Renormalization in quantum field theory and the Riemann-Hilbert problem

Abstract

We show that renormalization in quantum field theory is a special instance of a general mathematical procedure of multiplicative extraction of finite values based on the Riemann-Hilbert problem. Given a loop $\gamma(z), | z |=1$ of elements of a complex Lie group G the general procedure is given by evaluation of $ \gamma_{+}(z)$ at z=0 after performing the Birkhoff decomposition $ \gamma(z)=\gamma_{-}(z)^{-1} \gamma_{+}(z)$ where $ \gamma_{\pm}(z) \in G$ are loops holomorphic in the inner and outer domains of the Riemann sphere (with $\gamma_{-}(\infty)=1$). We show that, using dimensional regularization, the bare data in quantum field theory delivers a loop (where z is now the deviation from 4 of the complex dimension) of elements of the decorated Butcher group (obtained using the Milnor-Moore theorem from the Kreimer Hopf algebra of renormalization) and that the above general procedure delivers the renormalized physical theory in the minimal substraction scheme.