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Random Operators and Stochastic Equations

Quantum Lévy white noise derivatives with their applications to implementation problem

Anis Riahi

Published Nov 8, 2014• 1 citation
Publisher

Abstract

Abstract On the basis of the quantum Lévy white noise theory we introduce the notion of creation and annihilation derivatives of Fock space operators and examine their fundamental properties, in particular, with respect to the Wick product. Next, we solve the implementation problem witch is reduced to a system of differential equations containing the quantum Lévy white noise derivatives.

DOI
10.1515/rose-2014-0021

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Abstract On the basis of the quantum Lévy white noise theory we introduce the notion of creation and annihilation derivatives of Fock space operators and examine their fundamental properties, in particular, with respect to the Wick product. Next, we solve the implementation problem witch is reduced to a system of differential equations containing the quantum Lévy white noise derivatives.
In this paper we introduce the quantum white noise (QWN) conservation operator N Q acting on nuclear algebra of white noise operators) endowed with the Wick product.Similarly to the classical case, we give a useful integral representation in terms of the QWNderivatives {D - t , D + t ; t ∈ R} for the QWN-conservation operator from which it follows that the QWN-conservation operator is a Wick derivation.Via this property, a relation with the Cauchy problem associated to the QWN-conservation operator and the Wick differential equation is worked out.
We study an implementation problem for quadratic functions of annihilation and creation operators on a boson field in terms of quantum white noise calculus. The implementation problem is shown to be equivalent to a linear differential equation for white noise operators containing quantum white noise derivatives. The solution is explicitly obtained and turns out to form a class of white noise operators including generalized Fourier–Gauss and Fourier–Mehler transforms, Bogoliubov transform, and a quantum extension of the Girsanov transform.
The theory of one-mode type Interacting Fock Space IFS allows us to construct the quantum decomposition associated with stochastic processes on ℜ with moments of any order. The problem to extend this result to processes without moments of any order is still open but the Araki–Woods–Parthasarathy–Schmidt characterization of Levy processes in terms of boson Fock spaces, canonically associated with the Levy–Khintchine functions of these processes, provides a quantum decomposition for them which is based on boson creations, annihilation and preservation operators rather than on their IFS counterparts. In order to compare the two quantum decompositions in their common domain of application i.e., the Levy processes with moments of all orders the first step is to give a precise formulation of the quantum decomposition for these processes and the analytical conditions of its validity. We show that these conditions distinguish three different notions of quantum decomposition of a Levy process on ℜ according to the existence of second or only first moments, or no moments at all. For the last class a multiplicative renormalization procedure is needed.