Abstract
We give an overview of the use of combinatorics in renormalization of gauge theories, using the Connes–Kreimer Hopf algebra. We prove some physical results in perturbative quantum gauge theories without relying on the formal manipulations involving path integrals. Instead, we take a perturbative series of Feynman graphs as a starting point. After a careful setup and counting of Feynman graphs, we study the structure of the renormalization Hopf algebra of gauge theories on the level of Green’s functions. This involves Slavnov– Taylor identities, described by Hopf ideals, and Dyson–Schwinger equations, described by Hochschild cocycles [7]. As a new result, we prove the Kreimer’s gauge theory theorem formulated in [14].
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