The Operator Product Expansion in Quantum Field Theory

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.