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

Abstract

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.

Polysymplectic formulation for BF gravity with Immirzi parameter

“…Now, we need to apply the consistency conditions on the primary constraint (n − 1)-forms in order to obtain a complete set of constraints that characterizes the CMPR model. In particular, the consistency conditions (19) applied to the constraint (n − 1)-forms (25a) and (25b) give rise to a set of relations for the Lagrange multipliers χ and λ,…”

“…where λ H ab ν and χ H a ν stand for Lagrange multiplier 1-forms enforcing the primary constraint (n−1)-forms (49). One may directly show that the consistency conditions (19) applied to the primary constraint (n − 1)-forms (49)…”

“…The CMPR action is also useful to realize equivalent formulations to some relevant BF gravity models commonly implemented within the spin foam formalism, as described in detail in [17]. At the classical level, the CMPR action has been analyzed from different perspectives, including the Lagrangian formalism in [14] and the Dirac-Hamiltonian approach in [18] (see also [19,20] for more details on the CMPR model and BF theories), which strongly relies on the Hamiltonian analysis applied to the Plebanski action developed in [21,22]. From our perspective, given the physical relevance and constraint structure of the model described by the CMPR action, it is a suitable candidate for its analysis under the geometric covariant Hamiltonian approach for field theories known as the polysymplectic formalism.…”

Canonical analysis with no second-class constraints of BF gravity with Immirzi parameter

“…However, the analysis introduced second-class constraints, which were later solved while nonmanifestly preserving Lorentz invariance in Ref. [10], obtaining a canonical formulation that leads to the Ashtekar-Barbero variables [11] in the time gauge. The structure of the canonical theory containing second- * merced@fis.cinvestav.mx † mcelada@matmor.unam.mx class constraints is actually quite similar to that obtained [12] for the Holst action [13], and in fact, by getting rid of those constraints in a manifestly Lorentzcovariant fashion, the results of both theories agree [14].…”

Faddeev–Jackiw quantization of four dimensional BF theory

“…In consideration with the commented above, either the classical or the quantum study of BF theories and their close relation with GR is at the present a frontier subject of study [22,23].…”