Generations: three prints, in colour

Abstract

We point out a somewhat mysterious appearance of SUc(3) representations, which exhibit the behaviour of three full generations of standard model particles. These representations are found in the Clifford algebra ℂl(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 ↦ − i. The entire result is achieved using only the eight-dimensional complex octonions as a single ingredient.