Mathematical Renormalization in Quantum Electrodynamics via Noncommutative Generating Series

Abstract

In order to push the study of solutions of nonlinear differential equations involved in quantum electrodynamics 1 , we focus here on combinatorial aspects of their renormalization at {0, 1, +∞}.

Harmonic sums and polylogarithms at non-positive multi-indices

“…In here, z 0 = 0 if ι 0 f (resp. ι 1 f ) exists 12 and else z 0 = 1. One can check easily that θ 0 + θ 1 = ∂ z and θ 0 ι 0 = θ 1 ι 1 = Id [12].…”

“…3. Combinatorial aspects of harmonic sums and polylogarithms 12 This can be made precise, remarking that C{Li w } w∈X * = C ⊗ C C{Li w } w∈X * and then study the integral on the basis z k…”

Kleene stars of the plane, polylogarithms and symmetries

“…Firstly, (from the algebraic point of view) these series are closed under shuffle products and the shuffle exponential of letters (and their linear combinations) is precisely their Kleene star 5 . Secondly, the growth of their coefficients is tame 6 [11,25,26]. [11,25,26] and as such their associated polylogarithms can be rightfully computed [20,23,24].…”

“…Secondly, the growth of their coefficients is tame 6 [11,25,26]. [11,25,26] and as such their associated polylogarithms can be rightfully computed [20,23,24].…”

“…. .y s r as follows [10,11,21,22] : Li y s 1 ...y s r = Li s 1 ,...,s r and Li − y s 1 ...y s r = Li −s 1 ,...,−s r . We will explain the whole project to extend Li • over a sub algebra of rational power series.…”