Abstract: I will present two distinct operator-theoretic settings, which are useful for describing (or defining) propagators associated with a scalar Klein-Gordon field on a Lorentzian manifold M. The first setting is based on the Hilbert space of square integrable functions on M and leads to the definition of an "operator-theoretic Feynman propagator". The second setting is based on the Krein space of solutions of the Klein-Gordon equation and leads to a rigorous definition of two-point functions associated to two (possibly different) states. When M admits a reasonable definition of "out" and "in" states, the operator-theoretic Feynman propagator from the first setup typically corresponds to the time-ordered two-point function from the "in" to the "out" state in the second setup.
After a general introduction, I will discuss several examples -- the one with the richest structure being de Sitter space.