Unified theory of elementary fermions and their interactions based on Clifford algebras

Abstract

Seven commuting elements of the Clifford algebra $Cl_{7,7}$ define seven binary eigenvalues that distinguish the $2^7=128$ states of 32 fermions, and determine their parity, electric charge and interactions. Three commuting elements of the sub-algebra $Cl_{3,3}$ define three binary quantum numbers that distinguish the eight states of lepton doublets. The Dirac equation is reformulated in terms of a Lorentz invariant operator which expresses the properties of these states in terms of Dirac 4-component spinors. Re-formulation of the Standard Model shows chiral symmetry breaking to be redundant. A $Cl_{3,3}$ sub-algebra of $Cl_{5,5}$ defines two additional binary quantum numbers that distinguish quarks and leptons, and describes the SU(3) gluons that produce the hadron substrate, explaining quark confinement. Finally, a $Cl_{3,3}$ sub-algebra of $Cl_{7,7}$ defines a further two binary quantum numbers that distinguish four fermion generations. The predicted fourth generation is shown to have no neutrino and a distinct substrate, suggesting that ordinary matter is confined and providing candidates for unconfined dark matter. Interactions between fermions in the first three generations are predicted, including those that produce flavour symmetry. Relationships are explored between the $Cl_{1,3}$ algebra and general relativity, and between $Cl_{5,5}$ and SO(32) string theory.