quantum field theory - Why are there no particles in conformal theories?

In Matt Strassler's recent post (here) he makes the statement that scale invariant (I assume he means conformally invariant, more generally) theories have no particles in them. What's the reason for this? What are the technical and heuristic explanations, apart from what he mentions in his post?

Answer

The precise statement should be that massless fields in conformal field theories in 3+1 dimensions are necessarily free. This result was first proved by Buchholz and Fredenhagen.

There are two modern proofs of this fact, one by Steven Weinberg (please see arXiv: hep-th/1210.3864v1) and the other by Yoh Tanimoto in the framework of algebraic quantum field theory. (Weinberg's proof is a generalization of an unpublished argument by Witten for the spin 0 fields). Weinberg proves that the massless fields neceaarily satisfy free field equations.

The freeness of the massless fields should not be confused with the freeness of the whole theory, as conformal field theories contain massive fields also, where there is no known freeness restriction on the massive fields.

The conformal group in 3+1 dimensions is $SU(2,2)/\mathbb{Z}_4$. Its positive energy representations have been classified by Mack in: All unitary ray representations of the conformal group SU(2,2) with positive energy.

These representations are parameterized by two $SL_2$ quantum numbers $(j_1, j_2)$ and when restricted to the Poincaré subgroup, they reducibly or irreducibly decompose into representations of mass $m$ and spin $s$.The massless representations are the two families $(j, 0)$ or $(0, j)$ which reduce to massless Poincaré multiplets.

It should be emphasized that the existence of a mass parameter does not contradict conformal invariance, because the dilation parameter can shrink the energy to zero and these states are also gapless.

As mentioned above, there is no known restriction on the freeness of the massive states, and there is a construction by Odzijewicz of a massive conformal particle and an attempt to describe its interaction with an external field. The mass of such a particle is not a constant of motion and can change as a consequence of the interaction.

Odzijewicz works in the single particle level and uses the Orbit Method to describe the dynamics of the massive conformal particles. Weinberg and the others work on the field level for the same irreducible representations of the conformal group. It would be interesting to see a unified treatment of the two approaches.