Relativistic wave equation for anyons

Construction of one-particle states as unitary representations of the Poincar\'e algebra in 2+1 dimensions shows that an anyon has one polarization state. However, for nonzero spin manifestly linear and covariant realizations of Lorentz transformations require more than one field component, and an infinite number is needed when the value of spin is not an integer or half-integer. We discuss the relation between these two aspects of Poincar\'e symmetry. In particular, we construct a relativistic equation for anyons where the number of physical polarizations is reduced to one by virtue of a gauge symmetry or equivalent constraint.