Two-dimensional stochastic Yang-Mills equations

量子场论（QFT）出现于上个世纪，是一种在最小尺度上描述自然基本定律的方法。它的目标是结合量子力学、经典场论和狭义相对论，它是今天我们最成功的科学理论之一。对QFT的研究是当今许多技术进步的背后，如半导体、GPS设备和激光。QFT进一步发展成为一门深奥的学科，激发了物理学和数学之间的相互作用。一个重要的量子理论，被称为杨-米尔斯理论，在标准模型中被用来描述粒子之间的电弱力和强力。尽管它的重要性和过去几十年的许多努力，严格定义和研究量子杨-米尔斯理论的问题仍然是开放的。这个问题已经变得如此突出，以致它现在已成为千年奖问题的一部分。与QFT有密切联系的数学领域是随机分析。随机分析研究随机系统，现在是一个重要的工具，在一系列学科，包括物理学，生物学和金融数学。一个叫做随机偏微分方程（SPDEs）的随机分析领域在过去十年中取得了革命性的进展。spde被用来描述许多复杂的系统，从种群动力学到晶体的生长。最近的突破为我们提供了研究这些方程的新方法，从而加深了我们对它们所描述的现象的理解。QFT和spde共同面临的核心困难之一是重整。在spde中，对重整化的需求可以理解为对方程或参考系的错误选择，以研究潜在的物理现象。例如，一个关于生长中的晶体在其平均高度周围波动的朴素方程没有考虑到晶体边界向上移动的速度。这种运动迫使我们从等式中减去“无穷大”。从数学上讲，这个问题来自于试图乘高振荡函数，这会导致出现各种无穷大。在spde的上下文中，重整化允许人们重新解释这些无穷大并使它们变得严格。这些同样的无限性困扰着QFT，使量子场的研究变得如此困难。本提案的目标是将spde的最新突破应用于量子杨-米尔斯场的研究。PI的工作最近朝着这个目标迈出了关键的第一步。该提案将侧重于二维时空的设置，这是我们宇宙的简化模型。我们计划使用的主要工具是随机量化，它在spde和QFT之间建立了正式的联系。由于重整化的困难，直到最近，随机量化的spde才变得适合分析。这一建议的主要成果之一是在两个维度上理解这些spde的长期行为。最终目标是，通过深入探索二维情况，我们将找到构建和研究高维量子YM场的方法。将spde的最新发展与量子杨-米尔斯理论相结合是一项重要而紧迫的任务，有可能在这两个领域都取得根本性的发现。一方面，它将在spde中开发许多适用于一系列问题的工具，另一方面，它有可能阐明QFT的数学基础。

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.