Dotted and Undotted Algebraic Spinor Fields in General Relativity

TL;DR

This paper explores the algebraic structure of dotted and undotted spinor fields in General Relativity using Clifford algebra, revealing the physical significance of certain derivative rules and discussing implications for unified field theories.

Contribution

It introduces a Clifford algebra framework for dotted and undotted spinor fields, clarifies the meaning of derivative rules, and examines their role in unified theories of gravity and electromagnetism.

Findings

Certain ad hoc derivative rules have important physical meaning.

Clifford algebra provides a clearer understanding of spinor field derivatives.

Implications for unified theories involving spinor and paravector fields.

Abstract

We investigate using Clifford algebra methods the theory of algebraic dotted and undotted spinor fields over a Lorentzian spacetime and their realizations as matrix spinor fields, which are the usual dotted and undotted two component spinor fields. We found that some ad hoc rules postulated for the covariant derivatives of Pauli sigma matrices and also for the Dirac gamma matrices in General Relativity cover important physical meaning, which is not apparent in the usual matrix presentation of the theory of two components dotted and undotted spinor fields. We also discuss some issues related to the the previous one and which appear in a proposed "unified" theory of gravitation and electromagnetism which use two components dotted and undotted spinor fields and also paravector fields, which are particular sections of the even subundle of the Clifford bundle of spacetime.