Octonion Internal Space Algebra for the Standard Model

Abstract

This paper surveys recent progress in our search for an appropriate internal space algebra for the standard model (SM) of particle physics. After a brief review of the existing approaches, we start with the Clifford algebras involving operators of left multiplication by octonions. A central role is played by a distinguished complex structure that implements the splitting of the octonions O=C⊕C3, which reflect the lepton-quark symmetry. Such a complex structure on the 32-dimensional space S of Cℓ10 Majorana spinors is generated by the Cℓ6(⊂Cℓ10) volume form, ω6=γ1⋯γ6, and is left invariant by the Pati–Salam subgroup of Spin(10), GPS=Spin(4)×Spin(6)/Z2. While the Spin(10) invariant volume form ω10=γ1…γ10 of Cℓ10 is known to split S on a complex basis into left and right chiral (semi)spinors, P=12(1−iω6) is interpreted as the projector on the 16-dimensional particle subspace (which annihilates the antiparticles).The standard model gauge group appears as the subgroup of GPS that preserves the sterile neutrino (which is identified with the Fock vacuum). The Z2-graded internal space algebra A is then included in the projected tensor product A⊂PCℓ10P=Cℓ4⊗Cℓ60. The Higgs field appears as the scalar term of a superconnection, an element of the odd part Cℓ41 of the first factor. The fact that the projection of Cℓ10 only involves the even part Cℓ60 of the second factor guarantees that the color symmetry remains unbroken. As an application, we express the ratio mHmW of the Higgs to the W boson masses in terms of the cosine of the theoretical Weinberg angle.

Keywords

• Clifford algebra
• composition algebra
• triality
• Jordan algebra
• complex structure
• superselection rules