Lemma: Action on scalar Cauchy data

Under the collar contract, the weak current commutator has action

Proof. First use the Noether representative (eq:jN). Its formal surface functional, solely for computing a weak commutator, is

The canonical commutators give its action on $\varphi$ as

Its action on $\pi$ is

The conformal Killing identities imply

Together with ${\partial }_0\pi =\Delta \varphi$, the latter expression is exactly ${\partial }_0(v_X\varphi )$.

This weak surface commutator is legitimate independently of a sharp-time quadratic operator: one current-field contraction leaves a linear field, and the equal-time distributions are the canonical delta distributions in (eq:CCR). They can equivalently be obtained by smooth time approximation before taking the weak commutator.

The commutator with the full surface current, as a function of the insertion coordinate, satisfies the free wave equation. Equation (eq:waveintertwine) shows that $v_X\varphi$ does too. The calculated Cauchy data therefore identify them by uniqueness of the scalar Cauchy problem. This argument is local in the causal shadow of the insertion. Replacing the full surface by $\chi$ leaves the commutator unchanged because the omitted support is spacelike separated from the insertion.

By Lemma lem:super, the improved and Noether currents differ on the surface by a spatial divergence. After multiplying by $\chi$, the difference is supported where $\nabla \chi \ne 0$. Its commutator also vanishes by the collar contract.

Finally average the weak surface commutator with $a$. It is the same $v_X\varphi$ for every surface time in the slab, and $\int a=1$. The resulting averaged operator is the well-defined operator (eq:q). This proves (eq:elementaryaction) on ${\mathcal{D}}_E$.