CAREER: Properties of Solutions to Singular Stochastic Partial Differential Equations from Quantum Field Theory

职业：量子场论奇异随机偏微分方程解的性质

• 批准号: 2044415
• 负责人: Hao Shen
• 金额: $ 42.85万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2044415&HistoricalAwards=false
• 关键词: CAREER; Properties; Solutions; Singular; Stochastic

This mathematics research project focuses on studying the small-scale behavior of some quantum field theory models in physics. The research aims to produce an accurate description and understanding of the ultraviolet divergence phenomenon. In the study of the standard model of elementary particles, for example, physicists often use Monte-Carlo simulation; the research carried out in this project aims to reveal why discrete simulation can accurately predict the behavior of a continuous quantum field, how quickly dynamical algorithms will converge to the quantity being observed, and to what degree a homogenized approximation can effectively describe the macroscopic nature of a many-body problem. The project also includes two educational components, focusing on rejuvenation of the Graduate Student Probability Conference (GSPC), which is entirely run by graduate students primarily from the US, and a Stochastic Partial Differential Equations (SPDE) education program that is vertically integrated across the full spectrum of academic research. The GSPC and the SPDE education program seek sustained long-term benefit to the future generations of the probability community. The project focuses on studying the properties of the solutions of some singular stochastic partial differential equations. In recent years, there has been rapid progress in the construction of the solutions of these equations, and the project aims to study the properties of the solutions that have been constructed. Specifically, the investigator will study the universal limits of the stochastic Yang-Mills equation and explore the connections between stochastic partial differential equations, large-N problems in quantum field theory, and the theory of mean field limits. Recently developed theory of solutions will play a central role. The PI aims to develop rigorous proofs for results that make practical predictions. These studies are anticipated to result in profound connections with physics and even to influence other natural sciences.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该数学研究项目重点研究物理学中一些量子场论模型的小尺度行为。该研究旨在准确描述和理解紫外线发散现象。例如，在基本粒子标准模型的研究中，物理学家经常使用蒙特卡罗模拟；该项目进行的研究旨在揭示为什么离散模拟可以准确预测连续量子场的行为，动态算法如何快速收敛到被观察的量，以及均匀化近似在多大程度上可以有效地描述量子场的宏观性质多体问题。该项目还包括两个教育部分，重点是研究生概率会议 (GSPC) 的复兴，该会议完全由主要来自美国的研究生管理，以及随机偏微分方程 (SPDE) 教育计划，该计划垂直整合到整个大学全方位的学术研究。 GSPC 和 SPDE 教育计划旨在为概率界的子孙后代寻求持续的长期利益。该项目重点研究一些奇异随机偏微分方程解的性质。近年来，这些方程解的构造取得了快速进展，该项目旨在研究已构造解的性质。具体来说，研究人员将研究随机杨-米尔斯方程的普遍极限，并探索随机偏微分方程、量子场论中的大N问题和平均场极限理论之间的联系。最近发展起来的解决方案理论将发挥核心作用。 PI 旨在为做出实际预测的结果提供严格的证明。这些研究预计将与物理学产生深远的联系，甚至影响其他自然科学。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Large N Limit of the O(N) Linear Sigma Model in 3D

3D 中 O(N) 线性西格玛模型的大 N 极限

• DOI: 10.1007/s00220-022-04414-w
• 发表时间: 2022-06
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Shen, Hao;Zhu, Rongchan;Zhu, Xiangchan
• 通讯作者: Zhu, Xiangchan

Langevin dynamic for the 2D Yang–Mills measure

二维 Yang–Mills 测量的 Langevin 动态

• DOI: 10.1007/s10240-022-00132-0
• 发表时间: 2022-06
• 期刊: Publications mathématiques de l'IHÉS
• 作者: Chandra, Ajay;Chevyrev, Ilya;Hairer, Martin;Shen, Hao
• 通讯作者: Shen, Hao

An SPDE approach to perturbation theory of Φ24: Asymptoticity and short distance behavior

π24 微扰理论的 SPDE 方法：渐近性和短距离行为

• DOI: 10.1214/22-aap1873
• 发表时间: 2023-08
• 期刊: The Annals of Applied Probability
• 作者: Shen, Hao;Zhu, Rongchan;Zhu, Xiangchan
• 通讯作者: Zhu, Xiangchan

A Stochastic Analysis Approach to Lattice Yang–Mills at Strong Coupling

强耦合下格子 Yang–Mills 的随机分析方法

• DOI: 10.1007/s00220-022-04609-1
• 发表时间: 2023-06
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Shen, Hao;Zhu, Rongchan;Zhu, Xiangchan
• 通讯作者: Zhu, Xiangchan

A stochastic PDE approach to large N problems in quantum field theory: A survey

量子场论中大 N 问题的随机偏微分方程方法：一项调查

• DOI: 10.1063/5.0089851
• 发表时间: 2022-08
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Shen; Hao
• 通讯作者: Hao