The Hamiltonian of Einstein affine-metric formulation of General Relativity

TL;DR

This paper demonstrates that the Hamiltonian formulation of Einstein's affine-metric (first order) General Relativity naturally recovers four-diffeomorphism invariance without field redefinitions, contrasting with the second order formulation.

Contribution

It clarifies the gauge invariance structure of the first order Einstein affine-metric formulation and emphasizes the correct algorithm for deriving diffeomorphism invariance.

Findings

First order Hamiltonian leads to natural four-diffeomorphism invariance.

The Castellani algorithm correctly identifies gauge invariance without redefinitions.

Differences in constraint treatment between first and second order formulations are analyzed.

Abstract

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.