Scale Invariance plus Unitarity Implies Conformal Invariance in Four Dimensions

Understand

We give a non-perturbative proof that any 4D unitary and Lorentz-invariant quantum field theory with a conserved scale current is in fact conformally invariant.

• We show that any scale invariant theory (unitary or not) must have either a vanishing anomaly for global scale transformations or an operator of spin 2 and dimension 2.
• Neither of these possibilities is allowed for unitary theories, proving the result.
• This is also a strong constraint on non-unitary Euclidean theories with scale but not conformal invariance, suggesting the conjecture that all such theories are free field theories.