New algebraic aspects of perturbative and non-perturbative Quantum Field Theory
Abstract
In this expository article we review recent advances in our understanding of the combinatorial and algebraic structure of perturbation theory in terms of Feynman graphs, and Dyson-Schwinger equations. Starting from Lie and Hopf algebras of Feynman graphs, perturbative renormalization is rephrased algebraically. The Hochschild cohomology of these Hopf algebras leads the way to Slavnov-Taylor identities and Dyson-Schwinger equations. We discuss recent progress in solving simple Dyson-Schwinger equations in the high energy sector using the algebraic machinery. Finally there is a short account on a relation to algebraic geometry and number theory: understanding Feynman integrals as periods of mixed (Tate) motives.
Keywords
Cite
@article{arxiv.0704.0232,
title = {New algebraic aspects of perturbative and non-perturbative Quantum Field Theory},
author = {Christoph Bergbauer and Dirk Kreimer},
journal= {arXiv preprint arXiv:0704.0232},
year = {2009}
}
Comments
15 pages, Contribution to the Proceedings of the ICMP 2006, 2 typos corrected, final version