Interaction Representation in Boltzmann Field Theory

TL;DR

This paper explores an interaction representation in Boltzmann field theory related to planar diagrams, demonstrating its uniqueness, deriving integral equations, performing renormalization, and validating results through numerical solutions.

Contribution

It introduces and analyzes a unique interaction representation involving rational functions, deriving integral equations, and applying renormalization in Boltzmann field theory.

Findings

The interaction representation is shown to be unique under natural assumptions.

Derived closed integral equations for correlation functions.

Numerical solutions match planar approximation for large coupling constants.

Abstract

We consider an interaction representation in the Boltzmann field theory. It describes the master field for a subclass of planar diagrams in matrix models, so called half-planar diagrams. This interaction representation was found in the previous paper by Accardi, Volovich and one of us (I.A.) and it has an unusual property that one deals with a rational function of the interaction Lagrangian instead of the ordinary exponential function. Here we study the interaction representation in more details and show that under natural assumptions this representation is in fact unique. We demonstrate that corresponding Schwinger-Dyson equations lead to a closed set of integral equations for two- and four-point correlation functions. Renormalization of the model is performed and renormalization group equations are obtained. Some model examples with discrete number of degrees of freedom are solved numerically. The solution for one degree of freedom is compared with the planar approximation for one matrix model. For large variety of coupling constant it reproduces the planar approximation with good accuracy.