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Non--Commutative (Quantum) Probability, Master Fields and Stochastic Bosonization

L. Accardi·Y. G. Lu·I. Volovich
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Abstract

In this report we discuss some results of non--commutative (quantum) probability theory relating the various notions of statistical independence and the associated quantum central limit theorems to different aspects of mathematics and physics including: $q$--deformed and free central limit theorems; the description of the master (i.e. central limit) field in matrix models along the recent Singer suggestion to relate it to Voiculescu's results on the freeness of the large $N$ limit of random matrices; quantum stochastic differential equations for the gauge master field in QCD; the theory of stochastic limits of quantum fields and its applications to stochastic bosonization of Fermi fields in any dimensions; new structures in QED such as a nonlinear modification of the Wigner semicircle law and the interacting Fock space: a natural explicit example of a self--interacting quantum field which exhibits the non crossing diagrams of the Wigner semicircle law.

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L. Accardi, Y. G. Lu, I. Volovich. 1994-12-31. Non--Commutative (Quantum) Probability, Master Fields and Stochastic Bosonization. https://arxiv.org/abs/hep-th/9412241

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