  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 an algebra of
elementary vertex integrals 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 generalizations of the model are pointed
out.

Quantum Algebra;High Energy Physics - Theory;Number Theory