Spinor Lie derivatives and Fermion stress-energies

TL;DR

This paper develops a comprehensive framework for defining Lie derivatives of spinor fields, resolving conflicts in the literature, and deriving Fermion stress-energies from the principle of general covariance.

Contribution

It clarifies the definition of spinor Lie derivatives by analyzing the Infeld-van der Waerden symbol, leading to a consistent derivation of Fermion stress-energies.

Findings

Resolved conflicts in spinor Lie derivative definitions.

Derived Fermion stress-energy tensor from covariance principles.

Enhanced understanding of spinor geometry in gravitational contexts.

Abstract

Stress-energies for Fermi fields are derived from the principle of general covariance. This is done by developing a notion of Lie derivatives of spinors along arbitrary vector fields. A substantial theory of such derivatives was first introduced by Kosmann; here I show how an apparent conflict in the literature on this is due to a difference in the definitions of spinors, and that tracking the Lie derivative of the Infeld-van der Waerden symbol, as well as the spinor fields under consideration, gives a fuller picture of the geometry and leads to the Fermion stress-energy. The differences in the definitions of spinors do not affect the results here, but could matter in certain quantum-gravity programs and for spinor transformations under discrete symmetries.