  The ``little group'' for massless particles (namely, the Lorentz
transformations $\Lambda$ 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 $\Lambda$ 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.
