Canonical analysis of Holst action without second-class constraints

TL;DR

This paper performs a canonical analysis of the Holst action for general relativity without second-class constraints, deriving covariant variables and exploring new canonical transformations that connect different formulations.

Contribution

It introduces a method to analyze the Holst action without second-class constraints and identifies new covariant canonical variables related to existing formalisms.

Findings

Derivation of phase space with only first-class constraints

Construction of new covariant canonical variables

Connection between covariant variables and Ashtekar-Barbero formalism

Abstract

We perform the canonical analysis of the Holst action for general relativity with a cosmological constant without introducing second-class constraints. Our approach consists in identifying the dynamical and nondynamical parts of the involved variables from the very outset. After integrating out the nondynamical variables associated with the connection, we obtain the description of phase space in terms of manifestly $SO(3,1)$ [or $SO(4)$, depending on the signature] covariant canonical variables and first-class constraints only. We impose the time gauge on them and show that the Ashtekar-Barbero formulation of general relativity emerges. Later, we discuss a family of canonical transformations that allows us to construct new $SO(3,1)$ [or $SO(4)$] covariant canonical variables for the phase space of the theory and compare them with the ones already reported in the literature, pointing out the presence of a set of canonical variables not considered before. Finally, we resort to the time gauge again and find that the theory, when written in terms of the new canonical variables, either collapses to the $SO(3)$ ADM formalism or to the Ashtekar-Barbero formalism with a rescaled Immirzi parameter.