Symmetries of Massive and Massless Neutrinos

TL;DR

This paper explores the internal symmetries of massive and massless neutrinos using Wigner's little groups, revealing how their geometries relate and how gauge invariance influences neutrino polarization.

Contribution

It demonstrates that the symmetry of massless neutrinos can be derived as a zero-mass limit of the massive case and clarifies the geometric interpretation of their internal symmetries.

Findings

Massless neutrino symmetry is an E(2)-like gauge symmetry.

Massive neutrino symmetry resembles O(3) geometry.

Neutrino polarization results from gauge invariance.

Abstract

Wigner's little groups are subgroups of the Lorentz group dictating the internal space-time symmetries of massive and massless particles. These little groups are like O(3) and E(2) for massive and massless particles respectively. While the geometry of the O(3) symmetry is familiar to us, the geometry of the flat plane cannot explain the E(2)-like symmetry for massless particles. However, the geometry of a circular cylinder can explain the symmetry with the helicity and gauge degrees of freedom. It is shown further that the symmetry of the massless particle can be obtained as a zero-mass limit of O(3)-like symmetry for massive particles. It is shown further that the polarization of massless neutrinos is a consequence of gauge invariance, while the symmetry of massive neutrinos is still like O(3).