Volume 47, Issue 11, November 2006
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Belltype inequalities for nonlocal resources
View Description Hide DescriptionWe present bipartite Belltype inequalities which allow the two partners to use some nonlocal resource. Such inequalities can only be violated if the parties use a resource which is more nonlocal than the one permitted by the inequality. We introduce a family of input nonlocal machines, which are generalizations of the wellknown PR (PopescuRohrlich) box. Then we construct Belltype inequalities that cannot be violated by strategies that use one of these new machines. Finally we discuss implications for the simulation of quantum states.

On the Weyl algebras for systems with semibounded and bounded configuration space
View Description Hide DescriptionWe define the Weyl algebra suitable to represent the quantization postulate for onedimensional systems whose configuration space is semibounded. It consists of a group of unitary operators and a semigroup of nonunitary isometries. We show that the spectrum is a halfline , with an arbitrary , and that the irreducible representations of the Weyl algebra with the same are equivalent. We also consider the case when the semigroup of translations is substituted with a semigroup of partial isometries of index 1 (particle confined to a segment of unit length). The uniqueness of the irreducible representations of the related Weyl algebra is proved also for this case by exploiting the result for the halfline.

Scattering and bound state Green’s functions on a plane via so(2,1) Lie algebra
View Description Hide DescriptionWe calculate the Green’s functions for the particlevortex system, for two anyons on a plane with and without a harmonic regulator and in a uniform magnetic field. These Green’s functions which describe scattering or bound states (depending on the specific potential in each case) are obtained exactly using an algebraic method related to the SO(2,1) Lie group. From these Green’s functions we obtain the corresponding wave functions and for the bound states we also find the energy spectra.

The Selberg trace formula for Dirac operators
View Description Hide DescriptionWe examine spectra of Dirac operators on compact hyperbolic surfaces. Particular attention is devoted to symmetry considerations, leading to nontrivial multiplicities of eigenvalues. The relation to spectra of MaaßLaplace operators is also exploited. Our main result is a Selberg trace formula for Dirac operators on hyperbolic surfaces.

Localization effects in a periodic quantum graph with magnetic field and spinorbit interaction
View Description Hide DescriptionA general technique for the study of magnetic Rashba Hamiltonians in quantum graphs is presented. We use this technique to show how manipulating the magnetic and spin parameters can be used to create localized states in a certain periodic graph ( lattice).

Basic properties of the currentcurrent correlation measure for random Schrödinger operators
View Description Hide DescriptionThe currentcurrent correlation measure plays a crucial role in the theory of conductivity for disordered systems. We prove a PasturShubintype formula for the currentcurrent correlation measure expressing it as a thermodynamic limit for random Schrödinger operators on the lattice and the continuum. We prove that the limit is independent of the selfadjoint boundary conditions and independent of a large family of expanding regions. We relate this finitevolume definition to the definition obtained by using the infinitevolume operators and the traceperunit volume.

Equilibrium and eigenfunctions estimates in the semiclassical regime
View Description Hide DescriptionWe establish eigenfunctions estimates, in the semiclassical regime, for critical energy levels associated to an isolated singularity. For Schrödinger operators, the asymptotic repartition of eigenvectors is the same as in the regular case, excepted in dimension one where a concentration at the critical point occurs. This principle extends to pseudodifferential operators and the limit measure is the Liouville measure as long as the singularity remains integrable.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


On Chern–Simons (super) gravity, Yang–Mills and polyvectorvalued gauge theories in Clifford spaces
View Description Hide DescriptionIt is shown how the Yang–Mills theory is a small sector of a algebra gauge theory and why the Chern–Simons (super) gravity theory can be embedded into a algebra gauge theory. These results may shed some light into the origins behind the hidden symmetry of supergravity. To finalize, we explain how the Clifford algebra gauge theory (that contains the Chern–Simons gravity action in , for example) can itself be embedded into a more fundamental polyvectorvalued gauge theory in Clifford spaces involving tensorial coordinates in addition to antisymmetric tensor gauge fields . The polyvectorvalued supersymmetric extension of this polyvector valued bosonic gauge theory in Clifford spaces may reveal more important features of a Cliffordalgebraic structure underlying , theory.

Foundations of a spacetime path formalism for relativistic quantum mechanics
View Description Hide DescriptionQuantum field theory is the traditional solution to the problems inherent in melding quantum mechanics with special relativity. However, it has also long been known that an alternative firstquantized formulation can be given for relativistic quantum mechanics, based on the parametrized paths of particles in spacetime. Because time is treated similarly to the three space coordinates, rather than as an evolution parameter, such a spacetime approach has proved particularly useful in the study of quantum gravity and cosmology. This paper shows how a spacetime path formalism can be considered to arise naturally from the fundamental principles of the Born probability rule, superposition, and Poincaré invariance. The resulting formalism can be seen as a foundation for a number of previous parametrized approaches in the literature, relating, in particular, “offshell” theories to traditional onshell quantum field theory. It reproduces the results of perturbative quantum field theory for free and interacting particles, but provides intriguing possibilities for a natural program for regularization and renormalization. Further, an important consequence of the formalism is that a clear probabilistic interpretation can be maintained throughout, with a natural reduction to nonrelativistic quantum mechanics.

Observers, observables, spinors, and the confusion of tongues
View Description Hide DescriptionChoquetBruhat was the first to give a proper physical definition of covariant spinors, taking into account the reference system and treating them as equivalence classes defined from the transformation laws of the representatives when the reference system is changed. Recently, Rodriguez et al.[ Int. J. Theor. Phys.35, 1849 (1996)] have adapted this procedure from covariant spinors to the case of algebraic and operator spinors. These approaches are restrained in the sense that the type of spinor is chosen from the beginning, and it does not admit a general formulation. In this paper, we present a unified definition that is valid for any type of the space of representation, being independent of its particular properties. In our formulation the three types of spinors appear as particular cases of the general definition. Moreover, we stick out the importance of the bilinear covariants in the definition of spinors. From this, we recognize a completely different kind of spinor, characterized by the different nature of their bilinears. The unnoticed difference between this last one, which we have called rightoperator spinors, and the previous (left)operator spinors has been motive of a long time discussion.

Dimensional reduction of SeibergWitten monopole equations, noncommutative supersymmetric field theories and Young diagrams
View Description Hide DescriptionWe investigate the SeibergWitten monopole equations on noncommutative (N.C.) at the large N.C. parameter limit, in terms of the equivariant cohomology. In other words, supersymmetric U(1) gauge theories with a hypermultiplet on N.C. are studied. It is known that after topological twisting partition functions of supersymmetric theories on N.C. are invariant under the N.C. parameter shift; then the partition functions can be calculated by its dimensional reduction. At the large N.C. parameter limit, the SeibergWitten monopole equations are reduced to ADHM equations with the Dirac equation reduced to the dimension. The equations are equivalent to the dimensional reduction of nonAbelian SeibergWitten monopole equations in . The solutions of the equations are also interpreted as a configuration of a brane antibrane system. The theory has global symmetries under torus actions originated in space rotations and gauge symmetries. We investigate the SeibergWitten monopole equations reduced to the dimension and the fixed point equations of the torus actions. We show that the Dirac equation reduced to the dimension is automatically satisfied when the fixed point equations and the ADHM equations are satisfied. Then, we find that the SeibergWitten equations reduced to the 0 dimension and fixed point equations of the torus action are equivalent to just the ADHM equations with the fixed point equations. For finite , it is known that the fixed points of the ADHM data are isolated and are classified by the Young diagrams. We also give a new proof of this statement by solving the ADHM equations and the fixed point equations concretely and by giving graphical interpretations of the field components and these equations.
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 GENERAL RELATIVITY AND GRAVITATION


Type I vacuum solutions with aligned Papapetrou fields: An intrinsic characterization
View Description Hide DescriptionWe show that Petrov type I vacuum solutions admitting a Killing vector whose Papapetrou field is aligned with a principal bivector of the Weyl tensor are the Kasner and Taub metrics, their counterpart with timelike orbits and their associated windmilllike solutions, as well as the Petrov homogeneous vacuum solution. We recover all these metrics by using an integration method based on an invariant classification which allows us to characterize every solution. In this way we obtain an intrinsic and explicit algorithm to identify them.

On existence of matter outside a static black hole
View Description Hide DescriptionIt is expected that matter composed of a perfect fluid cannot be at rest outside of a black hole if the spacetime is asymptotically flat and static (nonrotating). However, there has not been rigorous proof for this expectation without assuming spherical symmetry. In this paper, we provide a proof of nonexistence of matter composed of a perfect fluid in static black holespacetimes under certain conditions, which can be interpreted as a relation between the stellar mass and the black hole mass.

Domain walls on the surface of stars
View Description Hide DescriptionWe study domain wall networks on the surface of stars in asymptotically flat or anti de Sitter spacetime. We provide numerical solutions for the whole phase space of the stable field configurations and find that the mass, radius, and particle number of the star is larger but the scalar field, responsible for the formation of the soliton, acquires smaller values when a domain wall network is entrapped on the star surface.

Propagating torsion in the Einstein frame
View Description Hide DescriptionThe Einstein–Cartan–Saa theory of torsion modifies the spacetime volume element so that it is compatible with the connection. The condition of connection compatibility gives constraints on torsion, which are also necessary for the consistence of torsion, minimal coupling, and electromagnetic gauge invariance. To solve the problem of positivity of energy associated with the torsionic scalar, we reformulate this theory in the Einstein conformal frame. In the presence of the electromagnetic field, we obtain the Hojman–Rosenbaum–Ryan–Shepley theory of propagating torsion with a different factor in the torsionic kinetic term.

Variational approach to Robertson–Walker spacetimes with homogeneous scalar fields
View Description Hide DescriptionExistence of solutions between prescribed configurations is proved for spatially flat Robertson–Walker spacetimes coupled with homogeneous scalar field sources, using a modified version of the Euler–Maupertuis least action variational principle. The solutions are obtained as limits of approximating variational problems, solved using the techniques originally introduced by Rabinowitz.
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 DYNAMICAL SYSTEMS


A new integrable equation with cuspons and W/Mshapepeaks solitons
View Description Hide DescriptionIn this paper, we propose a new completely integrable wave equation: , . The equation is derived from the two dimensional Euler equation and is proven to have Lax pair and biHamiltonian structures. This equation possesses new cusp solitons—cuspons, instead of regular peakons with speed . Through investigating the equation, we develop a new kind of soliton solutions—“W/M”shapepeaks solitons. There exist no smooth solitons for this integrable water wave equation.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


The inverse problem for sixdimensional codimension two nilradical lie algebras
View Description Hide DescriptionAdo’s theorem asserts that every real Lie Algebra of dimension has a faithful representation as a subalgebra of for some . The theorem offers no practical information about the size of in relation to and in principle may be very large compared to . This article is concerned with finding representations for a certain class of sixdimensional Lie algebras, specifically, real, indecomposable algebras that have a codimension two nilradical. These algebras were classified by Turkowski and comprise of 40 cases, some of which contain up to four parameters. Linear representations are found for each algebra in these classes: More precisely, a matrix Lie group is given whose Lie algebra corresponds to each algebra in Turkowski’s list and can be found by differentiating and evaluating at the identity element of the group. In addition a basis for the rightinvariant vector fields that are dual to the MaurerCartan forms are given thereby providing an effective realization of Lie’s third theorem. The geodesic spray of the canonical symmetric connection for each of the 40 linear Lie group is given. Thereafter the inverse problem of the calculus of variations for each of the geodesic sprays is investigated. In all cases it is determined whether the spray is of EulerLagrange type and in the affirmative case at least one concrete Lagrangian is written down. In none of the cases is there a Lagrangian of metric type.

On the Hamiltonian reduction of geodesic motion on SU(3) to SU(3)/SU(2)
View Description Hide DescriptionThe reduced Hamiltonian system on is derived from a Riemannian geodesic motion on the SU(3) group manifold parametrized by the generalized Euler angles and endowed with a biinvariant metric. Our calculations show that the metric defined by the derived reduced Hamiltonian flow on the orbit space is not isometric or even geodesically equivalent to the standard Riemannian metric on the fivesphere embedded into .
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 METHODS OF MATHEMATICAL PHYSICS


Nonequilibrium Glaubertype dynamics in continuum
View Description Hide DescriptionWe construct the nonequilibrium Glauber dynamics as a Markov process in configuration space for an infinite particle system in continuum with a general class of initial distributions. This class we define in terms of correlation functions bounds and it is preserved during the Markov evolution we constructed.
