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arXiv · 2601.09013

Dirac Sources for Nonmetricity and Torsion in Metric-affine Gravity

James T. Wheeler
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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.

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James T. Wheeler. 2026-01-13. Dirac Sources for Nonmetricity and Torsion in Metric-affine Gravity. https://arxiv.org/abs/2601.09013

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