Bergbauer, Christoph; Kreimer, Dirk New algebraic aspects of perturbative and non-perturbative quantum field theory. (English) Zbl 1175.81154 Sidoravičius, Vladas (ed.), New trends in mathematical physics. Selected contributions of the XVth international congress on mathematical physics, Rio de Janeiro, Brazil, August 5–11, 2006. Dordrecht: Springer (ISBN 978-90-481-2809-9/hbk; 978-90-481-2810-5/ebook). 45-58 (2009). Summary: 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.For the entire collection see [Zbl 1170.81006]. Cited in 3 Documents MSC: 81T15 Perturbative methods of renormalization applied to problems in quantum field theory 81T18 Feynman diagrams 16T05 Hopf algebras and their applications 16W25 Derivations, actions of Lie algebras 81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry PDFBibTeX XMLCite \textit{C. Bergbauer} and \textit{D. Kreimer}, in: New trends in mathematical physics. Selected contributions of the XVth international congress on mathematical physics, Rio de Janeiro, Brazil, August 5--11, 2006. Dordrecht: Springer. 45--58 (2009; Zbl 1175.81154) Full Text: DOI arXiv