Volume 51, Issue 11, November 2010
 ARTICLES

 Quantum Mechanics (General and Nonrelativistic)

Quantum integrals and anhomomorphic logics
View Description Hide DescriptionThe basic arena for a probabilistic structure is a set of events. Corresponding to is a dual structure of coevents. We call an anhomomorphic logic and the coevents are given by “truth functions” from to the twoelement Boolean algebra. One of the main goals of a physical theory is to describe physical reality and a coevent provides such a description in the sense that an event “actually occurs” if and only if ϕ(A) = 1. The quantum integral over an event A with respect to a coevent ϕ is defined and its properties are treated. Integrals with respect to various coevents are computed. Quantum systems are frequently described by a quantum measure μ which gives the propensity μ(A) that an event A occurs. For , if ϕ(A) = 0 whenever μ(A) = 0 we say that ϕ is preclusive. Preclusivity is a reality filter because it eliminates coevents that do not describe a possible reality for the system. A quantum measure that can be represented as a quantum integral with respect to a coevent ϕ is said to 1generate ϕ. This gives a stronger reality filter than preclusivity. What we believe to be a more general filter is defined in terms of a double quantum integral and is called 2generation. We show that there are quantum measures that 2generate coevents, but do not 1generate coevents. Examples also show that there are coevents that are 2generated but not 1generated. For simplicity only finite systems are considered.

Twist deformation of rotationally invariant quantum mechanics
View Description Hide DescriptionNoncommutative quantum mechanics in 3D is investigated in the framework of an abelian Drinfeld twist which deforms a given Hopf algebrastructure. Composite operators (of coordinates and momenta) entering the Hamiltonian have to be reinterpreted as primitive elements of a dynamical Lie algebra which could be either finite (for the harmonic oscillator) or infinite (in the general case). The deformed brackets of the deformed angular momenta close the so(3) algebra. On the other hand, undeformed rotationally invariant operators can become, under deformation, anomalous (the anomaly vanishes when the deformation parameter goes to zero). The deformed operators, Taylorexpanded in the deformation parameter, can be selected to minimize the anomaly. We present the deformations (and their anomalies) of undeformed rotationally invariant operators corresponding to the harmonic oscillator (quadratic potential), the anharmonic oscillator (quartic potential), and the Coulomb potential.

The structure of strongly additive states and Markov triplets on the CAR algebra
View Description Hide DescriptionWe find a characterization of states satisfying equality in strong subadditivity of entropy and of Markov triplets on the CAR algebra. For even states, a more detailed structure of the density matrix is given.

Jumpdiffusion unravelling of a nonMarkovian generalized Lindblad master equation
View Description Hide DescriptionThe “correlatedprojection technique” has been successfully applied to derive a large class of highly nonMarkovian dynamics, the so called nonMarkovian generalized Lindbladtype equations or Lindblad rate equations. In this article, general unravelings are presented for these equations, described in terms of jumpdiffusion stochastic differential equations for wave functions. We show also that the proposed unraveling can be interpreted in terms of measurements continuous in time but with some conceptual restrictions. The main point in the measurement interpretation is that the structure itself of the underlying mathematical theory poses restrictions on what can be considered as observable and what is not; such restrictions can be seen as the effect of some kind of superselection rule. Finally, we develop a concrete example and discuss possible effects on the heterodyne spectrum of a twolevel system due to a structured thermallike bath with memory.
 Quantum Information and Computation

All maximally entangled fourqubit states
View Description Hide DescriptionWe find an operational interpretation for the 4tangle as a type of residual entanglement, somewhat similar to the interpretation of the 3tangle. Using this remarkable interpretation, we are able to find the class of maximally entangled fourqubits states which is characterized by four real parameters. The states in the class are maximally entangled in the sense that their average bipartite entanglement with respect to all possible bipartite cuts is maximal. We show that while all the states in the class maximize the average tangle, there are only a few states in the class that maximize the average Tsillas or Renyi αentropy of entanglement. Quite remarkably, we find that up to local unitaries, there exists two unique states, one maximizing the average αTsallis entropy of entanglement for all α ⩾ 2, while the other maximizing it for all 0 < α ⩽ 2 (including the vonNeumann case of α = 1). Furthermore, among the maximally entangled four qubits states, there are only three maximally entangled states that have the property that for two, out of the three bipartite cuts consisting of twoqubits verses twoqubits, the entanglement is 2 ebits and for the remaining bipartite cut the entanglement between the two groups of two qubits is 1 ebit. The unique three maximally entangled states are the three cluster states that are related by a swap operator. We also show that the cluster states are the only states (up to local unitaries) that maximize the average αRenyi entropy of entanglement for all α ⩾ 2.
 Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

The relativistic treatment of spin0 particles under the rotating Morse oscillator
View Description Hide DescriptionWe present the energy eigenvalues and corresponding normalized eigenfunctions of the relativistic spin0 particles by solving the Klein–Gordon equation. Analytical forms for the energy eigenvalues and eigenfunctions have been derived by using Pekeris approximation to the centrifugal term within the framework of the asymptotic iteration method for the equal vector and scalar rotating Morse oscillator. The eigenvalue equation results in a transcendental form, in which the numerical values are presented in atomic units for arbitrary n and ℓ quantum states.

Supersymmetry algebra cohomology. II. Primitive elements in 2 and 3 dimensions
View Description Hide DescriptionThe primitive elements of the supersymmetryalgebra cohomology as defined in a companion paper are computed exhaustively for standard supersymmetryalgebras in dimensions D = 2 and D = 3, for all signatures (t, D − t) and all numbers N of sets of supersymmetries.

Euler characteristic of coherent sheaves on simplicial torics via the Stanley–Reisner ring
View Description Hide DescriptionWe combine work of Cox on the total coordinate ring of a toric variety and results of Eisenbud–Mustaţǎ–Stillman and Mustaţǎ on cohomology of toric and monomial ideals to obtain a formula for computing for a Weil divisor D on a complete simplicial toric variety X _{Σ}. The main point is to use Alexander duality to pass from the toric irrelevant ideal, which appears in the computation of , to the Stanley–Reisner ideal of Σ, which is used in defining the Chow ring of X _{Σ}.
 General Relativity and Gravitation

Perturbations of the Kerr black hole and the boundness of linear waves
View Description Hide DescriptionArtificial black holes (also called acoustic or optical black holes) are the black holes for the linear wave equation describing the wave propagation in a moving medium. They attracted a considerable interest of physicists who study them to better understand the black holes in general relativity. We consider the case of stationary axisymmetric metrics and we show that the Kerrblack hole is not stable under perturbations in the class of all axisymmetric metrics. We describe families of axisymmetric metrics having black holes that are the perturbations of the Kerrblack hole. We also show that the ergosphere can be determined by boundary measurements. Finally, we prove the uniform boundness of the solution in the exterior of the black hole when the event horizon coincides with the ergosphere.

A simple proof of Birkhoff's theorem for cosmological constant
View Description Hide DescriptionWe provide a simple, unified proof of Birkhoff’s theorem for the vacuum and cosmological constant case, emphasizing its local nature. We discuss its implications for the maximal analytic extensions of Schwarzschild, Schwarzschild(–anti)de Sitter, and Nariai spacetimes. In particular, we note that the maximal analytic extensions of extremal and overextremal Schwarzschild–de Sitter spacetimes exhibit no static region. Hence the common belief that Birkhoff’s theorem implies staticity is false for the case of positive cosmological constant. Instead, the correct point of view is that generalized Birkhoff’s theorems are local uniqueness theorems whose corollary is that locally spherically symmetric solutions of Einstein’s equations exhibit an additional local Killing vector field.
 Classical Mechanics and Classical Fields

Twodimensional Blasius viscous flow of a powerlaw fluid over a semiinfinite flat plane
View Description Hide DescriptionAnalytic results are obtained for the similarity equation governing the twodimensional Blasius viscousflow of a powerlaw fluid over a semiinfinite flat plane via Taylor series for small values of the independent similarity variable. Then, an analytic perturbative procedure is used to determine an approximate solution that exhibits the correct asymptotic behavior. This perturbation method allows for the computation of the shear stress at the wall, something which is impossible with a Taylor series approach. It is found that the perturbation solutions converge sufficiently rapidly; indeed, a first order approximation gives qualitatively accurate results. Furthermore, we employ the perturbation method to deduce the influence of the powerlaw index, n, on the obtained similarity solutions.

Collective coordinate variable for solitonpotential system in sineGordon model
View Description Hide DescriptionA collective coordinate variable for adding a space dependent potential to the sineGordon model is presented. Interaction of solitons with a delta function potential barrier and also a delta function potential well is investigated. A majority of the interactive characters are derived analytically. We find that the behavior of a solitonic solution is similar to a point particle which is moved under the influence of a complicated effective potential. The effective potential is a function of the initial field conditions and parameters of added potential.
 Statistical Physics

Theory of macroscopic fluctuations in systems of particles, interacting with hydrodynamic and gaslike media
View Description Hide DescriptionOur work offers a stochastic approach for description of long wavefluctuations in systems of particles interacting with hydrodynamic media. The approach is based on the averaging of nonlinear dynamic equations over random initial conditions for these equations. Random character of the initial conditions for motion equations causes the development of long wavefluctuations of description parameters in the system. Within the framework of our approach we derived evolution equations of long wale fluctuations in the system for both fluctuationkinetic and fluctuationhydrodynamic stages of evolution. We also found the class of description parameter transformations which does not change the dynamic equation structure. Dynamics of pair correlations has been studied. The “long hydrodynamic tails” theory for system of particles interacting with the hydrodynamic medium has been built. The system considered can be a model of neutron transport in hydrodynamic media without their multiplication and capture. The possibility of usage of our results for experimental evidence of long hydrodynamic tails in neutron scattering experiments is discussed.

Random close packing in a granular model
View Description Hide DescriptionWe introduce a twodimensional lattice model of granular matter. Using a combination of proof and simulation we demonstrate an order/disorder phase transition in the model, to which we associate the granular phenomenon of random close packing. We use Peierls contours to prove that the model is sensitive to boundary conditions at high density and Markov chain Monte Carlo simulation to show it is insensitive at low density.

Gibbs random graphs on point processes
View Description Hide DescriptionConsider a discrete locally finite subset Γ of and the complete graph (Γ, E), with vertices Γ and edges E. We consider Gibbs measures on the set of subgraphs with vertices Γ and edges E′⊂E. The Gibbs interaction acts between open edges having a vertex in common. We study percolationproperties of the Gibbs distribution of the graph ensemble. The main results concern percolationproperties of the open edges in two cases: (a) when Γ is sampled from a homogeneous Poisson process; and (b) for a fixed Γ with sufficiently sparse points.
 Methods of Mathematical Physics

Darboux transformations for a generalized Dirac equation in two dimensions
View Description Hide DescriptionWe construct explicit Darboux transformations for a generalized, twodimensional Dirac equation. Our results complement and generalize former findings for Dirac equations in two and three spatial dimensions. We show that as a particular case, our Darboux transformations are applicable to the twodimensional Dirac equation in cylindrical coordinates and give several examples.

A geometric interpretation of prolongation by means of connections
View Description Hide DescriptionA geometric interpretation of prolongation can be formulated by using the theory of connections. A fiber bundle can be established which is composed of a base manifold and variables which span a prolongation space. A particular connection is introduced in terms of these coordinates. This provides a very different way of viewing the technique and for introducing prolongation algebras as well as generating integrable equations in a novel way.

Decomposition results for Gram matrix determinants
View Description Hide DescriptionWe study the Gram matrix determinants for the groups S _{ n }, O _{ n }, B _{ n }, H _{ n }, for their free versions , and for the halfliberated versions . We first collect all the known computations of such determinants, along with complete and simplified proofs, and with generalizations where needed. We conjecture that all these determinants decompose as D = ∏_{π}φ(π), with product over all associated partitions.

A closed formula for the barrier transmission coefficient in quaternionic quantum mechanics
View Description Hide DescriptionIn this paper, we analyze, by using a matrix approach, the dynamics of a nonrelativistic particle in presence of a quaternionic potential barrier. The matrix method used to solve the quaternionic Schrödinger equation allows us to obtain a closed formula for the transmission coefficient. Up to now, in quaternionic quantum mechanics, almost every discussion on the dynamics of nonrelativistic particle was motivated by or evolved from numerical studies. A closed formula for the transmission coefficient stimulates an analysis of qualitative differences between complex and quaternionic quantum mechanics and by using the stationary phase method, gives the possibility to discuss transmission times.

Skin effect with arbitrary specularity in Maxwellian plasma
View Description Hide DescriptionThe problem of the skin effect with arbitrary specularity in Maxwellian plasma with specular–diffuse boundary conditions is solved. A new analytical method is developed that makes it possible to obtain a solution up to an arbitrary degree of accuracy. The method is based on the idea of symmetric continuation of not only the electric field, but also electron distribution function. The solution is obtained in a form of von Neumann series.