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Under the state–operator correspondence, let $|O_{ij}⟩$ be the state of the antisymmetric conformal primary operator $O_{ij}$. In radial quantization $P_i^{†}=K_i$, while primarity gives $K_i|O_{jk}⟩=0$. The norm of the level-one descendant obtained by taking a divergence is therefore determined by the conformal algebra commutatorUsing the two-form action of $M_{ab}$ gives, up to a positive normalization,Positivity of norm yields the two-form conformal unitarity boundAt saturation the descendant is null, and the operator obeys the conservation equation ${\partial }^i{O}_{ij}=0$.

In $d=3$, Hodge duality turns the two-form into the vector primary $V_k={ϵ}_{kij}{O}^{ij}/2$. The vector divergence descendant has norm proportional to $\Delta -d+1=\Delta -2$. Consequently the stronger bound israther than the formal two-form value $d-2=1$.

At spacelike separation $x_{12}^2>0$, conformal symmetry fixes the scalar-primary two-point function toApplying the d'Alembertian at $x_2$ givesaway from contact terms. Thus, up to a real multiplicative constant, $F_0\propto (x_{12}^2{)}^{-\Delta -1}$. The scalar conformal unitarity bound makes the coefficient nonnegative and makes it vanish when the bound is saturated, as expected for the null free-field equation of motion.

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.