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:

But Weinberg doesn't want to do more representation theory than necessary. Since these representations do not lead to causal quantum fields, he refers to experience to be able to take a shortcut.

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 some higher derivative string theories give rise to particles belonging to the continuous spin representation: For the excluded case of imaginary mass (tachyons) see the references in The paper itself constructs a free causal theory without vacuum state, with strange physical properties.

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