Interacting Fock Spaces and Gaussianization of Probability Measures (original) (raw)

Characterization of Probability Measures Through the Canonically Associated Interacting Fock Spaces

Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2004

We continue our program of coding the whole information of a probability measure into a set of commutation relations canonically associated to it by presenting some characterization theorems for the symmetry and factorizability of a probability measure on ℝd in terms of the canonically associated interacting creation, annihilation and number operators.

Interacting Fock Spaces and Orthogonal Polynomials in several variables

We extend to polynomials in several variables the Accardi-Bozeiko canonical isomorphism between 1-mode interacting Fock spaces and orthogonal polynomials in one variable. This gives a constructive rule to write down easily the quantum decomposition, as a sum of creation, annihilation and number operators, of an arbitrary vector valued random variable with moments of any order. In the multi-mode case not all interacting Fock spaces are canonically isomorphic to spaces of orthogonal polynomials. We characterize those which enjoy this property in terms of a sequence of quadratic commutation relations among finite dimensional matrices.

Identification of the theory of multidimensional orthogonal polynomials with the theory of symmetric interacting Fock spaces with finite dimensional one particle space

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 exactly 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 Hermitean matrices of the same dimension. Moreover, in this identification, the multi-dimensional extension of the compatibility condition for the positive Jacobi sequence becomes the condition which guarantees the existence of the creator in an interacting Fock space. The above result opens the way to the program of a purely algebraic classification of probability measures on R d with finite moments of any order. In this classification the usual Boson Fock space over C d is characterized by the fact that the positive Jacobi sequence is made up of identity matrices and the real Jacobi sequences are identically zero. The quantum decomposition of classical real valued random variables with all moments is one of the main ingredients in the proof.

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

Infinite Dimensional Analysis, Quantum Probability and Related Topics

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 [Formula: see text] with moments of any order and more generally of states on the polynomial algebra on [Formula: see text]. The quantum decomposition of classical real-valued random variables with all moments is one of the main ingredients in the proof.

The quantum moment problem for a classical random variable and a classification of interacting Fock spaces

2018

The fact that any classical random variable with all moments has a canonical 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 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 with respect to the other ones. Theorem 3 of the present paper provides such a criterium. Theorem 5 gives a sufficient condition which reduces the problem to the verification of a 4-th moment conditions (see (91)) which is simpler than the verification of the necessary and sufficient conditions of Theorem 3. Theorem 3 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.

Segal-Bargmann Transforms of One-mode Interacting Fock Spaces Associated with Gaussian and Poisson Measures

2001

Let mug\mu_{g}mug and mup\mu_{p}mup denote the Gaussian and Poisson measures on BbbR{\Bbb R}BbbR, respectively. We show that there exists a unique measure widetildemug\widetilde{\mu}_{g}widetildemug on BbbC{\Bbb C}BbbC such that under the Segal-Bargmann transform SmugS_{\mu_g}Smug the space L2(BbbR,mug)L^2({\Bbb R},\mu_g)L2(BbbR,mug) is isomorphic to the space calHL2(BbbC,widetildemug){\cal H}L^2({\Bbb C}, \widetilde{\mu}_{g})calHL2(BbbC,widetildemug) of analytic L2L^2L2-functions on BbbC{\Bbb C}BbbC with respect to widetildemug\widetilde{\mu}_{g}widetildemug. We also introduce the