Are octonions necessary to the Standard Model?

Abstract

There have been a number of claims, going back to the 1970s, that the Standard Model of particle physics, based on fermions and antifermions, might be derived from an octonion algebra. The emergence of SU(3), SU(2) and U(1) groups in octonion-based structures is suggestive of the symmetries of the Standard Model, but octonions themselves are an unsatisfactory model for physical application because they are antiassociative and consequently not a group. Instead, the ‘octonion’ models have to be based on adjoint algebras, such as left - or right-multiplied octonions, which can be seen to have group-like properties. The most promising of these candidates is the complexified left-multiplied octonion algebra, because it reduces, in effect, to Cl(6), which has been identified by one of us (PR) in a number of previous publications as the basic structure for the entire foundation of physics, as well as the algebra required for the Standard Model and the Dirac equation. Though this algebra has long been shown by PR as equivalent to using a complexified left-multiplied or ‘broken’ octonion, it doesn’t need to be derived in this way, as its real origins are in the respective real, complex, quaternion and complexified quaternion algebras of the fundamental parameters of mass, time, charge and space. The ‘broken’ octonion, however, does have value in leading to the higher (and equally broken) symmetries, such as E8, which incorporate fermions, with their two spin states, along with gauge bosons and vacuum states into a unified scheme.