Lemma: Transposition

For $c\in \mathcal{D}$, $h\in \mathcal{B}$, and every smooth $u$ (not necessarily a solution),

If $u$ is a solution, the left side is $\int c{\rho }_{1,g}[h;u]{\mathrm{d}}^{4}x$ and the right side is ${\mathrm{ev}}_u{R}_1(V_{1,g}[h];F_c)$. Thus the notation $-⟨c,{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{\mathsf{L}}_{1,g}[h]\hat{\varphi }⟩$ denotes the element $\hat{\varphi }(d_c[h])\in {{\mathfrak{A}}_0}^{(1)}$, characterized by its classical evaluations; it does not denote the application of ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$ to an unsmeared quantum field.

Proof. ${\mathsf{L}}_{1,g}[h]u$ is smooth with support in the compact set $\mathrm{supp}g$, so ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{\mathsf{L}}_{1,g}[h]u$ is defined and smooth, and $⟨c,{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}w⟩=⟨{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c,w⟩$ for compact $w$. Then $⟨{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c,{\mathsf{L}}_{1,g}[h]u⟩=⟨{\mathsf{L}}_{1,g}[h]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c,u⟩$ by formal self-adjointness of ${\mathsf{L}}_{1,g}[h]$, legitimate because its coefficients are compactly supported and ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c$, $u$ are smooth. Hence $-⟨c,{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{\mathsf{L}}_{1,g}[h]u⟩=⟨-{\mathsf{L}}_{1,g}[h]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c,u⟩=⟨d_c[h],u⟩$. The rest is Definition def:ev and ${\rho }_{1,g}[h;u]=-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{\mathsf{L}}_{1,g}[h]u$.