Symmetries of Massive and Massless Neutrinos

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).