Barbero's formulation from a $BF$-type action with the Immirzi parameter

TL;DR

This paper derives the Ashtekar-Barbero variables for general relativity starting from a constrained real BF-type action with the Immirzi parameter, explicitly solving second-class constraints and using the time gauge.

Contribution

It provides a derivation of Ashtekar-Barbero variables directly from a BF-type action including the Immirzi parameter, without relying on tetrad formalism.

Findings

Explicit solution of second-class constraints

Derivation of Ashtekar-Barbero variables from BF formalism

Variables expressed solely in terms of B-fields

Abstract

Starting from a constrained real $BF$-type action for general relativity that includes both the Immirzi parameter and the cosmological constant, we obtain the Ashtekar-Barbero variables used in the canonical approach to the quantization of the gravitational field. This is accomplished by explicitly solving the second-class constraints resulting from the Hamiltonian analysis of the considered action, and later imposing the time gauge. All throughout this work the tetrad formalism is left aside, obtaining the Ashtekar-Barbero variables entirely in terms of the $B$-fields that define the action.