The Operator Product Expansion in Quantum Field Theory

TL;DR

This work argues that quantum field theories, including nonlinear and curved-spacetime cases, can be rigorously formulated from their local operator product expansions (OPEs)—capturing singular behavior and finite trends of field products as points coalesce. It develops a covariant algebraic framework with axioms (C1–C9) for OPE coefficients, defines field dimensions via scaling degrees, and derives flow relations that propagate OPE data with couplings, enabling construction of interacting theories from free ones without reliance on Feynman diagrams. The text also discusses OPEs in conformal QFTs and the conformal bootstrap, and demonstrates a curved-spacetime generalization of the PCT theorem, tying together locality, covariance, and symmetry in a unified scheme. Together, these results position OPEs as the central organizing principle for QFT in curved spacetimes, with implications for dynamics, renormalization, and fundamental symmetries.

Abstract

Operator product expansions (OPEs) in quantum field theory (QFT) provide an asymptotic relation between products of local fields defined at points $x_1, \dots, x_n$ and local fields at point $y$ in the limit $x_1, \dots, x_n \to y$. They thereby capture in a precise way the singular behavior of products of quantum fields at a point as well as their ``finite trends.'' In this article, we shall review the fundamental properties of OPEs and their role in the formulation of interacting QFT in curved spacetime, the ``flow relations'' in coupling parameters satisfied by the OPE coefficients, the role of OPEs in conformal field theories, and the manner in which general theorems -- specifically, the PCT theorem -- can be formulated using OPEs in a curved spacetime setting.