$d$-Dimensional orthogonal polynomials, the quantum decomposition of a classical random variable and symmetric interacting Fock spaces over $\mathbb C^d$

Abstract:
The notion of interacting Fock spaces (IFS) was introduced by Accardi, Lu and Volovich in the paper [1] which was, in its turn, motivated by the stochastic limit of quantum electrodynamics.
In 1998 Accardi and Bozeiko proved the identification of the theory of orthogonal polynomials in one variable with the theory of one-mode IFS [2].
The problem of developing a satisfactory theory of multi-dimensional orthogonal polynomials is much more difficult and has remained open for several decades. In fact, already in the 1953 monograph [3] the authors complain that “…There does not seem to be an extensive general theory of orthogonal polynomials in several variables …”
The root of this problem is related to the fact that the multi-dimensional extensions of Favard's theorem existing, up to now, in the literature were not satisfactory for several reasons that will be discussed in the talk.
In a recent joint paper with A. Barhoumi and A. Dhahri we have proved that the theory of orthogonal polynomials in $d$ variables ($d \in \mathbb N$) can be canonically identified to the theory of symmetric interacting Fock spaces over $\mathbb C^d$.
An essential tool for the proof is the notion of quantum decomposition of a vector valued random variable which is nothing but the interpretation of the tri-diagonal Jacobi relations in terms of creation, preservation and annihilation operators.
In this identification the commutativity of the coordinates imposes non trivial commutation relations, between the creation, preservation and annihilation operators, which are uniquely determined by the generalized principal Jacobi sequence.
In the case of the Gaussian (or of the Poisson distribution, which has the same principal Jacobi sequence) one recovers the usual Heisenberg commutation relations.
Thus quantum mechanics emerges not as a generalization of classical probability but as its natural prolongation.

Language: English

References

L. Accardi, Y. G. Lu, I. Volovich, “The QED Hilbert module and Interacting Fock spaces”, IIAS reports 1997-008, Publications of IIAS, Kyoto, 1997

L. Accardi L., M. Bożejko, “Interacting Fock spaces and Gaussianization of probability measures”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 01:4 (1998), 663–670

A. Erdely, W. Magnus, F. Oberhettinger, F. Tricomi, Higher transcendental functions, v. 2, McGraw–Hill, 1953