The Operator Product Expansion in Quantum Field Theory

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.

20pp; article prepared for Encyclopedia of Mathematical Physics