A note on the Holst action, the time gauge, and the Barbero-Immirzi parameter

TL;DR

This paper reviews the Hamiltonian analysis of the Holst action in the time gauge, clarifying the role of the Barbero-Immirzi parameter and proposing a 3D gravity formulation to explore its quantum significance.

Contribution

It provides a detailed Hamiltonian analysis of the Holst action, highlighting conditions for the Barbero-Immirzi parameter's classical irrelevance and introducing a 3D gravity model with explicit parameter dependence.

Findings

Identifies covariant torsion components linked to the Barbero-Immirzi parameter.

Shows conditions under which the classical theory is independent of the Barbero-Immirzi parameter.

Proposes a 3D gravity formulation to study the parameter's quantum role.

Abstract

In this note, we review the canonical analysis of the Holst action in the time gauge, with a special emphasis on the Hamiltonian equations of motion and the fixation of the Lagrange multipliers. This enables us to identify at the Hamiltonian level the various components of the covariant torsion tensor, which have to be vanishing in order for the classical theory not to depend upon the Barbero-Immirzi parameter. We also introduce a formulation of three-dimensional gravity with an explicit phase space dependency on the Barbero-Immirzi parameter as a potential way to investigate its fate and relevance in the quantum theory.