Proposition: Conditional preservation of local affine Ward actions

Let $F$ be a declared local field polynomial on $\mathcal{E}$. Choose $h\in {C_c}^{\infty }(\mathbb{R})$ and a spatial cutoff $\chi \in {C_c}^{\infty }({\mathbb{R}}^3)$ equal to one on the relevant spatial neighborhood. Suppose that:

1. the current smears and their commutators with $F$ are defined on $\mathcal{E}$;

2. the support of $h(t)\nabla \chi (\mathbf{x})$ is spacelike separated from the localization of $F$;

3. the superpotential smears on that collar commute with $F$ on $\mathcal{E}$.

Then the affine-current smears formed from $T$ and $t$ have the same commutator with $F$.

Proof. By Lemma lem:affine, their difference is

Because $B_{00}=0$, the time derivative of $h$ contributes nothing. Every remaining term is supported on the spacelike collar. Item three makes its commutator with $F$ vanish.