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

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.

22 pages. Published version. Choice of gauge at the begining of section II.B. clarified. Published in Gen. Rel. Grav. (2013)

References in corpus (8)

• LQG vertex with finite Immirzi parameter
• Flipped spinfoam vertex and loop gravity
• The Barbero-Immirzi Parameter as a Scalar Field: K-Inflation from Loop Quantum Gravity?
• Interaction of the Barbero--Immirzi Field with Matter and Pseudo-Scalar Perturbations
• Canonical Gravity with Fermions
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