Generations: Three Prints, in Colour

Abstract

We point out a somewhat mysterious appearance of $S{U}_c(3)$ 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, $\ast$: $i\mapsto -i$. The entire result is achieved using only the eight-dimensional complex octonions as a single ingredient.

5 pages, 3 figures

References in corpus (7)

• Exceptional Deconfinement in G(2) Gauge Theory
• Color Screening, Casimir Scaling, and Domain Structure in G(2) and SU(N) Gauge Theories
• G_2 gauge theory at finite temperature
• Casimir scaling in G(2) lattice gauge theory
• Chiral symmetry and spectral properties of the Dirac operator in G2 Yang-Mills Theory
• Division Algebras; Spinors; Idempotents; The Algebraic Structure of Reality
• An Exceptional Electroweak Model

Cited by in corpus (2)

• Three fermion generations with two unbroken gauge symmetries from the complex sedenions
• The $C\ell (8)$ algebra of three fermion generations with spin and full internal symmetries