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International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 3, Pages 127-148

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 Hindawi Publishing Corporation. 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.


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.