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Methods of Functional Analysis and Topology

Higher powers of $q$-deformed white noise

Luigi Accardi, Andreas Boukas

Published Sep 1, 2006• 12 citations
OpenAlex

Abstract

We introduce the renormalized powers of q-deformed white noise, for any q in the open interval (−1, 1), and we extend to them the no–go theorem recently proved by Accardi–Boukas–Franz in the Boson case. The surprising fact is that the lower bound (6.5), which defines the obstruction to the positivity of the sesquilinear form, uniquely determined by the renormalized commutation relations, is independent of q in the half-open interval (−1, 1], thus including the Boson case. The exceptional value q = −1, corresponding to the Fermion case, is dealt with in the last section of the paper where we prove that the argument used to prove the no–go theorem for q 6= 0 does not extend to this case.

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Higher powers of qqq-deformed white noise
We introduce the renormalized powers of q-deformed white noise, for any q in the open interval (−1, 1), and we extend to them the no–go theorem recently proved by Accardi–Boukas–Franz in the Boson case. The surprising fact is that the lower bound (6.5), which defines the obstruction to the positivity of the sesquilinear form, uniquely determined by the renormalized commutation relations, is independent of q in the half-open interval (−1, 1], thus including the Boson case. The exceptional value q = −1, corresponding to the Fermion case, is dealt with in the last section of the paper where we prove that the argument used to prove the no–go theorem for q 6= 0 does not extend to this case.
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).
A rigorous mathematical formulation of higher powers of quantum white noises is given on the basis of the most recent theory of white noise distributions due to Cochran, Kuo and Sengupta. The renormalized quantum Itô formula due to Accardi, Lu and Volovich is derived from the renormalized product formula based on integral kernel operators on white noise functions. During the discussion, the analytic characterization of operator symbols and the expansion theorem for a white noise operator in terms of integral kernel operators are established.
In this paper, a characterization theorem for the S -transform of infinite dimensional distributions of noncommutative white noise corresponding to the p , q -deformed quantum oscillator algebra is investigated. We derive a unitary operator U between the noncommutative L 2 -space and the p , q -Fock space which serves to give the construction of a white noise Gel’fand triple. Next, a general characterization theorem is proven for the space of p , q -Gaussian white noise distributions in terms of new spaces of p , q -entire functions with certain growth rates determined by Young functions and a suitable p , q -exponential map.