Volume 51, Issue 11, November 2010
Index of content:
- Quantum Mechanics (General and Nonrelativistic)
51(2010); http://dx.doi.org/10.1063/1.3511333View Description Hide Description
The 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 two-element 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 1-generate 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 2-generation. We show that there are quantum measures that 2-generate coevents, but do not 1-generate coevents. Examples also show that there are coevents that are 2-generated but not 1-generated. For simplicity only finite systems are considered.
51(2010); http://dx.doi.org/10.1063/1.3506406View Description Hide Description
Noncommutative 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, Taylor-expanded 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.
51(2010); http://dx.doi.org/10.1063/1.3516474View Description Hide Description
51(2010); http://dx.doi.org/10.1063/1.3514539View Description Hide Description
The “correlated-projection technique” has been successfully applied to derive a large class of highly non-Markovian dynamics, the so called non-Markovian generalized Lindblad-type equations or Lindblad rate equations. In this article, general unravelings are presented for these equations, described in terms of jump-diffusion 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 two-level system due to a structured thermal-like bath with memory.
- Quantum Information and Computation
51(2010); http://dx.doi.org/10.1063/1.3511477View Description Hide Description
We find an operational interpretation for the 4-tangle as a type of residual entanglement, somewhat similar to the interpretation of the 3-tangle. Using this remarkable interpretation, we are able to find the class of maximally entangled four-qubits 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 α-entropy of entanglement. Quite remarkably, we find that up to local unitaries, there exists two unique states, one maximizing the average α-Tsallis entropy of entanglement for all α ⩾ 2, while the other maximizing it for all 0 < α ⩽ 2 (including the von-Neumann 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 two-qubits verses two-qubits, 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 α-Renyi entropy of entanglement for all α ⩾ 2.
- Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)
51(2010); http://dx.doi.org/10.1063/1.3503413View Description Hide Description
We present the energy eigenvalues and corresponding normalized eigenfunctions of the relativistic spin-0 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.
51(2010); http://dx.doi.org/10.1063/1.3515845View Description Hide Description
The 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.
51(2010); http://dx.doi.org/10.1063/1.3506404View Description Hide Description
We 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
51(2010); http://dx.doi.org/10.1063/1.3503297View Description Hide Description
Artificial 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.
51(2010); http://dx.doi.org/10.1063/1.3503447View Description Hide Description
We 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
51(2010); http://dx.doi.org/10.1063/1.3503774View Description Hide Description
Analytic results are obtained for the similarity equation governing the two-dimensional Blasius viscousflow of a power-law fluid over a semi-infinite 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 power-law index, n, on the obtained similarity solutions.
51(2010); http://dx.doi.org/10.1063/1.3511337View Description Hide Description
A collective coordinate variable for adding a space dependent potential to the sine-Gordon 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 media51(2010); http://dx.doi.org/10.1063/1.3505828View Description Hide Description
Our 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 fluctuation-kinetic and fluctuation-hydrodynamic 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.
51(2010); http://dx.doi.org/10.1063/1.3511359View Description Hide Description
We introduce a two-dimensional 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.
51(2010); http://dx.doi.org/10.1063/1.3514605View Description Hide Description
Consider 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 sub-graphs 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
51(2010); http://dx.doi.org/10.1063/1.3505127View Description Hide Description
We construct explicit Darboux transformations for a generalized, two-dimensional 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 two-dimensional Dirac equation in cylindrical coordinates and give several examples.
51(2010); http://dx.doi.org/10.1063/1.3504172View Description Hide Description
A 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.
51(2010); http://dx.doi.org/10.1063/1.3511332View Description Hide Description
We study the Gram matrix determinants for the groups S n , O n , B n , H n , for their free versions , and for the half-liberated 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 = ∏πφ(π), with product over all associated partitions.
51(2010); http://dx.doi.org/10.1063/1.3504165View Description Hide Description
In 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.
51(2010); http://dx.doi.org/10.1063/1.3503768View Description Hide Description
The 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.