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 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)-1γ+(z) where γ+/-(z)inG are loops holomorphic in the inner and outer domains of the Riemann sphere (with γ-(∞) = 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.
- Publication:
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Journal of High Energy Physics
- Pub Date:
- September 1999
- DOI:
- 10.1088/1126-6708/1999/09/024
- arXiv:
- arXiv:hep-th/9909126
- Bibcode:
- 1999JHEP...09..024C
- Keywords:
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- High Energy Physics - Theory;
- High Energy Physics - Phenomenology;
- Mathematical Physics;
- Mathematics - Quantum Algebra
- E-Print:
- 8 pages, plain LaTeX