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Mathematica Applicata

Generalized Fock Spaces Based on the Chaotic Representation

Ying Wu

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

The chaotic representation is crucial to the white noise theory.Consider the n-particle inner products in current generalized Fock spaces which the chaotic representation of functionals based on,we derive the relations between the two inner products as well as between the two generalized Fock spaces.

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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.
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Interacting systems of particles with generalized statistics are considered on both classical and quantum level. It is shown that all possible quantum states and corresponding processes can be represented in terms of certain specific categories. The corresponding Fock space representation is discussed. The problem of existence of well--defined scalar product is considered. It is shown that commutation relations corresponding to a system with generalized statistics can be constructed from relations corresponding to Bolzman statistics.
We describe the "no-go" theorems recently obtained by Accardi-Boukas-Franz in [\\cite{1}] for the Boson case, and by Accardi-Boukas in [\\cite{2}] for the qqq-deformed case, on the issue of the existence of a common Fock space representation of the renormalized powers of quantum white noise (RPWN).
By means of infinite-dimensional nuclear spaces, we generalize important results on the representation of the Weyl commutation relations. For this purpose, we construct a new nuclear Lie group generalizing the groups introduced by Parthasarathy [An Introduction to Quantum Stochastic Calculus (Birkhäuser, 1992)] and Gelfand–Vilenkin [Generalized Functions (Academic Press, 1964)] (see Ref. 15). Then we give an explicit construction of Weyl representations generated from a non-Fock representation. Moreover, we characterize all these Weyl representations in quantum white noise setting.