Renormalized powers of quantum white noise

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

Luigi Accardi, Andreas Boukas

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

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.

Un Cig Ji, Nobuaki Obata

Dong Myung Chung, Un Cig Ji, Nobuaki Obata

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.

Luigi Accardi, Andreas Boukas, Uwe Franz

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.