Dirac Sources for Nonmetricity and Torsion in Metric-affine Gravity
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.
Cite
@article{arxiv.2601.09013,
title = {Dirac Sources for Nonmetricity and Torsion in Metric-affine Gravity},
author = {James T. Wheeler},
journal= {arXiv preprint arXiv:2601.09013},
year = {2026}
}
Comments
Revised some sentences for readability and eliminated unneeded indices. Added a concluding summery. The central results are unchanged