The order-$\hbar^0$ initial-value conjecture for semiclassical gravity

Let $\mathcal C$ be a 3-manifold with $\hbar$-dependent initial metric $g^{\rm in}_{ij}$ and extrinsic curvature $K^{\rm in}_{ij}$, and let $\omega_{\mathcal C}$ be an $\hbar$-dependent state whose initial two-point functions are smooth through order $\hbar^0$. Order-$\hbar^0$ initial-value conjecture. (a) If the constraint equations hold through order $\hbar^0$, there exists a physical solution $((\mathcal M,g_{ab}),\omega)$, unique through order $\hbar^0$, inducing the prescribed data and state on $\mathcal C$. (b) Given the order-$\hbar^0$ parts of the initial two-point functions $\omega_{\mathcal C}(\varphi\varphi')$, $\omega_{\mathcal C}(\pi\varphi')$, and $\omega_{\mathcal C}(\varphi\pi')$, there exists a unique spacetime and a bisolution $G_0$ of the Klein–Gordon equation inducing these data and the prescribed $g^{\rm in}_{ij}$ and $K^{\rm in}_{ij}$; the full state may be taken as $\omega_{\mathcal C}\circ\operatorname{iso}_{\mathcal C}^{-1}$. Here the order-$\hbar^0$ stress tensor is defined from $G_0$ by the stated point-splitting prescription. This is presented as a simplified approximate theory and remains conjectural.