Moments and Commutators of Probability Measures

Abstract

Let a 0 , a − and a + be the preservation, annihilation, and creation operators of a probability measure µ on R d , respectively. The operators a 0 and [a − , a + ] are proven to uniquely determine the moments of µ. We discuss the question: "What conditions must two families of operators satisfy, in order to ensure the existence of a probability measure, having finite moments of any order, so that, its associated preservation operators and commutators between the annihilation and creation operators are the given families of operators?" For the case d = 1, a satisfactory answer to this question is obtained as a simple condition in terms of the Szegö-Jacobi parameters. For the multidimensional case, we give some necessary conditions for the answer to this question. We also give a table with the associated preservation and commutator between the annihilation and creation operators, for some of the classic probability measures on R.

On the multi-dimensional Favard lemma

“…It is clear that P n = P * n , n≥0 P n = 1 and P n P m = δ nm P n for all n, m ∈ N. Moreover, for any 1 ≤ j ≤ d and any n ∈ N, one has X j P n = P n+1 X j P n + P n X j P n + P n−1 X j P n (4) with the convention that P −1] = 0. The identity (4) is called the symmetric Jacobi relation.…”

Itô Calculus and Quantum White Noise Calculus

“…The first result in this direction was proved by von Waldenfels in the Bose and Fermi case [voWaGi78], [voWa78] and about 20 years later, with the introduction of the notion of interacting Fock space [AcLuVo97b], this principle became a quite universal principle of probability theory and opened the way to the program of a full algebraic classification of probability measures. This is a quite interesting direction, and is also deeply related to the main topic of the present paper, stochastic and white noise calculus, but we will not discuss this connection and we refer the interested reader to [AcBo98], [AcKuSt02], [AcKuSt05a].…”

Extensions of Quantum Theory Canonically Associated to Classical Probability Measures

“…However some discoveries of the past 15 years [1,6,7,3,4,5], show that the whole quantum theory, including quantum fields, is not a generalization, but rather a deeper level of classical probability. The fecundity of these new ideas is testified by the fact that they have allowed to solve a multiplicity of longstanding open problems both in classical probability and in the theory of orthogonal polynomials.…”