Showing papers in "International Journal of Geometric Methods in Modern Physics in 2007"

Introduction to modified gravity and gravitational alternative for dark energy

Abstract: We review various modified gravities considered as gravitational alternative for dark energy. Specifically, we consider the versions of f(R), f(G) or f(R, G) gravity, model with non-linear gravitational coupling or string-inspired model with Gauss-Bonnet-dilaton coupling in the late universe where they lead to cosmic speed-up. It is shown that some of such theories may pass the Solar System tests. On the same time, it is demonstrated that they have quite rich cosmological structure: they may naturally describe the effective (cosmological constant, quintessence or phantom) late-time era with a possible transition from decceleration to acceleration thanks to gravitational terms which increase with scalar curvature decrease. The possibility to explain the coincidence problem as the manifestation of the universe expansion in such models is mentioned. The late (phantom or quintessence) universe filled with dark fluid with inhomogeneous equation of state (where inhomogeneous terms are originated from the modifi...

Fourth order gravity: Equations, history, and applications to cosmology

Abstract: The field equations following from a Lagrangian L(R) will be deduced and solved for special cases. If L is a non-linear function of the curvature scalar, then these equations are of fourth order in the metric. In the introduction, we present the history of these equations beginning with the paper of H. Weyl from 1918, who first discussed them as alternative to Einstein's theory. In the third part, we give details about the cosmic no hair theorem, i.e. the details of how within fourth order gravity with L= R + R2, the inflationary phase of cosmic evolution turns out to be a transient attractor. Finally, the Bicknell theorem, i.e. the conformal relation from fourth order gravity to scalar-tensor theory, will be shortly presented.

Quantum gravity and higher curvature actions

Abstract: Effective equations are often useful to extract physical information from quantum theories without having to face all technical and conceptual difficulties. One can then describe aspects of the quantum system by equations of classical type, which correct the classical equations by modified coefficients and higher derivative terms. In gravity, for instance, one expects terms with higher powers of curvature. Such higher derivative formulations are discussed here with an emphasis on the role of degrees of freedom and on differences between Lagrangian and Hamiltonian treatments. A general scheme is then provided which allows one to compute effective equations perturbatively in a Hamiltonian formalism. Here, one can expand effective equations around any quantum state and not just a perturbative vacuum. This is particularly useful in situations of quantum gravity or cosmology where perturbations only around vacuum states would be too restrictive. The discussion also demonstrates the number of free parameters expected in effective equations, used to determine the physical situation being approximated, as well as the role of classical symmetries such as Lorentz transformation properties in effective equations. An appendix collects information on effective correction terms expected from loop quantum gravity and string theory.

Dark matter and dark energy as a effects of Modified Gravity

Abstract: We explain the effect of dark matter (flat rotation curve) using modified gravitational dynamics. We investigate in this context a low energy limit of generalized general relativity with a nonlinear Lagrangian , where R is the (generalized) Ricci scalar and n is parameter estimated from SNIa data. We estimate parameter β in modified gravitational potential . Then we compare value of β obtained from SNIa data with β parameter evaluated from the best fitted rotation curve. We find β ≃ 0.7 which becomes in good agreement with an observation of spiral galaxies rotation curve. We also find preferred value of Ωm,0 from the combined analysis of supernovae data and baryon oscillation peak. We argue that although amount of "dark energy" (of non-substantial origin) is consistent with SNIa data and flat curves of spiral galaxies are reproduces in the framework of modified Einstein's equation we still need substantial dark matter. For comparison predictions of the model with predictions of the ΛCDM concordance model we apply the Akaike and Bayesian information criteria of model selection.

Parametric nonholonomic frame transforms and exact solutions in gravity

Abstract: A generalized geometric method is developed for constructing exact solutions of gravitational field equations in Einstein theory and generalizations. First, we apply the formalism of nonholonomic frame deformations (formally considered for nonholonomic manifolds and Finsler spaces) when the gravitational field equations transform into systems of nonlinear partial differential equations which can be integrated in general form. The new classes of solutions are defined by generic off-diagonal metrics depending on integration functions on one, two and three (or three and four) variables if we consider four (or five) dimensional spacetimes. Second, we use a general scheme when one (two) parameter families of exact solutions are defined by any source-free solutions of Einstein's equations with one (two) Killing vector field(s). A successive iteration procedure results in new classes of solutions characterized by an infinite number of parameters for a non-Abelian group involving arbitrary functions on one variable. Five classes of exact off-diagonal solutions are constructed in vacuum Einstein and in string gravity describing solitonic pp-wave interactions. We explore possible physical consequences of such solutions derived from primary Schwarzschild or pp-wave metrics.

Quantization of the Riemann Zeta-Function and Cosmology

Abstract: Quantization of the Riemann zeta-function is proposed. We treat the Riemann zeta-function as a symbol of a pseudodifferential operator and study the corresponding classical and quantum field theories. This approach is motivated by the theory of p-adic strings and by recent works on stringy cosmological models. We show that the Lagrangian for the zeta-function field is equivalent to the sum of the Klein–Gordon Lagrangians with masses defined by the zeros of the Riemann zeta-function. Quantization of the mathematics of Fermat–Wiles and the Langlands program is indicated. The Beilinson conjectures on the values of L-functions of motives are interpreted as dealing with the cosmological constant problem. Possible cosmological applications of the zeta-function field theory are discussed.

"Minimal geometric data" approach to Dirac algebra, spinor groups and field theories

Abstract: The first three sections contain an updated, not-so-short account of a partly original approach to spinor geometry and field theories introduced by Jadczyk and myself [3–5]; it is based on an intrinsic treatment of 2-spinor geometry in which the needed background structures do not need to be assumed, but rather arise naturally from a unique geometric datum: a vector bundle with complex 2-dimensional fibers over a real 4-dimensional manifold. The following two sections deal with Dirac algebra and 4-spinor groups in terms of two spinors, showing various aspects of spinor geometry from a different perspective. The last section examines particle momenta in 2-spinor terms and the bundle structure of 4-spinor space over momentum space.

On the relation between mass and charge: a pure geometric approach

Abstract: A new solution to the field equations of the generalized field theory, constructed by Mikhail and Wanas in 1977, has been obtained. The geometric structure used, in the present application, is an absolute parallelism (AP)-space with spherical symmetry (type FIGI). The solution obtained represents a generalized field outside a charged massive central body. Two schemes have been used to get the physical meaning of the solution: The first is related to the metric of the Riemannian space associated with the AP-structure. The second is connected to a covariant scheme known as Type Analysis. It is shown that the dependence on both schemes for interpreting the results obtained, is better than the dependence on the metric of the Riemannian space associated with the AP-structure. In general, if we consider the solution obtained as representing a geometric model for an elementary charged particle, then the results of the present work can be summarized in the following points. (i) It is shown that the mass of the particle is made up of two contributions: the first is the gravitational contribution, and the second is the contribution due to the existence of charge. (ii) The model allows for the existence of a charged particle whose mass is completely electromagnetic in origin. (iii) The model prevents the existence of a charged massless particle. (iv) The electromagnetic contribution, to the mass, is independent of the sign of the electric charge. (v) It is shown that the mass of the electron (or a positron) is purely made of its charge.

Dark energy models toward observational tests and data

Abstract: A huge amount of good quality astrophysical data converges towards the picture of a spatially flat universe undergoing the today observed phase of accelerated expansion. This new observational trend is commonly addressed as Precision Cosmology. Despite of the quality of astrophysical surveys, the nature of dark energy dominating the matter-energy content of the universe is still unknown and a lot of different scenarios are viable candidates to explain cosmic acceleration. Methods to test these cosmological models are based on distance measurements and lookback time toward astronomical objects used as standard candles. I discuss the characterizing parameters and constraints of three different classes of dark energy models pointing out the related degeneracy problem which is the signal that more data at low (z ~ 0 ÷ 1), medium (1 < z < 10) and high (10 < z < 1000) redshift are needed to definitively select realistic models.

The Chrono-geometrical structure of special and general relativity: A Re-visitation of canonical geometrodynamics

Abstract: A modern re-visitation of the consequences of the lack of an intrinsic notion of instantaneous 3-space in relativistic theories leads to a reformulation of their kinematical basis emphasizing the role of non-inertial frames centered on an arbitrary accelerated observer. In special relativity the exigence of predictability implies the adoption of the 3 + 1 point of view, which leads to a well posed initial value problem for field equations in a framework where the change of the convention of synchronization of distant clocks is realized by means of a gauge transformation. This point of view is also at the heart of the canonical approach to metric and tetrad gravity in globally hyperbolic asymptotically flat space-times, where the use of Shanmugadhasan canonical transformations allows the separation of the physical degrees of freedom of the gravitational field (the tidal effects) from the arbitrary gauge variables. Since a global vision of the equivalence principle implies that only global non-inertial frames can exist in general relativity, the gauge variables are naturally interpreted as generalized relativistic inertial effects, which have to be fixed to get a deterministic evolution in a given non-inertial frame. As a consequence, in each Einstein's space-time in this class the whole chrono-geometrical structure, including also the clock synchronization convention, is dynamically determined and a new approach to the Hole Argument leads to the conclusion that "gravitational field" and "space-time" are two faces of the same entity. This view allows to get a classical scenario for the unification of the four interactions in a scheme suited to the description of the solar system or our galaxy with a deparametrization to special relativity and the subsequent possibility to take the non-relativistic limit.

Nonlinear realization of the local conform-affine symmetry group for gravity in the composite fiber bundle formalism

Abstract: A gauge theory of gravity based on a nonlinear realization (NLR) of the local Conform-Affine (CA) group of symmetry transformations is presented. The coframe fields and gauge connections of the theory are obtained. The tetrads and Lorentz group metric are used to induce a spacetime metric. The inhomogenously transforming (under the Lorentz group) connection coefficients serve as gravitational gauge potentials used to define covariant derivatives accommodating minimal coupling of matter and gauge fields. On the other hand, the tensor valued connection forms serve as auxiliary dynamical fields associated with the dilation, special conformal and deformational (shear) degrees of freedom inherent in the bundle manifold. The bundle curvature of the theory is determined. Boundary topological invariants are constructed. They serve as a prototype (source free) gravitational Lagrangian. The Bianchi identities, covariant field equations and gauge currents are obtained.

Reduction procedures in classical and quantum mechanics

Abstract: We present, in a pedagogical style, many instances of reduction procedures appearing in a variety of physical situations, both classical and quantum. We concentrate on the essential aspects of any reduction procedure, both in the algebraic and geometrical setting, elucidating the analogies and the differences between the classical and the quantum situations.

a CHERN-SIMONS e8 Gauge Theory of Gravity in D = 15, Grand Unification and Generalized Gravity in Clifford Spaces

Abstract: A novel Chern–Simons E8 gauge theory of gravity in D = 15 based on an octicE8 invariant expression in D = 16 (recently constructed by Cederwall and Palmkvist) is developed. A grand unification model of gravity with the other forces is very plausible within the framework of a supersymmetric extension (to incorporate spacetime fermions) of this Chern–Simons E8 gauge theory. We review the construction showing why the ordinary 11D Chern–Simons gravity theory (based on the Anti de Sitter group) can be embedded into a Clifford-algebra valued gauge theory and that an E8 Yang–Mills field theory is a small sector of a Clifford (16) algebra gauge theory. An E8 gauge bundle formulation was instrumental in understanding the topological part of the 11-dim M-theory partition function. The nature of this 11-dim E8 gauge theory remains unknown. We hope that the Chern–Simons E8 gauge theory of gravity in D = 15 advanced in this work may shed some light into solving this problem after a dimensional reduction.

Hopf algebras in dynamical systems theory

Abstract: The theory of exact and of approximate solutions for non-autonomous linear differential equations forms a wide field with strong ties to physics and applied problems. This paper is meant as a stepping stone for an exploration of this long-established theme, through the tinted glasses of a (Hopf and Rota–Baxter) algebraic point of view. By reviewing, reformulating and strengthening known results, we give evidence for the claim that the use of Hopf algebra allows for a refined analysis of differential equations. We revisit the renowned Campbell–Baker–Hausdorff–Dynkin formula by the modern approach involving Lie idempotents. Approximate solutions to differential equations involve, on the one hand, series of iterated integrals solving the corresponding integral equations; on the other hand, exponential solutions. Equating those solutions yields identities among products of iterated Riemann integrals. Now, the Riemann integral satisfies the integration-by-parts rule with the Leibniz rule for derivations as its partner; and skewderivations generalize derivations. Thus, we seek an algebraic theory of integration, with the Rota–Baxter relation replacing the classical rule. The methods to deal with noncommutativity are especially highlighted. We find new identities, allowing for an extensive embedding of Dyson–Chen series of time- or path-ordered products (of generalized integration operators); of the corresponding Magnus expansion; and of their relations, into the unified algebraic setting of Rota–Baxter maps and their inverse skewderivations. This picture clarifies the approximate solutions to generalized integral equations corresponding to non-autonomous linear (skew) differential equations.

Isotropic subbundles of tm ⊕ t*m

Abstract: We define integrable, big-isotropic structures on a manifold M as subbundles E ⊆ TM ⊕ T*M that are isotropic with respect to the natural, neutral metric (pairing) g of TM ⊕ T*M and are closed by Courant brackets (this also implies that [E, E⊥g] ⊆ E⊥g). We give the interpretation of such a structure by objects of M, we discuss the local geometry of the structure and we give a reduction theorem.

Pseudo-Riemannian JACOBI-VIDEV Manifolds

Abstract: We exhibit several families of Jacobi–Videv pseudo-Riemannian manifolds which are not Einstein. We also exhibit Jacobi–Videv algebraic curvature tensors where the Ricci operator defines an almost complex structure.

Graded Infinite Order Jet Manifolds

Abstract: The relevant material on differential calculus on graded infinite order jet manifolds and its cohomology is summarized. This mathematics provides the adequate formulation of Lagrangian theories of even and odd variables on smooth manifolds in terms of the Grassmann-graded variational bicomplex.

Conformal structures and twistors in the paravector model of spacetime

Abstract: Some properties of the Clifford algebras and are presented, and three isomorphisms between the Dirac–Clifford algebra and are exhibited, in order to construct conformal maps and twistors, using the paravector model of spacetime. The isomorphism between the twistor space inner product isometry group SU(2,2) and the group $pin+(2,4) is also investigated, in the light of a suitable isomorphism between and . After reviewing the conformal spacetime structure, conformal maps are described in Minkowski spacetime as the twisted adjoint representation of $pin+(2,4), acting on paravectors. Twistors are then presented via the paravector model of Clifford algebras and related to conformal maps in the Clifford algebra over the Lorentzian ℝ4,1 spacetime. We construct twistors in Minkowski spacetime as algebraic spinors associated with the Dirac–Clifford algebra using one lower spacetime dimension than standard Clifford algebra formulations, since for this purpose, the Clifford algebra over ℝ4,1 is also used to describe conformal maps, instead of ℝ2,4. Our formalism sheds some new light on the use of the paravector model and generalizations.

Quantum hall droplets on disc and effective Wess-Zumino-Witten action for edge states

Abstract: We algebraically analyze the quantum Hall effect of a system of particles living on the disc B1 in the presence of a uniform magnetic field B. For this, we identify the non-compact disc with the coset space SU(1,1)/U(1). This allows us to use the geometric quantization in order to get the wavefunctions as the Wigner -functions satisfying a suitable constraint. We show that the corresponding Hamiltonian coincides with the Maass Laplacian. Restricting to the lowest Landau level, we introduce the noncommutative geometry through the star product. Also we discuss the state density behavior as well as the excitation potential of the quantum Hall droplet. We show that the edge excitations are described by an effective Wess–Zumino–Witten action for a strong magnetic field and discuss their nature. We finally show that LLL wavefunctions are intelligent states.

The Kerr Theorem and Multi-Particle KERR-SCHILD Solutions

Abstract: We discuss an extended version of the Kerr theorem which allows one to construct the exact solutions of the Einstein–Maxwell field equations from a holomorphic generating function F of twistor variables. The exact multi-particle Kerr–Schild solutions are obtained from generating function of the form , where Fi are partial generating functions for the spinning particles i = 1 ⋯ k. Solutions have an unusual multi-sheeted structure. Twistorial structures of the ith and jth particles do not feel each other, forming a type of its internal space. Gravitational and electromagnetic interaction of the particles occurs via the light-like singular twistor lines. As a result, each particle turns out to be "dressed" by singular pp-strings connecting it to other particles. We argue that this solution may have a relation to quantum theory.

Geometric quantization of Hamiltonian actions of Lie algebroids and Lie groupoids

Abstract: We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose, we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kahler quantization suitable for this setting. We proceed by defining a Marsden–Weinstein quotient for our setting and prove a "quantization commutes with reduction" theorem. We explain how our geometric quantization procedure relates to a possible orbit method for Lie groupoids. Our theory encompasses the geometric quantization of symplectic manifolds, Hamiltonian Lie algebra actions, actions of bundles of Lie groups, and foliations, as well as some general constructions from differential geometry.

More on the isomorphism su(2) ⊗ su(2) ≅ so(4)

Abstract: In this paper, we revisit the isomorphism SU(2) ⊗ SU(2) ≅ SO(4) to apply to some subjects in Quantum Computation and Mathematical Physics. The unitary matrix Q by Makhlin giving the isomorphism as an adjoint action is studied and generalized from a different point of view. Some problems are also presented. In particular, the homogeneous manifold SU(2n)/SO(2n) which characterizes entanglements in the case of n = 2 is studied, and a clear-cut calculation of the universal Yang–Mills action in (hep-th/0602204) is given for the abelian case.

Topology change and signature change in non-linear first-order gravity

Abstract: We show that different topologies of a space-time manifold and different signatures of its metric can be encompassed into a single Lagrangian formalism, provided one adopts the first-order (Palatini) formulation and relies on nonlinear Lagrangians, that were earlier shown to produce, in the generic case, universality of Einstein field equations and of Komar's energy-momentum complex as well. An example in Relativistic Cosmology is provided.

Geometric Algebras and Extensors

Abstract: This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce the key algebraic tools for the development of our program, namely the euclidean geometrical algebra of multivectors and the theory of its deformations leading to metric geometric algebras and some special types of extensors. Those tools permit obtaining, the remarkable golden formula relating calculations in with easier ones in (e.g. a noticeable relation between the Hodge star operators associated to G and GE). Several useful examples are worked in details for the purpose of transmitting the "tricks of the trade".

Relativity Groupoid Instead of Relativity Group

Abstract: In 1908, Minkowski [13] used space-like binary velocity-field of a medium, relative to an observer. In 1974, Hestenes introduced, within a Clifford algebra, an axiomatic binary relative velocity as a Minkowski bivector [7, 8]. We propose to consider binary relative velocity as a traceless nilpotent endomorphism in an operator algebra. Any concept of a binary axiomatic relative velocity made possible the replacement of the Lorentz relativity group by the relativity groupoid. The relativity groupoid is a category of massive bodies in mutual relative motions, where a binary relative velocity is interpreted as a categorical morphism with the associative addition. This associative addition is to be contrasted with non-associative addition of (ternary) relative velocities in isometric special relativity (loop structure). We consider an algebra of many time-plus-space splits, as an operator algebra generated by idempotents. The kinematics of relativity groupoid is ruled by associative Frobenius operator algebra, whereas the dynamics of categorical relativity needs the non-associative Frolicher–Richardson operator algebra. The Lorentz covariance is the cornerstone of physical theory. Observer-dependence within relativity groupoid, and the Lorentz-covariance within the Lorentz relativity group, are different concepts. Laws of Physics could be observer-free, rather than Lorentz-invariant.

Smooth changes of frame and prolongations of vector fields

Abstract: We discuss how prolongations of vector fields, and hence symmetries of differential equations, are affected by changes of the reference frame in the dependent variables space depending on independent variables. We devise a relation with the recently developed theory of μ-prolongations of vector fields and μ-symmetries of differential equations.

QUADRATIC AND HOMOGENEOUS HAMILTON-POISSON SYSTEM ON (so(3))

Abstract: We study some geometrical and dynamical properties of a quadratic and homogeneous Hamilton–Poisson system defined on the dual of the Lie algebra so(3) with its minus-Lie-Poisson structure.

Multivector and extensor fields on smooth manifolds

Abstract: The main objective of this paper (second in a series of four) is to show how the Clifford and extensor algebras methods introduced in a previous paper of the series are indeed powerful tools for performing sophisticated calculations appearing in the study of the differential geometry of a n-dimensional manifold M of arbitrary topology, supporting a metric field g (of given signature (p,q)) and an arbitrary connection ∇. Specifically, we deal here with the theory of multivector and extensor fields on M. Our approach does not suffer the problems of earlier attempts which are restricted to vector manifolds. It is based on the existence of canonical algebraic structures over the canonical (vector) space associated to a local chart (Uo, ϕo) of a given atlas of M. The key concepts of a-directional ordinary derivatives of multivector and extensor fields are defined and their properties studied. Also, we recall the Lie algebra of smooth vector fields in our formalism, the concept of Hestenes derivatives and present some illustrative applications.

Levi–civita connection on almost commutative algebras

Abstract: Recently we introduced a new definition of metrics on almost commutative algebras. In this paper, we propose a coherent notion of compatible linear connection with respect to any almost commutative tensor and show that to every metric there corresponds a unique torsion-free compatible connection. This connection is called the Levi–Civita connection of the associated metric.

From peierls brackets to a generalized moyal bracket for type-i gauge theories

Abstract: In the space-of-histories approach to gauge fields and their quantization, the Maxwell, Yang–Mills and gravitational field are well known to share the property of being type-I theories, i.e. Lie brackets of the vector fields which leave the action functional invariant are linear combinations of such vector fields, with coefficients of linear combination given by structure constants. The corresponding gauge-field operator in the functional integral for the in-out amplitude is an invertible second-order differential operator. For such an operator, we consider advanced and retarded Green functions giving rise to a Peierls bracket among group-invariant functionals. Our Peierls bracket is a Poisson bracket on the space of all group-invariant functionals in two cases only: either the gauge-fixing is arbitrary but the gauge fields lie on the dynamical sub-space; or the gauge-fixing is a linear functional of gauge fields, which are generic points of the space of histories. In both cases, the resulting Peierls bracket is proved to be gauge-invariant by exploiting the manifestly covariant formalism. Moreover, on quantization, a gauge-invariant Moyal bracket is defined that reduces to iħ times the Peierls bracket to lowest order in ħ.