Previously we have shown that the algebra

given by the tensor product of the complex exceptional Jordan \(J_3 [{\mathbb {C}}\otimes {\mathbb {O}}]\) and the complex Clifford algebra \(C \ell (4,{\mathbb {C}})\), can describe all of the spinorial degrees of freedom of three generations of fermions in four-space-time dimensions. We extend our construction to show that it also includes the degrees of freedom of three sets of pairs of complex scalar Higgs-doublets \(\{{\mathbf {H}}^{(m)}_L, {\mathbf {H}}^{(m)}_R\}; m = 1,2,3\), and their \(\mathrm {CPT}\) conjugates. Furthermore, a close inspection of the fermion structure of each generation reveals that it fits naturally with the sixteen complex-dimensional representation of the internal left/right symmetric gauge group \(G_{LR} = \mathrm {SU}(3)_C \times \mathrm {SU}(2)_L \times \mathrm {SU}(2)_R \times \mathrm {U}(1)\). It is reviewed how the latter group emerges from the intersection of \(\mathrm {SO}(10)\) and \(\mathrm {SU}(3) \times \mathrm {SU}(3) \times \mathrm {SU}(3)\) in \(E_6\). In the concluding remarks we briefly discuss the role that the extra Higgs fields may have as dark matter candidates; the construction of Chern–Simons-like matrix cubic actions; hexaquarks; supersymmetry and Clifford bundles over the complex-octonionic projective plane \(({\mathbb {C}}\otimes {\mathbb {O}}) {\mathbb {P}}^2\) whose isometry group is \(E_6\).

之前我们已经证明代数

由复数异常 Jordan \(J_3 [{\mathbb {C}}\otimes {\mathbb {O}}]\)和复数 Clifford 代数\(C \ell (4,{\mathbb { C}})\)，可以描述四时空维度上三代费米子的所有旋向自由度。我们扩展我们的构造以表明它还包括三组复标量希格斯双峰对的自由度\(\{{\mathbf {H}}^{(m)}_L, {\mathbf {H} }^{(m)}_R\}; m = 1,2,3\)，以及它们的\(\mathrm {CPT}\)共轭。此外，仔细检查每一代的费米子结构，发现它与内部左/右对称规范群的十六个复维表示自然吻合\(G_{LR} = \mathrm {SU}(3)_C \times \mathrm {SU}(2)_L \times \mathrm {SU}(2)_R \times \mathrm {U}(1)\) . 回顾了后一组如何从\(\mathrm {SO}(10)\)和\(\mathrm {SU}(3) \times \mathrm {SU}(3) \times \mathrm { SU}(3)\)在 \(E_6\) 中。在结束语中，我们简要讨论了额外的希格斯场可能作为暗物质候选者的作用；类陈-西蒙斯矩阵三次动作的构造；六夸克；超对称和 Clifford 丛在复八元投影平面\(({\mathbb {C}}\otimes {\mathbb {O}}) {\mathbb {P}}^2\)其等距群是 \(E_6\ )。