Volume 47, Issue 5, May 2006
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Quantum entanglement and geometry of determinantal varieties
View Description Hide DescriptionQuantum entanglement was first recognized as a feature of quantum mechanics in the famous paper of Einstein, Podolsky, and Rosen. Recently it has been realized that quantum entanglement is a key ingredient in quantum computation, quantum communication, and quantum cryptography. In this paper, we introduce algebraic sets, which are determinantal varieties in the complex projective spaces or the products of complex projective spaces, for the mixed states on bipartite or multipartite quantum systems as their invariants under local unitary transformations. These invariants are naturally arised from the physical consideration of measuring mixed states by separable pure states. Our construction has applications in the following important topics in quantum information theory: (1) separability criterion, it is proved that the algebraic sets must be a union of the linear subspaces if the mixed states are separable; (2) simulation of Hamiltonians, it is proved that the simulation of semipositive Hamiltonians of the same rank implies the projective isomorphisms of the corresponding algebraic sets; (3) construction of bound entangled mixed states, examples of the entangled mixed states which are invariant under partial transpositions (thus PPT bound entanglement) are constructed systematically from our new separability criterion.

Observables of angular momentum as observables on the Fedosov quantized sphere
View Description Hide DescriptionIn this paper we construct quantum mechanical observables of a single free particle that lives on the surface of the twosphere by implementing the Fedosov ∗formalism. The Fedosov ∗ is a generalization of the Moyal star product on an arbitrary symplectic manifold. After their construction we show that they obey the standard angular momentum commutation relations in ordinary nonrelativistic quantum mechanics. The purpose of this paper is threefold. One is to find an exact, nonperturbative solution of these observables. The other is to verify that the commutation relations of these observables correspond to angular momentum commutation relations. The last is to show a more general computation of the observables in Fedosov ∗formalism; essentially an undeformation of Fedosov’s algorithm.

Semiclassical expansion of Wigner functions
View Description Hide DescriptionWe show that the time evolved Wigner function of a quantum particle under the action of a smooth potential can be formally expanded in powers of , where each term of the expansion can be computed in terms of the corresponding classical flow. Moreover the solution can be approximated by the order truncation with an error .

Fisher information of dimensional hydrogenic systems in position and momentum spaces
View Description Hide DescriptionThe spreading of the quantummechanical probability distribution density of dimensional hydrogenic orbitals is quantitatively determined by means of the local informationtheoretic quantity of Fisher in both position and momentum spaces. The Fisher information is found in closed form in terms of the quantum numbers of the orbital.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Combinatorics of point functions via Hopf algebra in quantum field theory
View Description Hide DescriptionWe use a coproduct on the timeordered algebra of field operators to derive simple relations between complete, connected and 1particle irreducible point functions. Compared to traditional functional methods our approach is much more intrinsic and leads to efficient algorithms suitable for concrete computations. It may also be used to efficiently perform tree level computations.

Investigation of the Nicole model
View Description Hide DescriptionWe study soliton solutions of the Nicole model—a nonlinear fourdimensional field theory consisting of the Lagrangian density to the noninteger power —using an ansatz within toroidal coordinates, which is indicated by the conformal symmetry of the static equations of motion. We calculate the soliton energies numerically and find that they grow linearly with the topological charge (Hopf index). Further we prove this behavior to hold exactly for the ansatz. On the other hand, for the full threedimensional system without symmetry reduction we prove a sublinear upper bound, analogously to the case of the Faddeev–Niemi model. It follows that symmetric solitons cannot be true minimizers of the energy for sufficiently large Hopf index, again in analogy to the Faddeev–Niemi model.

The twoloop massless model in nontranslational invariant domain
View Description Hide DescriptionWe study the massless scalar field theory in a fourdimensional Euclidean space, where all but one of the coordinates are unbounded. We are considering Dirichlet boundary conditions in two hyperplanes, breaking the translation invariance of the system. We show how to implement the perturbative renormalization up to twoloop level of the theory. First, analyzing the full two and fourpoint functions at the oneloop level, we show that the bulk counterterms are sufficient to render the theory finite. Meanwhile, at the twoloop level, we must also introduce surface counterterms in the bare Lagrangian in order to make finite the full two and also fourpoint Schwinger functions.

On the quantum mechanics for one photon
View Description Hide DescriptionThis paper revisits the quantum mechanics for one photon from the modern viewpoint and by the geometrical method. Especially, besides the ordinary (rectangular) momentum representation, we provide an explicit derivation for the other two important representations, called the cylindrically symmetrical representation and the spherically symmetrical representation, respectively. These other two representations are relevant to some current photon experiments in quantum optics. In addition, the latter is useful for us to extract the information on the quantized black holes. The framework and approach presented here are also applicable to other particles with arbitrary mass and spin, such as the particle with spin .

Noncommutative geometry and the standard model vacuum
View Description Hide DescriptionThe space of Dirac operators for the ConnesChamseddine spectral action for the standard model of particle physics coupled to gravity is studied. The model is extended by including righthanded neutrino states, and the reality axiom is not assumed. The possibility of allowing more general fluctuations than the inner fluctuations of the vacuum is proposed. The maximal case of all possible fluctuations is studied by considering the equations of motion for the vacuum. While there are interesting nontrivial vacua with Majoranatype mass terms for the leptons, the conclusion is that the equations are too restrictive to allow solutions with the standard model mass matrix.
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 GENERAL RELATIVITY AND GRAVITATION


Motion in KaluzaKlein type theories
View Description Hide DescriptionPath and path deviation equations for charged, spinning and spinning charged objects in different versions of KaluzaKlein (KK) theory using a modified Bazanski Lagrangian have been derived. The significance of motion in five dimensions, especially for a charged spinning object, has been examined. We have also extended the modified Bazanski approach to derive the path and path deviation equations of a test particle in a version of nonsymmetric KK theory.

Gravitational radiation, vorticity and the electric and magnetic part of Weyl tensor
View Description Hide DescriptionThe electric and the magnetic part of the Weyl tensor, as well as the invariants obtained from them, are calculated for the Bondi vacuum metric. One of the invariants vanishes identically and the other only exhibits contributions from terms of the Weyl tensor containing the static part of the field. It is shown that the necessary and sufficient condition for the spacetime to be purely electric is that such spacetime be static. It is also shown that the vanishing of the electric part implies Minkowski spacetime. Unlike the electric part, the magnetic part does not contain contributions from the static field. Finally a speculation about the link between the vorticity of world lines of observers at rest in a Bondi frame, and gravitational radiation, is presented.

Local Cauchy problem for the nonlinear Dirac and DiracKleinGordon equations on Kerr space–time
View Description Hide DescriptionWe prove the local existence of solutions of nonlinear Dirac and DiracKleinGordon equations in Kerr metric with regular Cauchy initial datas.

The determination of all syzygies for the dependent polynomial invariants of the Riemann tensor. II. Mixed invariants of even degree in the Ricci spinor
View Description Hide DescriptionWe continue our analysis of the polynomial invariants of the Riemann tensor in a fourdimensional Lorentzian space. We concentrate on the mixed invariants of even degree in the Ricci spinor and show how, using constructive graphtheoretic methods, arbitrary scalar contractions between copies of the Weyl spinor , its conjugate and an even number of Ricci spinors can be expressed in terms of paired contractions between these spinors. This leads to an algorithm for the explicit expression of dependent invariants as polynomials of members of the complete set. Finally, we rigorously prove that the complete set as given by Sneddon [J. Math. Phys.39, 1659–1679 (1998)] for this case is both complete and minimal.
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 DYNAMICAL SYSTEMS


Uniform attractors for nonautonomous incompressible nonNewtonian fluid with locally uniform integrable external forces
View Description Hide DescriptionThis paper discusses the long time behavior of solutions for a twodimensional (2D) nonautonomous incompressible nonNewtonian fluid in 2D bounded domains. When the external force is locally uniform integrable (see Definition 1.1) in , the authors obtain the existence and structure of uniform attractors in space for the family of processes associated with the fluid. Moreover, when is properly small, they provide some interesting corollaries.
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 STATISTICAL PHYSICS


A new energy method for the Boltzmann equation
View Description Hide DescriptionAn energy method for the Boltzmann equation was proposed by Liu, Yang, and Yu [Physica D188, 178–192 (2004)] based on the decomposition of the Boltzmann equation and its solution around the local Maxwellian. The main idea is to rewrite the Boltzmann equation as a fluidtype dynamics system with the nonfluid component appearing in the source terms, coupled with an equation for the time evolution of the nonfluid component. In this paper, we will elaborate this method and our main observation is that the microscopic projection of the local Maxwellian with respect to a given global Maxwellian is not linear but quadratic. Based on this and by analyzing the fluidtype system using the analytic techniques for the system of conservation laws, we can indeed control the conserved quantities , and of the Boltzmann equation by the microscopic projection of the solution of the Boltzmann equation with respect to the global Maxwellian, which is sufficient to deduce the energy estimates for the solution of the Boltzmann equation. The main purpose here is to show that there is no need to perform two sets of energy estimates with respect to the local and a global Maxwellian as in the previous works. In fact, one set of energy estimates with respect to the global Maxwellian is sufficient for closing the energy estimates. Therefore, it not only simplifies the analysis in the previous works, but also shed some light on the stability analysis in some complicated systems, such as the VlasovPoissonBoltzmann and VlasovMaxwellBoltzmann systems.
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 METHODS OF MATHEMATICAL PHYSICS


Semidirect sums of Lie algebras and discrete integrable couplings
View Description Hide DescriptionA relation between semidirect sums of Lie algebras and integrable couplings of lattice equations is established, and a practicable way to construct integrable couplings is further proposed. An application of the resulting general theory to the generalized Toda spectral problem yields two classes of integrable couplings for the generalized Toda hierarchy of lattice equations. The construction of integrable couplings using semidirect sums of Lie algebras provides a good source of information on complete classification of integrable lattice equations.

Tzitzéica transformation is a dressing action
View Description Hide DescriptionWe classify the simplest rational elements in a twisted loop group, and prove that dressing actions of them, on proper indefinite affine spheres, give the classical Tzitzéica transformation and its dual. We also give the group point of view of the permutability theorem, construct complex Tzitzéica transformations, and discuss the group structure for these transformations.

Gravitation theory in a fractal spacetime
View Description Hide DescriptionAssimilating the physical spacetime with a fractal, a general theory is built. For a fractal dimension , the virtual geodesics of this spacetime implies a generalized Schrödinger type equation. Subsequently, a geometric formulation of the gravitation theory on a fractalspacetime is given. Then, a connection is introduced on a tangent bundle, the connection coefficients, the Riemann curvature tensor and the Einstein field equation are calculated. It results, by means of a dilation operator, the equivalence of this model with quantum Einstein gravity.

Hilbert spaces built on a similarity and on dynamical renormalization
View Description Hide DescriptionWe develop a Hilbertspace framework for a number of general multiscale problems from dynamics. The aim is to identify a spectral theory for a class of systems based on iterations of a noninvertible endomorphism. We are motivated by the more familiar approach to wavelettheory which starts with the twotoone endomorphism in the onetorus , a wavelet filter, and an associated transfer operator. This leads to a scaling function and a corresponding closed subspace in the Hilbert space. Using the dyadic scaling on the line , one has a nested family of closed subspaces, , with trivial intersection, and with dense union in . More generally, we achieve the same outcome, but in different Hilbert spaces, for a class of nonlinear problems. In fact, we see that the geometry of scales of subspaces in Hilbert space is ubiquitous in the analysis of multiscale problems, e.g., martingales, complex iteration dynamical systems, graphiterated function systems of affine type, and subshifts in symbolic dynamics. We develop a general framework for these examples which starts with a fixed endomorphism (i.e., generalizing ) in a compact metric space. It is assumed that is onto, and finitetoone.

Jacobi identity for vertex algebras in higher dimensions
View Description Hide DescriptionVertex algebras in higher dimensions, introduced previously by Nikolov, provide an algebraic framework for investigating axiomatic quantum field theory with global conformal invariance. We develop further the theory of such vertex algebras by introducing formal calculus techniques and investigating the notion of polylocal fields. We derive a Jacobi identity which together with the vacuum axiom can be taken as an equivalent definition of vertex algebra.
