Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 354 / 1 / a

Let $|\mathcal{O}⟩$ be the state associated with a scalar conformal primary operator of scaling dimension $\Delta$. In radial quantization,and $P_a^{†}=-K_a$. The norm of a level-one conformal descendant isUnitarity first gives $\Delta \ge 0$. If $\Delta =0$, every $P_a|\mathcal{O}⟩$ is null, so the local operator is translation invariant and belongs to the identity conformal family. Excluding the identity therefore gives $\Delta >0$.

Now consider the scalar level-two descendant $P^2|\mathcal{O}⟩$. The conformal algebra and the scalar-primary conditions giveApplying the second $K_a$ and summing over $a$ yieldsPositivity of this norm, together with $\Delta >0$, proves the scalar conformal unitarity boundAt equality the level-two descendant is null; in position space this is the free scalar equation of motion.