Propagators in curved spacetimes from operator theory

We discuss two distinct operator-theoretic settings useful for describing (or defining) propagators associated with a scalar Klein-Gordon field on a Lorentzian manifold $M$. Typically, we assume that $M$ is globally hyperbolic. The term propagator here refers to any Green function or bisolution of the Klein-Gordon equation pertinent to Quantum Field Theory. The off-shell setting is based on the Hilbert space $L^2(M)$. It leads to the definition of the operator-theoretic Feynman and anti-Feynman propagators, which often coincide with the so-called in-out Feynman and out-in anti-Feynman propagator. On some special spacetimes, the sum of the operator-theoretic Feynman and anti-Feynman propagator equals the sum of the forward and backward propagator. This is always true on static stable spacetimes and, curiously, in some other cases as well. The on-shell setting is based on the Krein space $\mathcal{W}_{\rm KG}$ of solutions of the Klein-Gordon equation. It allows us to define 2-point functions associated to two, possibly distinct, Fock states as the Klein-Gordon kernels of projectors onto maximal uniformly positive subspaces of $\mathcal{W}_{\rm KG}$. After a general discussion, we review a number of examples. We start with static and asymptotically static spacetimes, which are especially well-suited for Quantum Field Theory. Then we discuss FLRW spacetimes, reducible by a mode decomposition to 1-dimensional Schr\"odinger operators. We compare various approaches to de Sitter space where, curiously, the off-shell approach gives non-physical propagators. Finally, we discuss the universal cover of anti-de Sitter spaces, where the on-shell approach may require boundary conditions, unlike the off-shell approach.