Relations between Clifford Algebra and Dirac Matrices in the Presence of Families

Abstract

The internal degrees of freedom of fermions are in the spin-charge-family theory described by the Clifford algebra objects, which are superposition of an odd number of γ a ’s. Arranged into irreducible representations of “eigenvectors” of the Cartan subalgebra of the Lorentz algebra S a b ( = i 2 γ a γ b | a ≠ b ) these objects form 2 d 2 − 1 families with 2 d 2 − 1 family members each. Family members of each family offer the description of all the observed quarks and leptons and antiquarks and antileptons, appearing in families. Families are reachable by S ˜ a b = 1 2 γ ˜ a γ ˜ b | a ≠ b . Creation operators, carrying the family member and family quantum numbers form the basis vectors. The action of the operators γ a ’s, S a b , γ ˜ a ’s and S ˜ a b , applying on the basis vectors, manifests as matrices. In this paper the basis vectors in d = ( 3 + 1 ) Clifford space are discussed, chosen in a way that the matrix representations of γ a and of S a b coincide for each family quantum number, determined by S ˜ a b , with the Dirac matrices. The appearance of charges in Clifford space is discussed by embedding d = ( 3 + 1 ) space into d = ( 5 + 1 ) -dimensional space. The achievements and predictions of the spin-charge-family theory is also shortly presented.