Jordan algebra approach to finite quantum geometry
Published on:
August 18, 2020
Abstract
The exceptional euclidean Jordan algebra J83, consisting of 3×3 hermitian octonionic matrices, appears to be tailor made for the internal space of the three generations of quarks and leptons. The maximal rank subgroup of the authomorphism group F4 of J83 that respects the lepton-quark splitting is (SU(3)c×SU(3)ew)/Z3. Its restriction to the special Jordan subalgebra J82⊂J83, associated with a single generation of fundamental fermions, is precisely the symmetry group S(U(3)×U(2)) of the Standard Model. The Euclidean extension H16(C)⊗H16(C) of J82, the subalgebra of hermitian matrices of the complexification of the associative envelope of J82, involves 32 primitive idempotents giving the states of the first generation fermions. The triality relating left and right Spin(8) spinors to 8-vectors corresponds to the Yukawa coupling of the Higgs boson to quarks and leptons.
DOI: https://doi.org/10.22323/1.376.0163
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