Fermions in the Ashtekar-Barbero connection formalism for arbitrary values of the Immirzi parameter

The Ashtekar-Barbero-Immirzi formulation of general relativity is extended to include spinor matter fields. Our formulation applies to generic values of the Immirzi parameter and reduces to the Ashtekar-Romano-Tate approach when the Immirzi parameter is taken equal to the imaginary unit. The dynamics of the gravity-fermions coupled system is described by the Holst plus Dirac action with a nonminimal coupling term. The nonminimal interaction together with the Holst modification to the Hilbert-Palatini action reconstruct the Nieh-Yan invariant, so that the effective action coming out is the one of Einstein-Cartan theory with a typical Fermi-like interaction term: in spite of the presence of spinor matter fields, the Immirzi parameter plays no role in the classical effective dynamics and results to be only a multiplicative factor in front of a total divergence. We reduce the total action of the theory to the sum of dynamically independent Ashtekar-Romano-Tate actions for self and anti-self-dual connections, with different weights depending on the Immirzi parameter. This allows to calculate the constraints of the complete theory in a simple way, it is only necessary to realize that the Barbero-Immirzi connection is a weighted sum of the self and anti-self-dual Ashtekar connections. Finally the obtained constraints for the separated action result to be polynomial in terms of the self and anti-self-dual connections, this could have implications in the inclusion of spinor matter in the framework of nonperturbative quantum gravity.