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 \(\mathcal {W}_\textrm{KG}\) of solutions of the Klein–Gordon equation. It allows us to define 2-point functions associated with two, possibly distinct, Fock states as the Klein–Gordon kernels of projectors onto maximal uniformly positive subspaces of \(\mathcal {W}_\textrm{KG}\) . 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 Schrödinger 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.
