Polar form of Dirac fields: implementing symmetries via Lie derivative

TL;DR

This work shows how the polar decomposition of Dirac spinors provides a covariant framework to study the Lie derivative along Killing vectors, separating symmetry requirements on the spinor from those on its bilinears. By expressing the spinor covariant derivative and the Lie derivative in polar form, the authors derive a key scalar condition that determines when strong (spinor) and weak (bilinear) invariance along a Killing vector are equivalent. They prove a no-go theorem: stationary spherical symmetry cannot be imposed on spinor fields via the Lie derivative, independent of tetrad choices or external interactions. The results constrain how spacetime symmetries can be implemented for spinor fields and highlight the role of the tensorial connections $P_{\mu}$ and $F_{ab\mu}$ in encoding geometric and gauge information within spinor dynamics.

Abstract

We consider the Lie derivative along Killing vector fields of the Dirac relativistic spinors: by using the polar decomposition we acquire the mean to study the implementation of symmetries on Dirac fields. Specifically, we will become able to examine under what conditions it is equivalent to impose a symmetry upon a spinor or only upon its observables. For one physical application, we discuss the role of the above analysis for the specific spherical symmetry, obtaining some no-go theorem regarding spinors and discussing the generality of our approach.