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 $\gamma^a$'s. Arranged into irreducible representations of "eigenvectors" of the Cartan subalgebra of the Lorentz algebra $S^{ab}$ $(= \frac{i}{2} \gamma^a \gamma^b|_{a \ne b})$ these objects form $2^{\frac{d}{2}-1}$ families with $2^{\frac{d}{2}-1}$ 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 $\tilde{S}^{ab}$ $=\frac{1}{2} \tilde{\gamma}^a \tilde{\gamma}^b|_{a \ne b}$. Creation operators, carrying the family member and family quantum numbers form the basic vectors. The action of the operators $\gamma^a$'s, $S^{ab}$, $\tilde{\gamma}^a$'s and $\tilde{S}^{ab}$, applying on the basic vectors, manifests as matrices. In this paper the basic vectors in $d=(3+1)$ Clifford space are discussed, chosen in a way that the matrix representations of $\gamma^a$ and of $S^{ab}$ coincide for each family quantum number, determined by $\tilde{S}^{ab} $, 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.