Single particle states in quantum field theory appear as discrete components in the spectrum of the Poincare group's action on the state space (i.e. in the decomposition of the Hilbert space of quantum states into irreducible representations of the Poincare group). Classification of irreducible unitary representations of the Poincare group leads to the notions of mass and spin.
Now, suppose we have a conformal QFT and are doing the same trick with the conformal group. Which irreducible representations do we have?
We still have the massless particles (at least I'm pretty sure we do although I don't immediately see the action of special conformal transformations). However, all representations for a given spin s and any mass m > 0 combine into a single irreducible representation.
What sort of physical object corresponds to this representation? Is it possible to construct a scattering theory for such objects? Is it possible to define unstable objects of this sort?