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Journal of Stochastic Analysis

Quantum Theories Associated to Increasing Hilbert Space Filtrations and Generalized Jacobi 3–Diagonal Relation

Luigi Accardi, Yun Gang Lu

Published Feb 14, 2021• 2 citations
Publisher

Abstract

We prove that the quantum decomposition of a classical random variable, or random field, is a very general phenomenon involving only an increasing filtration of Hilbert spaces and a family of Hermitean operators increasing by 1 the filtration.The creation, annihilation and preservation operators (CAP operators), defining the quantum decomposition of these Hermitean operators, satisfy commutation relations that generalize those of usual quantum mechanics.In fact there are two types of commutation relations (Type I and Type II).In Type I commutation relations the commutator is given by an operator-valued sesqui-linear form.The case when this operator-valued sesqui-linear form is scalar valued (multiples of the identity) characterizes the non-relativistic free Bose field and the associated commutation relations reduce to the Heisenberg ones.Type II commutation relations did not appear up to now because they are identically satisfied when the probability distribution of the random field is a product measure.In this sense they encode information on the self-interaction of the random field.

DOI
10.31390/josa.2.1.04

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We prove that the quantum decomposition of a classical random variable, or random field, is a very general phenomenon involving only an increasing filtration of Hilbert spaces and a family of Hermitean operators increasing by 1 the filtration.The creation, annihilation and preservation operators (CAP operators), defining the quantum decomposition of these Hermitean operators, satisfy commutation relations that generalize those of usual quantum mechanics.In fact there are two types of commutation relations (Type I and Type II).In Type I commutation relations the commutator is given by an operator-valued sesqui-linear form.The case when this operator-valued sesqui-linear form is scalar valued (multiples of the identity) characterizes the non-relativistic free Bose field and the associated commutation relations reduce to the Heisenberg ones.Type II commutation relations did not appear up to now because they are identically satisfied when the probability distribution of the random field is a product measure.In this sense they encode information on the self-interaction of the random field.
We survey some results obtained in the last 25 years to illustrate the following probabilistic quantization statement: quantum probability is not a generalization of classical probability but a deeper level of it. Generalizing the classical theory of orthogonal polynomials, one can show that any random field XXX admitting all the moments can be represented as the sum of three operators, which are natural extensions of the creation, annihilation, and preservation operators in the usual boson Fock quantum theory. These operators generate a noncommutative ∗*∗-algebra on which the quantum extension of the expected value relative to the probability distribution of the field XXX induces a quantum state. So, any classical algebraic probability space generates a quantum space with its own commutation relations. The Heisenberg commutation relation characterizes the classical fields, while the new type of commutation relations (of type II) appears in non-Gaussian cases. The same machinery, but applied to Bernoulli fields, leads to Fermi-Dirac anticommutation relations (see the introduction).
We survey some results obtained in the last 25 years to illustrate the following probabilistic quantization statement: quantum probability is not a generalization of classical probability but a deeper level of it. Generalizing the classical theory of orthogonal polynomials, one can show that any random field XXX admitting all the moments can be represented as the sum of three operators, which are natural extensions of the creation, annihilation, and preservation operators in the usual boson Fock quantum theory. These operators generate a noncommutative ∗*∗-algebra on which the quantum extension of the expected value relative to the probability distribution of the field XXX induces a quantum state. So, any classical algebraic probability space generates a quantum space with its own commutation relations. The Heisenberg commutation relation characterizes the classical fields, while the new type of commutation relations (of type II) appears in non-Gaussian cases. The same machinery, but applied to Bernoulli fields, leads to Fermi--Dirac anticommutation relations (see the introduction).
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 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.
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.
For classical field theories with probabilistic initial conditions the classical field observables are an idealization. Their arbitrarily precise values poorly reflect the characteristic uncertainty in the presence of substantial fluctuations. We propose to employ observables based on fluctuating fields. In terms of these "statistical observables" the probabilistic classical field theory becomes a quantum field theory. Non-commuting operators are associated to observables. The quantum rules follow from the laws for classical probabilities. The ``quantum'' is part of the ``classical''. A regularized functional integral guarantees the unitarity of the quantum field theory. We discuss in detail the classical relativistic Klein-Gordon equation with interactions.