Partial Differential Equations and Quantum States in Curved Spacetimes

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

Automated summary

"This review paper discusses the relationship between recent advances in partial differential equations theory and their applications to quantum field theory on curved spacetimes. It focuses on hyperbolic propagators and the construction of physically admissible quantum states, known as Hadamard states, on globally hyperbolic spacetimes. The paper reviews the concept of a propagator, its construction on Riemannian and Lorentzian manifolds, and the connection between Hadamard states and hyperbolic propagators through the wavefront set."