# Feynman graph representation of the perturbation series for general functional measures

August 20, 2004

A representation of the perturbation series of a general functional measure
is given in terms of generalized Feynman graphs and -rules. The graphical
calculus is applied to certain functional measures of Lévy type. A graphical
notion of Wick ordering is introduced and is compared with orthogonal
decompositions of the Wiener-Itô-Segal type. It is also shown that the linked
cluster theorem for Feynman graphs extends to generalized Feynman graphs. We
perturbatively prove existence of the thermodynamic limit for the free energy
density and the moment functions. The results are applied to the gas of charged
microscopic or mesoscopic particles - neutral in average - in $d=2$
dimensions generating a static field $\phi$ with quadratic energy density
giving rise to a pair interaction. The pressure function for this system is
calculated up to fourth order. We also discuss the subtraction of
logarithmically divergent self-energy terms for a gas of only one particle type
by a local counterterm of first order.

Keywords:

perturbation theory, Feynman diagrams, Wick ordering