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

TL;DR

The paper investigates the Okubo (Okubonion) algebra $\mathcal{O}$ as a novel algebraic framework for modeling QCD, contrasting it with the conventional octonion algebra $\mathbb{O}$. It shows that $\mathbb{O}$ and $\mathcal{O}$ yield different, inequivalent $\text{SU}(3)$ subgroups of $\text{Spin}(8)$—namely $\text{SU}(3)_{\mathbb{O}}$ and $\text{SU}(3)_{\mathcal{O}}$—which do not share a common $\text{SU}(2)$ substructure, ruling out a unified Magic Star interpretation for both. The authors propose interpreting $\mathcal{O}$ as the gluon sector, with $\mathcal{O}$ furnishing the adjoint $\mathbf{8}$ of $\text{SU}(3)_{\mathcal{O}}$, while octonions continue to inform quark representations via $\text{SU}(3)_{\mathbb{O}}$; the non‑unital and non‑alternative nature of $\mathcal{O}$ may connect to non‑perturbative QCD features. The work emphasizes that $\text{Aut}(\mathcal{O})=\text{SU}(3)_{\mathcal{O}}$ is smaller than $\text{G}_{2(-14)}=\text{Aut}(\mathbb{O})$, suggesting a complementary, potentially more economical algebraic route to modeling strong interactions, while acknowledging the conjectural status and open questions about spacetime symmetries and Poincaré representations.

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.

Figures (2)

• Figure 1.1: On the left: octonionic multiplication tables for the basis $\left\{ e_{0}=1,e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7}\right\}$. On the right: a mnemonic representation on the Fano plane of the same octonionic multiplication rule with the equivalence with the Dickson notation $\left\{ 1,\text{i},\text{j},\text{k},\text{l},\text{il},\text{jl},\text{kl}\right\}$ according to Co46.
• Figure 2.1: On the left : strong force symmetry pattern, pertaining to the $\boldsymbol{3}$ color states of a quark $q$, and the corresponding $\overline{\boldsymbol{3}}$ states of its anti-quark $\overline{q}$, along with the $\boldsymbol{8}$ gluon color states (with $g^{3}$ and $g^{8}$ coinciding in the center of the diagram, taken from Lisi). On the right : “ Magic Star” projection of the root lattice of $\mathfrak{g}_{2(-14)}$MukaiMagicStar, with the plane of the sheet being defined by the two Cartan generators of $\mathfrak{g}_{2(-14)}$ itself. Despite the fact that the Cartans of $\mathfrak{g}_{2(-14)}$ (on the right) and of $\mathfrak{su}(3)$ (on the left) coincide, the strong force pattern in the l.h.s. cannot be interpreted as the “ Magic Star” projection of $\mathfrak{g}_{2(-14)}$, and vice versa. This is ultimately due to the fact that the Lie groups SU$\left( 3\right) _{\mathcal{O}}$ (pertaining to the l.h.s) and SU$\left( 3\right) _{\mathbb{O}}$ (pertaining to the r.h.s.) are totally different subgroups of SO$\left( \mathbb{O}\right) \simeq$Spin$\left( 8\right) \simeq$SO$\left( \mathcal{O}\right)$; see (\ref{['diff']}), and the discussion at the end of Sec. 4.