  In the first part of our paper we analyze bisolutions and inverses of
(non-autonomous) evolution equations. We are mostly interested in
pseudo-unitary evolutions on Krein spaces, which naturally arise in linear
Quantum Field Theory. We prove that with boundary conditions given by a maximal
positive and maximal negative space we can associate an inverse, which can be
viewed as a generalization of the usual Feynman propagator. In the context of
globally hyperbolic manifolds, the Feynman propagator turns out to be a
distinguished inverse of the Klein-Gordon operator. Within the formalism of
Quantum Field Theory on curved spacetimes, the Feynman propagator yields the
expectation values of time-ordered products of fields between the in and out
vacuum --the basic ingredient for Feynman diagrams.
