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Solving Bogoliubov's recursion in renormalisation using a simple algebraic identity

Thursday 10th April 2008 - 15:30 to 16:15
INI Seminar Room 1

The Bogoliubov recursion is a particular procedure appearing in the process of renormalization in perturbative quantum field theory. In this talk we present a theory of functional identities for noncommutative Rota-Baxter algebras which is shown to encode, among others, this process in the context of Connes-Kreimer's Hopf algebra of renormalization. Glen Baxter originally introduced the notion of commutative Rota-Baxter algebra to give a more conceptual proof of Spitzer's identity known from fluctuation theory. In the commutative case these identities can be understood as deriving from the theory of symmetric functions. We show that an analogous property holds for noncommutative Rota-Baxter algebras. That is, we show that functional identities in the noncommutative setting can be derived from the theory of noncommutative symmetric functions. Lie idempotents, and particularly the Dynkin idempotent play a crucial role in the process. As an application we present a closed formula for Bogoliubov's recursion in the context of Connes-Kreimer's Hopf algebra of renormalization.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons