Combinatorial structures in quantum field theory

量子场论中的组合结构

• 批准号: RGPIN-2019-04412
• 负责人: Yeats, Karen
• 金额: $ 1.89万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=737611
• 关键词: Combinatorial; structures; quantum; field; theory

Quantum field theories (QFTs) give the most precise descriptions known of the fundamental particles which make up the universe. Perturbative QFT is the approach whereby field theoretical quantities are expanded as infinite sums of Feynman integrals. However, each individual integral is difficult; there are many that we have no way to evaluate, even numerically. Furthermore, the infinite series of Feynman integrals are typically divergent. To extract meaning from them we need to understand the recursive structure of perturbation theory. The overarching goal of my research program is to understand these matters using a combinatorial aesthetic and toolbox, and to develop new combinatorial and physical tools that lead to new insights. There are two large scale objectives within this overarching goal. First is to understand patterns in the values of Feynman graphs using properties of the graphs themselves. Second is to understand the recursive structure of QFT, particularly the Dyson-Schwinger equations (the quantum equations of motion), so as to rigorously and effectively resum graphs using combinatorial insights. These large scale objectives will be approached via a number of parallel short term objectives with the participation of students and collaborators. I have an excellent track record in this area and have trained 18 students who have gone on both in industry and academia. There remain many accessible yet fundamental questions that we will tackle in the coming years in order to advance the state of the art in both discrete mathematics and mathematical physics. Students will benefit by learning and developing new mathematics and working interdisciplinarily. Success in this program will produce both exciting mathematical results arising from the physics and useful physical results achieved using mathematics. The beauty and power of the program come from the two-way interaction between physics and pure mathematics. Achieving the first objective would shortcut the intricate computation of Feynman integrals and thus reveal currently inaccessible consequences of physical theories, including those which describe our world. Partial progress will help us understand the structure of QFT, and hence of our world, and also enrich our understanding of related mathematical objects, such as multiple zeta values, as well as feeding back into graph theory with new problems and new results. Achieving the second objective would give a rigorous underpinning to perturbative QFT and a rigorous link to the non-perturbative world. Partial results let us obtain better approximations to physically meaningful quantities and functions, and show us qualitative properties of our systems, as well as yielding rich combinatorial problems. The time is right for this program because related areas, such as resurgence theory, are maturing to the point that they are ready to be woven in, increasing the power of my program to answer fundamental questions.

量子场论（QFT）给出了构成宇宙的基本粒子的最精确的描述。微扰QFT是将场论量展开为费曼积分的无限和的方法。然而，每一个单独的积分都是困难的;有许多积分我们无法计算，甚至无法用数值计算。此外，费曼积分的无穷级数通常是发散的。为了从中提取意义，我们需要理解微扰论的递归结构。我的研究计划的总体目标是使用组合美学和工具箱来理解这些问题，并开发新的组合和物理工具，以获得新的见解。在这个总体目标中有两个大规模的目标。首先是利用图本身的性质来理解费曼图的值的模式。其次是理解QFT的递归结构，特别是Dyson-Schwinger方程（量子运动方程），以便使用组合见解严格有效地恢复图形。这些大规模的目标将通过一些平行的短期目标与学生和合作者的参与。我在这方面有着出色的记录，已经培训了18名学生，他们在工业界和学术界都有出色的表现。在未来的几年里，我们将解决许多可访问但基本的问题，以推进离散数学和数学物理的最新发展。学生将受益于学习和发展新的数学和跨学科的工作。在这个程序的成功将产生两个令人兴奋的数学结果所产生的物理和有用的物理结果使用数学。该程序的美丽和力量来自物理学和纯数学之间的双向互动。实现第一个目标将使费曼积分的复杂计算变得快捷，从而揭示物理理论目前无法实现的结果，包括那些描述我们世界的理论。部分进展将帮助我们理解QFT的结构，从而理解我们的世界，也丰富了我们对相关数学对象的理解，例如多重zeta值，以及反馈到图论中的新问题和新结果。实现第二个目标将为微扰QFT提供严格的基础，并与非微扰世界建立严格的联系。部分结果让我们获得更好的近似物理意义的量和功能，并向我们展示了我们的系统的定性性质，以及产生丰富的组合问题。这个项目的时机是正确的，因为相关的领域，如复苏理论，正在成熟到他们已经准备好被编织的地步，增加了我的项目回答基本问题的能力。