A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion

Abstract

The Bogoliubov recursion is a particular procedure appearing in the process of renormalization in perturbative quantum field theory. It provides convergent expressions for otherwise divergent integrals. We develop here 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. Our results generalize the seminal Cartier–Rota theory of classical Spitzer-type identities for commutative Rota–Baxter algebras. In the classical, commutative, case these identities can be understood as deriving from the theory of symmetric functions. Here 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. Their action on the pro-unipotent groups such as those of perturbative renormalization is described in detail along the way.