Proposition: The one-contraction identity

For the pair (A1)–(A2),

and for $V^{\prime }=-\mathsf{L}\phi$ the right-hand side equals $-F_{\mathsf{L}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c}(\phi )$. In particular, for $V=V_{1,g}[h]$ it equals $F_{d_c[h]}(\phi )$, whose class is $R_1(V_{1,g}[h];F_c)$. The identity is independent of ${\Delta }_{+}$ and of $\hslash$. Moreover ${\mathcal{C}}_{\mathrm{F}}(V,F_c)={\mathcal{C}}_{\mathrm{F}}(F_c,V)$.

Proof. By Lemma lem:wick and Lemma lem:DF(v), ${\mathcal{C}}_{\mathrm{F}}(V,F_c)-V\star F_c=\hslash ⟨V^{\prime },({\Delta }_{\mathrm{F}}-{\Delta }_{+})c⟩=-i\hslash ⟨V^{\prime },{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c⟩$. Multiplying by $i/\hslash$ gives $⟨V^{\prime },{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c⟩$, since $i\cdot (-i)=1$. The second equality is ${{\Delta }_S}^{\mathrm{A}}=-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$. The third is transposition, $⟨V^{\prime },{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c⟩=⟨{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{V}^{\prime },c⟩=-⟨c,{{\Delta }_S}^{\mathrm{R}}{V}^{\prime }⟩$, legitimate because $V^{\prime }$ is compactly supported and smooth. For $V^{\prime }=-\mathsf{L}\phi$: $⟨-\mathsf{L}\phi ,{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c⟩=-⟨\phi ,\mathsf{L}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c⟩$ by formal self-adjointness of $\mathsf{L}$ with compact coefficients and smoothness of ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c$; this is $-F_{\mathsf{L}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c}(\phi )$, and with $\mathsf{L}={\mathsf{L}}_{1,g}[h]$ it is $F_{d_c[h]}(\phi )$. Independence of ${\Delta }_{+}$ and $\hslash$ is visible in the result. Symmetry: by Lemma lem:DF(i), $⟨c,{\Delta }_{\mathrm{F}}{V}^{\prime }⟩=⟨{{\Delta }_{\mathrm{F}}}^{\mathsf{T}}c,V^{\prime }⟩=⟨V^{\prime },{\Delta }_{\mathrm{F}}c⟩$, and $V{F}_c=F_{c}V$ pointwise.