Wigner's little group and Berry's phase for massless particles

The 'little group' for massless particles (namely, the Lorentz transformations A that leave a null vector invariant) is isomorphic to the Euclidean group E2: translations and rotations in a plane. We show how to obtain explicitly the rotation angle of E2 as a function of A and we relate that angle to Berry's topological phase. Some particles admit both signs of helicity, and it is then possible to define a reduced density matrix for their polarization. However, that density matrix is physically meaningless because it has no transformation law under the Lorentz group, even under ordinary rotations.