A solvable algebra for massless fermions

TL;DR

This paper characterizes the stabilizer group of massless particle states as the Borel subgroup of the Lorentz group, revealing the structure of chiral states and their transformations, and discusses implications for nonphysical spin operators.

Contribution

It identifies the stabilizer group of massless particles as the Borel subgroup and explores its role in chiral states and Lorentz transformations.

Findings

Massless particle states are governed by the Borel subgroup of the Lorentz group.

Chiral states are associated with maximal solvable subgroups of order two.

The spin-flip contribution may relate to nonphysical spin operators.

Abstract

We derive the stabiliser group of the four-vector, also known as Wigner's little group, in case of massless particle states, as the maximal solvable subgroup of the proper orthochronous Lorentz group of dimension four, known as the Borel subgroup. In the absence of mass, particle states are disentangled into left- and right-handed chiral states, governed by the maximal solvable subgroups ${\rm sol}_2^\pm$ of order two. Induced Lorentz transformations are constructed and applied to general representations of particle states. Finally, in our conclusions it is argued how the spin-flip contribution might be closely related to the occurrence of nonphysical spin operators.