Dirac Sources for Nonmetricity and Torsion in Metric-affine Gravity

TL;DR

This paper introduces a novel way to couple Dirac spinor fields to metric-affine gravity, resulting in new sources for torsion and nonmetricity by expanding the gauge algebra in Clifford algebra bases.

Contribution

It presents a method to relate the Lie algebra gl(4) to Clifford algebras, enabling Dirac fields to generate torsion and nonmetricity in metric-affine gravity.

Findings

New Dirac sources for torsion and nonmetricity identified.

Clifford algebra basis used to relate gauge algebra to spinor fields.

Method provides a clear coupling mechanism between Dirac fields and geometric structures.

Abstract

Metric-affine gravity (GL(4) gauge theory) in 4-dimensions is coupled to a spacetime Dirac source field using the isomorphisms of the Lie algebra gl(4) to the Clifford algebras Cl(3,1) and Cl(2,2). A simple transformation relates the generators of Cl(3,1) to a real representation of Cl(2,2), while the real representation of Cl(2,2) serves directly as a basis for the Lie algebra gl(4). Therefore, although GL(4) does not contain a spinor representation of the Lorentz group, expanding its Lie algebra in the Cl(2,2) basis gives a Clifford valued connection with well-defined coupling to Dirac spinors. Variation of the expansion coefficients gives new Dirac sources for both torsion and nonmetricity, separated by identifying the so(3,1) basis within the gl(4) basis.