Volume 52, Issue 6, June 2011
Index of content:
 ARTICLES


Quantum Mechanics (General and Nonrelativistic)

Scattering in an external electric field asymptotically constant in time
View Description Hide DescriptionWe show the asymptotic completeness for twobody quantum systems in an external electric field asymptotically nonzero constant in time. One of the main ingredients of this paper is to give some propagation estimates for physical propagators generated by timedependent Hamiltonians which govern the systems under consideration.

The simplicity of perfect atoms: Degeneracies in supersymmetric hydrogen
View Description Hide DescriptionSupersymmetric QED hydrogenlike bound states are remarkably similar to nonsupersymmetric hydrogen, including an accidental degeneracy of the fine structure and is broken by the Lamb shift. This article classifies the states, calculates the leading order spectrum, and illustrates the results in several limits. The relation to other nonrelativistic bound states is explored.

Quantum mechanics on
View Description Hide DescriptionQuantum mechanics with positions in and momenta in is considered. Displacement operators and coherent states,parity operators, Wigner and Weyl functions, and time evolution are discussed. The restriction of the formalism to certain finite subspaces is equivalent to Good's factorization of quantum mechanics on .

The nobinding regime of the PauliFierz model
View Description Hide DescriptionThe PauliFierz model H(α) in nonrelativistic quantum electrodynamics is considered. The external potential V is sufficiently shallow and the dipole approximation is assumed. It is proven that there exist constants 0 < α_{−} < α_{+} such that H(α) has no ground state for α < α_{−}, which complements an earlier result stating that there is a ground state for α > α_{+}. We develop a suitable extension of the BirmanSchwinger argument. Moreover, for any given δ > 0 examples of potentials V are provided such that α_{+} − α_{−} < δ.

Snyder noncommutativity and pseudoHermitian Hamiltonians from a Jordanian twist
View Description Hide DescriptionNonrelativistic quantum mechanics and conformal quantum mechanics are deformed through a Jordanian twist. The deformed space coordinates satisfy the Snyder noncommutativity. The resulting deformed Hamiltonians are pseudoHermitian Hamiltonians of the type discussed by Mostafazadeh. The quantization scheme makes use of the socalled “unfolded formalism” discussed in previous works. A Hopf algebrastructure, compatible with the physical interpretation of the coproduct, is introduced for the universal enveloping algebra of a suitably chosen dynamical Lie algebra (the Hamiltonian is contained among its generators). The multiparticle sector, uniquely determined by the deformed twoparticle Hamiltonian, is composed of bosonic particles.

Wave functions of logperiodic oscillators
View Description Hide DescriptionWe use the Lewis and Riesenfeld invariant method [J. Math. Phys.10, 1458 (1969)]10.1063/1.1664991 and a unitary transformation to obtain the exact Schrödinger wave functions for timedependent harmonic oscillators exhibiting logperiodictype behavior. For each oscillator we calculate the quantum fluctuations in the coordinate and momentum as well as the quantum correlations between the coordinate and momentum. We observe that the oscillator with m = m _{0} t/t _{0} and ω = ω_{0} t _{0}/t, which exhibits an exact logperiodic oscillation, behaves as the harmonic oscillator with m and ω constant.

Quantum mechanics without an equation of motion
View Description Hide DescriptionWe propose a formulation of quantum mechanics for a finite level system whose potential function is not realizable and/or analytic solution of the wave equation is not feasible. The system's wavefunction is written as an infinite sum in a complete set of square integrable functions. Interaction in the theory is introduced in function space by a real finite tridiagonal symmetric matrix. The expansion coefficients of the wavefunction satisfy a threeterm recursion relation incorporating the parameters of the interaction. Information about the structure and dynamics of the system is contained in the scattering matrix, which is defined in the usual way. The bound state energy spectrum (system's structure) is finite. Apart from the 2M − 1 dimensionless parameters of the interaction matrix, whose rank is M, the theory has one additional scale parameter. In the development, we utilize the kinematic tools of the Jmatrix method.

Quantum Information and Computation

Generating random density matrices
View Description Hide DescriptionWe study various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems. Structured ensembles of random pure states, invariant with respect to local unitary transformations are introduced. To analyze statistical properties of quantum entanglement in bipartite systems we analyze the distribution of Schmidt coefficients of random pure states. Such a distribution is derived in the case of a superposition of k random maximally entangled states. For another ensemble, obtained by performing selective measurements in a maximally entangled basis on a multipartite system, we show that this distribution is given by the FussCatalan law and find the average entanglemententropy. A more general class of structured ensembles proposed, containing also the case of Bures, forms an extension of the standard ensemble of structureless random pure states, described asymptotically, as N → ∞, by the MarchenkoPastur distribution.

Capacities of Grassmann channels
View Description Hide DescriptionA new class of quantum channels called Grassmann channels is introduced and their classical and quantum capacity is calculated. The channel class appears in a study of the twomode squeezing operator constructed from operators satisfying the fermionic algebra. We compare Grassmann channels with the channels induced by the bosonic twomode squeezing operator. Among other results, we challenge the relevance of calculating entanglement measures to assess or compare the ability of bosonic and fermionic states to send quantum information to uniformly accelerated frames.

Upper continuity bounds on the relative qentropy for q > 1
View Description Hide DescriptionGeneralized entropies and relative entropies are the subject of active research. Similar to the standard relative entropy, the relative qentropy is generally unbounded for q > 1. Upper bounds on the quantum relative qentropy in terms of norm distances between its arguments are obtained in finitedimensional context. These bounds characterize a continuity property in the sense of Fannes.

Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

Relativistic quantum mechanics and relativistic entanglement in the restframe instant form of dynamics
View Description Hide DescriptionA new formulation of relativistic quantum mechanics is proposed in the framework of the restframe instant form of dynamics, where the worldlines of the particles are parametrized in terms of the FokkerPryce center of inertia and of Wignercovariant relative 3coordinates inside the instantaneous Wigner 3spaces, and where there is a decoupled (noncovariant and nonlocal) canonical relativistic center of mass. This approach: (a) allows us to make a consistent quantization in every inertial frame; (b) leads to a description of both bound and scattering states; (c) offers new insights on the relativistic localization problem; (d) leads to a nonrelativistic limit with a HamiltonJacobi treatment of the Newton center of mass; (e) clarifies nonlocal aspects (spatial nonseparability) of relativistic entanglement connected with Lorentz signature and not present in its nonrelativistic treatment.

Twocenter black holes dualityinvariants for stu model and its lowerrank descendants
View Description Hide DescriptionWe classify 2center extremal black hole charge configurations through dualityinvariant homogeneous polynomials, which are the generalization of the unique invariant quartic polynomial for singlecenter black holes based on homogeneous symmetric cubic special Kä hler geometries. A crucial role is played by a horizontal symmetry group, which classifies invariants for pcenter black holes. For p = 2, a (spin 2) quintet of quartic invariants emerge. We provide the minimal set of independent invariants for the rank3 , d = 4 stumodel, and for its lowerrank descendants, namely, the rank2 st ^{2} and rank1 t ^{3}models; these models, respectively, exhibit seven, six, and five independent invariants. We also derive the polynomial relations among these and other duality invariants. In particular, the symplectic product of two charge vectors is not independent from the quartic quintet in the t ^{3}model, but rather it satisfies a degree16 relation, corresponding to a quartic equation for the square of the symplectic product itself.

General Relativity and Gravitation

Gravity from the extension of spatial diffeomorphisms
View Description Hide DescriptionThe possibility of the extension of spatial diffeomorphisms to a larger family of symmetries in a class of classical field theories is studied. The generator of the additional local symmetry contains a quadratic kinetic term and a potential term which can be a general (not necessarily local) functional of the metric. From the perspective of the foundation of Einstein's gravity our results are positive: The extended constraint algebra is either that of Einstein's gravity or ultralocal gravity. If our goal is a simple modification of Einstein's gravity that for example makes it perturbatively renormalizable, as has recently been suggested, then our results show that there is no such theory within this class.

Classical Mechanics and Classical Fields

Thomasforbidden particle capture
View Description Hide DescriptionAt high energies, in particlecapture processes between ions and atoms, classical kinematic requirements show that generally doublecollision Thomas processes dominate. However, for certain massratios these processes are kinematically forbidden. This paper explores the possibility of capture for such processes by triple or higher order collision processes.

Statistical Physics

Kinetic equations for classical and quantum Brownian particles and eigenfunction expansions as generalized functions
View Description Hide DescriptionWe study momentum relaxation processes of a classical and a quantum Brownian particle by considering the eigenvalue problem of the collision operators in the kinetic equations. The collision operators are antiHermitian with an appropriate inner product defined by an integral with a weight factor given by the inverse of the equilibrium distribution function. Owing to the weight factor, the norm of a momentum distribution function is infinite, if the distribution is characterized by a temperature higher than a threshold temperature determined by the environmental temperature. Although the eigenfunction expansion of a given distribution function with an infinite norm does not converge to a function in the Hilbert space, it has a legitimate meaning as a generalized function and defines a linear functional. We introduce an Hfunction through the norm which directly reflect the spectral properties of the collision operators. When the norm of the momentum distribution function diverges, the Hfunction reduces to a simple form characterized by only the value at the accumulation point of the spectrum of the collision operator.

A counterexample against the Lesche stability of a generic entropy functional
View Description Hide DescriptionWe provide a counterexample to show that the generic form of entropy is not always stable against small variation of probability distribution (Lesche stability) even if g is concave function on [0, 1] and of class on ]0, 1]. Our conclusion is that the stability of such a generic functional needs more hypotheses on the property of the function g, or in other words, the stability of entropy cannot be discussed at this formal stage.

Spinspin correlation functions of the qvalencebondsolid state of an integer spin model
View Description Hide DescriptionWe consider the valencebondsolid ground state of the qdeformed higherspin Affleck, Kennedy, Lieb, and Tasaki model (qVBS state) with q real. We investigate the eigenvalues and eigenvectors of a matrix (G matrix), which is constructed from the matrix product representation of the qVBS state. We compute the longitudinal and transverse spinspin correlation functions, and determine the correlation amplitudes and correlation lengths.

Binary jumps in continuum. I. Equilibrium processes and their scaling limits
View Description Hide DescriptionLet Γ denote the space of all locally finite subsets (configurations) in . A stochastic dynamics of binary jumps in continuum is a Markov process on Γ in which pairs of particles simultaneously hop over . In this paper, we study an equilibrium dynamics of binary jumps for which a Poisson measure is a symmetrizing (and hence invariant) measure. The existence and uniqueness of the corresponding stochastic dynamics are shown. We next prove the main result of this paper: a big class of dynamics of binary jumps converge, in a diffusive scaling limit, to a dynamics of interacting Brownian particles. We also study another scaling limit, which leads us to a spatial birthanddeath process in continuum. A remarkable property of the limiting dynamics is that its generator possesses a spectral gap, a property which is hopeless to expect from the initial dynamics of binary jumps.

Methods of Mathematical Physics

Counting the exponents of single transfer matrices
View Description Hide DescriptionThe eigenvalueequation of a band or a block tridiagonal matrix, the tight binding model for a crystal, a molecule, or a particle in a lattice with random potential or hopping amplitudes, and other problems lead to threeterm recursive relations for (multicomponent) amplitudes. Amplitudes n steps apart are linearly related by a transfer matrix, which is the product of n matrices. Its exponents describe the decay lengths of the amplitudes. A formula is obtained for the counting function of the exponents, based on a duality relation and the Argument Principle for the zeros of analytic functions. It involves the corner blocks of the inverse of the associated Hamiltonian matrix. As an illustration, numerical evaluations of the counting function of quasi 1D Anderson model are shown.

Moyal multiplier algebras of the test function spaces of type S
View Description Hide DescriptionThe Gel'fandShilov spaces of type S are considered as topological algebras with respect to the Moyal star product and their corresponding algebras of multipliers are defined and investigated. In contrast to the wellstudied case of Schwartz's space S, these multipliers are allowed to have nonpolynomial growth or infinite order singularities. The Moyal multiplication is thereby extended to certain classes of ultradistributions, hyperfunctions, and analytic functionals. The main theorem of the paper characterizes those elements of the dual of a given test function space that are the Moyal multipliers of this space. The smallest nontrivial Fourierinvariant space in the scale of Stype spaces is shown to play a special role, because its corresponding Moyal multiplier algebra contains the largest algebra of functions for which the power series defining their star products are absolutely convergent. Furthermore, it contains analogous algebras associated with coneshaped regions, which can be used to formulate a causality condition in quantum field theory on noncommutative spacetime.
