Jordan algebra approach to finite quantum geometry

The exceptional euclidean Jordan algebra $J_3^8$, consisting of $3\times 3$ 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 $F_4$ of $J_3^8$ that respects the lepton-quark splitting is $(SU(3)_c\times SU(3)_{ew})/\mathbb{Z}_3$. Its restriction to the special Jordan subalgebra $J_2^8\subset J_3^8$, associated with a single generation of fundamental fermions, is precisely the symmetry group $S(U(3)\times U(2))$ of the Standard Model. The Euclidean extension $\mathcal{H}_{16}(\mathbb{C})\otimes \mathcal{H}_{16}(\mathbb{C})$ of $J_2^8$, the subalgebra of hermitian matrices of the complexification of the associative envelope of $J_2^8$, 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.