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

. 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.