Identification of the theory of orthogonal polynomials in d-indeterminates with the theory of 3-diagonal symmetric interacting Fock spaces on ℂd

Abstract

The identification mentioned in the title allows a formulation of the multidimensional Favard lemma different from the ones currently used in the literature and which parallels the original 1-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 positive definite kernels. The above result opens the way to the program of a purely algebraic classification of probability measures on ${ℝ}^d$ with moments of any order and more generally of states on the polynomial algebra on ${ℝ}^d$. The quantum decomposition of classical real-valued random variables with all moments is one of the main ingredients in the proof.