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

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.

• The paper explores using the non-unital, non-associative Okubo algebra (O) as an alternative mathematical structure for modeling Quantum Chromodynamics (QCD).
• It highlights that the Okubo algebra, residing in the adjoint representation (8) of SU(3), matches the exact gauge symmetry of QCD, suggesting a role in understanding gluonic interactions.
• Unlike octonions, the Okubo framework provides a minimalistic SU(3) structure without requiring extensions to larger symmetry groups, potentially offering insights into non-perturbative QCD phenomena.

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

The paper explores the potential application of the Okubo algebra, specifically the Okubonions, within the framework of quantum chromodynamics (QCD). Quantum chromodynamics, a cornerstone of the Standard Model (SM) of particle physics, describes the interactions between quarks and gluons. The gauge group of QCD is the special unitary group SU(3), responsible for color charge, an intrinsic property of quarks and gluons.

Okubo Algebra: A Distinct Structural Framework

The Okubo algebra, denoted as $\mathcal{O}$, is a non-unital, non-associative, and division algebra that stands in contrast to the well-known octonions $\mathbb{O}$, another eight-dimensional, non-associative division algebra. While octonions are alternative and unital, Okubonions lack a unit element and are non-alternative, placing them in a unique category of mathematical structures. The automorphism group of Okubonions is SU(3), and they reside in the adjoint representation $\mathbf{8}$ of this group, suggesting a potential role in modeling gluons, the force carriers in QCD.

The Role of Algebras in Physics

The paper considers the question of whether the unique algebraic properties of Okubonions can explain some of the peculiar phenomena in QCD, such as asymptotic freedom and color confinement, which are fundamental characteristics of the strong interaction. Particularly, while quarks are associated with the octonions, which possess unital symmetry and mirror the complexities of fermionic color charges in quarks, the intriguing properties of Okubonions might play a crucial role in understanding gluonic interactions.

Implications and Distinctions

Okubonions inhabit the adjoint representation of SU(3) and are consequently linked to the eight gluons facilitating QCD interactions. Unlike the octonions, whose symmetries align with G2(-14) and thus admit potential embedding into broader symmetry frameworks, the Okubo algebra presents a minimalistic structure via its SU(3) automorphic properties—matching the exact gauge symmetry of QCD without requiring extensions to grand unified theories (GUTs) such as G2(-14) that demand additional particle states.

Complementary to Octonions

The paper highlights that while octonions and Okubonions can furnish complementary approaches to modeling QCD, they are mutually exclusive when considered simultaneously due to their independent group theoretical embeddings within Spin(8). The Okubo framework does not share a common SU(2) subgroup with the octonion-based SU(3) symmetries, leading to a non-overlapping ideological perspective in the context of algebraic QCD modeling.

Future Directions

The research suggests that Okubonions, with their distinctive non-unital and non-associative properties, might hold the key to elucidating non-perturbative QCD phenomena, though detailed mechanisms remain to be further explored. The theoretical novelty of employing non-unital algebras such as $\mathcal{O}$ proposes a fertile ground for novel strategies in theoretical physics, particularly pertaining to the underlying algebraic and geometric frameworks of fundamental interactions.

In summary, this work positions the Okubo algebra as a compelling alternative structure for QCD modeling, showcasing its potential to enrich our understanding of particle physics while emphasizing the distinct algebraic landscape it offers compared to conventional approaches relying on octonions.