An Evolution Equation Approach to Linear Quantum Field Theory

Abstract

In this paper we describe the construction of various propagators based on an abstract theory of (non-autonomous) evolution equations on Hilbertizable and Krein spaces. We introduce a notion of asymptotically complementary pairs of subspaces, and in one of our central results show that this property is automatically satisfied for certain pairs in Krein spaces. An application of this theory to the Klein-Gordon field in spacetimes with an asymptotically stationary future and past leads to a rigorous construction of a distinguished Feynman propagator. After quantization, 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.