The operator product expansion for perturbative quantum field theory in curved spacetime

TL;DR

This paper develops an algorithm to construct the operator product expansion for perturbative quantum field theory in curved spacetime, ensuring its validity at short distances and establishing key properties of the expansion coefficients.

Contribution

It introduces a novel algorithm for the Wilson OPE in curved spacetime, with detailed properties and an example implementation, advancing the understanding of quantum fields in curved backgrounds.

Findings

The OPE remainder vanishes at short distances in Hadamard states.

OPE coefficients depend locally and covariantly on the metric and couplings.

OPE coefficients satisfy associativity, renormalization group, and scaling properties.

Abstract

We present an algorithm for constructing the Wilson operator product expansion (OPE) for perturbative interacting quantum field theory in general Lorentzian curved spacetimes, to arbitrary orders in perturbation theory. The remainder in this expansion is shown to go to zero at short distances in the sense of expectation values in arbitrary Hadamard states. We also establish a number of general properties of the OPE coefficients: (a) they only depend (locally and covariantly) upon the spacetime metric and coupling constants, (b) they satisfy an associativity property, (c) they satisfy a renormalization group equation, (d) they satisfy a certain microlocal wave front set condition, (e) they possess a ``scaling expansion''. The latter means that each OPE coefficient can be written as a sum of terms, each of which is the product of a curvature polynomial at a spacetime point, times a Lorentz invariant Minkowski distribution in the tangent space of that point. The algorithm is illustrated in an example.