Wave functions for free, massless particles and nonintegrable representations of sl(2,c)

A class of representations of the Lie algebra sl(2,c) is discussed. These representations are operator-irreducible and su(2)-integrable, but are not all integrable to representations of the group SL(2,c). Free, massless fields and wave functions belonging to such index-space representations are considered, and the possible invariant helicities are given in each case. It is shown that the nonintegrability of the sl(2,c) representations does not preclude the possibility of realizing the appropriate unitary representations of the Poincaré group ISL(2,c). This permits the removal of discrepancies between the results of Bender, Frishman et al., and Simon et al. In particular, it is concluded that the free electromagnetic potential, in the radiation gauge, belongs to an infinite-dimensional representation of sl(2,c) which is not integrable. The relationships between free, massless fields and wave functions, having the same invariant helicity but belonging to different representations of sl(2,c), are discussed, and the corresponding generalization of a result due to Weinberg is obtained.