Volume 46, Issue 1, January 2005
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Analysis of superoscillatory wave functions
View Description Hide DescriptionSurprisingly, differentiable functions are able to oscillate arbitrarily faster than their highest Fourier component would suggest. The phenomenon is called superoscillation. Recently, a practical method for calculating superoscillatory functions was presented and it was shown that superoscillatory quantum mechanical wave functions should exhibit a number of counterintuitive physical effects. Following up on this work, we here present more general methods which allow the calculation of superoscillatory wave functions with customdesigned physical properties. We give concrete examples and we prove results about the limits to superoscillatory behavior. We also give a simple and intuitive new explanation for the exponential computational cost of superoscillations.

Contextual logic for quantum systems
View Description Hide DescriptionIn this work we build a quantum logic that allows us to refer to physical magnitudes pertaining to different contexts from a fixed one without the contradictions with quantum mechanics expressed in nogo theorems. This logic arises from considering a sheaf over a topological space associated with the Boolean sublattices of the ortholattice of closed subspaces of the Hilbert space of the physical system. Different from standard quantum logics, the contextual logic maintains a distributive lattice structure and a good definition of implication as a residue of the conjunction.

Two and three dimensional Hamiltonians with generalized and ordinary shape invariance symmetry
View Description Hide DescriptionTwo and three dimensional Hamiltonians with generalized and ordinary shape invariance symmetry have been obtained by Fourier transforming over some coordinates of the SU(3) Casimir operator defined on symmetric space. It is shown that the generalized shape invariance of the two dimensional Hamiltonian is equivalent to SU(3) symmetry while in the three dimensional one, the ordinary shape invariance is equivalent to contracted SU(3) and there is one to one correspondence between the representations of the generalized shape invariance symmetry of the two (three) dimensional Hamiltonian and SU(3) [contracted SU(3)] Verma bases.

Implementation of groupcovariant positive operator valued measures by orthogonal measurements
View Description Hide DescriptionWe consider groupcovariant positive operator valued measures (POVMs) on a finite dimensional quantum system. Following Neumark’s theorem a POVM can be implemented by an orthogonal measurement on a larger system. Accordingly, our goal is to find a quantum circuit implementation of a given groupcovariant POVM which uses the symmetry of the POVM. Based on representation theory of the symmetry group we develop a general approach for the implementation of groupcovariant POVMs which consist of rankone operators. The construction relies on a method to decompose matrices that intertwine two representations of a finite group. We give several examples for which the resulting quantum circuits are efficient. In particular, we obtain efficient quantum circuits for a class of POVMs generated by Weyl–Heisenberg groups. These circuits allow to implement an approximative simultaneous measurement of the position and crystal momentum of a particle moving on a cyclic chain.

Remarks on the spectrum of the Neumann problem with magnetic field in the halfspace
View Description Hide DescriptionWe consider a Schrödinger operator with a constant magnetic field in a onehalf threedimensional space, with Neumanntype boundary conditions. It is known from the works by Lu–Pan and Helffer–Morame that the lower bound of its spectrum is less than , the intensity of the magnetic field, provided that the magnetic field is not normal to the boundary. We prove that the spectrum under is a finite set of eigenvalues (each of infinite multiplicity). In the case when the angle between the magnetic field and the boundary is small, we give a sharp asymptotic expansion of the number of these eigenvalues.

Wigner–Weyl isomorphism for quantum mechanics on Lie groups
View Description Hide DescriptionThe Wigner–Weyl isomorphism for quantum mechanics on a compact simple Lie group is developed in detail. Several features are shown to arise which have no counterparts in the familiar Cartesian case. Notable among these is the notion of a semiquantized phase space, a structure on which the Weyl symbols of operators turn out to be naturally defined and, figuratively speaking, located midway between the classical phase space and the Hilbert space of square integrable functions on . General expressions for the star product for Weyl symbols are presented and explicitly worked out for the angleangular momentum case.

Schrödinger problems for surfaces of revolution—the finite cylinder as a test example
View Description Hide DescriptionA set of ordinary differential equations is derived employing the method of differentiable forms so as to describe the quantum mechanics of a particle constrained to move on a general twodimensional surface of revolution. Eigenvalues and eigenstates are calculated quasianalytically in the case of a finite cylinder (finite along the axis) and compared with the eigenvalues and eigenstates of a full threedimensional Schrödinger problem corresponding to a hollow cylinder in the limit where the inner and outer radii approach each other. Good agreement between the two models is obtained for a relative difference less than 20% in inner and outer radii.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Quantum integrability of bosonic massive Thirring model in continuum
View Description Hide DescriptionBy using a variant of the quantum inverse scattering method, commutation relations between all elements of the quantum monodromy matrix of the bosonic massive Thirring (BMT) model are obtained. Using those relations, the quantum integrability of BMT model is established and the matrix of twobody scattering between the corresponding quasiparticles has been obtained. It is observed that for some special values of the coupling constant, there exists an upper bound on the number of quasiparticles that can form a quantumsoliton state of the BMT model. We also calculate the binding energy for a soliton state of the quantum BMT model.

The de Sitter relativistic top theory
View Description Hide DescriptionWe discuss the relativistic top theory from the point of view of the de Sitter (or anti–de Sitter) group. Our treatment rests on the Hanson–Regge spherical relativistic top Lagrangian formulation. We propose an alternative method for studying spinning objects via Kaluza–Klein theory. In particular, we derive the relativistic top equations of motion starting with the geodesic equation for a point particle in dimensions. We compare our approach with Fukuyama’s formulation of spinning objects, which is also based on Kaluza–Klein theory. We also report a generalization of our approach to a dimensional theory.

No selfinteraction for twocolumn massless fields
View Description Hide DescriptionWe investigate the problem of introducing consistent selfcouplings in free theories for mixed tensorgauge fields whose symmetry properties are characterized by Young diagrams made of two columns of arbitrary (but different) lengths. We prove that, in flat space, these theories admit no local, Poincaréinvariant, smooth, selfinteracting deformation with at most two derivatives in the Lagrangian. Relaxing the derivative and Lorentzinvariance assumptions, there still is no deformation that modifies the gauge algebra, and in most cases no deformation that alters the gauge transformations. Our approach is based on a BecchiRouetStoraTyutin (BRST) cohomology deformation procedure.

Critical behavior of the compactified theory
View Description Hide DescriptionWe investigate the critical behavior of the component Euclidean model, in the large limit, in three situations: confined between two parallel planes a distance apart from one another; confined to an infinitely long cylinder having a square transversal section of area ; and to a cubic box of volume . Taking the mass term in the form , we retrieve Ginzburg–Landau models which are supposed to describe samples of a material undergoing a phase transition, respectively, in the form of a film, a wire and of a grain, whose bulk transition temperature is known. We obtain equations for the critical temperature as functions of and of , and determine the limiting sizes sustaining the transition.

Uniqueness of the topological multivortex solution in the selfdual Chern–Simons theory
View Description Hide DescriptionWe establish a uniqueness result for the topological multivortex solution to the selfdual equations of the Abelian relativistic selfdual Chern–Simons–Higgs model. We prove that the topological multivortex solution is unique if the Chern–Simons coupling parameter is sufficiently small. We also establish a uniqueness result for sufficiently large.

 GENERAL RELATIVITY AND GRAVITATION


Weak field reduction in teleparallel coframe gravity: Vacuum case
View Description Hide DescriptionThe teleparallel coframe gravity may be viewed as a generalization of the standard GR. A coframe (a field of four independent 1forms) is considered, in this approach, to be a basic dynamical variable. The metric tensor is treated as a secondary structure. The general Lagrangian, quadratic in the first order derivatives of the coframe field is not unique. It involves three dimensionless free parameters. We consider a weak field approximation of the general coframe teleparallel model. In the linear approximation, the field variable, the coframe, is covariantly reduced to the superposition of the symmetric and antisymmetric field. We require this reduction to be preserved on the levels of the Lagrangian, of the field equations, and of the conserved currents. This occurs if and only if the pure Yang–Millstype term is removed from the Lagrangian. The absence of this term is known to be necessary and sufficient for the existence of the viable (Schwarzschild) sphericalsymmetric solution. Moreover, the same condition guarantees the absence of ghosts and tachyons in particle content of the theory. The condition above is shown recently to be necessary for a welldefined Hamiltonian formulation of the model. Here we derive the same condition in the Lagrangian formulation by means of the weak field reduction.

The wave equation on the Schwarzschild metric II. Local decay for the spin2 Regge–Wheeler equation
View Description Hide DescriptionOddtype spin2 perturbations of Einstein’s equation can be reduced to the scalar Regge–Wheeler equation. We show that the weighted norms of solutions are in of time and space. This result uses commutator methods and applies uniformly to all relevant spherical harmonics.

Necessary and sufficient conditions for dimensional conformal Einstein spaces via dimensionally dependent identities
View Description Hide DescriptionListing has recently extended results of Kozameh, Newman, and Tod for fourdimensional space–times and presented a set of necessary and sufficient conditions for a metric to be locally conformally equivalent to an Einstein metric in all semiRiemannian spaces of dimension —subject to a nondegeneracy restriction on the Weyl tensor. By exploiting dimensionally dependent identities we demonstrate how to construct two alternative versions of these necessary and sufficient conditions which we believe will be useful in applications. The fourdimensional case is discussed in detail and examples are also given in five and six dimensions.

On the eigenvalues of the Chandrasekhar–Page angular equation
View Description Hide DescriptionIn this paper we study for a given azimuthal quantum number the eigenvalues of the Chandrasekhar–Page angular equation with respect to the parameters and , where is the angular momentum per unit mass of a black hole, is the rest mass of the Dirac particle and is the energy of the particle (as measured at infinity). For this purpose, a selfadjoint holomorphic operator family associated to this eigenvalue problem is considered. At first we prove that for fixed the spectrum of is discrete and that its eigenvalues depend analytically on . Moreover, it will be shown that the eigenvalues satisfy a first order partial differential equation with respect to and , whose characteristic equations can be reduced to a Painlevé III equation. In addition, we derive a power series expansion for the eigenvalues in terms of and , and we give a recurrence relation for their coefficients. Further, it will be proved that for fixed the eigenvalues of are the zeros of a holomorphic function which is defined by a relatively simple limit formula. Finally, we discuss the problem if there exists a closed expression for the eigenvalues of the Chandrasekhar–Page angular equation.

 DYNAMICAL SYSTEMS


Integrable Hamiltonian systems with vector potentials
View Description Hide DescriptionWe investigate integrable twodimensional Hamiltonian systems with scalar and vector potentials, admitting second invariants which are linear or quadratic in the momenta. In the case of a linear second invariant, we provide some examples of weakly integrable systems. In the case of a quadratic second invariant, we recover the classical strongly integrable systems in Cartesian and polar coordinates and provide some new examples of integrable systems in parabolic and elliptical coordinates.

The difficulty of symplectic analysis with second class systems
View Description Hide DescriptionUsing the basic concepts of the chain by chain method we show that the symplectic analysis, which was claimed to be equivalent to the usual Dirac method, fails when second class constraints are present. We propose a modification in symplectic analysis that solves the problem.

A model for Hopfions on the space–time
View Description Hide DescriptionWe construct static and time dependent exact soliton solutions for a theory of scalar fields taking values on a wide class of two dimensional target spaces, and defined on the four dimensional space–time . The construction is based on an ansatz built out of special coordinates on . The requirement for finite energy introduce boundary conditions that determine an infinite discrete spectrum of frequencies for the oscillating solutions. For the case where the target space is the sphere , we obtain static soliton solutions with nontrivial Hopf topological charges. In addition, such Hopfions can oscillate in time, preserving their topological Hopf charge, with any of the frequencies belonging to that infinite discrete spectrum.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Diractype approach for consistent discretizations of classical constrained theories
View Description Hide DescriptionWe analyze the canonical treatment of classical constrained mechanical systems formulated with a discrete time. We prove that under very general conditions, it is possible to introduce nonsingular canonical transformations that preserve the constraint surface and the Poisson or Dirac bracket structure. The conditions for the preservation of the constraints are more stringent than in the continuous case and as a consequence some of the continuum constraints become second class upon discretization and need to be solved by fixing their associated Lagrange multipliers. The gauge invariance of the discrete theory is encoded in a set of arbitrary functions that appear in the generating function of the evolution equations. The resulting scheme is general enough to accommodate the treatment of field theories on the lattice. This paper attempts to clarify and put on sounder footing a discretization technique that has already been used to treat a variety of systems, including Yang–Mills theories, BF theory, and general relativity on the lattice.
