Sedenions, Clifford Algebras, and Three Fermion Generations: A Focused Review

Abstract

The existence of three fermion generations remains one of the unexplained structural features of the Standard Model. This article reviews an algebraic framework in which an intrinsic $S_3$ family symmetry relates three gauge-equivalent fermion sectors. The framework is motivated by division- and Clifford-algebra constructions in which the complex octonions and $\mathbb{C}\ell (6)$ organise the colour and electromagnetic quantum numbers of one generation. Continuing the Cayley--Dickson sequence, the sedenions provide an intrinsic $S_3$ automorphism structure, while their complexified left-multiplication operators generate an associative algebra isomorphic to $\mathbb{C}\ell (8)$. In the $\mathbb{C}\ell (8)$ formulation, the order-three family action generates three linearly independent fermion sectors while leaving a single $\mathfrak{su}(3{)}_C\oplus \mathfrak{u}(1{)}_{\mathrm{em}}$ gauge algebra. Extension to $\mathbb{C}\ell (10)$ incorporates a single $\mathfrak{su}(3{)}_C\oplus \mathfrak{su}(2{)}_L\oplus \mathfrak{u}(1{)}_Y$ gauge algebra, with each generation including a right-handed neutrino sterile under the SM gauge interactions. We place the construction in context by comparing it with representative algebraic and family-symmetry approaches to three generations, including triality-based proposals, and clarify why the family symmetry used here is distinct from standard $\mathrm{Spin}(8)$ triality despite the common appearance of $\mathbb{C}\ell (8)$ and $S_3$. The framework remains algebraic and representation theoretic rather than a complete dynamical theory; family-symmetry breaking, realistic fermion masses and mixing, and a dynamical account of the gauge and matter sectors remain open problems.

arXiv comment59 pages, 1 figure. Review article