Minimal fields of canonical dimensionality are free

TL;DR

In scale-invariant relativistic field theories, fields with specific Lorentz representations and dimensionality are necessarily free, implying that massless particles in conformal theories are also free, based on symmetry constraints.

Contribution

The paper proves that certain fields with specific Lorentz and dimensional properties are inherently free in scale-invariant theories, highlighting a fundamental link between symmetry and field dynamics.

Findings

Fields with (j,0) or (0,j) representations and dimensionality d=j+1 are free in scale-invariant theories.

Massless particles in conformal field theories must belong to these specific representations and are therefore free.

The proof does not rely on conformal invariance, but the results are particularly relevant within conformal theories.

Abstract

It is shown that in a scale-invariant relativistic field theory, any field $\psi_n$ 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.