Partial Differential Equations and Quantum States in Curved Spacetimes

In this review paper, we discuss the relation between recent advances in theory of partial differential equations and their applications to quantum field on curved spacetimes. particular, focus hyperbolic propagators role they play construction physically admissible states—the so-called Hadamard states—on globally We will notion a propagator how it can be constructed an explicit invariant fashion, first Riemannian manifold then Lorentzian spacetime. Finally, recall state relate latter via wavefront set, subset cotangent bundle capturing information about singularities distribution.