Dirac Sources for Nonmetricity and Torsion in Metric-affine Gravity

TL;DR

This work demonstrates that Dirac spinors can source both torsion and nonmetricity in metric-affine gravity by exploiting the isomorphisms $\mathfrak{gl}(4,\mathbb{R})\simeq \mathfrak{Cl}(3,1)$ and $\mathfrak{Cl}(2,2)$. By expanding the GL(4) connection in a Clifford basis and separating the $SO(3,1)$ (torsion) part from the nonmetric (nonmetricity) part, the authors derive explicit Dirac bilinear sources for all components of torsion and nonmetricity, organizing them via a real $Cl(2,2)$ basis and a Lorentz subalgebra projection. The resulting field equations couple the gravitational sector to 24 torsion components and 40 nonmetricity components through Dirac currents, with vacuum solutions recovering vanishing torsion and specific nonmetricity reductions. Special-case analyses illustrate particle–antiparticle differences in the nonmetric sector, highlighting the physical richness of Dirac couplings in metric-affine gravity and providing a concrete framework for exploring hypermomentum sourced by spinor fields.

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.