Generations: Three Prints, in Colour

TL;DR

This paper reveals how the algebraic structure of complex octonions and Clifford algebra $ ext{Cl}(6)$ naturally encodes the three generations of standard model fermions, including their color symmetry and antiparticles.

Contribution

It demonstrates that the complex octonions and Clifford algebra $ ext{Cl}(6)$ inherently produce the structure of three fermion generations in the standard model.

Findings

The algebraic structure partitions into three generations of fermions.

$SU(3)$ generators are identified within the algebra.

The model reproduces the chromodynamic structure of standard model fermions.

Abstract

We point out a somewhat mysterious appearance of $SU_c(3)$ representations, which exhibit the behaviour of three full generations of standard model particles. These representations are found in the Clifford algebra $\mathbb{C}l(6)$, arising from the complex octonions. In this paper, we explain how this 64-complex-dimensional space comes about. With the algebra in place, we then identify generators of $SU(3)$ within it. These $SU(3)$ generators then act to partition the remaining part of the 64-dimensional Clifford algebra into six triplets, six singlets, and their antiparticles. That is, the algebra mirrors the chromodynamic structure of exactly three generations of the standard model's fermions. Passing from particle to antiparticle, or vice versa, requires nothing more than effecting the complex conjugate, $*$: $i \mapsto -i$. The entire result is achieved using only the eight-dimensional complex octonions as a single ingredient.