Fermions without Vierbeins in Curved Space-Time

TL;DR

This paper presents a novel formulation of spinor fields in curved space-time that avoids vierbeins, using a general approach with a broader local symmetry group, leading to new insights into Dirac equations in curved geometries.

Contribution

It introduces a vierbein-free method for formulating spinor fields in curved space-time with invariance under the full GL(4,C) group, expanding the symmetry considerations beyond local Lorentz invariance.

Findings

Derived the Dirac equation without vierbeins.

Established invariance under GL(4,C) transformations.

Computed the energy-momentum tensor for the new formulation.

Abstract

A general formulation of spinor fields in Riemannian space-time is given without using vierbeins. The space-time dependence of the Dirac matrices required by the anticommutation relation {\gamma_{\mu},\gamma_{\nu}}=2g_{\mu\nu} determines the spin connection. The action is invariant under any local spin base transformations in the 32 parameter group Gl(4,c) and not just under local Lorentz transformations. The Dirac equation and the energy-momentum tensor are computed from the action.