Structure Group and Fermion-Mass-Term in General Nonlocality

TL;DR

This paper explores the role of the conformal group in resolving key issues in nonlocality, specifically the minimal group replacing GL(4,C) and the generation of fermion mass terms, linking these to physical phenomena like magnetic flux and angular momentum.

Contribution

It identifies the conformal group as the key to addressing the minimal group problem and fermion mass generation in nonlocality, providing a new theoretical framework.

Findings

Conformal group links the minimal group and fermion mass term.

Mass generation is related to chromo-magnetic flux and angular momentum.

The approach explains the physical origin of fermion masses in strong interactions.

Abstract

In our previous work [J. Math. Phys. 49, 033513 (2008)] two problems remain to be resolved. One is that we lack a minimal group to replace GL(4,C), the other is that the Equation of Motion (EoM) for fermion has no mass term. After careful investigation we find these two problems are linked by conformal group, a subgroup of GL(4,C) group. The Weyl group, a subgroup of conformal group, can bring about the running of mass, charge etc. while making it responsible for the transformation of interaction vertex. However, once concerning the generation of the mass term in EoM, we have to resort to the whole conformal group, in which the generators $K_\mu $ play a crucial role in making vacuum vary from space-like (or light-cone-like)to time-like. Physically the starting points are our previous conclusion, $\vec E^2-\vec B^2\neq 0$ for massive bosons, and the two-photon process yielding $e^+ e^-$ pair. Finally we get to the conclusion that the mass term of strong interaction is linearly relevant to (chromo-)magnetic flux as well as angular momentum.