Proposition: Restricted causal factorization

Let $S$ contain the compact coefficient support of $L$. Then:

1. if $S\cap J_{\eta }^{-}(\mathrm{supp}c)=\varnothing$,

   $${\mathcal{C}}_{\mathrm{F}}(V,F_c)=V\star F_c;$$

2. if $S\cap J_{\eta }^{+}(\mathrm{supp}c)=\varnothing$,

   $${\mathcal{C}}_{\mathrm{F}}(V,F_c)=F_c\star V.$$

For the physical vertex, the retarded map vanishes under condition (i).

Proof. The first difference is $-i\hslash ⟨V^{\prime },{\mathsf{E}}_{\mathrm{A}}c⟩$, and its supports are disjoint under (i). For (ii), use

and ${\Delta }_{\mathrm{F}}-{\Delta }_{+}^{\mathrm{T}}=-i{\mathsf{E}}_{\mathrm{R}}$. The supports are again disjoint.