The Lie Algebra of Quantum Fields

本项目研究微扰量子场论的李代数结构。特别地，它将研究费曼图的插入-消除李代数的结构。这些代数具有丰富的代数结构，这是理解量子运动方程结构及其重整化的基础。特别地，我们将研究它与李代数gl（无穷）的关系及其表示理论。量子场论奠定了我们对自然基本定律的理解，以及对相变和凝聚态物质研究的许多材料科学。这些理论通常被视为小的可观测参数的摄动序列。这种级数的项的计算是计算技术中最先进的领域之一。这些计算在其代数结构上与现代数学有着深远的联系。本项目将继续研究这些联系，以有利于计算实践和理论的概念基础。

The project investigates the Lie algebraic structures of perturbative quantum field theory. In particular, it will investigate the structure of the insertion-elimination Lie algebra of Feynman graphs. These algebras have a rich algebraic structure that is fundamental in understanding the structure of the quantum equations of motion and their renormalization. In particular, the relations to the Lie algebra gl(infinity) and its representation theory will be investigated. Quantum field theories underlie our understanding of the fundamental laws of nature, as well as much of materials science, in the study of phase transitions and condensed matter. These theories are typically treated as perturbation series in small observable parameters. Computation of the terms in such a series is one of the most advanced areas of computational technique. The computations have in their algebraic structures far-reaching connections to modern mathematics. This project will continue to investigate these connections to the benefit of both computational practice and conceptual foundations of the theory.