Showing papers in "International Journal of Geometric Methods in Modern Physics in 2012"

Pseudo z symmetric riemannian manifolds with harmonic curvature tensors

Abstract: In this paper we introduce a new notion of Z-tensor and a new kind of Riemannian manifold that generalize the concept of both pseudo Ricci symmetric manifold and pseudo projective Ricci symmetric manifold. Here the Z-tensor is a general notion of the Einstein gravitational tensor in General Relativity. Such a new class of manifolds with Z-tensor is named pseudoZ symmetric manifold and denoted by (PZS)n. Various properties of such an n-dimensional manifold are studied, especially focusing the cases with harmonic curvature tensors giving the conditions of closeness of the associated one-form. We study (PZS)n manifolds with harmonic conformal and quasi-conformal curvature tensor. We also show the closeness of the associated 1-form when the (PZS)n manifold becomes pseudo Ricci symmetric in the sense of Deszcz (see [A. Derdzinsky and C. L. Shen, Codazzi tensor fields, curvature and Pontryagin forms, Proc. London Math. Soc.47(3) (1983) 15–26; R. Deszcz, On pseudo symmetric spaces, Bull. Soc. Math. Belg. Ser. A44 (1992) 1–34]). Finally, we study some properties of (PZS)4 spacetime manifolds.

A 3-dimensional sasakian metric as a yamabe soliton

Abstract: If a 3-dimensional Sasakian metric on a complete manifold (M, g) is a Yamabe soliton, then we show that g has constant scalar curvature, and the flow vector field V is Killing. We further show that, either M has constant curvature 1, or V is an infinitesimal automorphism of the contact metric structure on M.

Natural connection with totally skew-symmetric torsion on almost contact manifolds with b-metric

Abstract: A natural connection with totally skew-symmetric torsion on almost contact manifolds with B-metric is constructed. The class of these manifolds, where the considered connection exists, is determined. Some curvature properties for this connection, when the corresponding curvature tensor has the properties of the curvature tensor for the Levi-Civita connection and the torsion tensor is parallel, are obtained.

An entropic picture of emergent quantum mechanics

Abstract: Quantum mechanics emerges a la Verlinde from a foliation of ℝ3 by holographic screens, when regarding the latter as entropy reservoirs that a particle can exchange entropy with. This entropy is quantized in units of Boltzmann's constant kB. The holographic screens can be treated thermodynamically as stretched membranes. On that side of a holographic screen where spacetime has already emerged, the energy representation of thermodynamics gives rise to the usual quantum mechanics. A knowledge of the different surface densities of entropy flow across all screens is equivalent to a knowledge of the quantum-mechanical wavefunction on ℝ3. The entropy representation of thermodynamics, as applied to a screen, can be used to describe quantum mechanics in the absence of spacetime, that is, quantum mechanics beyond a holographic screen, where spacetime has not yet emerged. Our approach can be regarded as a formal derivation of Planck's constant ℏ from Boltzmann's constant kB.

Canonical metrics on Cartan–Hartogs domains.

Abstract: In this paper we address two problems concerning a family of domains MΩ(μ) ⊂ ℂn, called Cartan–Hartogs domains, endowed with a natural Kahler metric g(μ). The first one is determining when the metric g(μ) is extremal (in the sense of Calabi), while the second one studies when the coefficient a2 in the Englis expansion of Rawnsley e-function associated to g(μ) is constant.

The Cauchy problem for f(R)-gravity: An Overview

Abstract: We review the Cauchy problem for f(R) theories of gravity, in metric and metric-affine formulations, pointing out analogies and differences with respect to General Relativity. The role of conformal transformations, effective scalar fields and sources in the field equations is discussed in view of the well-formulation and the well-position of the problem. Finally, criteria of viability of the f(R)-models are considered according to the various matter fields acting as sources.

Integration of semidirect product lie 2-algebras

Abstract: The semidirect product of a Lie algebra and a 2-term representation up to homotopy is a Lie 2-algebra. Such Lie 2-algebras include many examples arising from the Courant algebroid appearing in generalized complex geometry. In this paper, we integrate such a Lie 2-algebra to a strict Lie 2-group in the finite-dimensional case.

A pure dirac's canonical analysis for four-dimensional bf theories

Abstract: We perform Dirac's canonical analysis for a four-dimensional BF and for a generalized four-dimensional BF theory depending on a connection valued in the Lie algebra of SO(3, 1). This analysis is developed by considering the corresponding complete set of variables that define these theories as dynamical, and we find out the relevant symmetries, the constraints, the extended Hamiltonian, the extended action, gauge transformations and the counting of physical degrees of freedom. The results obtained are compared with other approaches found in the literature.

A hamilton–jacobi theory for singular lagrangian systems in the skinner and rusk setting

Abstract: We develop a Hamilton–Jacobi theory for singular Lagrangian systems in the Skinner–Rusk formalism. Comparisons with the Hamilton–Jacobi problem in the Lagrangian and Hamiltonian settings are discussed.

Recurrent z forms on riemannian and kaehler manifolds

Abstract: In this paper, we introduce a new kind of Riemannian manifold that generalize the concept of weakly Z-symmetric and pseudo-Z-symmetric manifolds. First a Z form associated to the Z tensor is defined. Then the notion of Z recurrent form is introduced. We take into consideration Riemannian manifolds in which the Z form is recurrent. This kind of manifold is named (ZRF)n. The main result of the paper is that the closedness property of the associated covector is achieved also for rank(Zkl) > 2. Thus the existence of a proper concircular vector in the conformally harmonic case and the form of the Ricci tensor are confirmed for(ZRF)n manifolds with rank(Zkl) > 2. This includes and enlarges the corresponding results already proven for pseudo-Z-symmetric (PZS)n and weakly Z-symmetric manifolds (WZS)n in the case of non-singular Z tensor. In the last sections we study special conformally flat (ZRF)n and give a brief account of Z recurrent forms on Kaehler manifolds.

Clifford modules and symmetries of topological insulators

Abstract: We complete the classification of symmetry constraints on gapped quadratic fermion hamiltonians proposed by Kitaev. The symmetry group is supposed compact and can include arbitrary unitary or antiunitary operators in the Fock space that conserve the algebra of quadratic observables. We analyze the multiplicity spaces of real irreducible representations of unitary symmetries in the Nambu space. The joint action of intertwining operators and antiunitary symmetries provides these spaces with the structure of Clifford module: we prove a one-to-one correspondence between the ten Altland–Zirnbauer symmetry classes of fermion systems and the ten Morita equivalence classes of real and complex Clifford algebras. The antiunitary operators, which occur in seven classes, are projectively represented in the Nambu space by unitary "chiral symmetries". The space of gapped symmetric hamiltonians is homotopically equivalent to the product of classifying spaces indexed by the dual object of the group of unitary symmetries.

On octonionic gravity, exceptional jordan strings and nonassociative ternary gauge field theories

Abstract: Novel nonassociative octonionic ternary gauge field theories are proposed based on a ternary bracket. This paves the way to the many physical applications of exceptional Jordan Strings/Membranes and Octonionic Gravity. The old octonionic gravity constructions based on the split octonion algebra Os (which strictly speaking is not a division algebra) is extended to the full fledged octonion division algebra O. A real-valued analog of the Einstein–Hilbert Lagrangian involving sums of all the possible contractions of the Ricci tensors plus their octonionic-complex conjugates is presented. A discussion follows of how to extract the Standard Model group (the gauge fields) from the internal part of the octonionic gravitational connection. The role of exceptional Jordan algebras, their automorphism and reduced structure groups which play the roles of the rotation and Lorentz groups is also re-examined. Finally, we construct (to our knowledge) generalized novel octonionic string and p-brane actions and raise the possibility that our generalized 3-brane action (based on a quartic product) in octonionic flat backgrounds of 7,8 octonionic dimensions may display an underlying E7, E8 symmetry, respectively. We conclude with some final remarks pertaining to the developments related to Jordan exceptional algebras, octonions, black-holes in string theory and quantum information theory.

3-lie multipliers on banach 3-lie algebras

Abstract: In this paper, we study the space of 3-Lie multipliers on Banach 3-Lie algebras. Moreover, we investigate a characterization of 3-Lie multipliers on commutative and without order Banach 3-Lie algebras. Finally, we establish the stability and superstability of 3-Lie multipliers.

Twisted topological structures related to M-branes II: Twisted Wu and Wuc structures

Abstract: Studying the topological aspects of M-branes in M-theory leads to various structures related to Wu classes. First we interpret Wu classes themselves as twisted classes and then define twisted notions of Wu structures. These generalize many known structures, including Pin- structures, twisted Spin structures in the sense of Distler–Freed–Moore, Wu-twisted differential cocycles appearing in the work of Belov–Moore, as well as ones introduced by the author, such as twisted Membrane and twisted Stringc structures. In addition, we introduce Wuc structures, which generalize Pinc structures, as well as their twisted versions. We show how these structures generalize and encode the usual structures defined via Stiefel–Whitney classes.

Internal symmetry groups of cubic algebras

Abstract: We investigate certain Z3-graded associative algebras with cubic Z3 invariant constitutive relations, introduced by one of us some time ago. The invariant forms on finite algebras of this type are given in the cases with two and three generators. We show how the Lorentz symmetry represented by the SL(2, C) group can be introduced without any notion of metric, just as the symmetry of Z3-graded cubic algebra and its constitutive relations. Its representation is found in terms of the Pauli matrices. The relationship of such algebraic constructions with quark states is also considered.

Geometric hamilton–jacobi field theory

Abstract: I review my proposal about how to extend the geometric Hamilton–Jacobi theory to higher derivative field theories on fiber bundles.

On the fermi–walker derivative and non-rotating frame

Abstract: In this study Fermi–Walker derivative and according to the derivative Fermi–Walker parallelism and non-rotating frame concepts are given for some frames. First, we get the Frenet frame, the Darboux frame, the Bishop frame for any curve in Euclid space. Fermi–Walker derivative and non-rotating frame being conditions are analyzed for each of the frames along the curve. Then we proved the Frenet frame is non-rotating frame along the plane curves. Darboux frame which is a non-rotating frame along the line of curvature. Then we proved the Bishop frame is a non-rotating frame along the all curves.

An Approximate Solution of the Jaynes-Cummings Model with Dissipation II : Another Approach

Abstract: In the preceding paper (arXiv: 1103.0329 [quant-ph]) we treated the Jaynes–Cummings model with dissipation and gave an approximate solution to the master equation for the density operator under the general setting by making use of the Zassenhaus expansion. However, to obtain a compact form of the approximate solution (which is in general complicated infinite series) is very hard when an initial condition is given. To overcome this difficulty we develop another approach and obtain a compact approximate solution when some initial condition is given.

Unfolding the quantum arnold transformation

Abstract: The recently proposed quantum Arnold transformation is revisited and extended and its relation with other methods is emphasized. Possible applications are also outlined.

Discrete second-order euler–poincaré equations: applications to optimal control

Abstract: In this paper we will discuss some new developments in the design of numerical methods for optimal control problems of Lagrangian systems on Lie groups. We will construct these geometric integrators using discrete variational calculus on Lie groups, deriving a discrete version of the second-order Euler–Lagrange equations. Interesting applications as, for instance, a discrete derivation of the Euler–Poincare equations for second-order Lagrangians and its application to optimal control of a rigid body, and of a Cosserat rod are shown at the end of the paper.

Dynamic rearrangement of vacuum and the phase transitions in the geometric structure of space-time

Abstract: It is shown that in the case of spontaneous breaking of the original gauge symmetry, a dynamic rearrangement of vacuum may lead to the formation of some anisotropic condensates. The appearance of such condensates causes the respective phase transitions in the geometric structure of space-time and creates a flat anisotropic, i.e. Finslerian event space. Actually there arises either a flat relativistically invariant Finslerian space with partially broken 3D isotropy, i.e. axially symmetric space, or a flat relativistically invariant Finslerian space with entirely broken 3D isotropy. The fact that any entirely anisotropic relativistically invariant Finslerian event space belongs to a 3-parameter family of such spaces gives rise to a fine structure of the respective geometric phase transitions. In the present paper the fine structure of the geometric phase transitions is studied by classifying all the metric states of the entirely anisotropic event space and the respective mass shell equations.

Weyl geometries and timelike geodesics

Abstract: In view of Ehlers–Pirani–Schild formalism, since 1972 Weyl geometries should be considered to be the most appropriate and complete framework to represent (relativistic) gravitational fields. We shall here show that in any given Lorentzian spacetime (M, g) that admits global timelike vector fields any such vector field u determines an essentially unique Weyl geometry ([g], Γ) such that u is Γ-geodesic (i.e. parallel with respect to Γ).

A natural connection on a basic class of riemannian product manifolds

Abstract: A Riemannian manifold M with an integrable almost product structure P is called a Riemannian product manifold. Our investigations are on the manifolds (M, P, g) of the largest class of Riemannian product manifolds, which is closed with respect to the group of conformal transformations of the metric g. This class is an analogue of the class of locally conformal Kahler manifolds in almost Hermitian geometry. In the present paper we study a natural connection D on (M, P, g) (i.e. DP = Dg = 0). We find necessary and sufficient conditions, the curvature tensor of D to have properties similar to the Kahler tensor in Hermitian geometry. We pay attention to the case when D has a parallel torsion. We establish that the Weyl tensors for the connection D and the Levi-Civita connection coincide as well as the invariance of the curvature tensor of D with respect to the usual conformal transformation. We consider the case when D is a flat connection. We construct an example of the considered manifold by a Lie group where D is a flat connection with non-parallel torsion.

Boundary effects in quantum physics

Abstract: We analyze the role of boundaries in the infrared behavior of quantum field theories. By means of a novel method we calculate the vacuum energy for a massless scalar field confined between two homogeneous parallel plates with the most general type of boundary properties. This allows the discrimination between boundary conditions which generate attractive or repulsive Casimir forces between the plates. In the interface between both regimes we find a very interesting family of boundary conditions which do not induce any type of Casimir force. We analyze the effect of the renormalization group flow on these boundary conditions. Even if the Casimirless conformally invariant conditions are physically unstable under renormalization group flow they emerge as a new set of conformally invariant boundary conditions which are anomaly free.

The physical foundations for the geometric structure of relativistic theories of gravitation: from general relativity to extended theories of gravity through ehlers–pirani–schild approach

Abstract: We discuss in a critical way the physical foundations of geometric structure of relativistic theories of gravity by the so-called Ehlers–Pirani–Schild formalism. This approach provides a natural interpretation of the observables showing how relate them to General Relativity and to a large class of Extended Theories of Gravity. In particular we show that, in such a formalism, geodesic and causal structures of space-time can be safely disentangled allowing a correct analysis in view of observations and experiment. As specific case, we take into account the case of f(R)-gravity.

SPIN FLUIDS IN BIANCHI-I f(R)-COSMOLOGY WITH TORSION

Abstract: We study Weyssenhoff spin fluids in Bianchi type-I cosmological models, within the framework of torsional f(R)-gravity; the resulting field equations are derived and discussed in both Jordan and Einstein frames, clarifying the role played by the spin and the non-linearity of the gravitational Lagrangian f(R) in generating the torsional dynamical contributions. The general conservation laws holding for f(R)-gravity with torsion are employed to provide the conditions needed to ensure the preservation of the Hamiltonian constraint and the consequent correct formulation of the associated initial value problem. Examples are eventually given.

A new lie–systems approach to second-order riccati equations

Abstract: This work presents a newly renovated approach to the analysis of second-order Riccati equations from the point of view of the theory of Lie systems. We show that these equations can be mapped into Lie systems through certain Legendre transforms. This result allows us to construct new superposition rules for studying second-order Riccati equations and to reduce their integration to solving (first-order) Riccati equations.

A theorem of solér, the theory of symmetry and quantum mechanics

Abstract: A classical problem in axiomatic quantum mechanics is deducing a Hilbert space realization for a quantum logic that admits a vector space coordinatization of the Piron–McLaren type. Our aim is to show how a theorem of M. Soler [Characterization of Hilbert spaces by orthomodular spaces, Comm. Algebra23 (1995) 219–243.] can be used to get a (partial) solution of this problem. We first derive a generalization of the Wigner theorem on symmetry transformations that holds already in the Piron–McLaren frame. Then we investigate which conditions on the quantum logic allow the use of Soler's theorem in order to obtain a Hilbert space solution for the coordinatization problem.

4d-polytopes and their dual polytopes of the coxeter group w(a4) represented by quaternions

Abstract: Four-dimensional A4 polytopes and their dual polytopes have been constructed as the orbits of the Coxeter–Weyl group W(A4) where the group elements and the vertices of the polytopes are represented by quaternions. Projection of an arbitrary W(A4) orbit into three dimensions is made using the subgroup W(A3). A generalization of the Catalan solids for 3D-polyhedra has been developed and dual polytopes of the uniform A4 polytopes have been constructed.

On a hamiltonian version of controls dynamic for a drift-free left invariant control system on g4

Abstract: Some dynamical and geometrical properties of controls dynamic for a drift-free left invariant control system from the Poisson geometry point of view are described. The integrability of such system are also studied.