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Infinite Dimensional Analysis Quantum Probability and Related Topics

RENORMALIZED POWERS OF QUANTUM WHITE NOISE

Luigi Accardi, Andreas Boukas, Uwe Franz

Published Mar 1, 2006• 33 citations
Publisher

Abstract

We prove some no-go theorems on the existence of a Fock representation of the *-Lie algebra generated by [Formula: see text], where [Formula: see text], bs are the Hida white noise densities. In particular we prove the nonexistence of such a representation for any *-Lie algebra containing [Formula: see text]. This drastic difference with the quadratic case proves the necessity of investigating different renormalization rules for the case of higher powers of white noise.

DOI
10.1142/s0219025706002263

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We prove some no-go theorems on the existence of a Fock representation of the *-Lie algebra generated by [Formula: see text], where [Formula: see text], bs are the Hida white noise densities. In particular we prove the nonexistence of such a representation for any *-Lie algebra containing [Formula: see text]. This drastic difference with the quadratic case proves the necessity of investigating different renormalization rules for the case of higher powers of white noise.
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).
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.
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.
White noise theory allows to formulate quantum white noises explicitly as elemental quantum stochastic processes. A traditional quantum stochastic differential equation of Itô type is brought into a normal-ordered white noise differential equation driven by lower powers of quantum white noises. The class of normal-ordered white noise differential equations covers quantum stochastic differential equations with highly singular noises such as higher powers or higher order derivatives of quantum white noises, which are far beyond the traditional Itô theory. For a general normal-ordered white noise differential equation unique existence of a solution is proved in the sense of white noise distribution. Its regularity properties are investigated by means of weighted Fock spaces interpolating spaces of white noise distributions and associated characterization theorems for S-transform and for operator symbols.