Left-Ideals, Dirac Fermions and SU(2)-Flavour

TL;DR

This paper explores the use of spacetime algebra's left ideals to describe fermions, linking geometric algebra concepts with Dirac formalism and clarifying the structure of chiral fermions.

Contribution

It introduces a matrix representation with bivector insertions that elucidates the relationship between spacetime algebra ideals and Dirac fermion components.

Findings

Left ideals describe massless chiral fermions as rotors.

The approach clarifies how Dirac matrices relate to spacetime algebra basis vectors.

Identification of fermion chiral components with algebra elements.

Abstract

In this paper I reconsider the use of the left ideals of the even-grade subalgebra of spacetime algebra to describe fermionic excitations. When interpreted as rotors the general elements of an even-grade left-ideal describe massless particles in chiral flavour doublets. To study the application of these ideas to the standard Dirac formalism I construct a $2 \times 2$-matrix representation with bivector insertions for the Dirac algebra. This algebra has four ideals, and this approach clarifies how the identification of Dirac $\g_{\mu}$-matrices with orthonormal basisvectors ${\bf e}_{\nu}$ annihilates half of the ideals. For one possible choice of this mapping the remaining ideals the chiral left- and righthanded components of the fermion coincide with the even- and odd elements of spacetime algebra.