  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.
