Canonical analysis ofn-dimensional Palatini action without second-class constraints

Abstract

We carry out the canonical analysis of the n-dimensional Palatini action with or without a cosmological constant (n ≥ 3) introducing neither second-class constraints nor resorting to any gauge fixing. This is accomplished by providing an expression for the spatial components of the connection that allows us to isolate the nondynamical variables present among them, which can later be eliminated from the action by using their own equation of motion. As a result, we obtain the description of the phase space of general relativity in terms of manifestly SO(n − 1, 1) [or SO(n)] covariant variables subject to first-class constraints only, with no second-class constraints arising during the process. Afterwards, we perform, at the covariant level, a canonical transformation to a set of variables in terms of which the above constraints take a simpler form. Finally, we impose the time gauge and make contact with the SO(n − 1) ADM formalism.I.

Lagrangian constraint analysis of first-order classical field theories with an application to gravity

“…As such, we shall study a manifestly first-order formulation of gravity based on N ¼ dðd þ 1Þ 2 =2 number of a priori independent degrees of freedom. Even though alternative manifestly first-order formulations do exist, such as the tetradic-Palatini action (for example, see [38] and its recent canonical study [39]), inconvenient subtleties to our aims arise in those frameworks due to their geometric construction. For instance, unlike the metric, vielbeine are not required to be invertible.…”

Straightforward Hamiltonian analysis of $BF$ gravity in $n$ dimensions

“…iii) We have shown that the BF -type action principles ( 2) and (32) have the same Hamiltonian formulation (involving only first-class constraints) that the n-dimensional Palatini action has [14,13]. This needs to be so, and not otherwise, because all three actions describe Einstein's general relativity.…”

“…Substituting them into the action (33) leads to the same intermediate action (10), which following again the approach of Ref. [13] produces the canonical formulation embodied in (26) and thus to the Hamiltonian formulation (30).…”

“…Now, following the same approach of Ref. [13], we realize that the term involving ∂ t A aIJ in (10) can be written as…”

“…One relevant aspect of the Hamiltonian formulations ( 26) and ( 30) is that they were achieved without involving the orthonormal frame of 1-forms (vielbein) e I . Moreover, the Hamiltonian formulations ( 26) and (30) are the same as the ones we obtain from the n-dimensional Palatini action by doing a suitable parametrization of the vielbein and the connection, and after integrating out the auxiliary fields involved [13] (see also [14] for a different approach).…”