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S. V. Kuzmin

Publications and source records attributed to S. V. Kuzmin.

32 records · Page 2Linked to original sources
arXiv2009

The Hamiltonian of Einstein affine-metric formulation of General Relativity

It is shown that the Hamiltonian of the Einstein affine-metric (first order) formulation of General Relativity (GR) leads to a constraint structure that allows the restoration of its unique gauge invariance, four-diffeomorphism, without the need of any field dependent redefinition of gauge parameters as is the case for the second order formulation. In the second order formulation of ADM gravity the need for such a redefinition is the result of the non-canonical change of variables [arXiv: 0809.0097]. For the first order formulation, the necessity of such a redefinition "to correspond to diffeomorphism invariance" (reported by Ghalati [arXiv: 0901.3344]) is just an artifact of using the Henneaux-Teitelboim-Zanelli ansatz [Nucl. Phys. B 332 (1990) 169], which is sensitive to the choice of linear combination of tertiary constraints. This ansatz cannot be used as an algorithm for finding a gauge invariance, which is a unique property of a physical system, and it should not be affected by different choices of linear combinations of non-primary first class constraints. The algorithm of Castellani [Ann. Phys. 143 (1982) 357] is free from such a deficiency and it leads directly to four-diffeomorphism invariance for first, as well as for second order Hamiltonian formulations of GR. The distinct role of primary first class constraints, the effect of considering different linear combinations of constraints, the canonical transformations of phase-space variables, and their interplay are discussed in some detail for Hamiltonians of the second and first order formulations of metric GR. The first order formulation of Einstein-Cartan theory, which is the classical background of Loop Quantum Gravity, is also discussed.

gr-qc↗
arXiv2009

The Hamiltonian formulation of N-bein, Einstein-Cartan, gravity in any dimension: the Progress Report (Extended version of a talk given on CAIMS-2009, June 11-14, London, Canada)

The Hamiltonian formulation of N-bein, Einstein-Cartan, gravity, using its first order form in any dimension higher than two, is analyzed. This Hamiltonian formulation allows to explicitly show where peculiarities of three dimensional case (\textit{A.M.Frolov et al, 0902.0856 [gr-qc]}) occur and make a conjecture, based on presented in this report results, that there is one general for \textit{all} dimensions characteristic of N-bein formulation of gravity: after elimination of second class constraints the algebra of Poisson brackets among remaining first class secondary constraints is the Poincare algebra and in all dimensions N-bein, Cartan-Einstein, gravity \textit{is the Poincare gauge theory}. The gauge symmetry corresponding to the algebra of first class constraints has two parameters - rotational (Lorentz) and translational. Translational invariance is common to all dimensions but some terms in general expressions for gauge transformations of N-beins and connections are zero in a particular, three dimensional, case. The proof of our conjecture is outlined in detail. Some straightforward but tedious calculations remain to be completed to call our conjecture - a theorem and will be reported later.

gr-qc↗
arXiv2008

Diffeomorphism Invariance in the Hamiltonian formulation of General Relativity

It is shown that when the Einstein-Hilbert Lagrangian is considered without any non-covariant modifications or change of variables, its Hamiltonian formulation leads to results consistent with principles of General Relativity. The first-class constraints of such a Hamiltonian formulation, with the metric tensor taken as a canonical variable, allow one to derive the generator of gauge transformations, which directly leads to diffeomorphism invariance. The given Hamiltonian formulation preserves general covariance of the transformations derivable from it. This characteristic should be used as the crucial consistency requirement that must be met by any Hamiltonian formulation of General Relativity.

gr-qc↗
arXiv2007

Two-dimensional metric and tetrad gravities as constrained second order systems

Using the Gitman-Lyakhovich-Tyutin generalization of the Ostrogradsky method for analyzing singular systems, we consider the Hamiltonian formulation of metric and tetrad gravities in two-dimensional Riemannian spacetime treating them as constrained higher-derivative theories. The algebraic structure of the Poisson brackets of the constraints and the corresponding gauge transformations are investigated in both cases.

hep-th↗
arXiv2006

A Canonical Analysis of the Einstein-Hilbert Action in First Order Form

Using the Dirac constraint formalism, we examine the canonical structure of the Einstein-Hilbert action $S_d = \frac{1}{16πG} \int d^dx \sqrt{-g} R$, treating the metric $g_{αβ}$ and the symmetric affine connection $Γ_{μν}^λ$ as independent variables. For $d > 2$ tertiary constraints naturally arise; if these are all first class, there are $d(d-3)$ independent variables in phase space, the same number that a symmetric tensor gauge field $ϕ_{μν}$ possesses. If $d = 2$, the Hamiltonian becomes a linear combination of first class constraints obeying an SO(2,1) algebra. These constraints ensure that there are no independent degrees of freedom. The transformation associated with the first class constraints is not a diffeomorphism when $d = 2$; it is characterized by a symmetric matrix $ξ_{μν}$. We also show that the canonical analysis is different if $h^{αβ} = \sqrt{-g} g^{αβ}$ is used in place of $g^{αβ}$ as a dynamical variable when $d = 2$, as in $d$ dimensions, $\det h^{αβ} = - (\sqrt{-g})^{d-2}$. A comparison with the formalism used in the ADM analysis of the Einstein-Hilbert action in first order form is made by applying this approach in the two dimensional case with $h^{αβ}$ and $Γ_{μν}^λ$ taken to be independent variables.

hep-th↗
arXiv2006

Comments on ``A note on first-order formalism and odd-derivative actions'' by S. Deser

We argue that the obstacles to having a first-order formalism for odd-derivative actions presented in a pedagogical note by Deser are based on examples which are not first-order forms of the original actions. The general derivation of an equivalent first-order form of the original second-order action is illustrated using the example of topologically massive electrodynamics (TME). The correct first-order formulations of the TME model keep intact the gauge invariance presented in its second-order form demonstrating that the gauge invariance is not lost in the Ostrogradsky process.

hep-th↗
arXiv2006

Comments on ``The Einstein-Hilbert Lagrangian Density in a 2-dimensional Spacetime is an Exact Differential'' by R. da Rocha and W.A. Rodrigues, Jr

We argue that the recent result of da Rocha and Rodrigues that in two dimensional spacetime the Lagrangian of tetrad gravity is an exact differential [1], despite the claim of the authors, neither proves the Jackiw conjecture [2], nor contradicts to the conclusion of [3]. This demonstrates that the tetrad formulation is different from the metric formulation of the Einstein-Hilbert action.

hep-th↗
arXiv2005

On Hamiltonian formulation of the Einstein-Hilbert action in two dimensions

It is shown that the well-known triviality of the Einstein field equations in two dimensions is not a sufficient condition for the Einstein-Hilbert action to be a total divergence, if the general covariance is to be preserved, that is, a coordinate system is not fixed. Consequently, a Hamiltonian formulation is possible without any modification of the two dimensional Einstein-Hilbert action. We find the resulting constraints and the corresponding gauge transfromations of the metric tensor.

hep-th↗
arXiv2005

Dirac and Lagrangian reductions in the canonical approach to the first order form of the Einstein-Hilbert action

It is shown that the Lagrangian reduction, in which solutions of equations of motion that do not involve time derivatives are used to eliminate variables, leads to results quite different from the standard Dirac treatment of the first order form of the Einstein-Hilbert action when the equations of motion correspond to the first class constraints. A form of the first order formulation of the Einstein-Hilbert action which is more suitable for the Dirac approach to constrained systems is presented. The Dirac and reduced approaches are compared and contrasted. This general discussion is illustrated by a simple model in which all constraints and the gauge transformations which correspond to first class constraints are completely worked out using both methods in order to demonstrate explicitly their differences. These results show an inconsistency in the previous treatment of the first order Einstein-Hilbert action which is likely responsible for problems with its canonical quantization.

hep-th↗
arXiv2005

Peculiarities of the Canonical Analysis of the First Order Form of the Einstein-Hilbert Action in Two Dimensions in Terms of the Metric Tensor or the Metric Density

The peculiarities of doing a canonical analysis of the first order formulation of the Einstein-Hilbert action in terms of either the metric tensor $g^{αβ}$ or the metric density $h^{αβ}= \sqrt{-g}g^{αβ}$ along with the affine connection are discussed. It is shown that the difference between using $g^{αβ}$ as opposed to $h^{αβ}$ appears only in two spacetime dimensions. Despite there being a different number of constraints in these two approaches, both formulations result in there being a local Poisson brackets algebra of constraints with field independent structure constants, closed off shell generators of gauge transformations and off shell invariance of the action. The formulation in terms of the metric tensor is analyzed in detail and compared with earlier results obtained using the metric density. The gauge transformations, obtained from the full set of first class constraints, are different from a diffeomorphism transformation in both cases.

hep-th↗
arXiv2005

A Canonical Approach to the Einstein-Hilbert Action in Two Spacetime Dimensions

The canonical structure of the Einstein-Hilbert Lagrange density $L=\sqrt{-g}R$ is examined in two spacetime dimensions, using the metric density $h^{μν}\equiv \sqrt{-g}g^{μν}$ and symmetric affine connection $Γ_{σβ}^λ$ as dynamical variables. The Hamiltonian reduces to a linear combination of three first class constraints with a local SO(2,1) algebra. The first class constraints are used to find a generator of gauge transformations that has a closed off-shell algebra and which leaves the Lagrangian and $\det (h^{μν})$ invariant. These transformations are distinct from diffeomorphism invariance, and are gauge transformations characterized by a symmetric matrix $ζ_{μν}$.

hep-th↗
arXiv2003

Influence of mass difference on dynamic properties of isotope mixtures

We present the results of a Molecular Dynamics computer simulations of a two component isotope mixture of Lennard-Jones particles, monodisperse in size but different in masses, at a fixed average density and temperature. We study changes in properties that result from mass heterogeneity, by measuring the pair distribution function, diffusion coefficient, velocity autocorrelation function, non-Gaussian and isotope effect parameters, as functions of the degree of mass difference. Our results show that if static properties are not influenced by a variation of mass variation, the dynamic properties are significantly affected and even exhibit the presence of ``critical'' values in the mass difference. We also demonstrate that our model gives a simple contra example to a recently proposed a universal scaling law for atomic diffusion in condensed matter [M. Dzugudov, Nature, v.381, 137 (1996)].

cond-mat↗