The Clifford algebra of nonrelativistic phase space and the concept of mass

Abstract

Prompted by a recent demonstration that the structure of a single quark–lepton generation may be understood via a Dirac-like linearization of the form p2 + x2, we analyse the corresponding Clifford algebra in some detail. After classifying all elements of this algebra according to their U(1) ⊗ SU(3) and SU(2) transformation properties, we identify the element which might be associated with the concept of lepton mass. This element is then transformed into a corresponding element for a single coloured quark. It is shown that—although none of the three thus obtained individual quark mass elements is rotationally invariant—the rotational invariance of the quark mass term is restored when the sum over quark colours is performed.