Proposition: Restricted causal factorization for the one pair

Let $V$ be as in (A1) with coefficient support $S=\mathrm{supp}\mathsf{L}\subset M$ compact, and $c\in \mathcal{D}$.

1. If $S\cap J^{-}(\mathrm{supp}c)=\varnothing$, then ${\mathcal{C}}_{\mathrm{F}}(V,F_c)=V\star F_c$, and $R_1(V;F_c)=0$ for $V=V_{1,g}[h]$.

2. If $S\cap J^{+}(\mathrm{supp}c)=\varnothing$, then ${\mathcal{C}}_{\mathrm{F}}(V,F_c)=F_c\star V$.

Proof. (i) ${\mathcal{C}}_{\mathrm{F}}(V,F_c)-V\star F_c=-i\hslash ⟨V^{\prime },{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c⟩$, and $\mathrm{supp}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c\subset J^{-}(\mathrm{supp}c)$ is disjoint from $\mathrm{supp}{V}^{\prime }\subset S$. The vanishing of $R_1$ is then (eq:check), or directly ${\mathsf{L}}_{1,g}[h]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c=0$. (ii) By Lemma lem:wick and Lemma lem:DF(v), ${\mathcal{C}}_{\mathrm{F}}(V,F_c)-F_c\star V=\hslash ⟨V^{\prime },({\Delta }_{\mathrm{F}}-{\Delta }_{+}^{\mathsf{T}})c⟩=-i\hslash ⟨V^{\prime },{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}c⟩$, and $\mathrm{supp}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}c\subset J^{+}(\mathrm{supp}c)$ is disjoint from $S$.