Collective potential for large-N Hamiltonian matrix models and free Fisher information

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. Noncommutative 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 crucial in determining the ground state. We find that this collective potential has a natural meaning in terms of noncommutative 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 noncommutative analog of the Cramer-Rao inequality.