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.
CITATION STYLE
Bergbauer, C., & Kreimer, D. (2009). New Algebraic Aspects of Perturbative and Non-perturbative Quantum Field Theory. In New Trends in Mathematical Physics (pp. 45–58). Springer Netherlands. https://doi.org/10.1007/978-90-481-2810-5_4
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