From Clifford Algebra of Nonrelativistic Phase Space to Quarks and Leptons of the Standard Model

Abstract: We review a recently proposed Clifford-algebra approach to elementary particles. We start with: (1) a philosophical background that motivates a maximally symmetric treatment of position and momentum variables, and: (2) an analysis of the minimal conceptual assumptions needed in quark mass extraction procedures. With these points in mind, a variation on Born's reciprocity argument provides us with an unorthodox view on the problem of mass. The idea of space quantization suggests then the linearization of the nonrelativistic quadratic form ${\bf p}^2 +{\bf x}^2$ with position and momentum satisfying standard commutation relations. This leads to the 64-dimensional Clifford algebra ${Cl}_{6,0}$ of nonrelativistic phase space within which one identifies the internal quantum numbers of a single Standard Model generation of elementary particles (i.e. weak isospin, hypercharge, and color). The relevant quantum numbers are naturally linked to the symmetries of macroscopic phase space. It is shown that the obtained phase-space-based description of elementary particles gives a subquark-less explanation of the celebrated Harari-Shupe rishon model. Finally, the concept of additivity is used to form novel suggestions as to how hadrons are constructed out of quarks and how macroscopically motivated invariances may be restored at the hadron level.