Sedenions、Clifford代数与三代费米子：一篇聚焦综述
Sedenions, Clifford Algebras, and Three Fermion Generations: A Focused Review
- Xi’an Jiaotong-Liverpool University（西交利物浦大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文综述了一个基于Sedenions和Clifford代数的代数框架，通过内在$S_3$族对称性生成三代费米子，并保留单一规范代数，但尚未提供完整动力学理论。
AI中文摘要：
三代费米子的存在仍是标准模型未解释的结构特征之一。本文回顾了一个代数框架，其中内在的$S_3$族对称性关联三个规范等价的费米子扇区。该框架由除法代数和Clifford代数构造所驱动，其中复八元数和$\mathbb{C}\ell(6)$组织了一代费米子的色和电磁量子数。延续Cayley–Dickson序列，Sedenions提供了内在的$S_3$自同构结构，而其复化左乘算子生成一个与$\mathbb{C}\ell(8)$同构的结合代数。在$\mathbb{C}\ell(8)$表述中，三阶族作用生成三个线性无关的费米子扇区，同时保留单一的$\mathfrak{su}(3)_C\oplus\mathfrak{u}(1)_{\mathrm{em}}$规范代数。扩展到$\mathbb{C}\ell(10)$则纳入单一的$\mathfrak{su}(3)_C\oplus\mathfrak{su}(2)_L\oplus\mathfrak{u}(1)_Y$规范代数，每一代包含一个在SM规范相互作用下为惰性的右手中微子。我们通过将该构造与具有代表性的代数和族对称性方法（包括基于triality的提议）进行比较来将其置于背景中，并阐明为何此处使用的族对称性不同于标准的$\operatorname{Spin}(8)$ triality，尽管$\mathbb{C}\ell(8)$和$S_3$共同出现。该框架仍是代数和表示论层面的，而非完整的动力学理论；族对称性破缺、现实的费米子质量和混合，以及规范与物质扇区的动力学描述仍是未解决的问题。
英文摘要：
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 $\operatorname{Spin}(8)$ 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.
