Octonions, E6, and Particle Physics

TL;DR

This paper explores the mathematical structure of octonions and the exceptional Jordan algebra, examining their connection to the Lie group E6 and proposing a symmetry-breaking scenario that models properties of leptons and potentially quarks.

Contribution

It introduces a novel approach linking octonionic algebra and E6 symmetry to particle physics, especially in modeling leptons and quarks.

Findings

Properties of octonions and the exceptional Jordan algebra are reviewed.

A specific real form of E6 preserving the Jordan algebra is analyzed.

A symmetry-breaking scenario within E6 is proposed to describe lepton properties.

Abstract

In 1934, Jordan et al. gave a necessary algebraic condition, the Jordan identity, for a sensible theory of quantum mechanics. All but one of the algebras that satisfy this condition can be described by Hermitian matrices over the complexes or quaternions. The remaining, exceptional Jordan algebra can be described by 3x3 Hermitian matrices over the octonions. We first review properties of the octonions and the exceptional Jordan algebra, including our previous work on the octonionic Jordan eigenvalue problem. We then examine a particular real, noncompact form of the Lie group E6, which preserves determinants in the exceptional Jordan algebra. Finally, we describe a possible symmetry-breaking scenario within E6: first choose one of the octonionic directions to be special, then choose one of the 2x2 submatrices inside the 3x3 matrices to be special. Making only these two choices, we are able to describe many properties of leptons in a natural way. We further speculate on the ways in which quarks might be similarly encoded.