Mastering the Master Field

TL;DR

This paper reviews non-commutative probability theory and applies it to construct the master field in large N matrix models, including coupled models, providing explicit constructions and equations of motion.

Contribution

It introduces a general framework for constructing the master field in any matrix model using non-commutative probability and Cuntz algebra, extending previous specific cases.

Findings

Explicit construction of the master gauge field in QCD2

Extension of techniques to coupled matrix models

Formulation of equations of motion for the master field

Abstract

The basic concepts of non-commutative probability theory are reviewed and applied to the large $N$ limit of matrix models. We argue that this is the appropriate framework for constructing the master field in terms of which large $N$ theories can be written. We explicitly construct the master field in a number of cases including QCD$_2$. There we both give an explicit construction of the master gauge field and construct master loop operators as well. Most important we extend these techniques to deal with the general matrix model, in which the matrices do not have independent distributions and are coupled. We can thus construct the master field for any matrix model, in a well defined Hilbert space, generated by a collection of creation and annihilation operators---one for each matrix variable---satisfying the Cuntz algebra. We also discuss the equations of motion obeyed by the master field.