# CFT and the Coleman-Mandula Theorem

The Coleman-Mandula theorem states that under certain seemingly-mild assumptions on the properties of the S matrix (roughly: one particle states are left invariant and the amplitudes are analytic in external momenta) the largest possible Lie algebra of symmetries of a (non-trivial) S matrix is given by Poincaré times an internal symmetry.
On the other hand, there are (interacting) field theories whose Lagrangians are symmetric under the conformal extension of the Poincaré group, and in some rare case this property is retained even at the quantum level.
Why (interacting) conformal invariant QFTs do not contradict the theorem? Is it possible to define an S matrix in these theories? I have read somewhere that they do not admit a particle interpretation, what does it mean exactly?

-