Collective potential for large N hamiltonian matrix models and free Fisher information

TL;DR

This paper develops a new formulation of large N matrix models using non-commutative probability theory, revealing a connection to free Fisher information and enabling variational analysis of complex models.

Contribution

It introduces a novel approach to large N matrix models via non-commutative variables, linking collective potential to free Fisher information and establishing a variational principle.

Findings

Derived a variational principle for large N matrix models.

Connected collective potential to free Fisher information.

Achieved accurate approximations for complex models.

Abstract

We formulate the planar `large N limit' of matrix models with a continuously infinite number of matrices directly in terms of U(N) invariant variables. Non-commutative probability theory, is found to be a good language to describe this formulation. The change of variables from matrix elements to invariants induces an extra term in the hamiltonian,which is crucual in determining the ground state. We find that this collective potential has a natural meaning in terms of non-commutative probability theory:it is the `free Fisher information' discovered by Voiculescu. This formulation allows us to find a variational principle for the classical theory described by such large N limits. We then use the variational principle to study models more complex than the one describing the quantum mechanics of a single hermitian matrix (i.e., go beyond the so called D=1 barrier). We carry out approximate variational calculations for a few models and find excellent agreement with known results where such comparisons are possible. We also discover a lower bound for the ground state by using the non-commutative analogue of the Cramer-Rao inequality.