Clifford Algebra of Nonrelativistic Phase Space and the Concept of Mass

TL;DR

This paper explores the Clifford algebra structure of nonrelativistic phase space, linking algebraic elements to the concept of mass for leptons and quarks, and demonstrating how quark mass invariance emerges from color summation.

Contribution

It provides a detailed classification of Clifford algebra elements related to particle properties and shows how quark mass invariance arises through color summation.

Findings

Identification of algebraic elements associated with lepton and quark masses.

Demonstration that quark mass invariance is restored when summing over colors.

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 p^2+x^2, we analyze the corresponding Clifford algebra in some detail. After classifying all elements of this algebra according to their U(1) x 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.