Fermions in Ashtekar-Barbero Connections Formalism for Arbitrary Values of the Immirzi Parameter

TL;DR

This paper extends the Ashtekar-Barbero formulation of General Relativity to include fermions for any Immirzi parameter, showing the classical dynamics are unaffected by the Immirzi parameter and simplifying the constraints for quantum gravity applications.

Contribution

It introduces a formulation that incorporates spinor matter fields for arbitrary Immirzi parameters, and simplifies the constraints by expressing them as sums of independent self and anti-self dual actions.

Findings

Immirzi parameter does not affect classical fermion dynamics.

Constraints become polynomial in self and anti-self dual connections.

The action reduces to a sum of independent Ashtekar-Romano-Tate actions.

Abstract

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.