The residues of quantum field theory - numbers we should know

Abstract

ABSTRACT. We discuss in an introductory manner structural similarities between the polylogarithm and Green functions in quantum field theory.

Noncommutative Geometry, Quantum Fields and Motives

“…• Kreimer showed in [190] that the period matrix (1.405) of the mixed Hodge-Tate structures associated to the polylogarithms can be seen as solutions of a Dyson-Schwinger equation, which reveals many of the fundamental structures of renormalization in quantum field theory.…”

Hopf algebras in renormalization theory: locality and Dyson–Schwinger equations from Hochschild cohomology

“…The presentation is by no means self-contained, and we refer the reader to the growing literature for more details [29,33,30,31,32,24].…”

“…for all k ≥ m, which is at the heart of a recursive determination of the above coefficients γ j (α) in (31). Here, F cm is the befooting operator…”

From Physics to Number Theory via Noncommutative Geometry

“…The appearance of multiple polylogarithms in the coefficients of divergences in QFT, discovered by Broadhurst and Kreimer ([11], [12]), as well as recent considerations of Kreimer on analogies between residues of quantum fields and variations of mixed Hodge-Tate structures associated to polylogarithms (cf. [80]), suggest the existence for the above category of equisingular flat bundles of suitable Hodge-Tate realizations given by a specific choice of Quantum Field Theory.…”

“…Hopf algebra structures based on rooted trees, that encode the combinatorics of Epstein-Glaser renormalization were developed by Bergbauer and Kreimer [5]. Kreimer developed an approach to the Dyson-Schwinger equation via a method of factorization in primitive graphs based on the Hochschild cohomology of the CK Hopf algebras of Feynman graphs ( [81], [82], [80], cf. also [13]).…”