[论文解读] Graphs in perturbation theory: Algebraic structure and asymptotics

Michael Borinsky|arXiv (Cornell University)|Jul 5, 2018

Advanced Combinatorial Mathematics参考文献 7被引用 14

一句话总结

本文将费曼图的霍普夫代数结构推广至一般图，并引入阶乘发散幂级数的微分环以分析渐近行为。该文为连通弦图和简单排列的所有阶渐近展开提供了闭式生成函数，并将这两种结构应用于零维量子场论（如φ³、φ⁴、QED和约旦理论）中所有阶渐近展开的计算。

ABSTRACT

This thesis provides an extension of the work of Dirk Kreimer and Alain Connes on the Hopf algebra structure of Feynman graphs and renormalization to general graphs. Additionally, an algebraic structure of the asymptotics of formal power series with factorial growth, which is compatible with the Hopf algebraic structure, is introduced. The Hopf algebraic structure on graphs permits the explicit enumeration of graphs with constraints for the allowed subgraphs. In the case of Feynman diagrams a lattice structure, which will be introduced, exposes additional unique properties for physical quantum field theories. The differential ring of factorially divergent power series allows the extraction of asymptotic results of implicitly defined power series with vanishing radius of convergence. Together both structures provide an algebraic formulation of large graphs with constraints on the allowed subgraphs. These structures are motivated by and used to analyze renormalized zero-dimensional quantum field theory at high orders in perturbation theory. As a pure application of the Hopf algebra structure, an Hopf algebraic interpretation of the Legendre transformation in quantum field theory is given. The differential ring of factorially divergent power series will be used to solve two asymptotic counting problems from combinatorics: The asymptotic number of connected chord diagrams and the number of simple permutations. For both asymptotic solutions, all order asymptotic expansions are provided as generating functions in closed form. Both structures are combined in an application to zero-dimensional quantum field theory. Various quantities are explicitly given asymptotically in the zero-dimensional version of $φ^3$, $φ^4$, QED, quenched QED and Yukawa theory with their all order asymptotic expansions.

研究动机与目标

• 将费曼图的霍普夫代数结构推广至任意图，使其在子图约束下可系统化枚举。
• 构建形式幂级数渐近行为的代数框架，其增长具有阶乘性，且与霍普夫代数结构相容。
• 将联合框架应用于提取零维量子场论中的所有阶渐近展开。
• 为量子场论中的勒让德变换提供霍普夫代数解释。
• 解决两个组合渐近计数问题：连通弦图与简单排列。

提出的方法

• 通过基于子图的递归分解，将费曼图的霍普夫代数适配至一般图。
• 在阶乘发散幂级数上引入微分环结构，以提取隐式定义级数的渐近行为。
• 利用霍普夫代数与微分环之间的相容性，分析具有子图约束的大型图。
• 通过计算重整化振幅的生成函数，将形式化应用于零维量子场论。
• 利用类似留数的运算，通过微分环推导出渐近展开的闭式生成函数。
• 在霍普夫代数框架内应用勒让德变换，以关联QFT中不同生成函数之间的关系。

实验结果

研究问题

• RQ1 如何将费曼图的霍普夫代数结构推广至具有子图约束的任意图？
• RQ2 何种代数结构控制具有阶乘发散的形式幂级数的渐近行为？
• RQ3 联合霍普夫代数与微分环框架能否为组合对象提供闭式渐近展开？
• RQ4 这些结构如何实现零维量子场论中所有阶渐近展开的计算？
• RQ5 量子场论中勒让德变换的霍普夫代数解释是什么？

主要发现

• 本文构建了一般图上的霍普夫代数，使得在子图约束下图的显式枚举成为可能。
• 引入了阶乘发散幂级数的微分环，使隐式定义级数的所有阶渐近展开得以提取。
• 连通弦图的渐近数量通过所有阶的闭式生成函数得以计算。
• 简单排列的数量通过所有阶展开的闭式生成函数得到渐近确定。
• 零维φ³、φ⁴、QED、屏蔽QED及约旦理论中重整化振幅的所有阶渐近展开被显式计算。
• 推导出勒让德变换的霍普夫代数表述，将其与QFT中的重整化群结构联系起来。