具有$S_3$家族对称性和未破缺规范对称性的三代费米子的$\u2102\ell(8)$代数实现
Algebraic realisation of three fermion generations with $S_3$ family and unbroken gauge symmetry from $\mathbb{C}\ell(8)$
AI总结：
本研究推广离散群$S_3$在复克利福德代数$\u2102\ell(8)$中的嵌入，构建出兼具$S_3$家族对称性、正确$U(1)_{em}$电荷分配且三代态线性无关的费米子代数实现模型。
摘要：
在前期工作基础上，我们通过纳入$U(1)_{em}$规范对称性，拓展了复克利福德代数$\u2102\ell(8)$内三代费米子的代数实现。$\u2102\ell(8)$代数对应从十六元数的凯莱-迪克森代数$\u212d$（的复化）到其自身的复线性映射代数。前期工作用被三阶$S_3$对称性置换的$SU(3)_C$色对称性表示三代费米子，但未能包含能为所有态赋予正确电荷的$U(1)$生成元，且三代态之间存在一定程度的线性相关性。通过将对应$\u212d$自同构的离散群$S_3$嵌入$\u2102\ell(8)$的方式推广，我们引入了一个$S_3$不变的$U(1)$，可正确分配电荷。第一代态由两个偶$\u2102\ell(8)$半旋量表示，这两个半旋量来自两个极小左理想，通过二阶$S_3$对称性相互关联。其余两代通过将三阶$S_3$对称性作用于第一代得到。在该模型中，规范对称性$SU(3)_C\times U(1)_{em}$是$S_3$不变的，且保持半旋量不变。由于将$\u212d$的$S_3$自同构推广嵌入$\u2102\ell(8)$，三代态现在是线性无关的。
英文摘要：
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 $U(1)_{em}$ 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, $SU(3)_C\times U(1)_{em}$, 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.
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