Volume 49, Issue 7, July 2008
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Weyl–Wigner formulation of noncommutative quantum mechanics
View Description Hide DescriptionWe address the phasespace formulation of a noncommutative extension of quantum mechanics in arbitrary dimension, displaying both spatial and momentum noncommutativities. By resorting to a covariant generalization of the Weyl–Wigner transform and to the Darboux map, we construct an isomorphism between the operator and the phasespace representations of the extended Heisenberg algebra. This map provides a systematic approach to derive the entire structure of noncommutative quantum mechanics in phase space. We construct the extended star product and Moyal bracket and propose a general definition of noncommutative states. We study the dynamical and eigenvalue equations of the theory and prove that the entire formalism is independent of the particular choice of the Darboux map. Our approach unifies and generalizes all the previous proposals for the phasespace formulation of noncommutative quantum mechanics. For concreteness we rederive these proposals by restricting our formalism to some twodimensional spaces.

Solution of the Cauchy problem for a timedependent Schrödinger equation
View Description Hide DescriptionWe construct an explicit solution of the Cauchy initial value problem for the dimensional Schrödinger equation with certain timedependent Hamiltonian operator of a modified oscillator. The dynamical symmetry of the harmonic oscillatorwave functions, Bargmann’s functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner–Pollaczek polynomials, a Hankeltype integral transform, and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as byproducts.

Deformations and central extensions of the antibracket superalgebra
View Description Hide DescriptionWe consider the antibracket superalgebras realized on the smooth Grassmannvalued functions with compact support in and with the grading inverse to Grassmannian parity. The deformations and central extensions of these superalgebras are found. The results differ from the case of the antibracket superalgebras realized on the polynomial functions.

Asymptotic distinguishability measures for shiftinvariant quasifree states of fermionic lattice systems
View Description Hide DescriptionWe apply the recent results of Hiai et al. [J. Math. Phys.49, 032112 (2008)] to the asymptotic hypothesis testing problem of locally faithful shiftinvariant quasifree states on a CAR algebra. We use a multivariate extension of Szegő’s theorem to show the existence of the mean Chernoff and Hoeffding bounds and the mean relative entropy and show that these quantities arise as the optimal error exponents in suitable settings.

Quantum Markov chains
View Description Hide DescriptionA new approach to quantum Markov chains is presented. We first define a transition operation matrix (TOM) as a matrix whose entries are completely positive maps whose column sums form a quantum operation. A quantum Markov chain is defined to be a pair where is a directed graph and is a TOM whose entry labels the edge from vertex to vertex . We think of the vertices of as sites that a quantum system can occupy and is the transition operation from site to site in one time step. The discrete dynamics of the system is obtained by iterating the TOM . We next consider a special type of TOM called a transition effect matrix. In this case, there are two types of dynamics, a state dynamics and an operator dynamics. Although these two types are not identical, they are statistically equivalent. We next give examples that illustrate various properties of quantum Markov chains. We conclude by showing that our formalism generalizes the usual framework for quantum random walks.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Mesonic eightfold way from dynamics and confinement in strongly coupled lattice quantum chromodynamics
View Description Hide DescriptionWe show the existence of all the 36 eightfold way mesons and determine their masses and dispersion curves exactly, from dynamical first principles such as directly from the quarkfluon dynamics. We also give a proof of confinement below the twomeson energy threshold. For this purpose, we consider an imaginary time functional integral representation of a 3+1 dimensional lattice QCDmodel with Wilson action, global and local symmetries. We work in the strong coupling regime, such that the hopping parameter is small and much larger than the plaquette coupling (). In the quantum mechanical physical Hilbert space, a FeynmanKac type representation for the twomeson correlation and its spectral representation are used to establish an exact rigorous connection between the complex momentum singularities of the twomeson truncated correlation and the energymomentum spectrum of the model. The total spin operator and its component are defined by using rotations about the spatial coordinate axes, and agree with the infinitesimal generators of the continuum for improper zeromomentum meson states. The mesons admit a labelling in terms of the quantum numbers of total isospin , the third component of total isospin, the component of total spin and quadratic Casimir for . With this labelling, the mesons can be organized into two sets of states, distinguished by the total spin . These two sets are identified with the nonet of pseudoscalar mesons and the three nonets of vector mesons. Within each nonet a further decomposition can be made using to obtain the singlet state and the eight members of the octet . By casting the problem of determination of the meson masses and dispersion curves into the framework of the the anaytic implicit function theorem, all the masses are found exactly and are given by convergent expansions in the parameters and . The masses are all of the form with and real analytic; for is jointly analytic in and . The masses of the vector mesons are independent of and are all equal within each octet. All isospin singlet masses are also equal for the vector mesons. For each nonet and , up to and including , the masses of the octet and the singlet are found to be equal. But there is a pseudoscalarvector meson mass splitting given by and the splitting persists for . For , the dispersion curves are all of the form , with . For the pseudoscalar mesons, is jointly analytic in and , for and small. We use some machinery from constructive field theory, such as the decoupling of hyperplane method, in order to reveal the gaugeinvariant eightfold way meson states and a correlation subtraction method to extend our spectral results to all , the subspace of generated by vectors with an even number of Grassmann variables, up to near the twomeson energy threshold of . Combining this result with a previously similar result for the baryon sector of the eightfold way, we show that the only spectrum in all ( being the odd subspace) below is given by the eightfold way mesons and baryons. Hence, we prove confinement up to near this energy threshold.
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 DYNAMICAL SYSTEMS


On the evolution of a reflection coefficient under the Korteweg–de Vries flow
View Description Hide DescriptionWe are concerned with the Korteweg–de Vries equation on the full line with real nondecaying initial profiles. We find the time evolution of a (relative) reflection coefficient. An inverse spectral formalism is also considered for certain mixed problems on the full line.

Integrable discrete systems on and related dispersionless systems
View Description Hide DescriptionA general framework for integrable discrete systems on , in particular, containing lattice soliton systems and their deformed analogs, is presented. The concept of regular grain structures on , generated by discrete oneparameter groups of diffeomorphisms, in terms of which one can define algebra of shift operators is introduced. Two integrable hierarchies of discrete chains together with biHamiltonian structures and their continuous limits are constructed. The inverse problem based on the deformation quantization scheme is considered.

Paths of Friedmann–Robertson–Walker brane models
View Description Hide DescriptionThe dynamics of braneworld models of dark energy is reviewed. We demonstrate that simple dark energybranemodels can be represented as twodimensional dynamical systems of a Newtonian type. Hence a fictitious particle moving in a potential well characterizes the model. We investigate the dynamics of the branemodels using methods of dynamical systems. The simple braneworld models can be successfully unified within a single scheme—an ensemble of branedark energymodels. We characterize generic models of this ensemble as well as exceptional ones using the notion of structural stability (instability). Then due to the Peixoto theorem we can characterize the class of generic branemodels. We show that the global dynamics of the generic branemodels of dark energy is topologically equivalent to the concordance model. We also demonstrate that the bouncing models or models in which acceleration of the universe is only transient phenomenon are nongeneric (or exceptional cases) in the ensemble. We argue that the adequate branemodel of dark energy should be a generic case in the ensemble of Friedmann–Robertson–Walker dynamical systems on the plane.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


An integrable coupled hybrid lattice and its exact solutions
View Description Hide DescriptionA coupled hybrid lattice system is proposed. The model can be considered as one of the discrete forms of the coupled integrable Korteweg–de Vries system which is a special case of some physically significant models. The Lax integrability of the coupled hybrid lattice system is proved. By using a simple function expansion approach, various novel types of exact solutions such as the kinksoliton lattice, the bellstaggered soliton lattice, the kinkkink breather lattice, and the solitonsoliton breather lattice, are found.
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 STATISTICAL PHYSICS


The large expansion in hyperbolic sigma models
View Description Hide DescriptionInvariant correlation functions for hyperbolic sigma models are investigated. The existence of a large asymptotic expansion is proven on finite lattices of dimension . The unique saddle point configuration is characterized by a negative gap vanishing at least like with the volume. Technical difficulties compared to the compact case are bypassed using horospherical coordinates and the matrixtree theorem.

Solutions for multidimensional fractional anomalous diffusion equations
View Description Hide DescriptionIn this paper, we investigate the solutions of a generalized fractional diffusionequation that extends some known diffusionequations by taking a spatial timedependent diffusion coefficient and dimensional case into account, which subjects to natural boundaries and the general initial condition. In our analysis, the presence of external force is also taken into account. We obtain explicit analytical expressions for the probability distribution and study the relation between our solutions and those obtained within the maximum entropy principle by using the Tsallis entropy.

Calculating effective resistances on underlying networks of association schemes
View Description Hide DescriptionRecently, in the work of Jafarizadeh et al. [J. Phys, A: Math. Theor.40, 4949 (2007); eprint arXiv:0705.2480], calculation of effective resistances on distanceregular networks was investigated, where in the first paper, the calculation was based on stratification and Stieltjes functions associated with the network, whereas in the latter one a recursive formula for effective resistances was given based on the Christoffel–Darboux identity. In this paper, evaluation of effective resistances on more general networks that are underlying networks of association schemes is considered, where by using the algebraic combinatoricstructures of association schemes such as stratification and Bose–Mesner algebras, an explicit formula for effective resistances on these networks is given in terms of the parameters of the corresponding association schemes. Moreover, we show that for particular underlying networks of association schemes with diameter such that the adjacency matrix possesses distinct eigenvalues, all of the other adjacency matrices , , 1 can be written as polynomials of , i.e., , where is not necessarily of degree . Then, we use this property for these particular networks and assume that all of the conductances except for one of them, say, , are zero to give a procedure for evfor a galuating effective resistances on these networks. The preference of this procedure is that one can evaluate effective resistances by using the structure of their Bose–Mesner algebra without any need to know the spectrum of the adjacency matrices.

Dynamic model and phase transitions for liquid helium
View Description Hide DescriptionThis article presents a phenomenological dynamic phase transition theory—modeling and analysis—for liquid helium4. First we derive a timedependent Ginzburg–Landau model for helium4 by (1) separating the superfluid and the normal fluid densities with the superfluid density given in terms of a wave function and (2) using a unified dynamical Ginzburg–Landau model. One the one hand, the analysis of this model leads to phase diagrams consistent with the classical ones for liquid helium4. On the other hand, it leads to predictions of (1) the existence of a metastable region , where both solid and liquid states are metastable and appear randomly depending on fluctuations and (2) the existence of a switch point on the curve, where the transitions changes types. It is hoped that these predictions will be useful for designing better physical experiments and lead to better understanding of the physical mechanism of superfluidity.
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 METHODS OF MATHEMATICAL PHYSICS


A new approach to electronspin resonance absorption line shape function
View Description Hide DescriptionWe derive a new electronspin resonance spectrum formula utilizing a quantumstatistical projection technique. The line shape factor appearing in the susceptibilitytensor contains the manybody effects for electrons more realistically. It thus becomes possible to explain, in an organized way, all electron transitions including spin conserved processes and spin flipped processes.

Harmonic oscillator chains as Wigner quantum systems: Periodic and fixed wall boundary conditions in solutions
View Description Hide DescriptionWe describe a quantum system consisting of a onedimensional linear chain of identical harmonic oscillators coupled by a nearest neighbor interaction. Two boundary conditions are taken into account: periodic boundary conditions (where the oscillator is coupled back to the first oscillator) and fixed wall boundary conditions (where the first oscillator and the oscillator are coupled to a fixed wall). The two systems are characterized by their Hamiltonian. For their quantization, we treat these systems as Wigner quantum systems (WQSs), allowing more solutions than just the canonical quantization solution. In this WQS approach, one is led to certain algebraic relations for operators (which are linear combinations of position and momentum operators) that should satisfy triple relations involving commutators and anticommutators. These triple relations have a solution in terms of the Lie superalgebra. We study a particular class of representations , the socalled ladder representations. For these representations, we determine the spectrum of the Hamiltonian and of the position operators (for both types of boundary conditions). Furthermore, we compute the eigenvectors of the position operators in terms of stationary states. This leads to explicit expressions for position probabilities of the oscillators in the chain. An analysis of the plots of such position probability distributions gives rise to some interesting observations. In particular, the physical behavior of the system as a WQS is very much in agreement with what one would expect from the classical case, except that all physical quantities (energy, position, and momentum of each oscillator) have a finite spectrum.

Longtime selfsimilar asymptotic of the macroscopic quantum models
View Description Hide DescriptionThe unipolar and bipolar macroscopic quantum models derived recently, for instance, in the area of charge transport are considered in spatial onedimensional whole space in the present paper. These models consist of nonlinear fourthorder parabolic equation for unipolar case or coupled nonlinear fourthorder parabolic system for bipolar case. We show for the first time the selfsimilarity property of the macroscopic quantum models in large time. Namely, we show that there exists a unique global strong solution with strictly positive density to the initial value problem of the macroscopic quantum models which tends to a selfsimilar wave (which is not the exact solution of the models) in large time at an algebraic timedecay rate.

The global solution of the dimensional long wave–short wave resonance interaction equation
View Description Hide DescriptionIn this paper, we prove the existence and uniqueness of the global smooth solution to the dimensional long wave–short wave resonance interactionequation.

The modeling of degenerate neck pinch singularities in Ricci flow by Bryant solitons
View Description Hide DescriptionIn earlier work, carrying out numerical simulations of the Ricci flows of families of rotationally symmetric geometries on , we have found strong support for the contention that (at least in the rotationally symmetric case) the Ricci flow for a “critical” initial geometry—one which is at the transition point between initial geometries (on ) whose volumenormalized Ricci flows develop a singular neck pinch, and other initial geometries whose volumenormalized Ricci flows converge to a round sphere—evolves into a “degenerate neck pinch.” That is, we have seen in this earlier work that the Ricci flows for the critical geometries become locally cylindrical in a neighborhood of the initial pinching and have the maximum amount of curvature at one or both of the poles. Here, we explore the behavior of these flows at the poles and find strong support for the conjecture that the Bryant steady solitons accurately model this polar flow.

A geometric description of tensor product decompositions in
View Description Hide DescriptionThe direct sum decomposition of tensor products for has many applications in physics, and the problem has been studied extensively. This has resulted in many decomposition methods, each with its advantages and disadvantages. The description given here is geometric in nature and it describes both the constituents of the direct sum and their multiplicities. In addition to providing decompositions of specific tensor products, this approach is very well suited to studying tensor products as the parameters vary and helping draw general conclusions. After a description and proof of the method, several consequences are discussed and proved. In particular, questions regarding multiplicities are considered.
