Showing papers in "International Journal of Geometric Methods in Modern Physics in 2008"
TL;DR: In this paper, the current status of Finsler-Lagrange geometry and generalizations is reviewed and a canonical scheme for geometrical objects on a (pseudo) Riemannian space are nonholonomically deformed into generalized Lagrange configurations on the same manifold.
Abstract: We review the current status of Finsler–Lagrange geometry and generalizations. The goal is to aid non-experts on Finsler spaces, but physicists and geometers skilled in general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. We also would like to orient mathematicians working in generalized Finsler and Kahler geometry and geometric mechanics how they could perform their results in order to be accepted by the community of "orthodox" physicists. Although the bulk of former models of Finsler–Lagrange spaces where elaborated on tangent bundles, the surprising result advocated in our works is that such locally anisotropic structures can be modeled equivalently on Riemann–Cartan spaces, even as exact solutions in Einstein and/or string gravity, if nonholonomic distributions and moving frames of references are introduced into consideration. We also propose a canonical scheme when geometrical objects on a (pseudo) Riemannian space are nonholonomically deformed into generalized Lagrange, or Finsler, configurations on the same manifold. Such canonical transforms are defined by the coefficients of a prime metric and generate target spaces as Lagrange structures, their models of almost Hermitian/Kahler, or nonholonomic Riemann spaces. Finally, we consider some classes of exact solutions in string and Einstein gravity modeling Lagrange–Finsler structures with solitonic pp-waves and speculate on their physical meaning.
115 citations
TL;DR: In this article, an inverse scattering approach to defects in classical integrable field theories is presented, where the contribution of the defect to all orders is explicitely identified in terms of a defect matrix.
Abstract: We present an inverse scattering approach to defects in classical integrable field theories. Integrability is proved systematically by constructing the generating function of the infinite set of modified integrals of motion. The contribution of the defect to all orders is explicitely identified in terms of a defect matrix. The underlying geometric picture is that those defects correspond to Backlund transformations localized at a given point. A classification of defect matrices as well as the corresponding defect conditions is performed. The method is applied to a collection of well-known integrable models and previous results are recovered (and extended) directly as special cases. Finally, a brief discussion of the classical r-matrix approach in this context shows the relation to inhomogeneous lattice models and the need to resort to lattice regularizations of integrable field theories with defects.
76 citations
TL;DR: In this paper, the authors review the classical and quantum foundations necessary to study field-theory approaches to quantum gravity, the passage from old to new unification in quantum field theory and canonical quantum gravity.
Abstract: Quantum gravity was born as that branch of modern theoretical physics that tries to unify its guiding principles, i.e., quantum mechanics and general relativity. Nowadays it is providing new insight into the unification of all fundamental interactions, while giving rise to new developments in mathematics. The various competing theories, e.g. string theory and loop quantum gravity, have still to be checked against observations. We review the classical and quantum foundations necessary to study field-theory approaches to quantum gravity, the passage from old to new unification in quantum field theory, canonical quantum gravity, the use of functional integrals, the properties of gravitational instantons, the use of spectral zeta-functions in the quantum theory of the universe, Hawking radiation, some theoretical achievements and some key experimental issues.
75 citations
TL;DR: In this paper, the f(R)-theories of gravity with torsion in the framework of -bundles are discussed. And some cosmological applications and relations between f(r)-gravity and scalar-tensor theories are given.
Abstract: We discuss the f(R)-theories of gravity with torsion in the framework of -bundles. Such an approach is particularly useful since the components of the torsion and curvature tensors can be chosen as fiber -coordinates on the bundles and then the symmetries and the conservation laws of the theory can be easily achieved. Field equations of f(R)-gravity are studied in empty space and in presence of various forms of matter as Dirac fields, Yang–Mills fields and spin perfect fluid. Such fields enlarge the jet bundles framework and characterize the dynamics. Finally we give some cosmological applications and discuss the relations between f(R)-gravity and scalar-tensor theories.
45 citations
TL;DR: A survey of the use of differential geometric formalisms to describe quantum mechanics can be found in this paper, where the authors analyze the Weyl-Wigner construction of quantum mechanics.
Abstract: In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schrodinger framework from this perspective and provide a description of the Weyl–Wigner construction. Finally, after reviewing the basics of the geometric formulation of quantum mechanics, we apply the methods presented to the most interesting cases of finite dimensional Hilbert spaces: those of two, three and four level systems (one qubit, one qutrit and two qubit systems). As a more practical application, we discuss the advantages that the geometric formulation of quantum mechanics can provide us with in the study of situations as the functional independence of entanglement witnesses.
37 citations
TL;DR: In this article, a multi-dimensional fractional action-like problem of the calculus of variations where fractional field theories and fractional differential Dirac operators are constructed is addressed.
Abstract: Fractional calculus has recently attracted considerable attention. In particular, various fractional differential equations are used to model nonlinear wave theory that arises in many different areas of physics such as Josephson junction theory, field theory, theory of lattices, etc. Thus one may expect fractional calculus, in particular fractional differential equations, plays an important role in quantum field theories which are expected to satisfy fractional generalization of Klein–Gordon and Dirac equations. Until now, in high-energy physics and quantum field theories the derivative operator has only been used in integer steps. In this paper, we want to extend the idea of differentiation to arbitrary non-integers steps. We will address multi-dimensional fractional action-like problems of the calculus of variations where fractional field theories and fractional differential Dirac operators are constructed.
36 citations
TL;DR: In this paper, the role played by local translational symmetry in the context of gauge theories of fundamental interactions is analyzed, with special attention being paid to their universal coupling to other variables, as well as their contributions to field equations and to conserved quantities.
Abstract: We analyze the role played by local translational symmetry in the context of gauge theories of fundamental interactions. Translational connections and fields are introduced, with special attention being paid to their universal coupling to other variables, as well as to their contributions to field equations and to conserved quantities.
35 citations
TL;DR: In this paper, the authors discuss ternary algebraic structures appearing in various domains of theoretical and mathematical physics, some of them are associative, and some are not associative.
Abstract: We discuss ternary algebraic structures appearing in various domains of theoretical and mathematical physics. Some of them are associative, and some are not. Their interesting and curious properties can be exploited in future applications to enlarged and generalized field theoretical models in the years to come. Many ideas presented here have been developed and clarified in countless discussions with Michel Dubois-Violette.
34 citations
TL;DR: In this article, the relative position of two von Neumann algebras in Hilbert space is measured and combined with the spectrum of the Dirac operator, giving a complete invariant of Riemannian geometry.
Abstract: We introduce an invariant of Riemannian geometry which measures the relative position of two von Neumann algebras in Hilbert space, and which, when combined with the spectrum of the Dirac operator, gives a complete invariant of Riemannian geometry. We show that the new invariant plays the same role with respect to the spectral invariant as the Cabibbo–Kobayashi–Maskawa mixing matrix in the Standard Model plays with respect to the list of masses of the quarks.
30 citations
TL;DR: In this paper, a standard anti-de Sitter gauge theory becomes spontaneously broken due to a suitable Higgs mechanism, and the resulting Lorentz invariant gravitational action includes the Hilbert-Einstein term of ordinary Einstein-Cartan gravity with cosmological constant, plus contributions quadratic in curvature and torsion, and a scalar Higgs sector.
Abstract: Due to a suitable Higgs mechanism, a standard Anti-de Sitter gauge theory becomes spontaneously broken. The resulting Lorentz invariant gravitational action includes the Hilbert–Einstein term of ordinary Einstein–Cartan gravity with cosmological constant, plus contributions quadratic in curvature and torsion, and a scalar Higgs sector.
25 citations
TL;DR: In this paper, a definition of space-time metric deformations on an n-dimensional manifold is given, and the deformations can be regarded as extended conformal transformations, which are related to the perturbation theory giving a natural picture by which gravitational waves are described by small deformations of the metric.
Abstract: A definition of space-time metric deformations on an n-dimensional manifold is given. We show that such deformations can be regarded as extended conformal transformations. In particular, their features can be related to the perturbation theory giving a natural picture by which gravitational waves are described by small deformations of the metric. As further result, deformations can be related to approximate Killing vectors (approximate symmetries) by which it is possible to parameterize the deformed region of a given manifold. The perspectives and some possible physical applications of such an approach are discussed.
TL;DR: In this article, the general solution of the quantum damped harmonic oscillator is given, and the solution is shown to be the same as that of the classical Dijkstra oscillator.
Abstract: In this paper the general solution of the quantum damped harmonic oscillator is given.
TL;DR: Algebraic entropy is given, which is a global index of complexity for dynamical systems with a rational evolution, and conjectured to be the logarithm of algebraic integer, with a limited range of values, still to be explored.
Abstract: We give the definition of algebraic entropy, which is a global index of complexity for dynamical systems with a rational evolution. We explain its geometrical meaning, and different methods, heuristic or exact to calculate this entropy. This quantity is a very good integrability detector. It also has remarkable properties, which make it an interesting object of study by itself. It is in particular conjectured to be the logarithm of algebraic integer, with a limited range of values, still to be explored.
TL;DR: In this paper, the relation between symmetries and conservation laws in gauge-natural field theories was investigated, and it was shown that a canonical spinor connection can be selected by the simple requirement of the global existence of canonical superpotentials for the Lagrangian describing the coupling of gravitational and Fermionic fields.
Abstract: We investigate canonical aspects concerning the relation between symmetries and conservation laws in gauge-natural field theories. In particular, we find that a canonical spinor connection can be selected by the simple requirement of the global existence of canonical superpotentials for the Lagrangian describing the coupling of gravitational and Fermionic fields. In fact, the naturality of a suitably defined variational Lagragian implies the existence of an associated energy-momentum conserved current. Such a current defines a Hamiltonian form in the corresponding phase space; we show that an associated Hamiltonian connection is canonically defined along the kernel of the generalized gauge-natural Jacobi morphism and uniquely characterizes the canonical spinor connection.
Abstract: In this paper, we study Absolute Parallelism (AP-) geometry on the tangent bundle TM of a manifold M. Accordingly, all geometric objects defined in this geometry are not only functions of the positional argument x, but also depend on the directional argument y. Moreover, many new geometric objects, which have no counterpart in the classical AP-geometry, emerge in this different framework. We refer to such a geometry as an Extended Absolute Parallelism (EAP-) geometry. The building blocks of the EAP-geometry are a nonlinear connection (assumed given a priori) and 2n linearly independent vector fields (of special form) defined globally on TM defining the parallelization. Four different d-connections are used to explore the properties of this geometry. Simple and compact formulae for the curvature tensors and the W-tensors of the four defined d-connections are obtained, expressed in terms of the torsion and the contortion tensors of the EAP-space. Further conditions are imposed on the canonical d-connection assuming that it is of Cartan type (resp. Berwald type). Important consequences of these assumptions are investigated. Finally, a special form of the canonical d-connection is studied under which the classical AP-geometry is recovered naturally from the EAP-geometry. Physical aspects of some of the geometric objects investigated are pointed out and possible physical implications of the EAP-space are discussed, including an outline of a generalized field theory on the tangent bundle TM of M.
TL;DR: In this paper, the authors show that if the scale of quantum gravity is of the order of few TeVs, proton-proton collisions at the LHC could lead to the formation of time machines (spacetime regions with closed timelike curves) which violate causality.
Abstract: Recently, black hole and brane production at CERN's Large Hadron Collider (LHC) has been widely discussed. We suggest that there is a possibility to test causality at the LHC. We argue that if the scale of quantum gravity is of the order of few TeVs, proton-proton collisions at the LHC could lead to the formation of time machines (spacetime regions with closed timelike curves) which violate causality. One model for the time machine is a traversable wormhole. We argue that the traversable wormhole production cross section at the LHC is of the same order as the cross section for the black hole production. Traversable wormholes assume violation of the null energy condition (NEC) and an exotic matter similar to the dark energy is required. Decay of the wormholes/time machines and signatures of time machine events at the LHC are discussed.
TL;DR: In this paper, the authors give some examples of almost para-hyperhermitian structures on the tangent bundle of an almost product manifold, on the product manifold M × ℝ, where M is a manifold endowed with a mixed 3-structure.
Abstract: In this paper we give some examples of almost para-hyperhermitian structures on the tangent bundle of an almost product manifold, on the product manifold M × ℝ, where M is a manifold endowed with a mixed 3-structure and on the circle bundle over a manifold with a mixed 3-structure.
TL;DR: In contrast with QFT, classical field theory can be formulated in strict mathematical terms of fibre bundles, graded manifolds and jet manifolds as discussed by the authors, and second noether theorems provide BRST extension of classical field theories by means of ghosts and antifields for the purpose of its quantization.
Abstract: In contrast with QFT, classical field theory can be formulated in strict mathematical terms of fibre bundles, graded manifolds and jet manifolds. Second Noether theorems provide BRST extension of this classical field theory by means of ghosts and antifields for the purpose of its quantization.
TL;DR: In this article, a formal solution for the heat kernel diagonal was proposed, which gives a generating function for the whole sequence of heat invariants, after a suitable regularization and analytical continuation.
Abstract: We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We argue that the obtained formal solution correctly reproduces the exact heat kernel diagonal after a suitable regularization and analytical continuation.
TL;DR: In this article, a complete description of three-dimensional Lorentzian manifolds with commuting curvature operators is given at the algebraic level and results are obtained at the differentiable setting for manifolds which additionally are assumed to be locally symmetric or homogeneous.
Abstract: Three-dimensional Lorentzian manifolds with commuting curvature operators are studied. A complete description is given at the algebraic level. Consequences are obtained at the differentiable setting for manifolds which additionally are assumed to be locally symmetric or homogeneous.
TL;DR: In this article, the 1st jet space of a relativistic particle is considered and the geometric conditions by which the Lorentz metric and a connection of the phase space yield contact and Jacobi structures are investigated.
Abstract: This paper is concerned with basic geometric properties of the phase space of a classical general relativistic particle, regarded as the 1st jet space of motions, i.e. as the 1st jet space of timelike 1-dimensional submanifolds of spacetime. This setting allows us to skip constraints. Our main goal is to determine the geometric conditions by which the Lorentz metric and a connection of the phase space yield contact and Jacobi structures. In particular, we specialize these conditions to the cases when the connection of the phase space is generated by the metric and an additional tensor. Indeed, the case generated by the metric and the electromagnetic field is included, as well.
TL;DR: In this paper, the Schwarzschild solution in EHG can be coordinate-transformed such that it is also a solution in CG, and the resulting solutions are non-perturbative in the asymptotic regime, and are reproduced in the massless limit of the massive solutions, hence no van Dam-Veltman-Zakharov discontinuity.
Abstract: In gravitational Higgs mechanism graviton components acquire mass via spontaneous diffeomorphism breaking by scalar vacuum expectation values. We point out that in the massless limit the resulting theory is not Einstein–Hilbert gravity (EHG) but constrained gravity (CG). Consequently, massive solutions in the massless limit must be compared to those in CG (as opposed to EHG). We discuss spherically symmetric solutions in this context. The Schwarzschild solution in EHG can be coordinate-transformed such that it is also a solution in CG. The resulting solutions are non-perturbative in the asymptotic regime, and are reproduced in the massless limit of asymptotic massive solutions, hence no van Dam–Veltman–Zakharov discontinuity. We point out that higher curvature terms must be included to obtain non-singular spherically symmetric massive solutions and discuss a suitable framework.
TL;DR: The parastatistics algebra is a superalgebra with even parafermi and odd parabose creation and annihilation operators as discussed by the authors, and it is shown that the states in this algebra are in one-to-one correspondence with the Super Semistandard Young Tableaux (SSYT) subject to further constraints.
Abstract: The parastatistics algebra is a superalgebra with (even) parafermi and (odd) parabose creation and annihilation operators. The states in the parastatistics Fock-like space are shown to be in one-to-one correspondence with the Super Semistandard Young Tableaux (SSYT) subject to further constraints. The deformation of the parastatistics algebra gives rise to a monoidal structure on the SSYT which is a super-counterpart of the plactic monoid.
TL;DR: In this paper, it was shown that a structure Lie group K of a principal bundle P → X is reducible to its closed subgroup H iff there exists a global section of the quotient bundle P/K → X.
Abstract: By virtue of the well-known theorem, a structure Lie group K of a principal bundle P → X is reducible to its closed subgroup H iff there exists a global section of the quotient bundle P/K → X. In gauge theory, such sections are treated as Higgs fields, exemplified by pseudo-Riemannian metrics on a base manifold X. Under some conditions, this theorem is extended to principal superbundles in the category of G-supermanifolds. Given a G-supermanifold M and a graded frame superbundle over M with a structure general linear supergroup, a reduction of this structure supergroup to an orthogonal-symplectic supersubgroup is associated to a supermetric on a G-supermanifold M.
TL;DR: In this paper, a number of additional constraints of algebraic character have been discussed, some of which have to be considered as part of the core of the BPHZ framework.
Abstract: Renormalized perturbation theory a la BPHZ can be founded on causality as analyzed by Epstein and Glaser in the seventies. Here, we list and discuss a number of additional constraints of algebraic character some of which have to be considered as parts of the core of the BPHZ framework.
TL;DR: In this paper, the authors considered an action of the group of curves in GL(2,ℝ) on the set of linear systems and therefore on the sets of Schrodinger equations in full similarity with the action on the Riccati equations considered in previous articles, and showed that both situations appear, e.g., in the usual problem of partner Hamiltonians in quantum mechanics.
Abstract: We consider an action of the group of curves in GL(2,ℝ) on the set of linear systems and therefore on the set of Schrodinger equations in full similarity with the action of the group of curves in SL(2,ℝ) on the set of Riccati equations considered in previous articles. We also consider the transformations defined by a first-order differential expression which carry solutions of a Schrodinger equation into solutions of another one. We find then two non-trivial situations: transformations which can be described by the previous transformation group, generalizing previous work by us, and transformations which are singular. We show that both situations appear, e.g., in the usual problem of partner Hamiltonians in quantum mechanics. We show that the difference Backlund algorithm, both in the finite and confluent versions, can be understood in terms of the above mentioned transformation group, the case of two exactly equal factorization energies being an instance of the singular case. We apply the generalized theorem relating three eigenfunctions of three different Hamiltonians to the generation of new potentials with a known (excited state) eigenfunction, starting from potentials of Coulomb, Morse and Rosen–Morse type. The potentials found are new and non-trivial.
TL;DR: Some derivation-based differential calculi which have been used to construct models of noncommutative gauge theories are presented and commented in this paper, and some comparisons between them are made.
Abstract: Some derivation-based differential calculi which have been used to construct models of noncommutative gauge theories are presented and commented. Some comparisons between them are made.
TL;DR: The existence of a modified Cliff(1,1)-structure compatible with an Osserman 0-model of signature (2,2) was shown in this paper, where the authors applied this algebraic result to certain classes of pseudo-Riemannian manifolds of signature 2,2.
Abstract: We show the existence of a modified Cliff(1,1)-structure compatible with an Osserman 0-model of signature (2,2). We then apply this algebraic result to certain classes of pseudo-Riemannian manifolds of signature (2,2). We obtain a new characterization of the Weyl curvature tensor of an (anti-)self-dual manifold and we prove some new results regarding (Jordan) Osserman manifolds.
TL;DR: In this paper, an optimal control problem for a drift-free controllable system on the Lie group SO(4) is discussed and some of its dynamical and geometrical properties are pointed out.
Abstract: An optimal control problem for a drift-free controllable system on the Lie group SO(4) is discussed and some of its dynamical and geometrical properties are pointed out.
TL;DR: In this paper, a basic treatment of lattices Γ in these groups is given, and periodic geodesics in these compact nilmanifolds, obtaining a complete calculation of the period spectrum of certain flat spaces.
Abstract: We give a basic treatment of lattices Γ in these groups. Certain tori TF and TB provide the model fiber and the base for a submersion of Γ\N. This submersion may not be pseudoriemannian in the usual sense, because the tori may be degenerate. We then begin the study of periodic geodesics in these compact nilmanifolds, obtaining a complete calculation of the period spectrum of certain flat spaces.