Renormalization in quantum field theory and the Riemann-Hilbert problem

Alain Connes; Dirk Kreimer

doi:10.1088/1126-6708/1999/09/024

paper

Renormalization in quantum field theory and the Riemann-Hilbert problem

arXiv:hep-th/9909126 · doi:10.1088/1126-6708/1999/09/024

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 $γ (z), ∣ z ∣ = 1$ of elements of a complex Lie group G the general procedure is given by evaluation of $γ_{+} (z)$ at z=0 after performing the Birkhoff decomposition $γ (z) = γ_{-} (z)^{- 1} γ_{+} (z)$ where $γ_{\pm} (z) \in G$ are loops holomorphic in the inner and outer domains of the Riemann sphere (with $γ_{-} (\infty) = 1$ ). 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.

8 pages, plain LaTeX

Renormalization in quantum field theory and the Riemann-Hilbert problem

Abstract

Cited by in corpus (13)