A note on spinor fields in spherical symmetry

TL;DR

This paper demonstrates that under spherical symmetry, Dirac equations have no solutions if the spinor fields are required to share the same symmetries as the spacetime, using a polar reformulation approach.

Contribution

It introduces a polar reformulation to analyze spinor fields and proves the non-existence of symmetric solutions in spherical symmetry.

Findings

No solutions of Dirac equations with symmetric spinors in spherical symmetry.

Polar reformulation provides a new perspective on spinor symmetry constraints.

Abstract

By employing the polar re-formulation, we show that there are no solutions of the Dirac equations in spherical symmetry when the spinor is required to satisfy the same symmetries as the space-time via the Lie derivative.