Feynman Propagators on Static Spacetimes

TL;DR

This paper analyzes the Feynman propagator for the Klein-Gordon equation on static spacetimes, establishing its relation to the resolvent and Wick rotation, and exploring its properties as an inverse of the operator.

Contribution

It demonstrates that the Feynman propagator can be viewed as the boundary value of the resolvent and as a limit of Wick rotated inverses, providing new insights into its mathematical structure.

Findings

Feynman propagator is the boundary value of the resolvent.

Feynman propagator is the limit of the Wick rotated inverse.

Klein-Gordon operator is essentially self-adjoint on compactly supported smooth functions.

Abstract

We consider the Klein-Gordon equation on a static spacetime and minimally coupled to a static electromagnetic potential. We show that it is essentially self-adjoint on $C_{\mathrm{c}}^\infty$. We discuss various distinguished inverses and bisolutions of the Klein-Gordon operator, focusing on the so-called Feynman propagator. We show that the Feynman propagator can be considered the boundary value of the resolvent of the Klein-Gordon operator, in the spirit of the limiting absorption principle known from the theory of Schr\"odinger operators. We also show that the Feynman propagator is the limit of the inverse of the Wick rotated Klein-Gordon operator.