Non-minimally coupled Dirac equation with torsion: Poincar\'e gauge theory of gravity with even and odd parity terms

TL;DR

This paper explores a non-minimally coupled Dirac field within Poincaré gauge gravity, analyzing torsion and curvature effects, including odd parity terms, and decomposes the equations into Einsteinian and post-Riemannian components.

Contribution

It introduces a comprehensive analysis of Dirac fields coupled to Poincaré gauge gravity with torsion, including odd parity terms, and derives explicit solutions for torsion and decomposes the field equations.

Findings

Derived the second field equation with respect to torsion.

Decomposed gravity and Dirac equations into Einsteinian and post-Riemannian parts.

Solved for torsion explicitly in the weak gravity approximation.

Abstract

We take a Dirac field non-minimally coupled to the gravitational field within the framework of the Poincar\'e gauge theory of gravity with torsion and curvature. We study the subcase of "weak" gravity, that is, the gravitational Lagrangian depends only linearly on the curvature and quadratically on the torsion. We include all pieces in curvature and torsion that are of odd parity. The second field equation of gravity is derived by varying the Lorentz connection. We solve it with respect to the torsion and decompose the first field equation of gravity and the Dirac equation into Einsteinian pieces and post-Riemannian terms.