The surface-Hadamard well-posedness conjecture for semiclassical electrodynamics

Let $\mathcal C$ be an initial constant-time surface, let $\mathbf E^{\rm in}$ and $\mathbf B^{\rm in}$ be initial electric and magnetic fields on $\mathcal C$, and let $\omega_{\mathcal C}$ be a surface Hadamard state on the CCR algebra satisfying the constraint equations $\nabla\cdot\mathbf E=4\pi\omega(\rho)$ and $\nabla\cdot\mathbf B=0$. Electrodynamical well-posedness conjecture. There exist unique fields $\mathbf E$ and $\mathbf B$ with the prescribed initial values and a state $\omega$ satisfying the full semiclassical Maxwell system, with $\omega\circ\operatorname{iso}_{\mathcal C}=\omega_{\mathcal C}$. Equivalently, in Coulomb gauge, prescribed initial $\mathbf E^{\rm in}$ and $\mathbf A^{\rm in}$ satisfying the Coulomb constraint should determine unique $\mathbf E$, $\mathbf A$, and state. This is proposed as the semiclassical electrodynamical analogue of the Yukawa initial-value conjecture; no proof is supplied.