Wigner’s Little Groups

Wigner’s little groups are subgroups of the Lorentz group dictating the internal space-time symmetries of massive and massless particles. The little group for the massive particle is like O(3) or the three-dimensional rotation group, and the little group for the massless particle is E(2) or the two-dimensional Euclidean group consisting of rotations and translations on a two-dimensional plane. 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 that the cylindrical group is like E(2) and thus like the little group for the massless particle. While Wigner discussed the O(3)-like little group for the massive particle at rest, it is possible to Lorentzboost this rotation matrix. It is shown further that the E(2)-like 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).