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Infinite Dimensional Analysis Quantum Probability and Related Topics

A NEW IDEA TO DEFINE THE δ-FUNCTION OF AN OBSERVABLE IN THE CONTEXT OF WHITE NOISE ANALYSIS

Caishi Wang

Published Nov 22, 2005• 3 citations
Publisher

Abstract

In this paper, we present a new idea to define the δ-function of an observable within the canonical framework of white noise analysis. We introduce a class of observables called δ-composable observables. We then define the δ-function of such an observable as the generalized operator determined uniquely by the spectral density of the observable. Surprisingly, we find that the δ-function of a δ-composable observable is always positive as a generalized operator. Finally, we show an example of a δ-composable observable and examine properties of its δ-function.

DOI
10.1142/s0219025705002165

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In this paper, we present a new idea to define the δ-function of an observable within the canonical framework of white noise analysis. We introduce a class of observables called δ-composable observables. We then define the δ-function of such an observable as the generalized operator determined uniquely by the spectral density of the observable. Surprisingly, we find that the δ-function of a δ-composable observable is always positive as a generalized operator. Finally, we show an example of a δ-composable observable and examine properties of its δ-function.
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.
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.