Noncommutative renormalization Hopf algebras

Abstract: In pQFT, the renormalization group acts on the Lagrangian as a group of formal diffeomorphisms in the powers of the coupling constant, by substitution of the bare coupling and multiplication by some renormalization factors built on the counterterms of divergent Feynman graphs.

For scalar theories, such groups are proalgebraic (functorial on the coefficients algebra) and are represented by Faà di Bruno types of Hopf algebras on graphs, called renormalization Hopf algebras. In this talk I review Connes-Kreimer's settings and comment on the improvements expected for the BPHZ formula which computes the counterterms of the graphs.

For non-scalar theories, Feynman graphs have matrix-valued amplitudes: even if the counterterms are scalar-valued, the renormalization group cannot be represented by a Hopf algebra in a functorial way, because associativity fails for the composition of series with non-commutative coefficients. Both commutative and noncommutative renormalization Hopf algebras can be defined, with different meanings. In this talk I explain in which sense the first ones are not functorial (hence not universal) and how the second ones require a functorial extension of proalgebraic groups to non-commutative algebras which can only be done as "non-associative" groups.

The talk is based on Connes-Kreimer's results (2000), on joint works with Christian Brouder (2000-2006) and on the recent paper doi.org/10.1016/j.aim.2019.04.053

mathematical physics

Audience: researchers in the topic

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Noncommutative Geometry and Physics

Series comments: Noncommutative geometry is a very general mathematical paradigm arising from quantum mechanics. As such, it permeates different branches of mathematics and physics.

The series of monthly talks accompanies the special issue of Journal of Physics A and is intended to present the many facets of the emergence of noncommutativity in physics.

Past seminars can be viewed on YouTube channel.

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