On the Generalized Einstein-Cartan Action with Fermions

TL;DR

This paper demonstrates that a generalized Einstein-Cartan action can be constructed to match the standard Einstein-Cartan theory coupled to fermions without dependence on the Immirzi parameter or torsion, establishing their equivalence.

Contribution

It shows how to derive the effective Einstein-Cartan action with fermions from a generalized Einstein action, eliminating the usual current-current interaction and parameter dependencies.

Findings

Effective Einstein-Cartan action without Immirzi parameter

No dependence on torsion in the effective action

Equivalence between Einstein-Cartan and general relativity with fermions

Abstract

From the freedom exhibited by the generalized Einstein action proposed in [1], we show that we can construct the standard effective Einstein-Cartan action coupled to the fermionic matter without the usual current-current interaction and therefore an effective action which does not depend neither on the Immirzi parameter nor on the torsion. This establishes the equivalence between the Einstein-Cartan theory and the theory of the general relativity minimally coupled to the fermionic matter.