The perturbative scalar massless propagator in Schwarzschild spacetime

TL;DR

This paper derives the weak field approximation of the massless scalar propagator in Schwarzschild spacetime, explicitly including curvature coupling, and confirms its consistency with known vacuum states and renormalized quantities.

Contribution

It provides a detailed derivation of the scalar propagator in Schwarzschild spacetime with curvature coupling, extending previous work by explicitly including direct curvature effects.

Findings

Propagator satisfies Hadamard's pattern.

Vacuum state aligns with perturbative Boulware vacuum.

Reproduces renormalized <φ^2(x)> for massless case.

Abstract

A short derivation is given of the weak gravitational field approximation to the scalar massless propagator in Schwarzschild spacetime obtained by Paszko using the path-integral approach. The contribution from the direct coupling of the quantum field to the scalar curvature is explictly included. The propagator complies with Hadamard's pattern, and the vacuum state is consistent with the perturbative version of the Boulware vacuum. The momentum space propagator is computed for massless or massive particles to the same perturbative order. The renormalized value of $\ < \phi^2(x)\ >$ for the massless case is reproduced.