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

TL;DR

This paper explores non-commutative quantum probability, linking various notions of independence and central limit theorems to mathematical physics, including quantum fields, matrix models, and stochastic bosonization.

Contribution

It introduces new connections between quantum probability, matrix models, and quantum field theory, including stochastic limits and modifications of classical laws.

Findings

Relation of master fields to Voiculescu's freeness results

Quantum stochastic differential equations for QCD fields

New structures in QED like nonlinear Wigner semicircle law

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.