Symmetries of Nonrelativistic Phase Space and the Structure of Quark-Lepton Generation

TL;DR

This paper explores the symmetries of nonrelativistic phase space using a Hamiltonian approach, revealing quantum structures and internal quantum numbers that correspond to quark-lepton generation properties without subparticles.

Contribution

It introduces a Dirac-like linearization of phase space invariance, reproduces the Harari-Shupe model algebraically, and links phase space symmetries to fundamental particle quantum numbers.

Findings

Reproduces the Harari-Shupe preon algebra without subparticles.

Identifies phase space symmetries with internal quantum numbers.

Suggests a new perspective on lepton and quark mass concepts.

Abstract

According to the Hamiltonian formalism, nonrelativistic phase space may be considered as an arena of physics, with momentum and position treated as independent variables. Invariance of x^2+p^2 constitutes then a natural generalization of ordinary rotational invariance. We consider Dirac-like linearization of this form, with position and momentum satisfying standard commutation relations. This leads to the identification of a quantum-level structure from which some phase space properties might emerge. Genuine rotations and reflections in phase space are tied to the existence of new quantum numbers, unrelated to ordinary 3D space. Their properties allow their identification with the internal quantum numbers characterising the structure of a single quark-lepton generation in the Standard Model. In particular, the algebraic structure of the Harari-Shupe preon model of fundamental particles is reproduced exactly and without invoking any subparticles. Analysis of the Clifford algebra of nonrelativistic phase space singles out an element which might be associated with the concept of lepton mass. This element is transformed into a corresponding element for a single coloured quark, leading to a generalization of the concept of mass and a different starting point for the discussion of quark unobservability.