Clifford algebra of the Standard Model

Abstract

We explore the Z2 graded product C`10 = C`4⊗ˆC`6 (introduced by Furey) as a finite quantum algebra of the Standard Model of particle physics. The gamma matrices generating C`10 are expressed in terms of left multiplication by the imaginary octonion units and the Pauli matrices. The subgroup of Spin(10) that fixes an imaginary unit (and thus allows to write O = C⊗C 3 expressing the quark-lepton splitting) is the Pati-Salam group GP S = Spin(4) × Spin(6)/Z2 ⊂ Spin(10). If we identify the preserved imaginary unit with the C`6 pseudoscalar ω6 = γ1...γ6, ω2 6 = −1 (cf. the talk of Furey and Hughes), then Pex = 1 2 (1 − iω6) will play the role of the projector on the extended particle subspace including the right-handed (sterile) neutrino. We express the generators of C`4 and C`6 in terms of fermionic oscillators aα, a∗ α, α = 1, 2 and bj , b∗ j , j = 1, 2, 3 describing flavour and colour, respectively. The internal space observable algebra (an analog of the algebra of real functions on space-time) is then defined as the Jordan subalgebra of hermitian elements of the complexified Clifford algebra C ⊗ C`10 that commute with the weak hypercharge 1 2 Y = 1 3 P3 j=1 b ∗ j bj − 1 2 P2 α=1 a ∗ αaα. We only distinguish particles from antiparticles if they have different eigenvalues of Y . Thus the sterile neutrino and antineutrino (with Y = 0) are allowed to mix into Majorana neutrinos. Restricting C`10 to the particle subspace which consists of leptons with Y < 0 and quarks with Y > 0 allows a natural definition of the Higgs field Φ, the scalar of Quillen’s superconnection, as an element of C`1 4, the odd part of the first factor in C`10. As an application we express the ratio mH mW of the Higgs and the W-boson masses in terms of the cosine of the theoretical Weinberg angle. The talk is based on the paper arXiv:2010.15621v3 a copy of which including minor corrections is attached.