Dirac equation in non-Riemannian geometries

TL;DR

This paper derives the Dirac equation within non-Riemannian geometries incorporating torsion and non-metricity, utilizing vielbein and Clifford algebra formalisms to achieve a clear, invariant formulation that clarifies previous ambiguities.

Contribution

It introduces a simplified, invariant derivation of the Dirac equation in geometries with torsion and non-metricity, enhancing understanding of these complex structures.

Findings

Derived a simple form of the Dirac equation in non-Riemannian geometries

Used index-free formalism for invariant object construction

Clarified previously obscure details in the literature

Abstract

We present the Dirac equation in a geometry with torsion and non-metricity balancing generality and simplicity as much as possible. In doing so, we use the vielbein formalism and the Clifford algebra. We also use an index-free formalism which allows us to construct objects that are totally invariant. It turns out that the previous apparatuses not only make possible a simple deduction of the Dirac equation but also allow us to exhibit some details that is generally obscure in the literature.