Triality, Biquaternion and Vector Representation of the Dirac Equation

TL;DR

This paper explores the triality properties of Dirac spinors, constructs biquaternion algebra, and demonstrates a vector-representation of the Dirac equation, revealing its self-dual nature and analyzing the massive term using non-integrable phases.

Contribution

It introduces a vector-representation of Dirac spinors and proves the self-duality of the massive Dirac equation within this framework.

Findings

Existence of a vector-representation for Dirac spinors

Self-duality of the massive Dirac equation in this representation

Application of non-integrable phases to analyze the massive term

Abstract

The triality properties of Dirac spinors are studied, including a construction of the algebra of (complexified) biquaternion. It is proved that there exists a vector-representation of Dirac spinors. The massive Dirac equation in the vector-representation is actually self-dual. The Dirac's idea of non-integrable phases is used to study the behavior of massive term.