The Operator Product Expansion in Quantum Field Theory

TL;DR

This paper reviews the fundamental properties and roles of operator product expansions in quantum field theory, emphasizing their applications in curved spacetime, conformal theories, and the formulation of key theorems.

Contribution

It provides a comprehensive review of OPEs, highlighting their significance in interacting QFT in curved spacetime and their use in formulating fundamental theorems.

Findings

Clarifies the role of OPEs in curved spacetime QFT

Explores the flow relations in coupling parameters

Demonstrates the use of OPEs in formulating the PCT theorem

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.