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Infinite Dimensional Analysis Quantum Probability and Related Topics

A characterization of probability measures in terms of semi-quantum operators

Gabriela Popa, Aurel I. Stan

Published Jun 1, 2019• 5 citations
Publisher

Abstract

A canonical definition of the joint semi-quantum operators of a finite family of random variables, having finite moments of all orders, is given first in terms of an existence and uniqueness theorem. Then two characterizations, one for the polynomially symmetric, and another for the polynomially factorizable probability measures, having finite moments of all orders, are presented.

DOI
10.1142/s0219025719500097

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A canonical definition of the joint semi-quantum operators of a finite family of random variables, having finite moments of all orders, is given first in terms of an existence and uniqueness theorem. Then two characterizations, one for the polynomially symmetric, and another for the polynomially factorizable probability measures, having finite moments of all orders, are presented.
We present first a set of commutator relationships involving the joint quantum, semi-quantum, and number operators generated by a finite family of random variables, having finite moments of all orders, and show how these commutators can be used to recover the joint quantum operators from the semi-quantum operators. We show that any linear operator defined on an algebra of polynomials or the polynomial random variables, generated by a finite family of random variables, having finite moments of all orders, can be written uniquely as an infinite sum of compositions of the multiplication operators, generated by these random variables, and the partial derivative operators. In the terms of this sum, each multiplication operator is placed to the left side of each partial derivative operator. We provide many examples concerning the decomposition of some classic operators.
This paper discusses the properties of the canonical quantum decomposition of the classical Pearson random variables. We show that this leads to the problem of representing the creation–annihilation–preservation (CAP) operators canonically associated to a real-valued random variable [Formula: see text] with all moments as (normally ordered) differential operators with polynomial coefficients — a problem already studied in the literature (see references in the introduction). We deduce a formula, for the polynomial coefficients in the representation of the CAP operators of [Formula: see text] as pseudo-differential operators, more explicit than the one existing in the literature. We give a new characterization of the Pearson distributions in terms of the Hermitianity of the associated Sturm–Liouville operators. In the second part of the paper, we introduce the notion of finite type random variable [Formula: see text] and characterize type-[Formula: see text] and type-[Formula: see text] real-valued random variables. We prove that a necessary condition for [Formula: see text] to be of finite type is the polynomial growth of the corresponding principal Jacobi sequence. This allows to single out three classes of random variables of infinite type and to prove that the Beta and the uniform distributions are of infinite type.
The fact that any classical random variable with all moments has a quantum decomposition allows to associate to it a family of quantum moments. On the other hand, a classical random variable may have several inequivalent quantum decompositions, which lead to the same classical, but different quantum moments. Even in the simplest Central Limit Theorems (CLT), i.e. those of Bernoulli type, there are examples in which the corresponding quantum moments converge to the canonical quantum moments of the associated classical random variable, and examples in which this is not the case. This poses the problem to find a constructive criterium that characterizes the quantum moments associated to the canonical quantum decomposition (which is unique) with respect to the other ones. Theorem [Formula: see text] of the present paper provides such a criterium. Theorem [Formula: see text] deals with the case when one knows a priori that the quantum moments come from a central limit theorem (the motivation of the present paper arose in this context). It gives only a sufficient condition, but simpler to verify than the necessary and sufficient conditions of Theorem [Formula: see text]. Theorem [Formula: see text] naturally leads to a classification of Interacting Fock Spaces (IFS) into three types. We construct examples showing that all these possibilities can effectively take place. On the way, we prove that all the best known deformations of Heisenberg commutation relations can be obtained as special cases of a general construction within the algebraic approach to the theory of orthogonal polynomials.