Concept of Lie Derivative of Spinor Fields A Geometric Motivated Approach

Abstract

In this paper using the Clifford bundle (${\mathcal{C}\ell(M,\mathtt{g})}$) and spin-Clifford bundle (${\mathcal{C}\ell_{\mathrm{Spin}_{1,3}^{e}} (M,\mathtt{g})}$) formalism, which allow to give a meaningful representative of a Dirac-Hestenes spinor field (even section of ${\mathcal{C}\ell_{\mathrm{Spin}_{1,3}^{e}}(M,\mathtt{g})}$) 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 dim M = 4, g is Lorentzian of signature (1, 3). Our Lie derivative, called the spinor Lie derivative (and denoted ${\overset{s}{\pounds}_{\boldsymbol{\xi}}}$) is given by nice formulas when applied to Clifford and spinor fields, and moreover ${\overset{s}{\pounds }_{{\xi}}{\boldsymbol {g}}=0}$ for any vector field ${\boldsymbol {\xi}}$ . We compare our definitions and results with the many others appearing in literature on the subject.