Do Small-mass Neutrinos participate in Gauge Transformations?

TL;DR

This paper explores how Wigner's little groups and SL(2,c) representations can distinguish between massive and massless neutrinos, showing that massive neutrinos cannot be gauge invariant.

Contribution

It introduces a method to separate gauge-dependent and gauge-invariant spinors for neutrinos using group theory and applies this to the Dirac equation.

Findings

Large components of Dirac spinors are gauge invariant.

Small components of Dirac spinors are gauge dependent.

Massive neutrinos cannot be invariant under gauge transformations.

Abstract

Neutrino oscillation experiments presently suggest that neutrinos have a small but finite mass. If neutrinos are to have mass, there should be a Lorentz frame in which they can be brought to rest. This paper discusses how Wigner's little groups can be used to distinguish between massive and massless particles. We derive a representation of the SL(2,c) group which separates out the two sets of spinors contained therein. One set is gauge dependent. The other set is gauge-invariant and represents polarized neutrinos. We show that a similar calculation can be done for the Dirac equation. In the large-momentum/zero-mass limit, the Dirac spinors can be separated into large and small components. The large components are gauge invariant, while the small components are not. These small components represent spin-$\frac{1}{2}$ non-zero mass particles. If we renormalize the large components, these gauge invariant spinors again represent the polarization of neutrinos. Massive neutrinos cannot be invariant under gauge transformations.