Self-Dual Action for Fermionic Fields and Gravitation

TL;DR

This paper explores the self-dual Einstein-Dirac theory, establishing its equivalence to the Einstein-Cartan-Sciama-Kibble-Dirac theory and highlighting the role of torsion in the equations of motion.

Contribution

It generalizes the Jacobson-Smolin proof to include fermionic fields, demonstrating the equivalence of self-dual and traditional gravitational actions with torsion.

Findings

Proves equivalence between self-dual Einstein-Dirac and Einstein-Cartan theories.

Shows torsion plays a key role in the self-dual framework.

Extends the understanding of fermionic fields in gravitational theories.

Abstract

This paper studies the self-dual Einstein-Dirac theory. A generalization is obtained of the Jacobson-Smolin proof of the equivalence between the self-dual and Palatini purely gravitational actions. Hence one proves equivalence of self-dual Einstein-Dirac theory to the Einstein-Cartan-Sciama-Kibble-Dirac theory. The Bianchi symmetry of the curvature, core of the proof, now contains a non-vanishing torsion. Thus, in the self-dual framework, the extra terms entering the equations of motion with respect to the standard Einstein-Dirac field equations, are neatly associated with torsion.