Collective Potential for Large-N Hamiltonian Matrix Models and Free Fisher Information

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.

YANGIAN SYMMETRIES OF MATRIX MODELS AND SPIN CHAINS: THE DILATATION OPERATOR OF ${\mathcal N} = 4$ SYM

“…By now there is a large literature devoted to these techniques and their applications to low dimensional string theory. When dealing with the quantum mechanics of several matrices, it is not clear how to make such direct approaches work (although extending the collective field theory formalism has been shown to hold a lot of promise [76,65,66]). However, one can proceed to isolate the operators and the states that dominate the large N limit, work out the algebra of observables obeyed by the dominant observables and continue to work within this restricted sector.…”

Noncommutative probability, matrix models, and quantum orbifold geometry

“…The large-N expansion of matrix models was found out to be a genus expansion [3]; the dual of the Feynman diagrams of the leading term may be treated as discretised string worldsheets with the topology of a sphere [4,5,6]. The commonality of random matrices in mathematics and physics triggers an interest in the relationship between noncommutative probability theory of type A and large-N matrix models [7,8,9,10,11,12]. It is now possible to think of notions of noncommutative probability in physical terms and describe the master field in algebraic terms.…”

Non-anomalous `Ward' identities to supplement large-Nmulti-matrix loop equations for correlations

“…Meanwhile, there has been a steady stream of developments in matrix models of which we cite a few examples. These include their connections to non-commutative probability theory [9,10,11,12,13,14], the study of multi-matrix symmetry algebras and their connections to spin chains [15,16], exact solutions [17] and their relation to CFT [18] and algebraic geometry and detailed studies of the loop equations [19,20]. Much of the existing literature deals with 1-matrix models or exact solutions for specific observables of carefully chosen multi-matrix models.…”