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

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.