  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.
