2019/05/20 by Adam B. Gillard, Gillard, Adam B., Niels G. Gresnigt +1
Chemistry · Mathematics · #Advanced Algebra and Geometry #Advanced NMR Techniques and Applications #Algebraic and Geometric Analysis #FOS: Physical sciences #General Physics (physics.gen-ph) #High Energy Physics - Theory (hep-th)
paper · pdf · doi:10.48550/arxiv.1906.05102
openalex publication_date 2019/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, the basis states of the minimal left ideals of the complex Clifford algebra Cℓ(8) are shown to contain three generations of Standard Model fermion states, with full Lorentzian, right and left chiral, weak isospin, spin, and electrocolor degrees of freedom. The left adjoint action algebra of Cℓ(8)≅ℂ(16) on its minimal left ideals contains the Dirac algebra, weak isopin and spin transformations. The right adjoint action algebra on the other hand encodes the electrocolor symmetries. These results extend earlier work in the literature that shows that the eight minimal left ideals of ℂ(8)≅ Cℓ(6) contain the quark and lepton states of one generation of fixed spin. Including spin degrees of freedom extends Cℓ(6) to Cℓ(8), which unlike Cℓ(6) admits a triality automorphism. It is this triality that underlies the extension from a single generation of fermions to exactly three generations.