Elementary particles have nonnegative mass and finite, discrete spin
Elementary particles must satisfy the principles of relativistic quantum field theory. This implies that they are described by nontrivial irreducible unitary representations of the Poincare group, compatible with a vacuum state.
Having a unitary representation of the Poincare group characterizes relativistic invariance. Irreducibility corresponds to the elementarity of the particle. The vacuum is excluded by forbidding the trivial representation.
Finally, causality requires the principle of locality, namely that commutators (or in case of fermions anticommutators) of the creation and annihilation fields at points with spacelike relative position must commute. Otherwise, the dynamics of distant points would be influenced in a superluminal way.
This rules out many of the irreducible unitary representations, leaving only those with nonnegative mass and finite spin.
Of the other irreducible unitary representations, all of which were
classified by Wigner in 1939, the massless continuous spin
representations are those most difficult to dismiss of.
On page 71 of his QFT book, Weinberg says that massless particles are not observed to have a continuous degree of freedom. Weinberg uses an empirical fact (''are not observed to have'') to eliminate this case in his analysis. He says that there are such representation, but that they are irrelevant as they don't match observation. One can eliminate the continuous spin representation also by causality arguments; but these arguments are lengthy:
By relaxing the assumptions, one can find certain almost acceptable variations of traditional quantum fields.
For the excluded case of zero mass and continuous spin (also referred to as infinite spin) see
Note that irreducibility is not necessary for causality. A generalized free causal field theory carrying a reducible representation is described in
Arnold Neumaier (Arnold.Neumaier@univie.ac.at) A theoretical physics FAQ