Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra

TL;DR

This paper shows how the complex octonions' algebraic structure can naturally produce three generations of quarks and leptons, along with gauge symmetries, potentially unifying standard model features within an eight-dimensional algebraic framework.

Contribution

It identifies a specific $su(3) imes u(1)$ action on complex octonions that reproduces three generations of particles and their gauge symmetries, extending previous work by including electric charge.

Findings

Revealed a $su(3) imes u(1)$ action splitting 64-dimensional space into standard model-like states.

Demonstrated the emergence of three generations of quarks and leptons from octonionic algebra.

Outlined a proposal to embed all standard model states and gauge bosons within the Clifford algebra $ ext{Cl}(8)$.

Abstract

A considerable amount of the standard model's three-generation structure can be realised from just the $8\hspace{.3mm}\mathbb{C}$-dimensional algebra of the complex octonions. Indeed, it is a little-known fact that the complex octonions can generate on their own a $64\hspace{.3mm}\mathbb{C}$-dimensional space. Here we identify an $su(3)\oplus u(1)$ action which splits this $64\hspace{.3mm}\mathbb{C}$-dimensional space into complexified generators of $SU(3)$, together with 48 states. These 48 states exhibit the behaviour of exactly three generations of quarks and leptons under the standard model's two unbroken gauge symmetries. This article builds on a previous one, [1], by incorporating electric charge. Finally, we close this discussion by outlining a proposal for how the standard model's full set of states might be identified within the left action maps of $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ (the Clifford algebra $\mathbb{C} l(8)$). Our aim is to include not only the standard model's three generations of quarks and leptons, but also its gauge bosons.