Yang–Mills for Probabilists

Abstract

The rigorous construction of quantum Yang–Mills theories, especially in dimension four, is one of the central open problems of mathematical physics. Construction of Euclidean Yang–Mills theories is the first step towards this goal. This article presents a formulation of some of the core aspects this problem as problems in probability theory. The presentation begins with an introduction to the basic setup of Euclidean Yang–Mills theories and lattice gauge theories. This is followed by a discussion of what is meant by a continuum limit of lattice gauge theories from the point of view of theoretical physicists. Some of the main issues are then posed as problems in probability. The article ends with a brief review of the mathematical literature.