Two-dimensional stochastic Yang-Mills equations

Quantum field theory (QFT) emerged in the last century as a way to describe the fundamental laws of nature at its smallest scales. Its objective is to combine quantum mechanics, classical field theory, and special relativity, and it is today one of our most successful scientific theories. The study of QFT is behind many of today's technological advances, such as semiconductors, GPS devices, and lasers. QFT has furthermore developed into a deep subject which stimulates much interaction between physics and mathematics.

An important class of quantum theories, known as Yang-Mills theories, are used in the Standard Model to describe electroweak and strong forces between particles. Despite its importance and many efforts over the past decades, the problem of rigorously defining and studying quantum Yang-Mills theory is still open. This problem has become so outstanding that it now constitutes a part of the Millennium Prize Problems.

An area of mathematics which has close connections to QFT is stochastic analysis. Stochastic analysis studies random systems and is now an important tool in a range of disciplines, including physics, biology, and financial mathematics. A field of stochastic analysis called stochastic partial differential equations (SPDEs) has seen revolutionary progress in the past decade. SPDEs are used to describe many complex systems, from population dynamics to the growth of crystals. Recent breakthroughs have given us new ways to study these equations, thus deepening our understanding of the phenomena that they describe.

One of the core difficulties that QFT and SPDEs share is renormalisation. In SPDEs, the need for renormalisation can be understood as an incorrect choice of equation, or reference frame, to study the underlying physical phenomenon. For example, a naive equation for the fluctuations of a growing crystal around its average height fails to take into account the speed at which the crystal's boundary moves upwards. This movement forces us to subtract 'infinity' from the equation. Mathematically, the problem arises from trying to multiply highly oscillatory functions, which causes various infinities to appear. In the context of SPDEs, renormalisation allows one to reinterpret these infinities and make them rigorous. These same infinities plague QFT and make the study of quantum fields so difficult.

The goal of this proposal is to apply recent breakthroughs in SPDEs to the study of quantum Yang-Mills fields. The PI's work has recently taken crucial first steps towards this aim. The proposal will focus on the setting of two-dimensional space-time, which is a simplified model of our universe. The main tool we plan to use is stochastic quantisation, which makes a formal connection between SPDEs and QFT. Due to the difficulties of renormalisation, only recently have the SPDEs in stochastic quantisation become amenable to analysis. One of the main outcomes of this proposal is to understand the long-time behaviour of these SPDEs in two dimensions. The ultimate goal is that, by exploring in depth the two-dimensional case, we will find ways to construct and study quantum YM fields in higher dimensions.

Merging recent developments in SPDEs with quantum Yang-Mills theory is an important and urgent task with potential to make fundamental discoveries in both fields. On the one hand, it will develop a number of tools in SPDEs applicable to a range of problems, and, on the other hand, it has potential to shed light on the mathematical foundations of QFT.