state lifetime shows remarkable deviations with respect to the result of the infinite size theory.

Relativistic effects and rigorous limits for discrete and continuoustime quantum walks
View Description Hide DescriptionThe mathematical relationship between the discretetime and continuoustime quantum walks and the onedimensional Dirac equation is explored by studying a class of solutions for each, expressed in terms of the generalized, regular, and modified Bessel functions, respectively. Rigorous limits connecting these solutions are established. In addition, new analytical and numerical results are presented for quantum walks and the Dirac equation, including entanglement, relativistic localization and wave packet spreading, and normal and anomalous Zitterbewegung.

Universal jointmeasurement uncertainty relation for error bars
View Description Hide DescriptionWe formulate and prove a new, universally valid uncertainty relation for the necessary error bar widths in any approximate joint measurement of position and momentum.

Quantum mechanics with respect to different reference frames
View Description Hide DescriptionGeometric (Schrödinger) quantization of nonrelativistic mechanics with respect to different reference frames is considered. In classical nonrelativistic mechanics, a reference frame is represented by a connection on a configuration space fibered over a time axis . Under quantization, it yields a connection on the quantum algebra of Schrödinger operators. The operators of energy with respect to different reference frames are examined.

Simple derivation of the vector addition coefficients
View Description Hide DescriptionTwo simple operators are constructed in the product space of two angular momenta. Powers of the operators, when acting on selected product vectors, produce vectors with definite total angular momentum and thereby directly yield the vector addition coefficients in a symmetric form.

Exact Heisenberg operator solutions for multiparticle quantum mechanics
View Description Hide DescriptionExact Heisenberg operator solutions for independent “sinusoidal coordinates” as many as the degree of freedom are derived for typical exactly solvable multiparticle quantum mechanical systems, the Calogero systems [J. Math. Phys.12, 419 (1971)] based on any root system. These Heisenberg operator solutions also present the explicit forms of the annihilationcreation operators for various quanta in the interacting multiparticle systems. At the same time they can be interpreted as multivariable generalization of the three term recurrence relations for multivariable orthogonal polynomials constituting the eigenfunctions.

No nonlocal box is universal
View Description Hide DescriptionWe show that standard nonlocal boxes, also known as PopescuRohrlich machines, are not sufficient to simulate any nonlocal correlations that do not allow signaling. This was known in the multipartite scenario, but we extend the result to the bipartite case. We then generalize this result further by showing that no finite set containing any finiteoutputalphabet nonlocal boxes can be a universal set for nonlocality.

Point interaction Hamiltonians in bounded domains
View Description Hide DescriptionMaking use of recent techniques in the theory of selfadjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a threedimensional bounded domain with regular boundaries. In the particular case of one point interaction acting in the center of a ball, we obtain an explicit representation of the point spectrum of the operator together with the corresponding eigenfunctions. These operators are used to build up a model system where the dynamics of a quantum particle depends on the state of a quantum bit.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Finite temperature Casimir effect for a massless fractional KleinGordon field with fractional Neumann conditions
View Description Hide DescriptionThis paper studies the Casimir effect due to fractional massless KleinGordon field confined to parallel plates. A new kind of boundary condition called fractional Neumann condition which involves vanishing fractional derivatives of the field is introduced. The fractional Neumann condition allows the interpolation of Dirichlet and Neumann conditions imposed on the two plates. There exists a transition value in the difference between the orders of the fractional Neumann conditions for which the Casimir force changes from attractive to repulsive. Low and high temperature limits of Casimir energy and pressure are obtained. For sufficiently high temperature, these quantities are dominated by terms independent of the boundary conditions. Finally, validity of the temperature inversion symmetry for various boundary conditions is discussed.

An approximate solution of DiracHulthén problem with pseudospin and spin symmetry for any state
View Description Hide DescriptionFor any spinorbit quantum number , the analytical solutions of the Dirac equation are presented for the Hulthén potential by applying an approximation to spinorbit coupling potential for the case of spin symmetry, , and pseudospin symmetry . The bound state energy eigenvalues and the corresponding spinors are obtained in the closed forms.

Full and partial gauge fixing
View Description Hide DescriptionGauge fixing may be done in different ways. We show that using the chain structure to describe a constrained system enables us to use either a full gauge, in which all gauged degrees of freedom are determined, or a partial gauge, in which some first class constraints remain as subsidiary conditions to be imposed on the solutions of the equations of motion. We also show that the number of constants of motion depends on the level in a constraint chain in which the gauge fixing condition is imposed. The relativistic point particle, electromagnetism, and the Polyakov string are discussed as examples and full or partial gauges are distinguished.

LandauGinzburg to CalabiYau dictionary for Dbranes
View Description Hide DescriptionBased on the work by Orlov (eprint arXiv:math.AG∕0503632), we give a precise recipe for mapping between Btype Dbranes in a LandauGinzburg orbifold model (or Gepner model) and the corresponding large radius CalabiYau manifold. The Dbranes in LandauGinzburgtheories correspond to matrix factorizations and the Dbranes on the CalabiYau manifolds are objects in the derived category. We give several examples including branes on quotient singularities associated with weighted projective spaces. We are able to confirm several conjectures and statements in the literature.

Charged particles in crossed and longitudinal electromagnetic fields and beam guides
View Description Hide DescriptionWe consider a class of electromagnetic fields that contains crossed fields combined with longitudinal electric and magnetic fields. We study the motion of a classical particle (solutions of the Lorentzequations) in such fields. Then, we present an analysis that allows one to decide which fields from the class act as a beam guide for charged particles, and we find some timeindependent and timedependent configurations with beam guiding properties. We demonstrate that the KleinGordon and Dirac equations with all the fields from the class can be solved exactly. We study these solutions, which were not known before, and prove that they form complete and orthogonal sets of functions.

Classical BecchiRouetStoraTyutin charge for nonlinear algebras
View Description Hide DescriptionWe study the construction of the classical nilpotent canonical BecchiRouetStoraTyutin (BRST) charge for the nonlinear gauge algebras, where a commutator (in terms of Poisson brackets) of the constraints is a finite order polynomial of the constraints. Such a polynomial is characterized by the coefficients forming a set of higher order structure constraints. Assuming the set of constraints to be linearly independent, we find the restrictions on the structure constraints when the nilpotent BRST charge can be written in a simple and universal form. In the case of quadratically nonlinear algebras, we find the expression for third order contribution in the ghost fields to the BRST charge without the use of any additional restrictions on the structure constants.
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 GENERAL RELATIVITY AND GRAVITATION


Areal foliation and asymptotically velocityterm dominated behavior in symmetric spacetimes with positive cosmological constant
View Description Hide DescriptionWe prove a global foliation result, using areal time, for symmetric spacetimes with a positive cosmological constant. We then find a class of solutions that exhibit asymptotically velocityterm dominated behavior near the singularity.

On some solutions of Maxwell’s equations in a Schwarzschild–de Sitter black hole
View Description Hide DescriptionWe determine the electrostatic potential due to a point charge in the Schwarzschild–de Sitter (SdS) spacetime as well as the magnetic potential due to a radial current in this background. In these two cases, an extremal SdS black hole scenario is assumed.
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 DYNAMICAL SYSTEMS


New integrable hierarchy, its parametric solutions, cuspons, onepeak solitons, and M/Wshape peak solitons
View Description Hide DescriptionIn this paper, we propose a new completely integrable hierarchy. Particularly in the hierarchy we draw two new soliton equations: (1) ; (2) , . The first one is the second positive member in the hierarchy while the second one is the second negative member in the hierarchy. Both equations can be derived from the twodimensional Euler equation by using the approximation procedure. All equations in the hierarchy are proven to have biHamiltonian operators and Lax pairs through solving a crucial matrix equation. Moreover, we develop parametric solutions of the entire hierarchy through constructing two kinds of constraints; one is for all negative members of the hierarchy on a symplectic submanifold, and the other is for all positive members in the standard symplectic space. The most interesting things are both equations possess new type of peaked solitons—continuous and piecewise smooth “W/Mshape peak” soliton solutions. In addition, we find new cusp solitons—cuspons for the second equation and onesinglepeak solitons for the first—which are also continuous and piecewise smooth but not in the regular type ( is a constant).

Dynamics of the Lorenz system having an invariant algebraic surface
View Description Hide DescriptionIn this paper, we characterize all dynamics of the Lorenz system , , having an invariant algebraic surface.

Asymptotic stability for a class of metriplectic systems
View Description Hide DescriptionUsing the framework of metriplectic systems on we will describe a constructive geometric method to add a dissipation term to a HamiltonPoisson system such that any solution starting in a neighborhood of a nonlinear stable equilibrium converges toward a certain invariant set. The dissipation term depends only on the Hamiltonian function and the Casimir functions.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Influence of nonholonomic constraints on variations, symplectic structure, and dynamics of mechanical systems
View Description Hide DescriptionBased on a serious analysis of the Frobenius integrability condition for affine differential constraints that mechanical systems are subject to the necessary and sufficient conditions for coincidence of three kinds of unfree variations, the existence of simple symplectic structure of the constraint submanifold and equivalence of nonholonomic and vakonomic dynamics for the constrained systems are, respectively, obtained, which are all related with the Frobenius integrability condition in their special forms. Two illustrative examples are presented to verify the results.
