The Lie Algebra of Quantum Fields

该项目研究微扰量子场论的李代数结构。特别地，它将研究Feynman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的结构及其重整化的基础。特别地，将研究与李代数gl(无穷)及其表示理论的关系。在相变和凝聚物质的研究中，量子场论奠定了我们对自然界基本规律的理解，以及许多材料科学的理解。这些理论通常被视为小可观测参数中的微扰级数。计算这类级数中的项是计算技术中最先进的领域之一。这些计算在它们的代数结构中与现代数学有着深远的联系。本项目将继续调查这些联系，以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.