Algebraic and Dirac-Hestenes Spinors and Spinor Fields

TL;DR

This paper critically examines the mathematical foundations of Hestenes' approach to Dirac spinor fields, proposing precise definitions for algebraic and Dirac-Hestenes spinor fields to clarify conceptual misunderstandings.

Contribution

It introduces rigorous definitions for algebraic and Dirac-Hestenes spinor fields as equivalence classes, addressing inconsistencies in previous formulations and setting the stage for a comprehensive geometric framework.

Findings

Identifies conflicts in Hestenes' original spinor definition with Fierz identities.

Proposes new definitions for algebraic and Dirac-Hestenes spinor fields.

Provides detailed Clifford algebra results to support the theoretical framework.

Abstract

Almost all presentations of Dirac theory in first or second quantization in Physics (and Mathematics) textbooks make use of covariant Dirac spinor fields. An exception is the presentation of that theory (first quantization) offered originally by Hestenes and now used by many authors. There, a new concept of spinor field (as a sum of non homogeneous even multivectors fields) is used. However, a carefully analysis (detailed below) shows that the original Hestenes definition cannot be correct since it conflicts with the meaning of the Fierz identities. In this paper we start a program dedicated to the examination of the mathematical and physical basis for a comprehensive definition of the objects used by Hestenes. In order to do that we give a preliminary definition of algebraic spinor fields (ASF) and Dirac-Hestenes spinor fields (DHSF) on Minkowski spacetime as some equivalence classes of well defined pairs of mathematical objects, one of the members of the pair being an even nonhomegeneous differential form. The necessity of our definitions are shown by a carefull analysis of possible formulations of Dirac theory and the meaning of the set of Fierz identities. We believe that the present paper clarifies some misunderstandings (past and recent) appearing on the literature of the subject. It will be followed by a sequel paper where definitive definitions of ASF and DHSF are given as appropriate sections a vector bundle called the left spin-Clifford bundle. The present paper contains also Appendices (A-E) which exhibits a truly useful collection of results concerning the theory of Clifford algebras (including many `tricks of the trade') necessary for the intelligibility of the text.