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

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

2001-11-28

arxiv.org/abs/hep-th/0111263v2

Int.J.Mod.Phys. A17 (2002) 2413-2444

Physics

High Energy Physics

High Energy Physics - Theory

45 pages with 1 figure, added references

Scientific paper

10.1142/S0217751X02010790

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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