Séminaire Physique mathématique ICJ

On a renormalization "loop" in pQFT

par Prof. Alessandra Frabetti

Europe/Paris
Zoom (Institut Camille Jordan)

Zoom

Institut Camille Jordan

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Description

In perturbative quantum field theory, the renormalization group is a group of formal diffeomorphisms in the powers of the coupling constant, acting on Green's functions and on the Lagrangian by substitution of the bare coupling constant and multiplication by some renormalization factors. The coefficients of these series are built on the counterterms of divergent Feynman graphs by means of the BPHZ formula.

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. In principle, then, the BPHZ formula can be pushed to the sum of Feynman graphs describing the integral coefficients and give rise to a recursive algorithm at each order of interacion.  

For non-scalar theories, Feynman graphs have matrix-valued amplitudes: even if the counterterms are scalar-valued, the renormalization group cannot be functorially represented by a Hopf algebra, because associativity fails for the composition of series with non-commutative coefficients. In this talk I discuss its functorial extention to matrix-valued series as a loop (a non-associative group), based on the joint work https://doi.org/10.1016/j.aim.2019.04.053 with Ivan P. Shestakov.
 

Organisé par

Nguyen-Viet Dang