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G. Sardanashvily

Publications and source records attributed to G. Sardanashvily.

At least 163 records · Page 9Linked to original sources
arXiv1997

Background Geometry in Gauge Gravitation Theory

Dirac fermion fields are responsible for spontaneous symmetry breaking in gauge gravitation theory because the spin structure associated with a tetrad field is not preserved under general covariant transformations. Two solutions of this problem can be suggested. (i) There exists the universal spin structure $S\to X$ such that any spin structure $S^h\to X$ associated with a tetrad field $h$ is a subbundle of the bundle $S\to X$. In this model, gravitational fields correspond to different tetrad (or metric) fields. (ii) A background tetrad field $h$ and the associated spin structure $S^h$ are fixed, while gravitational fields are identified with additional tensor fields $q^\la{}_\m$ describing deviations $\wt h^\la_a=q^\la{}_\m h^\m_a$ of $h$. One can think of $\wt h$ as being effective tetrad fields. We show that there exist gauge transformations which keep the background tetrad field $h$ and act on the effective fields by the general covariant transformation law. We come to Logunov's Relativistic Theory of Gravity generalized to dynamic connections and fermion fields.

gr-qc↗
arXiv1997

Universal Spin Structure in Gauge Gravitation Theory

Building on the universal covering group of the general linear group, we introduce the composite spinor bundle whose subbundles are Lorentz spin structures associated with different gravitational fields. General covariant transformations of this composite spinor bundle are canonically defined.

gr-qc↗
arXiv1997

Differential Geometry of Time-Dependent Mechanics

The usual formulations of time-dependent mechanics start from a given splitting $Y=R\times M$ of the coordinate bundle $Y\to R$. From physical viewpoint, this splitting means that a reference frame has been chosen. Obviously, such a splitting is broken under reference frame transformations and time-dependent canonical transformations. Our goal is to formulate time-dependent mechanics in gauge-invariant form, i.e., independently of any reference frame. The main ingredient in this formulation is a connection on the bundle $Y\to R$ which describes an arbitrary reference frame. We emphasize the following peculiarities of this approach to time-dependent mechanics. A phase space does not admit any canonical contact or presymplectic structure which would be preserved under reference frame transformations, whereas the canonical Poisson structure is degenerate. A Hamiltonian fails to be a function on a phase space. In particular, it can not participate in a Poisson bracket so that the evolution equation is not reduced to the Poisson bracket. This fact becomes relevant to the quantization procedure. Hamiltonian and Lagrangian formulations of time-dependent mechanics are not equivalent. A degenerate Lagrangian admits a set of associated Hamiltonians, none of which describes the whole mechanical system given by this Lagrangian.

dg-ga↗
arXiv1996

Energy-Momentum in Gauge Gravitation Theory

Building on the first variational formula of the calculus of variations, one can derive the energy-momentum conservation laws from the condition of the Lie derivative of gravitation Lagrangians along vector fields corresponding to generators of general covariant transformations to be equal to zero. The goal is to construct these vector fields. In gauge gravitation theory, the difficulty arises because of fermion fields. General covariant transformations fail to preserve the Dirac spin structure $S_h$ on a world manifold $X$ which is associated with a certain tetrad field $h$. We introduce the universal Dirac spin structure $S\toΣ\to X$ such that, given a tetrad field $h:X\to Σ$, the restriction of $S$ to $h(X)$ is isomorphic to $S_h$. The canonical lift of vector fields on $X$ onto $S$ is constructed. We discover the corresponding stress-energy-momentum conservation law. The gravitational model in the presence of a background spin structure also is examined.

gr-qc↗
arXiv1995

Stress-Energy-Momentum Tensors in Lagrangian Field Theory. Part 2. Gravitational Superpotential

Our investigation of differential conservation laws in Lagrangian field theory is based on the first variational formula which provides the canonical decomposition of the Lie derivative of a Lagrangian density by a projectable vector field on a bundle (Part 1: gr-qc/9510061). If a Lagrangian density is invariant under a certain class of bundle isomorphisms, its Lie derivative by the associated vector fields vanishes and the corresponding differential conservation laws take place. If these vector fields depend on derivatives of parameters of bundle transformations, the conserved current reduces to a superpotential. This Part of the work is devoted to gravitational superpotentials. The invariance of a gravitational Lagrangian density under general covariant transformations leads to the stress-energy-momentum conservation law where the energy-momentum flow of gravity reduces to the corresponding generalized Komar superpotential. The associated energy-momentum (pseudo) tensor can be defined and calculated on solutions of metric and affine-metric gravitational models.

gr-qc↗
arXiv1995

Stress-Energy-Momentum of Affine-Metric Gravity. Generalized Komar Superportential

In case of the Einstein's gravitation theory and its first order Palatini reformulation, the stress-energy-momentum of gravity has been proved to reduce to the Komar superpotential. We generalize this result to the affine-metric theory of gravity in case of general connections and arbitrary Lagrangian densities invariant under general covariant transformations. In this case, the stress-energy-momentum of gravity comes to the generalized Komar superpotential depending on a Lagrangian density in a precise way.

gr-qc↗
arXiv1995

Stress-Energy-Momentum Tensors in Lagrangian Field Theory. Part 1. Superpotentials

Differential conservation laws in Lagrangian field theory are usually related to symmetries of a Lagrangian density and are obtained if the Lie derivative of a Lagrangian density by a certain class of vector fields on a fiber bundle vanishes. However, only two field models meet this property in fact. In gauge theory of exact internal symmetries, the Lie derivative by vertical vector fields corresponding to gauge transformations is equal to zero. The corresponding Nöether current is reduced to a superpotential that provides invariance of the Nöether conservation law under gauge transformations. In the gravitation theory, we meet the phenomenon of "hidden energy". Only the superpotential part of energy-momentum of gravity and matter is observed when the general covariant transformations are exact. Other parts of energy-momentum display themselves if the invariance under general covariance transformations is broken, e.g., by a background world metric. In this case, the Lie derivatives of Lagrangian densities by vector fields which call into play the stress-energy-momentum tensors fail to be equal to zero in general. We base our analysis of differential conservation laws on the canonical decomposition of the Lie derivative of a Lagrangian density $L$ by a projectable vector field on a bundle and with respect to different Lepagian equivalents of $L$. Different Lepagian equivalents lead to conserved quantities which differ from each other in superpotential terms. We have different stress-energy-momentum tensors depending on different lifts of vector fields on a base onto a bundle. Moreover, different solutions of the same Euler-Lagrange equations may require different energy-momentum tensors. We show that different stress-energy-momentum tensors differ from each other in Nöether currents. As

gr-qc↗
arXiv1995

Fermions in Gravitation Theory

In gravitation theory, a fermion field must be regarded only in a pair with a certain tetrad gravitational field. These pairs can be represented by sections of the composite spinor bundle $S\to\Si\to X^4$ where values of gravitational fields play the role of parameter coordinates, besides the familiar world coordinates.

gr-qc↗
arXiv1995

Stress-Energy-Momentum Tensors in Constraint Field Theories

One has not any conventional energy-momentum conservation law in Lagrangian field theory, but relations involving different stress-energy-momentum tensors associated with different connections. It is not obvious how to choose the true energy-momentum tensor. This problem is solved in the framework of the multimomentum Hamiltonian formalism which provides the adequate description of constraint field systems. The goal is that, for different solutions of the same constraint field model, one should choose different stress-energy-momentum tensors in general. Gauge theory illustrates this result. The stress-energy-momentum tensors of affine-metric gravity are examined.

gr-qc↗
arXiv1995

Composite Spinor Bundles in Gravitation Theory

In gravitation theory, the realistic fermion matter is described by spinor bundles associated with the cotangent bundle of a world manifold $X$. In this case, the Dirac operator can be introduced. There is the 1:1 correspondence between these spinor bundles and the tetrad gravitational fields represented by sections of the quotient $\Si$ of the linear frame bundle over $X$ by the Lorentz group. The key point lies in the fact that different tetrad fields imply nonequivalent representations of cotangent vectors to $X$ by the Dirac's matrices. It follows that a fermion field must be regarded only in a pair with a certain tetrad field. These pairs can be represented by sections of the composite spinor bundle $S\to\Si\to X$ where values of tetrad fields play the role of parameter coordinates, besides the familiar world coordinates.

gr-qc↗
arXiv1995

Energy-Momentum Conservation Laws in Affine-Metric Gravitation Theory

The Lagrangian formulation of field theory does not provide any universal energy-momentum conservation law in order to analize that in gravitation theory. In Lagrangian field theory, we get different identities involving different stress energy-momentum tensors which however are not conserved, otherwise in the covariant multimomentum Hamiltonian formalism. In the framework of this formalism, we have the fundamental identity whose restriction to a constraint space can be treated the energy-momentum transformation law. This identity remains true also for gravity. Thus, the tools are at hand to investigate the energy-momentum conservation laws in gravitation theory. The key point consists in the feature of a metric gravitational field whose canonical momenta on the constraint space are equal to zero.

gr-qc↗
arXiv1994

Differential Geometry of Composite Fibred Manifolds

In classical field theory, the composite fibred manifolds Y -> Z -> X provides the adequate mathematical formulation of gauge models with broken symmetries, e.g., the gauge gravitation theory. This work is devoted to connections on composite fibred manifolds. In particular, we get the horizontal splitting of the vertical tangent bundle of a composite fibred manifold, besides the familiar one of its tangent bundle. This splitting defines the modified covariant differential and implies the special fashion of Lagrangian densities of field models on composite manifolds. The spinor composite bundles are examined.

dg-ga↗
arXiv1994

Energy-Momentum Conservation Laws in Hamiltonian Field Theory

In the Lagrangian field theory, one gets different identities for different stress energy-momentum tensors, e.g., canonical energy-momentum tensors. Moreover, these identities are not conservation laws of the above-mentioned energy-momentum tensors in general. In the framework of the multimomentum Hamiltonian formalism, we have the fundamental identity whose restriction to a constraint space can be treated the energy-momentum conservation law. In standard field models, this appears the metric energy-momentum conservation law.

gr-qc↗
arXiv1994

Five Lectures on the Jet Methods in Field Theory

The fibre bundle formulation of gauge theory is generally accepted. The jet manifold machinery completes this formulation and provides the adequate mathematical description of dynamics of fields represented by sections of fibre bundles. Theory of differential operators, Lagrangian and Hamiltonian formalisms on bundles have utilized widely the language of jet manifolds. Moreover, when not restricted to principal connections, differential geometry also is phrased in jet terms. However, this powerful tool remains almost unknown to physicists. These Lectures give introduction to jet manifolds, Lagrangian and Hamiltonian formalisms in jet manifolds and their application to a number of fundamental field models.

hep-th↗
arXiv1994

Gravity as a Higgs Field. III. Nongravitional Deviations of Gravitational Fields

In Parts I and II of the work (gr-qc/9405013, 9407032), we have shown that gravity is {\it sui generis} a Higgs field corresponding to spontaneous symmetry breaking when the fermion matter admits only the Lorentz subgroup of world symmetries of the geometric arena. From the mathematical viewpoint, the Higgs nature of gravity issues from the fact that different gravitational fields are responsible for nonequivalent representations of cotangent vectors to a world manifold by $γ$-matrices on spinor bundles. It follows that gravitational fields fail to form an affine space modelled on a linear space of deviations of some background field. In other words, even weak gravitational fields do not satisfy the superposition principle and, in particular, can not be quantized by usual methods. At the same time, one can examine superposable deviations $σ$ of a gravitational field $h$ so that $h+σ$ fails to be a gravitational field. These deviations are provided with the adequate mathematical description in the framework of the affine group gauge theory in dislocated manifolds, and their Lagrangian densities differ from the familiar gravitational ones. They make contribution to the standard gravitational effects, e.g., modify Newton's gravitational potential.

gr-qc↗
arXiv1994

Gauge Gravitation Theory. What is the Geometry of the World?

When joined the unified gauge picture of fundamental interactions, the gravitation theory leads to geometry of a space-time which is far from simplicity of pseudo-Riemannian geometry of Einstein's General Relativity. This is geometry of the affine-metric composite dislocated manifolds. The goal is modification of the familiar equations of a gravitational field and entirely the new equations of its deviations. In the present brief, we do not detail the mathematics, but discuss the reasons why it is just this geometry. The major physical underlying reason lies in spontaneous symmetry breaking when the fermion matter admits only the Lorentz subgroup of world symmetries of the geometric arena.

gr-qc↗
arXiv1994

True Functional Integrals in Algebraic Quantum Field Theory

The familiar generating functionals in QFT fail to be true measures since the Lebesgue measure in infinite-dimensional spaces is not defined in general. The problem lies in constructing representations of topological $^*$-algebras of quantum fields which are not normed. We restrict our consideration only to chronological forms on quantum field algebras. In this case, since chronological forms of boson fields are symmetric, the algebra of quantum fields can be replaced with the commutative tenzor algebra of the corresponding infinite-dimensional nuclear space $Φ$. This is the enveloping algebra of the abelian Lie group of translations in $Φ$. The generating functions of unitary representations of this group play the role of Euclidean generating functionals in algebraic QFT. They are the Fourier transforms of measures in the dual $Φ'$ to the space $Φ$. By analogy with the case of boson fields, the corresponding anticommutative algebra of fermion fields is defined to be the algebra of functions taking their values into an infinite Grassman algebra.

hep-th↗