Symmetries of Field Theories on the Noncommutative Plane

We report new developments concerning the symmetry properties and their actions on special solutions allowed by certain field theory models on the noncommutative plane. In particular, we seek Galilean-invariant models. The analysis indicates that this requirement strongly restricts the admissible interactions. Moreover, if a scalar field is coupled to a gauge field, then a geometric phase emerges for vortexlike solutions transformed by Galilean boosts.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 64–73, July, 2005.     

The physical interpretation of partial traces: Two nonstandard views

Mixed states are introduced in physics to express our ignorance about the actual state of a physical system and are represented in standard quantum mechanics by density operators. Such operators also appear if we consider a (pure) entangled state of a compound system Ω and take partial traces on the projection operator representing it. But because the coefficients in the convex sums expressing them never bear the ignorance interpretation in this case, they represent not mixed states (proper mixtures) but improper mixtures of the subsystems. Hence, states cannot be attributed to the subsystems of a compound physical system in an entangled state (the subentity problem). We discuss two alternative proposals that can be developed in the Brussels and the Lecce approaches. We firstly summarize the general framework provided by the Brussels approach, which suggests that improper mixtures can be regarded as new pure states. We then show that improper mixtures can also be regarded as true (but nonpure) states according to the Lecce approach. Despite their different terminologies, the two proposals seem compatible. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 248–264, August, 2007.     

Static Vortex Solutions in a Planar Particle System Obeying a Generalized Exclusion Principle

We consider a planar particle system obeying a generalized Pauli exclusion principle. In the mean field approximation, this system is described by a Schrödinger equation we recently introduced, containing a complex nonlinearity. The particle number, the total energy, and the angular momentum are conserved in such a system. We consider vortexlike stationary solutions of the form and write the differential equation for the vortex shape. We find an analytic solution of this equation and obtain a closed expression for the vortex profile. We investigate some mean properties and, in particular, calculate the energy spectrum and angular momentum of the vortex.     

Linear dependence of stationary distributions in ergodic Markov decision processes

In ergodic MDPs we consider stationary distributions of policies that coincide in all but n states, in which one of two possible actions is chosen. We give conditions and formulas for linear dependence of the stationary distributions of n+2 such policies, and show some results about combinations and mixtures of policies.     

Echo spectroscopy and atom optics billiards

We discuss a recently demonstrated type of microwave spectroscopy of trapped ultra-cold atoms known as “echo spectroscopy” [Phys. Rev. Lett., 2003;90:023001[1]–[4]]. Echo spectroscopy can serve as an extremely sensitive experimental tool for investigating quantum dynamics of trapped atoms even when a large number of states are thermally populated. We show numerical results for the stability of eigenstates of an atom-optics billiard of the Bunimovich type, and discuss its behavior under different types of perturbations. Finally, we propose to use special geometrical constructions to make a dephasing free dipole trap.     

Parapositronium atom in the field of annihilation and optical photons as a nonlinear quantum system

A parapositronium atom in an optical laser field is described beyond the perturbation theory framework by a closed system of Heisenberg equations on operators of atoms and photons. Wwe consider the annihilation of the parapositronium atom, which starts from one or two quantum states; optical quantum transitions between these states are caused by one or two optical photons. Mean occupation numbers of these states are governed by a system of two nonlinear equations. We investigated particular stationary and nonstationary solutions of this system and find that annihilation photons substantially affect the annihilation process. We show that definite optical laser radiation may stabilize the parapositronium atom and make its lifetime hundreds of times longer than the lifetime of the free parapositronium atom in the 1s state. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 1, pp. 148–168, July, 2000.     

Representation and interpretation of quantum mixtures in the ESR model

The interpretation of mixtures is problematic in quantum mechanics (QM) because physical properties are nonobjective in this theory. An extended semantic realism model was recently developed, restoring objectivity by reinterpreting quantum probabilities as conditional on detection and embodying the QM mathematical formalism in a broader noncontextual (hence local) framework. In this model, each generalized observable is represented by a family of positive operator-valued measures parameterized by the pure states of the considered physical system Ω. We here propose a new proof that each proper mixture is represented by a family of density operators parameterized by the macroscopic properties characterizing Ω. We then show that this representation implies some predictions differing from the QM predictions and avoids the problems following from the standard QM representation of proper mixtures. We also recall that the state transformations induced by idealized nondestructive measurements can be obtained using a nontrivial generalization of the Lüders postulate.     

Adaptive grid refinement for a model of two confined and interacting atoms

We have applied adaptive grid refinement to solve a two-dimensional Schrödinger equation in order to study the feasibility of a quantum computer based on extremely-cold neutral alkali-metal atoms. Qubits are implemented as motional states of an atom trapped in a single well of an optical lattice of counter-propagating laser beams. Quantum gates are constructed by bringing two atoms together in a single well leaving the interaction between the atoms to cause entanglement. For special geometries of the optical lattices and thus shape of the wells, quantifying the entanglement reduces to solving for selected eigenfunctions of a Schrödinger equation that contains a two-dimensional Laplacian, a trapping potential that describes the optical well, and a short-ranged interaction potential. The desired eigenfunctions correspond to eigenvalues that are deep in the interior of the spectrum where the trapping potential becomes significant. The spatial range of the interaction potential is three orders of magnitude smaller than the spatial range of the trapping potential, necessitating the use of adaptive grid refinement.     

Visible light emission in collisions between O5+ ions and He atoms

Visible light emission spectra during collisions between O⁵⁺ ions and He atoms in the range of 200–800 nm at different projectile energies (4.06–5.31 keV-u⁻¹) have been measured. Emission spectra show that there are three channels of excitation in the O⁵⁺-He collision system: (i) single-electron capture into excited states; (ii) double-electron capture into excited states; (iii) direct excitation of target atoms. There are transitions between states with comparably high quantum numbersnl. The absolute emission cross-sections of every spectral line are calculated. The relations of these cross-sections with the energy of the projectiles are studied. Project supported by the National Natural Science Foundation of China and Chinese Association for Atomic and Molecular Dada.     

Travelling waves of phase transitions in porous media

Liquid–vapour phase changes for a fluid flow through a porous medium are considered; in particular, the model allows for phase mixtures and includes an equilibrium pressure. Existence and uniqueness of travelling waves is established in a wide range of situations; the end states may be formed either by pure phases or mixtures; in the latter case the pressure equals the equilibrium pressure. A formal asymptotic analysis for vanishing relaxation time is carried out to show that the friction and reaction source terms have smoothing effect when the pressure is close to the equilibrium pressure and pure phases are avoided.     

Gibbs paradox, liquid phase as an alternative to the bose condensate, and homogeneous mixtures of new ideal gases

In this paper, relations for the liquid phase of a new ideal gas are given. The points of the spinodal of the liquid phase (i.e., the endpoints of metastable states) in the domains of positive and negative pressures are defined. Relations for the critical values of homogeneous mixtures are presented. We study the phase transition and critical indices from the point of view of the geometric quantization of thermodynamics.     

A (3, 3)-homogeneous quantum logic with 18 atoms. I

A quantum logic is called (m, n)-homogeneous if any its atom is contained exactly in m maximal (with respect to inclusion) orthogonal sets of atoms (we call them blocks), and every block contains exactly n elements. We enumerate atoms by natural numbers. For each block {i, j, k} we use the abbreviation i-j-k. Every such logic has the following 7 initial blocks B ₁, ..., B ₇: 1-2-4-5, 1-6-7, 2-8-9, 2-10-11, 3-12-13, and 3-14-15. For an 18-atom logic the arrangements the rest atoms 16, 17, and 18 is important. We consider the case when they form a loop of order in one of layers composed of initial blocks, for example, l ₄: 3-14-15, 15-16-17, 17-18-13, and 13-12-3. We prove that there exist (up to isomorphism) only 5 such logics, and describe pure states and automorphism groups for them.     

The final σ-algebra of an inhomogeneous Markov chain with a finite number of states

It is proved that the final -algebra in the case of an inhomogeneous Markov chain with a finite number of states n is generated by a finite number ( n) of atoms. The atoms are characterized from the point of view of the behavior of trajectories of the chain. Sufficient conditions are given (in the case of a countable number of states) that there should exist an unique atom at infinity.Translated from Matematicheskie Zametki, Vol. 12, No. 3, pp. 295–302, September, 1972.     

Two-particle scattering in a periodic medium

We consider a two-particle Schrödinger difference operator with a periodic potential perturbed by an exponentially decaying interaction potential for particles on a one-dimensional lattice. We obtain rigorous results for the two-particle scattering problem in the case of a small interaction and low velocities. Here, as in other quasi-one-dimensional models, small interactions can significantly affect the scattering pattern. In particular, we find the probability that the velocities of two particles in a periodic medium (e.g., they can be ultracold atoms in a one-dimensional optical lattice) change their signs during a collision. This probability increases as the relative velocity decreases and also as the absolute value of the matrix element between single-particle unperturbed Bloch states increases.     

A characterization for mixtures of semi-Markov processes

Mixtures of recurrent semi-Markov processes are characterized through a partial exchangeability condition of the array of successor states and holding times. A stronger invariance condition on the joint law of successor states and holding times leads to mixtures of Markov laws.     

Adaptive Mixture Modeling Metropolis Methods for Bayesian Analysis of Nonlinear State-Space Models

We describe a strategy for Markov chain Monte Carlo analysis of nonlinear, non-Gaussian state-space models involving batch analysis for inference on dynamic, latent state variables and fixed model parameters. The key innovation is a Metropolis–Hastings method for the time series of state variables based on sequential approximation of filtering and smoothing densities using normal mixtures. These mixtures are propagated through the nonlinearities using an accurate, local mixture approximation method, and we use a regenerating procedure to deal with potential degeneracy of mixture components. This provides accurate, direct approximations to sequential filtering and retrospective smoothing distributions, and hence a useful construction of global Metropolis proposal distributions for simulation of posteriors for the set of states. This analysis is embedded within a Gibbs sampler to include uncertain fixed parameters. We give an example motivated by an application in systems biology. Supplemental materials provide an example based on a stochastic volatility model as well as MATLAB code.     

Semiclassical modeling of the states and spectra of para helium singlets with an algebraic formula for excited states

Highly accurate energy states and spectra are deduced for para helium singlets from a new semiclassical model of two-electron atoms. The required computer methodology is simple and economical, with no computation requiring more than one second on a DEC-20 mainframe. An explicit algebraic formula results for states in which only one electron is excited.     

Modeling the dynamics of entangled quantum states

In the study of association of a diatomic molecule, we consider the atoms as quantum particles in an entangle quantum state. The proposed model of association is based on the selection of such states on the principle of their density in the configuration space, which is in exact agreement with the selection by the amplitude of two-particle states. The result of one step of such a selection is presented that was calculated using the collective behavior method. Original Russian Text K.S. Arakelov, Yu.I. Ozhigov, 2008, published in Vestnik Moskovskogo Universiteta. Vychislitel’naya Matematika i Kibernetika, 2008, No. 4, pp. 34–38.     

On the lattice of stratified principal L-topologies

We investigate the lattice structure of the set of all stratified principal L-topologies on a given set X. It proves that the lattice of stratified principal L-topologies S p(X) has atoms and dual atoms if and only if L has atoms and dual atoms respectively. Moreover, it is complete and semi-complemented. We also discuss some other properties of the lattice.