Existence of Local Covariant Time Ordered Products of Quantum Fields in Curved Spacetime

TL;DR

This paper proves the existence of local, covariant time ordered products for quantum fields in curved spacetime, completing the theoretical foundation for perturbative quantum field theory in such backgrounds.

Contribution

It establishes the existence and properties of time ordered products of Wick polynomials in curved spacetime, extending previous uniqueness results.

Findings

Constructed local covariant time ordered products satisfying key hypotheses.

Derived a scaling expansion expressing time ordered products in terms of curvature polynomials.

Confirmed the renormalizability and consistency of perturbative expansions in curved spacetime.

Abstract

We establish the existence of local, covariant time ordered products of local Wick polynomials for a free scalar field in curved spacetime. Our time ordered products satisfy all of the hypotheses of our previous uniqueness theorem, so our construction essentially completes the analysis of the existence, uniqueness and renormalizability of the perturbative expansion for nonlinear quantum field theories in curved spacetime. As a byproduct of our analysis, we derive a scaling expansion of the time ordered products about the total diagonal that expresses them as a sum of products of polynomials in the curvature times Lorentz invariant distributions, plus a remainder term of arbitrary low scaling degree.