The massive Feynman propagator on asymptotically Minkowski spacetimes

TL;DR

This paper constructs a Feynman propagator for the massive Klein-Gordon equation on asymptotically Minkowski spacetimes, establishing its Fredholm properties and parametrix construction with boundary conditions at infinity.

Contribution

It introduces a novel framework for defining Feynman and anti-Feynman scattering data and constructs a Feynman parametrix with compact remainders for the Klein-Gordon operator.

Findings

Established Fredholm property of the Klein-Gordon operator with boundary conditions.

Constructed a parametrix that is also a Feynman parametrix in the sense of Duistermaat and Hörmander.

Proved the operator's properties on asymptotically Minkowski spacetimes.

Abstract

We consider the massive Klein-Gordon equation on asymptotically Minkowski spacetimes, in the sense that the manifold is $R^{1+d}$ and the metric approaches that of Minkowski space at infinity in a short-range way (jointly in time and space variables). In this setup we define Feynman and anti-Feynman scattering data and prove the Fredholm property of the Klein-Gordon operator with the associated Atiyah-Patodi-Singer type boundary conditions at infinite times. We then construct a parametrix (with compact remainder terms) for the Fredholm problem and prove that it is also a Feynman parametrix in the sense of Duistermaat and H\"ormander.