The natural background fermion

For $\xi \in {\mathfrak{g}}_U$, the retarded inverse ${G_{\mathrm{gh}}}^{\mathrm{R}}=-{{\mathsf{E}}_0}^{\mathrm{R}}$ gives the analogous first-order response identity

Indeed, ${\mathcal{L}}_{\xi }c$ is compactly supported and ${G_{\mathrm{gh}}}^{\mathrm{R}}{P_{\mathrm{gh}}}^{(0)}{\mathcal{L}}_{\xi }c={\mathcal{L}}_{\xi }c$. Equation (eq:ghostjet) is a retarded equation identity. It does not establish a nonlinear interacting quantum differential.