Planar Approximation as Two-Field Boltzmann Theory

TL;DR

This paper introduces a modified two-field Boltzmann theory framework to efficiently sum planar diagrams in large N matrix field theories, advancing the understanding of master fields in quantum field theory.

Contribution

It develops a novel two-field Boltzmannian representation for the master field, enabling recursive equations for planar diagram summation.

Findings

Representation simplifies the summation of planar diagrams.

Half-planar diagrams are isolated in the new formalism.

Potential for recursive integral-differential equations for large N theories.

Abstract

A modified interaction representation for the master field describing connected $SU(N)$-invariant Wightman's functions in the large $N$ limit of matrix fields is constructed. This construction is based on the representation of the master field in terms of Boltzmannian field theory found before. In the modified interaction representation we deal with two scalar Boltzmann fields ({\it up} and {\it down} fields). For up and down fields only half-planar diagrams contribute and this could help to write down a recursive set of non-linear integral-differential equations summing up planar diagrams.