Quantum Field Theory on Curved Backgrounds, I. The Euclidean Functional Integral

TL;DR

This paper develops a rigorous mathematical framework for Euclidean quantum field theory on curved backgrounds, extending Osterwalder-Schrader quantization to curved spacetimes with symmetries, ensuring well-defined operators and reflection positivity.

Contribution

It generalizes Euclidean quantum field theory construction to curved backgrounds, introducing sharp-time localization and analyzing symmetry-induced operators.

Findings

Constructed Euclidean QFT on curved backgrounds with reflection positivity.

Established conditions for classical symmetries to induce quantum operators.

Demonstrated the use of sharp-time localization for operator construction.

Abstract

We give a mathematical construction of Euclidean quantum field theory on certain curved backgrounds. We focus on generalizing Osterwalder-Schrader quantization, as these methods have proved useful to establish estimates for interacting fields on flat space-times. In this picture, a static Killing vector generates translations in Euclidean time, and physical positivity is played by positivity under reflection of Euclidean time. We discuss the quantization of flows which correspond to classical space-time symmetries, and give a general set of conditions which imply that broad classes of operators in the classical picture give rise to well-defined operators on the quantum-field Hilbert space. In particular, Killing fields on spatial sections give rise to unitary groups on the quantum-field Hilbert space, and corresponding densely-defined self-adjoint generators. We construct the Schr\"odinger representation using a method which involves localizing certain integrals over the full manifold to integrals over a codimension-one submanifold. This method is called sharp-time localization, and implies reflection positivity.