Combinatorial structures in quantum field theory

量子场论(QFT)对构成宇宙的基本粒子给出了已知最精确的描述。微扰量子傅里叶变换是一种将场理论量展开为费曼积分的无穷和的方法。然而，每一个单独的积分都是困难的；有许多我们无法评估的，即使是数字。此外，费曼积分的无穷级数通常是发散的。要从中提取意义，我们需要理解微扰理论的递归结构。我的研究计划的首要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，从而获得新的见解。在这一总体目标中有两个大规模目标。首先是使用图本身的属性来理解费曼图值中的模式。二是了解QFT的递归结构，特别是Dyson-Schwinger方程(量子运动方程)，以便严格而有效地使用组合洞察力来恢复图形。这些大规模的目标将通过学生和合作者参与的多个平行的短期目标来实现。我在这一领域有着出色的记录，并培养了18名在工业界和学术界继续深造的学生。在未来的几年里，我们仍将解决许多容易理解但又基本的问题，以促进离散数学和数学物理的最新发展。学生将从学习和发展新的数学以及跨学科工作中受益。这一计划的成功将产生令人兴奋的数学结果，以及利用数学获得的有用的物理结果。该程序的美和力量来自于物理和纯数学之间的双向互动。实现第一个目标将简化费曼积分的复杂计算，从而揭示目前无法理解的物理理论的结果，包括那些描述我们世界的理论。部分进展将帮助我们了解QFT的结构，从而了解我们世界的结构，也将丰富我们对相关数学对象的理解，例如多个Zeta值，以及用新问题和新结果反馈到图论中。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。部分结果使我们能够更好地逼近有物理意义的量和函数，并向我们展示系统的定性性质，以及产生丰富的组合问题。这个项目的时机恰到好处，因为相关领域，如复兴理论，正在成熟到可以被编入的地步，增加了我的项目回答基本问题的能力。

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.