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1 MOMENTS AND q-COMMUTATORS OF NONCOMMUTATIVE RANDOM VECTORS

Wojciech Matysiak

Published Aug 17, 2016• 6 citations
OpenAlex

Abstract

A method for computing the mixed moments of (not necessarily commutative) random vectors from the first-order moments, the q-commutators between the annihilation and creation operators, and the q-commutators between the annihilation and preservation operators, is presented. The method is illustrated by a relevant characterization of q-Gaussian vectors.

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A method for computing the mixed moments of (not necessarily commutative) random vectors from the first-order moments, the q-commutators between the annihilation and creation operators, and the q-commutators between the annihilation and preservation operators, is presented. The method is illustrated by a relevant characterization of q-Gaussian vectors.
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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).