Volume 52, Issue 6, June 2011
Index of content:
- Quantum Mechanics (General and Nonrelativistic)
52(2011); http://dx.doi.org/10.1063/1.3592600View Description Hide Description
We show the asymptotic completeness for two-body quantum systems in an external electric field asymptotically non-zero constant in time. One of the main ingredients of this paper is to give some propagation estimates for physical propagators generated by time-dependent Hamiltonians which govern the systems under consideration.
52(2011); http://dx.doi.org/10.1063/1.3570676View Description Hide Description
Supersymmetric QED hydrogen-like 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.
52(2011); http://dx.doi.org/10.1063/1.3597555View Description Hide Description
Quantum 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 .
52(2011); http://dx.doi.org/10.1063/1.3598465View Description Hide Description
The Pauli-Fierz 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 Birman-Schwinger argument. Moreover, for any given δ > 0 examples of potentials V are provided such that α+ − α− < δ.
52(2011); http://dx.doi.org/10.1063/1.3602075View Description Hide Description
Nonrelativistic 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 pseudo-Hermitian Hamiltonians of the type discussed by Mostafazadeh. The quantization scheme makes use of the so-called “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 multi-particle sector, uniquely determined by the deformed two-particle Hamiltonian, is composed of bosonic particles.
52(2011); http://dx.doi.org/10.1063/1.3601739View Description Hide Description
We 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 time-dependent harmonic oscillators exhibiting log-periodic-type 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 = ω0 t 0/t, which exhibits an exact log-periodic oscillation, behaves as the harmonic oscillator with m and ω constant.
52(2011); http://dx.doi.org/10.1063/1.3602278View Description Hide Description
We 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 three-term 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 J-matrix method.
- Quantum Information and Computation
52(2011); http://dx.doi.org/10.1063/1.3595693View Description Hide Description
We 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 bi-partite 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 multi-partite system, we show that this distribution is given by the Fuss-Catalan 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 Marchenko-Pastur distribution.
52(2011); http://dx.doi.org/10.1063/1.3597233View Description Hide Description
A 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 two-mode squeezing operator constructed from operators satisfying the fermionic algebra. We compare Grassmann channels with the channels induced by the bosonic two-mode 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.
52(2011); http://dx.doi.org/10.1063/1.3600535View Description Hide Description
Generalized entropies and relative entropies are the subject of active research. Similar to the standard relative entropy, the relative q-entropy is generally unbounded for q > 1. Upper bounds on the quantum relative q-entropy in terms of norm distances between its arguments are obtained in finite-dimensional 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 rest-frame instant form of dynamics52(2011); http://dx.doi.org/10.1063/1.3591131View Description Hide Description
A new formulation of relativistic quantum mechanics is proposed in the framework of the rest-frame instant form of dynamics, where the world-lines of the particles are parametrized in terms of the Fokker-Pryce center of inertia and of Wigner-covariant relative 3-coordinates inside the instantaneous Wigner 3-spaces, and where there is a decoupled (non-covariant and non-local) 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 non-relativistic limit with a Hamilton-Jacobi treatment of the Newton center of mass; (e) clarifies non-local aspects (spatial non-separability) of relativistic entanglement connected with Lorentz signature and not present in its non-relativistic treatment.
52(2011); http://dx.doi.org/10.1063/1.3589319View Description Hide Description
We classify 2-center extremal black hole charge configurations through duality-invariant homogeneous polynomials, which are the generalization of the unique invariant quartic polynomial for single-center 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 p-center black holes. For p = 2, a (spin 2) quintet of quartic invariants emerge. We provide the minimal set of independent invariants for the rank-3 , d = 4 stumodel, and for its lower-rank descendants, namely, the rank-2 st 2 and rank-1 t 3models; 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 3model, but rather it satisfies a degree-16 relation, corresponding to a quartic equation for the square of the symplectic product itself.
- General Relativity and Gravitation
52(2011); http://dx.doi.org/10.1063/1.3596173View Description Hide Description
The 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
52(2011); http://dx.doi.org/10.1063/1.3598418View Description Hide Description
At high energies, in particle-capture processes between ions and atoms, classical kinematic requirements show that generally double-collision Thomas processes dominate. However, for certain mass-ratios 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 functions52(2011); http://dx.doi.org/10.1063/1.3598428View Description Hide Description
We 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 anti-Hermitian 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 H-function through the norm which directly reflect the spectral properties of the collision operators. When the norm of the momentum distribution function diverges, the H-function reduces to a simple form characterized by only the value at the accumulation point of the spectrum of the collision operator.
52(2011); http://dx.doi.org/10.1063/1.3589845View Description Hide Description
We 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.
52(2011); http://dx.doi.org/10.1063/1.3598424View Description Hide Description
We consider the valence-bond-solid ground state of the q-deformed higher-spin Affleck, Kennedy, Lieb, and Tasaki model (q-VBS 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 q-VBS state. We compute the longitudinal and transverse spin-spin correlation functions, and determine the correlation amplitudes and correlation lengths.
52(2011); http://dx.doi.org/10.1063/1.3601118View Description Hide Description
Let 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 birth-and-death 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
52(2011); http://dx.doi.org/10.1063/1.3594654View Description Hide Description
The 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 three-term 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.
52(2011); http://dx.doi.org/10.1063/1.3596172View Description Hide Description
The Gel'fand-Shilov 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 well-studied 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 Fourier-invariant space in the scale of S-type 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 cone-shaped regions, which can be used to formulate a causality condition in quantum field theory on noncommutative space-time.