  In this Colloqium Lecture (by one of the authors (D.S)) a thorough
presentation of the authors' research on the subjects, stated in the title, is
given. By quite laborious mathematics it is explained how one can handle
systems in which each Heisenberg commutation relation is deformed separately.
For Hilbert space realizability a detailed determinant computations (extending
Zagier's one-parameter formulas) are carried out. The inversion problem of the
associated Gram matrices on Fock weight spaces is completely solved (Extended
Zagier's conjecture) and a counterexample (for $n=8$) to the original Zagier's
conjecture is presented in detail.
