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

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dc.creatorKiriushcheva, N.
dc.creatorKuzmin, S. V.
dc.creatorMcKeon, D. G. C.
dc.date2006-09-29
dc.date.accessioned2026-07-07T11:27:47Z
dc.date.available2026-07-07T11:27:47Z
dc.descriptionUsing 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.
dc.description21 pages, published in Int. J. Mod. Phys. A, Vol. 21, 3401-3420 (2006)
dc.identifierhttps://arxiv.org/abs/hep-th/0609219
dc.identifierhttp://arxiv.org/abs/hep-th/0609219
dc.identifierInt.J.Mod.Phys.A21:3401-3420,2006
dc.identifierdoi:10.1142/S0217751X06029545
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/196247
dc.subjectHigh Energy Physics - Theory
dc.titleA Canonical Analysis of the Einstein-Hilbert Action in First Order Form
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