The S-matrix in Schr\"odinger Representation for Curved Spacetimes in General Boundary Quantum Field Theory

TL;DR

This paper develops a method to compute the S-matrix for interacting scalar fields in various curved spacetimes using the General Boundary Formulation, extending previous flat and simple curved spacetime results.

Contribution

It provides a general expression for the S-matrix and Feynman propagator in curved spacetimes within the GBF framework, including new spacetime region types.

Findings

Derived the Feynman propagator in curved spacetimes.

Expressed the S-matrix as a limit of GBF amplitudes for different regions.

Extended previous flat spacetime results to a broad class of curved geometries.

Abstract

We use the General Boundary Formulation (GBF) of Quantum Field Theory to compute the S-matrix for a general interacting scalar field in a wide class of curved spacetimes. As a by-product we obtain the general expression of the Feynman propagator for the scalar field, defined in the following three types of spacetime regions. First, there are the familiar interval regions (e.g.~a time interval times all of space). Second, we consider the rod hypercylinder regions (all of time times a solid ball in space). Third, the tube hypercylinders (all of time times a solid shell in space) are related to interval regions, and result from removing a smaller rod from a concentric larger one. Using the Schr\"odinger representation for the quantum states combined with Feynman's path integral quantization, we obtain the S-matrix as the asymptotic limit of the GBF amplitude associated with finite interval and rod regions. For interval regions, whose boundary consists of two Cauchy surfaces, the asymptotic GBF-amplitude becomes the standard S-matrix. Our work generalizes previous results (obtained in Minkowski, Rindler, de Sitter, and Anti de Sitter spacetimes) to a wide class of curved spacetimes.