Concept of Lie Derivative of Spinor Fields. A Geometric Motivated Approach

TL;DR

This paper introduces a geometrically motivated definition of the Lie derivative for spinor fields using Clifford algebra formalism, ensuring compatibility with Lorentzian structures and providing clear formulas for applications.

Contribution

It presents a new, geometrically motivated definition of the Lie derivative of spinor fields within Clifford bundle formalism, aligning with Lorentzian geometry and offering explicit formulas.

Findings

The spinor Lie derivative preserves the metric g.

Formulas for the Lie derivative of Clifford and spinor fields are explicitly derived.

Comparison with existing definitions shows advantages of the new approach.

Abstract

In this paper using the Clifford bundle (Cl(M,g)) and spin-Clifford bundle (Cl_{Spin_{1,3}^{e}}(M,g)) formalism, which permit to give a meaningfull representative of a Dirac-Hestenes spinor field (even section of Cl_{Spin_{1,3}^{e}}(M,g)) in the Clifford bundle , we give a geometrical motivated definition for the Lie derivative of spinor fields in a Lorentzian structure (M,g) where M is a manifold such that dimM =4, g is Lorentzian of signature (1,3). Our Lie derivative, called the spinor Lie derivative (and denoted {\pounds}_{{\xi}}) is given by nice formulas when applied to Clifford and spinor fields, and moreoverl {\pounds}_{{\xi}}g=0 for any vector field {\xi}. We compare our definitions and results with the many others appearing in literature on the subject.