Volume 49, Issue 3, March 2008
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Solitary wave dynamics in timedependent potentials
View Description Hide DescriptionThe long time dynamics of solitary wave solutions of the nonlinear Schrödinger equation in timedependent external potentials is rigorously studied. To set the stage, the wellposedness of the Cauchy problem for a generalized nonautonomous nonlinear Schrödinger equation with timedependent nonlinearities and potential is established. Afterward, the dynamics of NLS solitary waves in timedependent potentials is studied. It is shown that in the spaceadiabatic regime where the external potential varies slowly in space compared to the size of the soliton, the dynamics of the center of the soliton is described by Hamilton’s equations, plus terms due to radiation damping. Finally, two physical applications are discussed: the first is adiabatic transportation of solitons and the second is the Mathieu instability of trapped solitons due to timeperiodic perturbations.

The divine clockwork: Bohr’s correspondence principle and Nelson’s stochastic mechanics for the atomic elliptic state
View Description Hide DescriptionWe consider the Bohr correspondence limit of the Schrödinger wave function for an atomic elliptic state. We analyze this limit in the context of Nelson’s stochastic mechanics, exposing an underlying deterministic dynamical system in which trajectories converge to Keplerian motion on an ellipse. This solves the long standing problem of obtaining Kepler’s laws of planetary motion in a quantum mechanical setting. In this quantum mechanical setting, local mild instabilities occur in the Keplerian orbit for eccentricities greater than which do not occur classically.

Rotations of occupied invariant subspaces in selfconsistent field calculations
View Description Hide DescriptionIn this article, the selfconsistent field (SCF) procedure as used in Hartree–Fock and Kohn–Sham calculations is viewed as a sequence of rotations of the socalled occupied invariant subspace of the potential and density matrices. Computational approximations are characterized as erroneous rotations of this subspace. Differences between subspaces are measured and controlled by the canonical angles between them. With this approach, a first step is taken toward a method where errors from computational approximations are rigorously controlled and threshold values are directly related to the accuracy of the current trial density, thus eliminating the use of ad hoc threshold values. Then, the use of computational resources can be kept down as much as possible without impairment of the SCF convergence.

The structure of classical extensions of quantum probability theory
View Description Hide DescriptionOn the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models, we prove that every such classical extension is essentially given by the socalled Misra–Bugajski reduction map. We consider how this map enables one to understand quantum mechanics as a reduced classical statistical theory on the projective Hilbert space as phase space and discuss features of the induced hiddenvariable model. Moreover, some relevant technical results on the topology and Borel structure of the projective Hilbert space are reviewed.

Quantum reference frames and the classification of rotationally invariant maps
View Description Hide DescriptionWe give a convenient representation for any map that is covariant with respect to an irreducible representation of , and use this representation to analyze the evolution of a quantum directional reference frame when it is exploited as a resource for performing quantum operations. We introduce the moments of a quantum reference frame, which serve as a complete description of its properties as a frame, and investigate how many times a quantum directional reference frame represented by a spin system can be used to perform a certain quantum operation with a given probability of success. We provide a considerable generalization of previous results on the degradation of a reference frame, from which follows a classification of the dynamics of spin system under the repeated action of any covariant map with respect to .

Spectral properties of the twophoton absorption and emission process
View Description Hide DescriptionThe quantum Markov semigroup of the twophoton absorption and emission process has two extremal normal invariant states. Starting from an arbitrary initial state it converges toward some convex combination of these states as time goes to infinity (approach to equilibrium). We compute the exact exponential rate of this convergence showing that it depends only on the emission rates. Moreover, we show that offdiagonal matrix elements of any initial state go to zero with an exponential rate which is smaller than the exponential rate of convergence of the diagonal part. In other words quantum features of a state survive longer than the relaxation time of its classical part.

Approximating a wavefunction as an unconstrained sum of Slater determinants
View Description Hide DescriptionThe wavefunction for the multiparticle Schrödinger equation is a function of many variables and satisfies an antisymmetry condition, so it is natural to approximate it as a sum of Slater determinants. Many current methods do so, but they impose additional structural constraints on the determinants, such as orthogonality between orbitals or an excitation pattern. We present a method without any such constraints, by which we hope to obtain much more efficient expansions and insight into the inherent structure of the wavefunction. We use an integral formulation of the problem, a Green’s functioniteration, and a fitting procedure based on the computational paradigm of separated representations. The core procedure is the construction and solution of a matrixintegral system derived from antisymmetric inner products involving the potential operators. We show how to construct and solve this system with computational complexity competitive with current methods.

Solutions for a Schrödinger equation with a nonlocal term
View Description Hide DescriptionWe obtain time dependent solutions for a Schröndiger equation in the presence of a nonlocal term by using the Green function approach. These solutions are compared with recent results obtained for the fractional Schrödinger equation as well as for the usual one. The nonlocal term incorporated in the Schrödinger equation may also be related to the spatial and time fractional derivative and introduces different regimes of spreading of the solution with the time evolution.

Fermionic quasifree states and maps in information theory
View Description Hide DescriptionThis paper and the results therein are geared toward building a basic toolbox for calculations in quantum informationtheory of quasifree fermionic systems. Various entropy and relative entropy measures are discussed. The main emphasis is on completely positive quasifree maps. The set of quasifree affine maps on the state space is determined and fully characterized in terms of operations on oneparticle subspaces. For a subclass of tracepreserving completely positive maps and for their duals, Choi matrices and Jamiolkowski states are discussed.

Supersymmetric associated vector coherent states and generalized Landau levels arising from twodimensional supersymmetry
View Description Hide DescriptionWe describe a method for constructing vector coherent states for quantum supersymmetric partner Hamiltonians. The method is then applied to such partner Hamiltonians arising from a generalization of the fractional quantum Hall effect. Explicit examples are worked out.

Hiatus perturbation for a singular Schrödinger operator with an interaction supported by a curve in
View Description Hide DescriptionWe consider Schrödinger operators in with a singular interaction supported by a finite curve . We present a proper definition of the operators and study their properties, in particular, we show that the discrete spectrum can be empty if is short enough. If it is not the case, we investigate properties of the eigenvalues in the situation when the curve has a hiatus of length . We derive an asymptotic expansion with the leading term which a multiple of .

Error exponents in hypothesis testing for correlated states on a spin chain
View Description Hide DescriptionWe study various error exponents in a binary hypothesis testing problem and extend recent results on the quantum Chernoff and Hoeffding bounds for product states to a setting when both the null hypothesis and the alternative hypothesis can be correlated states on a spin chain. Our results apply to states satisfying a certain factorization property; typical examples are the global Gibbs states of translationinvariant finiterange interactions as well as certain finitely correlated states.

Geometry of sets of quantum maps: A generic positive map acting on a highdimensional system is not completely positive
View Description Hide DescriptionWe investigate the set (a) of positive, trace preserving maps acting on density matrices of size and a sequence of its nested subsets: the sets of maps which are (b) decomposable, (c) completely positive, and (d) extended by identity impose positive partial transpose and (e) are superpositive. Working with the Hilbert–Schmidt (Euclidean) measure, we derive tight explicit twosided bounds for the volumes of all five sets. A sample consequence is the fact that, as increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiołkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on the systematic use of duality to derive quantitative estimates and on various tools of classical convexity, highdimensional probability, and geometry of Banach spaces, some of which are not standard.

Maps on states preserving the relative entropy
View Description Hide DescriptionLet be a finite dimensional Hilbert space. The aim of this short note is to prove that every bijective map on the space of all density operators on which preserves the relative entropy is of the form with some unitary or antiunitary operator on .
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


On the PoissonLie plurality of boundary conditions
View Description Hide DescriptionConditions for the gluing matrix defining consistent boundary conditions of twodimensional nonlinear models are analyzed and reformulated. Transformation properties of the rightinvariant fields under the PoissonLie plurality are used to derive a formula for the transformation of the boundary conditions. Examples of transformation of branes in two and three dimensions are presented. We investigate obstacles arising in this procedure and propose possible solutions.

Numerical Calabi–Yau metrics
View Description Hide DescriptionWe develop numerical methods for approximating Ricci flat metrics on Calabi–Yau hypersurfaces in projective spaces. Our approach is based on finding balanced metrics and builds on recent theoretical work by Donaldson. We illustrate our methods in detail for a one parameter family of quintics. We also suggest several ways to extend our results.

Topological photon
View Description Hide DescriptionWe associate intrinsic energy equal to with the spin angular momentum of photon, and propose a topological model based on orbifold in space and tifold in time as topological obstructions. The model is substantiated using vector wavefield disclinations. The physical photon is suggested to be a particlelike topological photon and a propagating wave such that the energy of photon is equally divided between spin energy and translational energy, corresponding to linear momentum of . The enigma of waveparticle duality finds natural resolution, and the proposed model gives new insights into the phenomena of interference and emission of radiation.

On the regularized fermionic projector of the vacuum
View Description Hide DescriptionWe construct families of fermionic projectors with spherically symmetric regularization, which satisfy the condition of a distributional product. The method is to analyze regularization tails with a power law or logarithmic scaling in composite expressions in the fermionic projector. The resulting regularizations break the Lorentz symmetry and give rise to a multilayer structure of the fermionic projector near the light cone. Furthermore, we construct regularizations which go beyond the distributional product in that they yield additional distributional contributions supported at the origin. The remaining freedom for the regularization parameters and the consequences for the normalization of the fermionic states are discussed.

Heat kernel estimates and spectral properties of a pseudorelativistic operator with magnetic field
View Description Hide DescriptionBased on the Mehler heat kernel of the Schrödinger operator for a free electron in a constant magnetic field, an estimate for the kernel of is derived, where represents the kinetic energy of a Dirac electron within the pseudorelativistic nopair Brown–Ravenhall model. This estimate is used to provide the bottom of the essential spectrum for the twoparticle Brown–Ravenhall operator, describing the motion of the electrons in a central Coulomb field and a constant magnetic field, if the central charge is restricted to .
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 GENERAL RELATIVITY AND GRAVITATION


“Massless” vector field in de Sitter universe
View Description Hide DescriptionWe proceed to the quantization of the massless vector field in the de Sitter (dS) space. This work is the natural continuation of a previous article devoted to the quantization of the dS massive vector field [J. P. Gazeau and M. V. Takook, J. Math. Phys.41, 5920 (2000); T. Garidi et al., ibid.43, 6379 (2002).] The term “massless” is used by reference to conformal invariance and propagation on the dS lightcone whereas “massive” refers to those dS fields which unambiguously contract to Minkowskian massive fields at zero curvature. Due to the combined occurrences of gauge invariance and indefinite metric, the covariant quantization of the massless vector field requires an indecomposable representation of the de Sitter group. We work with the gauge fixing corresponding to the simplest Gupta–Bleuler structure. The field operator is defined with the help of coordinateindependent de Sitter waves (the modes). The latter are simple to manipulate and most adapted to group theoretical approaches. The physical states characterized by the divergencelessness condition are, for instance, easy to identify. The whole construction is based on analyticity requirements in the complexified pseudoRiemannian manifold for the modes and the twopoint function.
