The surface-Hadamard well-posedness conjecture for semiclassical gravity

Let $\mathcal C$ be a 3-manifold with classical initial data $(g^{\rm in}_{ij},\dot g^{\rm in}_{ij},g^{(2)\,\rm in}_{ij},g^{(3)\,\rm in}_{ij})$, or $(g^{\rm in}_{ij},\dot g^{\rm in}_{ij})$ for physical solutions, and let $\omega_{\mathcal C}$ be a state on its CCR algebra satisfying the appropriate surface Hadamard condition. Semiclassical-gravity well-posedness conjecture. If the initial data and $\omega_{\mathcal C}$ satisfy the gravitational constraint equations, there exists a unique solution $((\mathcal M,g_{ab}),\omega)$ of the semiclassical Einstein and Klein–Gordon equations such that $\mathcal C$ is a Cauchy surface, the prescribed classical data are the restrictions of $g_{ab}$ and its first three time derivatives (or first time derivative for physical solutions), and $\omega_{\mathcal C}=\omega\circ\operatorname{iso}_{\mathcal C}$. This is the proposed full initial-value formulation; its existence and uniqueness remain open.