Standard-model symmetry in complexified spacetime algebra

TL;DR

This paper explores the complexified spacetime algebra's structure, revealing its isomorphism with certain symmetry groups relevant to particle physics, and identifies multiple copies of these groups within the algebra.

Contribution

It demonstrates the isomorphism between subgroups of complexified spacetime algebra and SU(3), and uncovers four distinct copies of the symmetry group within the algebra.

Findings

Subgroups of CSTA are isomorphic to SU(3).

The symmetry group is $U(1) imes SU(2) imes SU(3)$.

There are four distinct copies of this group within CSTA.

Abstract

Complexified spacetime algebra is defined as the geometric (Clifford) algebra of spacetime with complex coefficients, isomorphic $\mathcal{G}_{1,4}$. By resorting to matrix representation by means of Dirac-Pauli gamma matrices, the paper demonstrates isomorphism between subgroups of CSTA and SU(3). It is shown that the symmetry group of those subgroups is indeed $U(1) \otimes SU(2) \otimes SU(3)$ and that there are 4 distinct copies of this group within CSTA.