An Evolution Equation Approach to Linear Quantum Field Theory

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.