Renormalization has been settled as a self-consistent approach to the treatment of short-distance singularities in the perturbative expansion of quantum field theories thanks to the work of Bogoliubov, Parasuik, Hepp, Zimmermann, and followers. Nevertheless, its intricate Hopf algebra combinatorics went unrecognized for a long time. In this talk we want to describe recent results devoted to the Hopf algebra structure of quantum field theory (QFT).
The talk will review the results obtained so far in a fairly informative style, emphasizing the underlying ideas and concepts. Furthermore, challenges for further investigation will be emphasized, as it more and more becomes clear that this Hopf algebra structure provides a fine tool for a better understanding of a correct mathematical formulation of QFT as well as for applications of concepts from operad theory to number theory. A review can be found in http://arXiv.org/ps/hep-th/0010059.
Return to: DMJ-IMRN Conference * Department of Mathematics * Duke University
Last modified: March 9, 2001