Two-dimensional stochastic Yang-Mills equations

二维随机 Yang-Mills 方程

基本信息

批准号：
EP/X015688/1
负责人：
Ilya Chevyrev
金额：
$ 39.14万
依托单位：
University of Edinburgh
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2023
资助国家：
英国
起止时间：
2023 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FX015688%2F1
关键词：
Two dimensional stochastic Yang Mills

项目摘要

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.

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。它的目标是将量子力学、经典场论和狭义相对论结合起来，它是当今我们最成功的科学理论之一。 QFT 研究是当今许多技术进步的基础，例如半导体、GPS 设备和激光器。 QFT还进一步发展成为一门深刻的学科，激发了物理和数学之间的大量相互作用。标准模型中使用了一类重要的量子理论，即杨-米尔斯理论来描述粒子之间的电弱力和强力。尽管量子杨米尔斯理论很重要并且在过去几十年里做出了许多努力，但严格定义和研究量子杨米尔斯理论的问题仍然悬而未决。这个问题已经变得如此突出，以至于它现在成为了千年奖问题的一部分。与 QFT 密切相关的数学领域是随机分析。随机分析研究随机系统，现在是物理、生物学和金融数学等一系列学科的重要工具。称为随机偏微分方程 (SPDE) 的随机分析领域在过去十年中取得了革命性的进展。 SPDE 用于描述许多复杂系统，从种群动态到晶体生长。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 共同的核心困难之一是重整化。在 SPDE 中，重整化的需要可以理解为研究潜在物理现象时对方程或参考系的错误选择。例如，生长晶体围绕其平均高度波动的简单方程未能考虑晶体边界向上移动的速度。这种运动迫使我们从方程中减去“无穷大”。从数学上讲，问题源于试图将高度振荡的函数相乘，这会导致出现各种无穷大。在 SPDE 的背景下，重整化允许人们重新解释这些无穷大并使它们变得严格。这些相同的无穷大困扰着 QFT，并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 的工作最近为实现这一目标迈出了关键的第一步。该提案将重点关注二维时空的设置，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，直到最近随机量化中的 SPDE 才变得易于分析。该提案的主要成果之一是了解这些 SPDE 在二维上的长期行为。最终的目标是，通过深入探索二维情况，我们将找到在更高维度构建和研究量子YM场的方法。将 SPDE 的最新进展与量子杨-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取得根本性的发现。一方面，它将开发许多适用于一系列问题的 SPDE 工具，另一方面，它有可能揭示 QFT 的数学基础。