Fractional helicity, Lorentz symmetry breaking, compactification and anyons

We construct the covariant, spinor sets of relativistic wave equations for a massless field on the basis of the two copies of the R-deformed Heisenberg algebra. For the finite-dimensional representations of the algebra they give a universal description of the states with integer and half-integer helicity. The infinite-dimensional representations correspond formally to the massless states with fractional (real) helicity. The solutions of the latter type, however, break down the (3+1)D Poincaré invariance to the (2+1)D Poincaré invariance, and via a compactification on a circle a consistent theory for massive anyons in d=2+1 is produced. A general analysis of the "helicity equation" shows that the (3+1)D Poincaré group has no massless irreducible representations with the trivial non-compact part of the little group constructed on the basis of the infinite-dimensional representations of sl(2, C) . This result is in contrast with the massive case where integer and half-integer spin states can be described on the basis of such representations, and means, in particular, that the (3+1)D Dirac positive energy covariant equations have no massless limit.