The surface-Hadamard well-posedness conjecture for the semiclassical Yukawa model

Let $\mathcal C$ be an initial constant-time surface, let $(\chi_{\rm in},\dot\chi_{\rm in})$ be classical initial data on $\mathcal C$, and let $\omega_{\mathcal C}$ be a state on the CCR algebra of $\mathcal C$ satisfying the surface Hadamard condition for those data. Yukawa well-posedness conjecture. There exists a unique solution $(\chi,\omega)$ of the semiclassical Yukawa equations, with $\chi$ a scalar function on Minkowski space and $\omega$ a state on the *-algebra of the quantum equation, such that the restrictions of $\chi$ and $\dot\chi$ to $\mathcal C$ equal the prescribed data and $\omega\circ\operatorname{iso}_{\mathcal C}=\omega_{\mathcal C}$. This is the proposed well-posed initial-value formulation for the model, conditional on the iterative surface-Hadamard construction; existence and uniqueness remain conjectural.