TY - JOUR
TI - On Jordan–Clifford Algebras, Three Fermion Generations with Higgs Fields and a $${{\mathrm {SU}(3) \times \mathrm {SU}(2)_L \times \mathrm {SU}(2)_R \times \mathrm {U}(1)}}$$Model
AU - Perelman, Carlos Castro
T2 - Advances in Applied Clifford Algebras
AB - Previously we have shown that the algebra $$\begin{aligned} J_3 [{\mathbb {C}}\otimes {\mathbb {O}}] \otimes C \ell (4,{\mathbb {C}}), \end{aligned}$$J3[C⊗O]⊗Cℓ(4,C),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 $$E_6$$.
DA - 2021/06/11/
PY - 2021
DO - 10.1007/s00006-021-01136-5
DP - Springer Link
VL - 31
IS - 3
SP - 53
J2 - Adv. Appl. Clifford Algebras
LA - en
SN - 1661-4909
UR - https://link.springer.com/article/10.1007/s00006-021-01136-5
Y2 - 2021/08/05/14:22:28
ER -