Existence of Local Covariant Time Ordered Products of Quantum Fields in Curved Spacetime
Stefan Hollands, Robert M. Wald
TL;DR
This work resolves the long-standing issue of constructing local covariant time-ordered products for nonlinear quantum fields in curved spacetime by reducing the problem to extending scalar distributions to the total diagonal. It employs a locally covariant Wick expansion around the diagonal and a scaling expansion that separates curvature-dependent terms from flat-space distributions, enabling a controlled extension while preserving locality, covariance, and microlocal properties. The resulting time-ordered products satisfy a full set of axioms (T1–T9), leading to a finite-parameter renormalization structure and establishing perturbative renormalizability for curved-spacetime quantum field theories. The framework paves the way for generalizations to massive fields, higher spins, and operator-product expansions, and suggests avenues toward an axiomatic, curved-spacetime quantum field theory.
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.