  The identification mentioned in the title allows a formulation of the multidi
mensional Favard Lemma different from the ones currently used in the literature
and which exactly parallels the original one dimensional formulation in the
sense that the positive Jacobi sequence is replaced by a sequence of positive
Hermitean (square) matrices and the real Jacobi sequence by a sequence of
Hermitean matri ces of the same dimension. Moreover, in this identification,
the multidimensional extension of the compatibility condition for the positive
Jacobi sequence becomes the condition which guarantees the existence of the
creator in an interacting Fock space. The above result opens the way to the
program of a purely algebraic clas sification of probability measures on
$\mathbb{R}^d$ with finite moments of any order. In this classification the
usual Boson Fock space over $\mathbb{C}^d$ is characterized by the fact that
the positive Jacobi sequence is made up of identity matrices and the real
Jacobi sequences are identically zero. The quantum decomposition of classical
real valued random variables with all moments is one of the main ingredients in
the proof.
