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

Identification of the theory of multidimensional orthogonal polynomials  with the theory of symmetric interacting Fock spaces with finite dimensional  one particle space