Eur. Phys. J. C (2024) 84: 1129
https://doi.org/10.1140/epjc/s10052-024-13476-0

Regular Article - Theoretical Physics

Algebraic realisation of three fermion generations with $S_3$ family and unbroken gauge symmetry from $\mathbb{C}\ell (8)$

Department of Physics, Xi’an Jiaotong-Liverpool University, 111 Ren’ai Rd, 215123, Suzhou, People’s Republic of China

^b niels.gresnigt@xjtlu.edu.cn

Received: 12 June 2024
Accepted: 10 October 2024
Published online: 30 October 2024

Abstract

Building on previous work, we extend an algebraic realisation of three fermion generations within the complex Clifford algebra $\mathbb{C}\ell (8)$ by incorporating a $U{(1)}_{\mathit{em}}$ gauge symmetry. The algebra $\mathbb{C}\ell (8)$ corresponds to the algebra of complex linear maps from the (complexification of the) Cayley–Dickson algebra of sedenions, $\mathbb{S}$, to itself. Previous work represented three generations of fermions with $SU{(3)}_C$ colour symmetry permuted by an $S_3$ symmetry of order-three, but failed to include a U(1) generator that assigns the correct electric charge to all states. Furthermore, the three generations suffered from a degree of linear dependence between states. By generalising the embedding of the discrete group $S_3$, corresponding to automorphisms of $\mathbb{S}$, into $\mathbb{C}\ell (8)$, we include an $S_3$-invariant U(1) that correctly assigns electric charge. First-generation states are represented in terms of two even $\mathbb{C}\ell (8)$ semi-spinors, obtained from two minimal left ideals, related to each other via the order-two $S_3$ symmetry. The remaining two generations are obtained by applying the $S_3$ symmetry of order-three to the first generation. In this model, the gauge symmetries, $SU{(3)}_C\times U{(1)}_{\mathit{em}}$, are $S_3$-invariant and preserve the semi-spinors. As a result of the generalised embedding of the $S_3$ automorphisms of $\mathbb{S}$ into $\mathbb{C}\ell (8)$, the three generations are now linearly independent.