Scale invariant Einstein-Cartan theory in three dimensions

TL;DR

This paper revisits the Einstein-Cartan theory in three dimensions by modifying the covariant derivative for spinors, reformulating it in Riemannian spacetime, and analyzing the implications of curvature and torsion.

Contribution

It introduces a modified covariant derivative for spinors in Einstein-Cartan theory and reformulates the theory in Riemannian spacetime with a self-consistent approach.

Findings

Reformulation of Einstein-Cartan theory in Riemannian spacetime.

Analytical solution for the affine connection.

Discussion of curvature and torsion effects.

Abstract

We retreat the well-known Einstein-Cartan theory by slightly modifying the covariant derivative of spinor field by investigating double cover of the Lorentz group. We first write the Lagrangian consisting of the Einstein-Hilbert term, Dirac term and a scalar field term in a non-Riemannian spacetime with curvature and torsion. Then by solving the affine connection analytically we reformulate the theory in the Riemannian spacetime in a self-consistent way. Finally we discuss our results and give future perspectives on the subject.