Canonical analysis of Holst action without second-class constraints

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.

arXiv commentPaper's title was changed, expanded analysis, notation was changed a bit, added references, corrected typos

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Cited by in corpus (13)

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• Euclidean self-dual gravity: Ashtekar variables without gauge fixing
• Hamiltonian analysis of fermions coupled to the Holst action
• Canonical analysis with no second-class constraints of $BF$ gravity with Immirzi parameter
• Linking the ADM formulation to other Hamiltonian formulations of general relativity
• Straightforward Hamiltonian analysis of $BF$ gravity in $n$ dimensions
• Fermions coupled to the Palatini action in $n$ dimensions
• Barbero-Immirzi connections and how to build them
• Generators of Local Lorentz Transformation in ADM-Vielbein Formalism of Gravitational Relativity
• Canonical analysis of $BF$ gravity in $n$ dimensions