Complex Geometry and Dirac Equation

TL;DR

This paper explores how complex geometry influences the Dirac equation within Clifford algebra, comparing two geometric interpretations of the imaginary unit and enabling translation between standard and geometric algebra formulations.

Contribution

It introduces a formalism that connects standard Dirac theory with its geometric algebra version, analyzing two complex geometries and their implications for the imaginary unit.

Findings

Two complex geometries, 23 and 1, are identified within multivector algebra.

A set of rules for translating between standard and geometric algebra formulations is developed.

Discussion on the double geometric interpretation of the imaginary unit i is provided.

Abstract

Complex geometry represents a fundamental ingredient in the formulation of the Dirac equation by the Clifford algebra. The choice of appropriate complex geometries is strictly related to the geometric interpretation of the complex imaginary unit $i=\sqrt{-1}$. We discuss {\em two} possibilities which appear in the multivector algebra approach: the $\sigma_{123}$ and $\sigma_{21}$ complex geometries. Our formalism permits to perform a set of rules which allows an immediate translation between the complex standard Dirac theory and its version within geometric algebra. The problem concerning a double geometric interpretation for the complex imaginary unit $i=\sqrt{-1}$ is also discussed.