Entropy of Operator-valued Random Variables: A Variational Principle for Large N Matrix Models

TL;DR

This paper introduces a variational principle for large N matrix models based on an entropy concept for non-commutative probability distributions, enabling new approximation methods.

Contribution

It formulates matrix models as a classical theory with an entropy-based action principle, providing a novel approach to solve these models.

Findings

Derived an explicit formula for the entropy of non-commutative distributions.

Established an action principle incorporating entropy for matrix models.

Achieved reasonable agreement with existing methods in simple cases.

Abstract

We show that, in 't Hooft's large N limit, matrix models can be formulated as a classical theory whose equations of motion are the factorized Schwinger--Dyson equations. We discover an action principle for this classical theory. This action contains a universal term describing the entropy of the non-commutative probability distributions. We show that this entropy is a nontrivial 1-cocycle of the non-commutative analogue of the diffeomorphism group and derive an explicit formula for it. The action principle allows us to solve matrix models using novel variational approximation methods; in the simple cases where comparisons with other methods are possible, we get reasonable agreement.