Nieh-Yan Invariant and Fermions in Ashtekar-Barbero-Immirzi Formalism

TL;DR

This paper demonstrates that adding a non-minimal term in the Ashtekar-Barbero-Immirzi formalism links the Immirzi parameter to the Nieh-Yan invariant, ensuring classical dynamics remain unaffected by the parameter.

Contribution

It introduces a non-minimal coupling approach that connects the Immirzi parameter to the Nieh-Yan invariant without altering classical Einstein-Cartan dynamics.

Findings

The non-minimal term reconstructs the Nieh-Yan invariant.

The Immirzi parameter does not influence classical dynamics.

Classical equations remain governed by Einstein-Cartan action.

Abstract

In order to introduce an interaction between gravity and fermions in the Ashtekar-Barbero-Immirzi formalism without affecting classical dynamics a non-minimal term is necessary. The non-minimal term together with the Holst modification to the Hilbert-Palatini action reconstruct the Nieh-Yan invariant. As a consequence the Immirzi parameter, differently from the minimal coupling approach, does not affect the classical dynamics, which is described by the Einstein-Cartan action.