Feynman graph representation of the perturbation series for general functional measures

TL;DR

This paper introduces a generalized Feynman graph framework for perturbation series of functional measures, applies it to Levy-type measures, and demonstrates its utility in statistical physics models, including particle gases.

Contribution

It develops a graphical calculus for general functional measures, extending Feynman graph techniques and proving the existence of thermodynamic limits in complex systems.

Findings

Extended Feynman graph representation for Levy measures

Proved thermodynamic limit existence for free energy and moments

Calculated pressure up to fourth order for a charged particle gas

Abstract

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 L\'evy type. A graphical notion of Wick ordering is introduced and is compared with orthogonal decompositions of the Wiener-It\^o-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.