Combinatorial approaches to quantum field theory

点击翻译按钮获取中文摘要

Quantum field theory describes our universe over an enormous range of orders of magnitude and at very high precision. It is also full of beautiful patterns and structures. The goal of this research program is increase understanding of quantum field theory by furthering our combinatorial understanding of the underlying patterns in Feynman graphs, Feynman integrals, and their combinations into physically meaningful quantities. In particular, this proposal seeks first to give a graph theoretical understanding of the connection between graphs and transcendental values through their Feynman integrals, and second to understand the recursive decomposition of Feynman graphs into subgraphs along with the analytic version of this decomposition, the Dyson-Schwinger equations. The outcome of this work will be a better understanding of the fundamental theories which make up our universe as well as rich and beautiful mathematical objects and results. This proposal will incorporate the training of students at all levels in both pure and applied skills. The work is collaborative, enhancing my own strengths with those of other researchers, to yield innovative results. This work is of value both as pure mathematics and as mathematical physics, and gives Canada a place in this cutting edge approach to quantum field theory.