New trends in quantum probability and quantum mechanics

The solution of the problem to generalize, to the multi-dimensional case, the Favard Lemma, which lies at the foundations of the theory of orthogonal polynomials in one real variable, was recently obtained in the joint paper [1]. This suggests a new approach to the theory of orthogonal polynomials, that emphasizes the fact that it generalizes in several, unexpected and non-trivial ways the usual mathematical structure of quantum mechanics and (in its infinite dimensional version) quantum field theory.
These generalizations will be briefly described and it will explained in what sense they extend the program of "non-linear quantization" whose first achievement was obtained in the paper [2].