  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 non-minimal coupling term. The non-minimal 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 non-perturbative quantum
gravity.
