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

L. Akant·G. S. Krishnaswami·S. G. Rajeev
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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.

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L. Akant, G. S. Krishnaswami, S. G. Rajeev. 2001-12-05. Entropy of Operator-valued Random Variables: A Variational Principle for Large N Matrix Models. https://doi.org/10.1142/s0217751x02010790

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