Clifford algebra implying three fermion generations revisited

Abstract

The author's idea of {\it algebraic compositeness} of fundamental particles, allowing to understand the existence in Nature of three fermion generations, is revisited. It is based on two postulates. i) For all fundamental particles of matter the Dirac square-root procedure $\sqrt{p^2}\to\Gamma^{(N)}\cdot p$ works, leading to a sequence $N=1,2,3,...$ of Dirac-type equations, where four Dirac-type matrices $\Gamma^{(N)}_\mu$ are embedded into a Clifford algebra {\it via} a Jacobi definition introducing four "centre-of-mass" and $(N-1)\times$four "relative" Dirac-type matrices. These define one "centre-of-mass" and $N-1$ "relative" Dirac bispinor indices. ii) The "centre-of-mass" Dirac bispinor index is coupled to the Standard Model gauge fields, while $N-1$ "relative" Dirac bispinor indices are all free undistinguishable physical objects obeying Fermi statistics with the Pauli principle which requires the full antisymmetry with respect to "relative" indices. This allows only for {\it three} $N = 1,3,5$ in the case of $N$ odd, implying the existence of {\it three and only three} generations of fundamental fermions.