The Klein-Gordon equation on asymptotically Minkowski spacetimes: the Feynman propagator

TL;DR

This paper develops a microlocal analysis framework for the Feynman propagator of the Klein-Gordon equation on asymptotically Minkowski spacetimes, establishing its global properties and Hadamard condition.

Contribution

It introduces a novel approach using Vasy's 3sc-calculus to analyze the Feynman propagator's properties in asymptotically static spacetimes.

Findings

Proves global spacetime mapping properties of the Feynman propagator.

Shows the Feynman propagator satisfies a microlocal Hadamard condition.

Realizes the Feynman propagator as an inverse between Sobolev spaces with regularity near radial points.

Abstract

We develop a theory of Feynman propagators for the massive Klein--Gordon equation with asymptotically static perturbations. Building on our previous work on the causal propagators, we employ a framework based on propagation of singularities estimates in Vasy's 3sc-calculus. We combine these estimates to prove global spacetime mapping properties for the Feynman propagator, and to show that it satisfies a microlocal Hadamard condition. We show that the Feynman propagator can be realized as the inverse of a mapping between appropriate $L^2$-based Sobolev spaces with additional regularity near the asymptotic sources of the Hamiltonian flow, realized as a family of radial points on a compactified spacetime.