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

The 'little group' for massless particles (namely, the Lorentz transformations Λ 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 Λ 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. (letter to the editor)