The Generalized Dirac Equation in the Metric Affine Spacetime

TL;DR

This paper derives a comprehensive form of the Dirac equation in metric affine spacetimes with curvature, torsion, and non-metricity, introducing new constraints and effects on spinor mass through two different formulation approaches.

Contribution

It presents the most general Dirac equation in non-Riemannian spacetimes, compares two formulation methods, and uncovers novel constraints and mass shift effects.

Findings

Derived a generalized Dirac equation including all Clifford algebra bases.

Established consistency and constraints between direct and variational approaches.

Discovered terms causing a mass shift depending on spinor handedness.

Abstract

We discuss the most general form of Dirac equation in the non$-$Riemannian spacetimes containing curvature, torsion and non$-$metricity. It includes all bases of the Clifford algebra $cl(1,3)$ within the spinor connection. We adopt two approaches. First, the generalized Dirac equation is directly formulated by applying the minimal coupling prescription to the original Dirac equation. It is referred to as the {\it direct Dirac equation} for seek of clarity and to preserve the tractability. Second, through the application of variational calculation to the original Dirac Lagrangian, the resulting Dirac equation is referred to as the {\it variational Dirac equation}. A consistency crosscheck is performed between these two approaches, leading to novel constraints on the arbitrary coupling constants appearing in the covariant derivative of spinor. Following short analysis on the generalized Dirac Lagrangian, it is observed that two of the novel terms give rise to a shift in the spinor mass by sensing its handedness.