Physics with non-unital algebras? An invitation to the Okubo algebra

TL;DR

This paper explores the potential role of the Okubo algebra, a non-unital algebra, in quantum chromodynamics, proposing it could model gluons and relate to key phenomena like confinement and asymptotic freedom.

Contribution

It introduces the Okubo algebra as a novel mathematical framework for modeling gluons in QCD, highlighting its differences from octonions and discussing its possible physical implications.

Findings

Okubo algebra is distinct from octonions and related to different SU(3) groups.

Okubonions may represent gluons in QCD.

Potential links between Okubonions and phenomena like confinement and asymptotic freedom.

Abstract

This paper presents some preliminary discussion on the possible relevance of the Okubonions, i.e. the real Okubo algebra $\mathcal{O}$, in quantum chromodynamics (QCD). The Okubo algebra lacks a unit element and sits in the adjoint representation of its automorphism group $\text{SU}_{\mathcal{O}}$, thus being fundamentally different from the better-known octonions $\mathbb{O}$. While these latter may represent quarks (and color singlets), the Okubonions are conjectured to represent the gluons, i.e. the gauge bosons of the QCD $\text{SU}(3)$ color symmetry. However, it is shown that the $\text{SU}(3)$ groups pertaining to Okubonions and octonions are distinct and inequivalent subgroups of $Spin(8)$ that share no common $\text{SU}(2)$ subgroup. The unusual properties of Okubonions may be related to peculiar QCD phenomena like asymptotic freedom and color confinement, though the actual mechanisms remain to be investigated.