Separate backgrounds and quantum fields

Use a free scalar $\mathbf{\varphi }$ and a free gauge-fixed symmetric tensor $\mathbf{r}$. The latter is not the classical background $h$. Its equations and commutator are those of ${\mathsf{P}}_{\mathrm{gr}}$:

The scalar commutator uses the scalar action Hessian $-P_m$, hence the causal inverse ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$.

Fix a block-diagonal Hadamard Wick matrix with zero scalar–graviton cross blocks. Its graviton block has the form

with the antisymmetric part chosen to match ${{\Delta }_{\mathrm{gr}}}^{\mathrm{R}}-{{\Delta }_{\mathrm{gr}}}^{\mathrm{A}}$. No positive physical state on this covariant graviton carrier is assumed. The Wick and microcausal framework is the usual free functional construction FR.

Write

When a metric argument is quantum, the notation means the corresponding admitted mixed kernel, not a scalar test evaluated at an operator-valued point.