Bialgebra of specified graphs and external structures
Dominique Manchon, Mohamed Belhaj Mohamed
TL;DR
This paper develops a Hopf algebra framework for specified Feynman graphs in quantum field theory, enabling algebraic renormalization techniques like Birkhoff decomposition for different schemes.
Contribution
It introduces a Hopf algebra structure on specified Feynman graphs and implements algebraic renormalization methods for quantum field theory.
Findings
Constructed a Hopf algebra on specified Feynman graphs.
Implemented Birkhoff decomposition for renormalization schemes.
Facilitated algebraic treatment of renormalization processes.
Abstract
We construct a Hopf algebra structure on the space of specified Feynman graphs of a quantum field theory. We introduce a convolution product and a semigroup of characters of this Hopf algebra with values in some suitable commutative algebra taking momenta into account. We then implement the renormalization described by A. Connes and D. Kreimer and the Birkhoff decomposition for two renormalization schemes: the minimal subtraction scheme and the Taylor expansion scheme.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Nonlinear Waves and Solitons