International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 3, Pages 127-148
doi:10.1155/S0161171202106028
A quantum field theoretical representation of Euler-Zagier sums
1Institut für Physik, Johannes-Gutenberg-Universität Mainz, Staudinger-Weg 7, Mainz D-55099, Germany
2Laboratoire d'Annecy-le-Vieux de Physique Théorique LAPTH, Chemin de Bellevue, BP 110, Annecy-le-Vieux CEDEX F-74941, France
Received 4 June 2001; Revised 16 January 2002
Copyright © 2002 Uwe Müller and Christian Schubert. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We establish a novel representation of arbitrary Euler-Zagier
sums in terms of weighted vacuum graphs. This representation uses
a toy quantum field theory with infinitely many propagators and
interaction vertices. The propagators involve Bernoulli
polynomials and Clausen functions to arbitrary orders. The
Feynman integrals of this model can be decomposed in terms of a
vertex algebra whose structure we investigate. We derive a large
class of relations between multiple zeta values, of arbitrary
lengths and weights, using only a certain set of graphical
manipulations on Feynman diagrams. Further uses and possible
generalisations of the model are pointed out.