Recursive construction of the operator product expansion in curved space

TL;DR

This paper develops a recursive method to compute operator product expansion coefficients in curved space using the quantum action principle, starting from free theory data, and demonstrates it with scalar fields in hyperbolic space.

Contribution

It introduces a recursive formula for OPE coefficients in curved space, extending the quantum action principle, and provides explicit calculations in hyperbolic space.

Findings

Derived a recursive formula for OPE coefficients in curved space.

Successfully computed scalar OPEs in hyperbolic space up to first order in interaction.

Provided explicit OPE coefficient calculations at second order in coupling.

Abstract

I derive a formula for the coupling-constant derivative of the coefficients of the operator product expansion (Wilson OPE coefficients) in an arbitrary curved space, as the natural extension of the quantum action principle. Expanding the coefficients themselves in powers of the coupling constants, this formula allows to compute them recursively to arbitrary order. As input, only the OPE coefficients in the free theory are needed, which are easily obtained using Wick's theorem. I illustrate the method by computing the OPE of two scalars $\phi$ in hyperbolic space (Euclidean Anti-de Sitter space) up to terms vanishing faster than the square of their separation to first order in the quartic interaction $g \phi^4$, as well as the OPE coefficient $\mathcal{C}^{\mathbb{1}}_{\phi \phi}$ at second order in $g$.