Volume 48, Issue 10, October 2007
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Generalized infimum and sequential product of quantum effects
View Description Hide DescriptionThe quantum effects for a physical system can be described by the set of positive operators on a complex Hilbert space that are bounded above by the identity operator . For , , the operation of sequential product was proposed as a model for sequential quantum measurements. A nice investigation of properties of the sequential product has been carried over [Gudder, S. and Nagy, G., “Sequential quantum measurements,”J. Math. Phys.42, 5212 (Year: 2001)]. In this note, we extend some results of this reference. In particular, a gap in the proof of Theorem 3.2 in this reference is overcome. In addition, some properties of generalized infimum are studied.

Role of scaling limits in the rigorous analysis of BoseEinstein condensation
View Description Hide DescriptionIn the context of the rigorous analysis of BoseEinstein condensation, recent achievements have been obtained in the form of asymptotic results when some appropriate scaling is performed in the Hamiltonian, and the limit of infinite number of particles is taken. In particular, two modified thermodynamic limits of infinite dilution turned out to provide an insight in this analysis, the socalled GrossPitaevskiĭ limit and the related ThomasFermi limit. Here such scalings are discussed with respect to their physical and mathematical motivations and to the currently known results obtained within this framework.

Rapidly rotating BoseEinstein condensates in homogeneous traps
View Description Hide DescriptionWe extend the results of a previous paper on the GrossPitaevskii description of rotating BoseEinstein condensates in twodimensional traps to confining potentials of the form , . Writing the coupling constant as , we study the limit . We derive rigorously the leading asymptotics of the ground state energy and the density profile when the rotation velocity tends to infinity as a power of . The case of asymptotically homogeneous potentials is also discussed.

Semiclassical coherentstate propagator for many spins
View Description Hide DescriptionWe obtain the semiclassical coherentstate propagator for a manyspin system with an arbitrary Hamiltonian.

Global well posedness for the GrossPitaevskii equation with an angular momentum rotational term in three dimensions
View Description Hide DescriptionIn this paper, we establish the global well posedness of the Cauchy problem for the GrossPitaevskii equation with an angular momentum rotational term in which the angular velocity is equal to the isotropic trapping frequency in the space .

The quantum harmonic oscillator on the sphere and the hyperbolic plane: dependent formalism, polar coordinates, and hypergeometric functions
View Description Hide DescriptionA nonlinear model representing the quantum harmonic oscillator on the sphere and the hyperbolic plane is solved in polar coordinates by making use of a curvaturedependent formalism. The curvature is considered as a parameter and then the radial Schrödinger equation becomes a dependent Gauss hypergeometric equation. The energy spectrum and the wave functions are exactly obtained in both the sphere and the hyperbolic plane . A comparative study between the spherical and the hyperbolic quantum results is presented.

Visualizing two qubits
View Description Hide DescriptionThe notions of entanglement witnesses and separable and entangled states for a two qubit system can be visualized in three dimensions using the SLOCC equivalence classes. This visualization preserves the duality relations between the various sets and allows us to give “proof by inspection” of a nonelementary result of Horodecki et al. [Phys. Lett. A223, 1–8 (1996)] that for two qubits, Peres PPT (positive partial transpose) test for separability is iff. We then show that the CHSH Bell inequalities can be visualized as circles and cylinders in the same diagram. This allows us to give a geometric proof of yet another result of Horodecki et al. [Phys. Lett. A200, 340–344 (1995)], which optimizes the violation of the CHSH Bell inequality. Finally, we give numerical evidence that, remarkably, allowing Alice and Bob to use three rather than two measurements each does not help them to distinguish any new entangled SLOCC equivalence class beyond the CHSH class.

Concurrence revisited
View Description Hide DescriptionConcurrence is a widely used entanglement measure of bipartite mixed states. We propose to consider the concurrence as being defined by a quadratic form on the space of selfadjoint operators. The square root of this form determines the values of the concurrence on the pure states, while the values on the mixed states are obtained as the largest convex extension, the convex roof. This viewpoint admits a generalization. Namely, the space of selfadjoint operators and the convex cone of positive semidefinite operators contained therein can be replaced by an arbitrary real vector space containing a convex cone. Then the concurrence is determined on the extreme rays of the cone by the square root of a quadratic form, and on the rest of the cone by the convex roof. We compute this generalized concurrence in the case when the cone is a second order cone. This enables us to compute the concurrence of arbitrary bipartite mixed states of rank 2. As an application, we compute the concurrences of the density matrices of all graphs with two edges or with three edges forming a triangle. We also consider the problem of maximizing the concurrence on the set of mixed states having a fixed spectrum.

A rigorous approach to the FeynmanVernon influence functional and its applications. I
View Description Hide DescriptionA rigorous representation of the FeynmanVernon influence functional used to describe open quantum systems is given, based on the theory of infinite dimensional oscillatory integrals. An application to the case of the density matrices describing the CaldeiraLeggett model of two quantum systems with a quadratic interaction is treated.

The von Neumann entropy asymptotics in multidimensional fermionic systems
View Description Hide DescriptionWe study the von Neumann entropy asymptotics of pure translationinvariant quasifree states of dimensional fermionic systems. It is shown that the entropic area law is violated by all these states: apart from the trivial cases, the entropy of a cubic subsystem with edge length cannot grow slower than . As for the upper bound of the entropy asymptotics, the zeroentropydensity property of these pure states is the only limit: it is proven that arbitrary fast subentropy growth is achievable.

Bounds for the adiabatic approximation with applications to quantum computation
View Description Hide DescriptionWe present straightforward proofs of estimates used in the adiabatic approximation. The gap dependence is analyzed explicitly. We apply the result to interpolating Hamiltonians of interest in quantum computing.

QuasiHermitian quantum mechanics in phase space
View Description Hide DescriptionWe investigate quasiHermitian quantum mechanics in phase space using standard deformation quantization methods: Groenewold star products and Wigner transforms. We focus on imaginary Liouville theory as a representative example where exact results are easily obtained. We emphasize spatially periodic solutions, compute various distribution functions and phasespace metrics, and explore the relationships between them.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


quaternions and deformed su(2) instantons
View Description Hide DescriptionWe construct (anti)instanton solutions of a wouldbe deformed su(2) YangMillstheory on the quantum Euclidean space [the covariant noncommutative space] by reinterpreting the function algebra on the latter as a quaternion bialgebra. Since the (anti)selfduality equations are covariant under the quantum group of deformed rotations, translations, and scale change, by applying the latter we can generate new solutions from the one centered at the origin and with unit size. We also construct multiinstanton solutions. As they depend on noncommuting parameters playing the roles of “sizes” and “coordinates of the centers” of the instantons, this indicates that the moduli space of a complete theory should be a noncommutative manifold. Similarly, gauge transformations should be allowed to depend on additional noncommutative parameters.

The Casimir effect for parallel plates revisited
View Description Hide DescriptionThe Casimir effect for a massless scalar field with Dirichlet and periodic boundary conditions (bc’s) on infinite parallel plates is revisited in the local quantum field theory (lqft) framework introduced by Kay [Phys. Rev. D20, 3052 (1979)]. The model displays a number of more realistic features than the ones he treated. In addition to local observables, as the energy density, we propose to consider intensive variables, such as the energy per unit area , as fundamental observables. Adopting this view, lqft rejects Dirichlet (the same result may be proved for Neumann or mixed) bc, and accepts periodic bc: in the former case diverges, in the latter it is finite, as is shown by an expression for the local energy density obtained from lqft through the use of the Poisson summation formula. Another way to see this uses methods from the Euler summation formula: in the proof of regularization independence of the energy per unit area, a regularizationdependent surface term arises upon use of Dirichlet bc, but not periodic bc. For the conformally invariant scalar quantum field, this surface term is absent due to the condition of zero trace of the energy momentum tensor, as remarked by De Witt [Phys. Rep.19, 295 (1975)]. The latter property does not hold in the application to the dark energy problem in cosmology, in which we argue that periodic bc might play a distinguished role.

Renormalization group and Ward identities for infrared QED4
View Description Hide DescriptionA regularized version of Euclidean QED4 in the Feynman gauge is considered, with a fixed ultraviolet cutoff, photon mass of the size of the cutoff, and any value, including zero, of the electron mass. We will prove that the Schwinger functions are expressed by convergent series for small values of the charge and verify the Ward identities, up to corrections which are small for momentum scales far from the ultraviolet cutoff.
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 GENERAL RELATIVITY AND GRAVITATION


Pseudosymmetry collineations
View Description Hide DescriptionThe pseudosymmetric spaces are a generalization of the spaces of constant sectional curvature in the sense that a scalarvalued curvature function, now depending on two planes, is assumed to be isotropic. From the study of the pseudosymmetry condition, a new set of tensors on the manifold receives attention and an invariance group on this set of tensors is introduced, the socalled pseudosymmetry collineations. This group is the infinitesimal counterpart of the pseudosymmetry condition. Their relationship with other types of transformations is discussed and, as an example, the pseudosymmetry collineations for the vacuum ppwaves are explicitly obtained.

Liouvillian perturbations of black holes
View Description Hide DescriptionWe apply the wellknown Kovacic algorithm to find closed form, i.e., Liouvillian solutions, to the differential equations governing perturbations of black holes. Our analysis includes the full gravitational perturbations of Schwarzschild and Kerr, the full gravitational and electromagnetic perturbations of ReissnerNordstrom, and specialized perturbations of the KerrNewman geometry. We also include the extreme geometries. We find all frequencies , in terms of black hole parameters and an integer , which allow Liouvillian perturbations. We display many classes of black hole parameter values and their corresponding Liouvillian perturbations, including new closedform perturbations of Kerr and ReissnerNordstrom. We also prove that the only type 1 Liouvillian perturbations of Schwarzschild are the known algebraically special ones and that type 2 Liouvillian solutions do not exist for extreme geometries. In cases where we do not prove the existence or nonexistence of Liouvillian perturbations we obtain sequences of Diophantine equations on which decidability rests.

Scalar fields coupled to gravity in dimensions: Static spherically symmetric stiff matter solutions
View Description Hide DescriptionEinstein’s field equations for a spherically symmetric metric and a massless scalar field source are reduced to a system effectively of second order in time for the metric function . The solutions of this system split into two classes that we called the positive and negative branches, corresponding to scalar fields with spacelike and timelike gradients. Static solutions for the positive branch are well known, but the proof that is a global attractor for the region , is given for completeness. The negative branch of static solutions have the interpretation of a perfect fluid with pressure equal to mass density, called “stiff matter.” The main result of the paper is the characterization of static solutions of the negative branch by phase plane analysis. We prove that the solution is a global attractor for the region where and . Two first integrals, one of which reducing to the solution in the static case, are also presented.

On the invariant symmetries of the metrics
View Description Hide DescriptionWe analyze the symmetries and other invariant qualities of the metrics (type D aligned EinsteinMaxwell solutions with cosmological constant whose Debever null principal directions determine shearfree geodesic null congruences). We recover some properties and deduce new ones about their isometry group and about their quadratic first integrals of the geodesic equation, and we analyze when these invariant symmetries characterize the family of metrics. We show that the subfamily of the KerrNUT solutions are those admitting a Papapetrou field aligned with the Weyl tensor.
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 DYNAMICAL SYSTEMS


A dynamical approximation for stochastic partial differential equations
View Description Hide DescriptionRandom invariant manifolds provide geometric structures for understanding stochastic dynamics. In this paper, a dynamical approximation estimate is derived for a class of stochastic partial differential equations, by showing that the random invariant manifold is almost surely asymptotically complete. The asymptotic dynamical behavior is thus described by a stochastic ordinary differential system on the random invariant manifold, under suitable conditions. As an application, stationary states (invariant measures) are considered for a class of stochastic hyperbolic partial differential equations.
