Volume 47, Issue 9, September 2006
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Is weak pseudoHermiticity weaker than pseudoHermiticity?
View Description Hide DescriptionFor a weakly pseudoHermitian linear operator, we give a spectral condition that ensures its pseudoHermiticity. This condition is always satisfied whenever the operator acts in a finitedimensional Hilbert space. Hence weak pseudoHermiticity and pseudoHermiticity are equivalent in finitedimensions. This equivalence extends to a much larger class of operators. Quantum systems whose Hamiltonian is selected from among these operators correspond to pseudoHermitian quantum systems possessing certain symmetries.

On the optimality of quantum encryption schemes
View Description Hide DescriptionIt is well known that bits of entropy are necessary and sufficient to perfectly encrypt bits (onetime pad). Even if we allow the encryption to be approximate, the amount of entropy needed does not asymptotically change. However, this is not the case when we are encrypting quantum bits. For the perfect encryption of quantum bits, bits of entropy are necessary and sufficient (quantum onetime pad), but for approximate encryption one asymptotically needs only bits of entropy. In this paper, we provide the optimal tradeoff between the approximation measure and the amount of classical entropy used in the encryption of single quantum bits. Then, we consider qubit encryption schemes which are a composition of independent singlequbit ones and provide the optimal schemes both in the 2 and the norm. Moreover, we provide a counterexample to show that the encryption scheme of AmbainisSmith [Proceedings of RANDOM ’04, pp. 249–260] based on smallbias sets does not work in the norm.

Sheaftheoretic representation of quantum measure algebras
View Description Hide DescriptionWe construct a sheaftheoretic representation of quantum probabilistic structures, in terms of covering systems of Boolean measurealgebras. These systems coordinatize quantum states by means of Boolean coefficients, interpreted as Boolean localization measures. The representation is based on the existence of a pair of adjoint functors between the category of presheaves of Boolean measurealgebras and the category of quantum measurealgebras. The sheaftheoretic semantic transition of quantum structures shifts their physical significance from the orthoposet axiomatization at the level of events, to the sheaftheoretic gluing conditions at the level of Boolean localization systems.

Galois quantum systems, irreducible polynomials and Riemann surfaces
View Description Hide DescriptionFinite quantum systems in which the position and momentum take values in the Galois field , are studied. Ideas from the subject of field extension are transferred in the context of quantum mechanics. The Frobenius automorphisms in Galois fields lead naturally to the “Frobenius formalism” in a quantum context. The Hilbert space splits into “Frobenius subspaces” which are labeled with the irreducible polynomials associated with the . The Frobenius maps transform unitarily the states of a Galois quantum system and leave fixed all states in some of its Galois subsystems (where the position and momentum take values in subfields of ). An analytic representation of these systems in the sheeted complex plane shows deeper links between Galois theory and Riemann surfaces.

Magnetic monopoles in quantum adiabatic dynamics and the immersion property of the control manifold
View Description Hide DescriptionIt is well known that the Berry phase of a cyclic adiabatic dynamical system appears formally as the flux of a magnetic field in the control parameter manifold. In this electromagnetic picture a level crossing appears as a Dirac magnetic monopole in this manifold. We make an extensive study of the magnetic monopole model of eigenvalue crossings. We show that the properties of the monopole magnetic field in the control manifold are determined by the immersion of the control manifold in a space given by the universal classifying theorem of fiber bundles. We give a detailed illustrative study of the simple but instructive case of a two level crossing of a system controlled by a twodimensional manifold.

Phase space quantum mechanics  Direct
View Description Hide DescriptionConventional approach to quantum mechanics in phase space, , is to take the operator based quantum mechanics of Schrödinger, or an equivalent, and assign a number function in phase space to it. We propose to begin with a higher level of abstraction, in which the independence and the symmetric role of and is maintained throughout, and at once arrive at phase space state functions. Upon reduction to the  or space the proposed formalism gives the conventional quantum mechanics, however, with a definite rule for ordering of factors of noncommuting observables. Further conceptual and practical merits of the formalism are demonstrated throughout the text.

Extremal covariant measurements
View Description Hide DescriptionWe characterize the extremal points of the convex set of quantum measurements that are covariant under a finitedimensional projective representation of a compact group, with action of the group on the measurement probability space which is generally nontransitive. In this case the POVM density is made of multiple orbits of positive operators, and, in the case of extremal measurements, we provide a bound for the number of orbits and for the rank of POVM elements. Two relevant applications are considered, concerning state discrimination with mutually unbiased bases and the maximization of the mutual information.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Pointwise analysis of scalar fields: A nonstandard approach
View Description Hide DescriptionA new nonstandardanalytical approach to quantum fields is presented, which gives a mathematical foundation for manipulating pointwisedefined quantum fields. In our approach, a field operator is not a standard operatorvalued distribution, but a nonstandard operatorvalued function. Then formal expressions containing, e.g., can be understood literally, and shown to be well defined. In the free field cases, we show that the Wightman functions are explicitly calculated with the pointwise field, without any regularization, e.g., Wick product. Our notion of pointwise fields is applied also to the path integral formalisms of scalar fields. We show that some of physicists’ naive expressions of Lagrangian path integral formulas can be rigorously justified.

Analyticity of the scattering amplitude, causality, and highenergy bounds in quantum field theory on noncommutative space–time
View Description Hide DescriptionIn the framework of quantum field theory on noncommutative space–time with the symmetry group , we prove that the JostLehmannDyson representation, based on the causality condition taken in connection with this symmetry, leads to the mere impossibility of drawing any conclusion on the analyticity of the scattering amplitude in , being the scattering angle. Discussions on the possible ways of obtaining highenergy bounds analogous to the FroissartMartin bound on the total cross section are also presented.

Higherdimensional knotlike topological defects in local nonAbelian topological tensor currents
View Description Hide DescriptionWe present the novel topological tensor currents to describe the infinitesimal thin higherdimensional topological defects in the local gauge theory. The topological quantization of defects and the inner structure of the currents are obtained. As the generalization of NielsenOlesen local field theory for Nambu string, the local gaugeinvariant Lagrangian and the motion equation of the higherdimensional topological defects are derived. Moreover, for closed defects, we study their important topological configuration, i.e., the higherdimensional knotlike structures. Using the topological tensor currents and their preimages, we construct a series of metric independent integrals and prove their gauge independence. Similar to the helicity integral characterizing onedimensional knotlike vortex filament, these topological invariants are evaluated to the generalized linking numbers of higherdimensional knotlike defects.

brane charges in Gepner models
View Description Hide DescriptionWe construct Gepner models in terms of coset conformal field theories and compute their twisted equivariant Ktheories. These classify the brane charges on the associated geometric backgrounds and therefore agree with the topological Ktheories. We show this agreement for various cases, in particular, the Fermat quintic.
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 GENERAL RELATIVITY AND GRAVITATION


Electromagnetic and gravitational interactions of the spinning particle
View Description Hide DescriptionWe consider the invariance of the spinning free particle Lagrangian under the global coordinate transformations for the classical model of the electron with internal degrees of freedom and obtain the conservation of the energymomentum, total angular momentum, and electric charge. The local gauge transformations give the electromagnetic and gravitational interactions of the spinning particle in the RiemannCartan space from the generalized spin connections. We show that the covariant constancy of the Dirac matrices gives; (i) the form invariance of the classical equations of motion, except the gravitational force terms in nongeodesic equation, (ii) the conservation of the electromagnetic current, (iii) the quantum Hamiltonian and equations of motion from the classical ones without the quantum ordering corrections, and (iv) the minimal coupling of the gravitation with the spinning particle in the Hamiltonian and in wave equations in the RiemannCartan space–time.

On the essential constants in Riemannian geometries
View Description Hide DescriptionIn the present work the problem of distinguishing between essential and spurious (i.e., absorbable) constants contained in a metric tensor field in a Riemannian geometry is considered. The contribution of the study is the presentation of a sufficient and necessary criterion, in terms of a covariant statement, which enables one to determine whether a constant is essential or not. It turns out that the problem of characterization is reduced to that of solving a system of partial differential equations of the first order. In any case, the metric tensor field is assumed to be smooth with respect to the constant to be tested. It should be stressed that the entire analysis is purely of local character.

Intersecting hypersurfaces in antide Sitter and Lovelock gravity
View Description Hide DescriptionColliding and intersecting hypersurfaces filled with matter (membranes) are studied in the Lovelock higher order curvature theory of gravity. Lovelock terms couple hypersurfaces of different dimensionalities, extending the range of possible intersection configurations. We restrict the study to constant curvature membranes in constant curvature antide Sitter (AdS) and dS background and consider their general intersections. This illustrates some key features which make the theory different from the Einstein gravity. Higher codimension membranes may lie at the intersection of codimension one hypersurfaces in Lovelock gravity; the hypersurfaces are located at the discontinuities of the first derivative of the metric, and they need not carry matter. The example of colliding membranes shows that general solutions can only be supported by (spacelike) matter at the collisionsurface, thus naturally conflicting with the dominant energy condition (DEC). The imposition of the DEC gives selection rules on the types of collision allowed. When the hypersurfaces do not carry matter, one gets a solitonlike configuration. Then, at the intersection one has a codimension two or higher membrane standing alone in AdSvacuum space–time without conical singularities. Another result is that if the number of intersecting hypersurfaces goes to infinity the limiting space–time is free of curvature singularities if the intersection is put at the boundary of each AdS bulk.
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 DYNAMICAL SYSTEMS


Meromorphic Lax representations of dimensional multiHamiltonian dispersionless systems
View Description Hide DescriptionRational Lax hierarchies introduced by Krichever are generalized. A systematic construction of infinite multiHamiltonian hierarchies and related conserved quantities is presented. The method is based on the classical matrix approach applied to Poisson algebras. A proof that Poisson operators constructed near different points of Laurent expansion of Lax functions are equal is given. All results are illustrated by several examples.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Map of discrete system into continuous
View Description Hide DescriptionContinuous limits of discrete systems with longrange interactions are considered. The map of discrete models into continuous medium models is defined. A wide class of longrange interactions that give the fractional equations in the continuous limit is discussed. The onedimensional systems of coupled oscillators for this type of longrange interactions are considered. The discrete equations of motion are mapped into the continuum equation with the Riesz fractional derivative.

Domain wall and periodic solutions of coupled models in an external field
View Description Hide DescriptionCoupled double well onedimensional potentials abound in both condensed matter physics and field theory. Here we provide an exhaustive set of exact periodic solutions of a coupled model in an external field in terms of elliptic functions (domain wall arrays) and obtain single domain wallsolutions in specific limits. We also calculate the energy and interaction between solitons for various solutions. Both topological and nontopological (e.g., some pulselike solutions in the presence of a conjugate field) domain walls are obtained. We relate some of these solutions to the recently observed magnetic domain walls in certain multiferroicmaterials and also in the field theory context wherever possible. Discrete analogs of these coupled models, relevant for structural transitions on a lattice, are also considered.
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 FLUIDS


Steady states of selfgravitating incompressible fluid in two dimensions
View Description Hide DescriptionIn this paper we develop a simple model for the steady state of twodimensional selfgravitating incompressible gas which is based on the hydrodynamicequations for stratified fluid. These equations are then reduced to a system of two equations for the mass density and the gravitational field. Analytical analysis and numerical solutions of these equations under different modeling assumptions (with special attention to the isothermal case) are then used to study the structure of the resulting steady state of the fluid.

Stokes flow in ellipsoidal geometry
View Description Hide DescriptionParticleincellmodels for Stokes flow through a relatively homogeneous swarm of particles are of substantial practical interest, because they provide a relatively simple platform for the analytical or semianalytical solution of heat and mass transport problems. Despite the fact that many practical applications involve relatively small particles (inorganic, organic, biological) with axisymmetric shapes, the general consideration consists of rigid particles of arbitrary shape. The present work is concerned with some interesting aspects of the theoretical analysis of creeping flow in ellipsoidal, hence nonaxisymmetric domains. More specifically, the low Reynolds number flow of a swarm of ellipsoidal particles in an otherwise quiescent Newtonian fluid, that move with constant uniform velocity in an arbitrary direction and rotate with an arbitrary constant angular velocity, is analyzed with an ellipsoidincell model. The solid internal ellipsoid represents a particle of the swarm. The external ellipsoid contains the ellipsoidal particle and the amount of fluid required to match the fluid volume fraction of the swarm. The nonslip flow condition on the surface of the solid ellipsoid is supplemented by the boundary conditions on the external ellipsoidal surface which are similar to those of the sphereincell model of Happel (selfsufficient in mechanical energy). This model requires zero normal velocity component and shear stress. The boundary value problem is solved with the aim of the potential representation theory. In particular, the Papkovich–Neuber complete differential representation of Stokes flow, valid for nonaxisymmetric geometries, is considered here, which provides the velocity and total pressure fields in terms of harmonic ellipsoidal eigenfunctions. The flexibility of the particular representation is demonstrated by imposing some conditions, which made the calculations possible. It turns out that the velocity of first degree, which represents the leading term of the series, is sufficient for most engineering applications, so long as the aspect ratios of the ellipsoids remains within moderate bounds. Analytical expressions for the leading terms of the velocity, the total pressure, the angular velocity, and the stress tensor fields are obtained. Corresponding results for the prolate and the oblate spheroid, the needle and the disk, as well as for the sphere are recovered as degenerate cases. Novel relations concerning the ellipsoidal harmonics are included in the Appendix.
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 METHODS OF MATHEMATICAL PHYSICS


Secondorder superintegrable systems in conformally flat spaces. V. Two and threedimensional quantum systems
View Description Hide DescriptionThis paper is the conclusion of a series that lays the groundwork for a structure and classification theory of secondorder superintegrable systems, both classical and quantum, in conformally flat spaces. For twodimensional and for conformally flat threedimensional spaces with nondegenerate potentials we have worked out the structure of the classical systems and shown that the quadratic algebra always closes at order 6. Here we describe the quantum analogs of these results. We show that, for nondegenerate potentials, each classical system has a unique quantum extension. We also correct an error in an earlier paper in the series (that does not alter the structure results) and we elucidate the distinction between superintegrable systems with bases of functionally linearly independent and functionally linearly dependent symmetries.
