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Infinite Dimensional Analysis Quantum Probability and Related Topics

Powers of quantum white noise derivatives

Aymen Ettaieb, Habib Ouerdiane, Hafedh M. S. Rguigui

Published Jul 30, 2014• 15 citations
Publisher

Abstract

We introduce a new operator obtained from the quantum white noise (QWN) derivatives which satisfies new important commutation relations generalizing those of the renormalized power white noise Lie algebra introduced by Accardi, Boukas and Franz (see Ref. 5) without using renormalization conditions.
DOI
10.1142/s0219025714500180

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Regarding a Fock space operator as a function of quantum white noise = ( at;a t ; t2 T ), we introduce its quantum white noise derivatives (qwn- derivatives) as a kind of functional derivatives with respect to at and a t . We prove that every white noise operator is dierentiable and the qwn-derivatives form a gen- eralized white noise operator eld on T. We show a relation between qwn-derivatives and generalized quantum stochastic integrals. We obtain a condition under which qwn-derivatives are dened pointwisely.
In this paper we introduce the quantum white noise (QWN) conservation operator N Q acting on nuclear algebra of white noise operators) endowed with the Wick product.Similarly to the classical case, we give a useful integral representation in terms of the QWNderivatives {D - t , D + t ; t ∈ R} for the QWN-conservation operator from which it follows that the QWN-conservation operator is a Wick derivation.Via this property, a relation with the Cauchy problem associated to the QWN-conservation operator and the Wick differential equation is worked out.