The Lie Algebra of Quantum Fields

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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.