Minimal fields of canonical dimensionality are free

Abstract

It is shown that in a scale-invariant relativistic field theory, any field $ψ_n$ belonging to the $(j,0)$ or $(0,j)$ representations of the Lorentz group and with dimensionality $d=j+1$ is a free field. For other field types there is no value of the dimensionality that guarantees that the field is free. Conformal invariance is not used in the proof of these results, but it gives them a special interest; as already known and as shown here in an appendix, the only fields in a conformal field theory that can describe massless particles belong to the $(j,0)$ or $(0,j)$ representations of the Lorentz group and have dimensionality $d=j+1$. Hence in conformal field theories massless particles are free.