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Journal of Mathematical Physics

A study of random variables in terms of the number operator

Aurel I. Stan, Gabriela Popa, Ruma Dutta

Published Mar 1, 2023• 4 citations
Publisher

Abstract

We show first how the joint semi-quantum and quantum operators of a finite family of random variables having finite moments of all orders can be recovered from their joint number operator. We then characterize the polynomially symmetric and polynomially factorizable random variables in terms of their joint number operator. Finally, we present the quantum decomposition of the number and quantum operators of the random variables whose orthogonal polynomials are the Gegenbauer polynomials.

DOI
10.1063/5.0124172

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We show first how the joint semi-quantum and quantum operators of a finite family of random variables having finite moments of all orders can be recovered from their joint number operator. We then characterize the polynomially symmetric and polynomially factorizable random variables in terms of their joint number operator. Finally, we present the quantum decomposition of the number and quantum operators of the random variables whose orthogonal polynomials are the Gegenbauer polynomials.
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.
The quantum decomposition of a classical random variable is one of the deep results of quantum probability: it shows that any classical random variable or stochastic process has a built-in non-commutative structure which is intrinsic and canonical, and not artificially put by hands. Up to now the technique to deduce the quantum decomposition has been based on the theory of interacting Fock spaces and on Jacobi's tri-diagonal relation for orthogonal polynomials. Therefore it requires the existence of moments of any order and cannot be applied to random variables without this property. The problem to find an analogue of the quantum decomposition for random variables without finite moments of any order remained open for about fifteen years and nobody had any idea of how such a decomposition could look like. In the present paper we prove that any infinitely divisible random variable has a quantum decomposition canonically associated to its Lévy–Khintchin triple. The analytical formulation of this result is based on Kolmogorov representation of these triples in terms of 1–cocycles (helices) in Hilbert spaces and on the Araki–Woods–Parthasarathy–Schmidt characterization of these representation in terms of Fock spaces. It distinguishes three classes of random variables: (i) with finite second moment; (ii) with finite first moment only; (iii) without any moment. The third class involves a new type of renormalization based on the associated Lévy–Khinchin function.
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.