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Interdisciplinary Information Sciences

Calculating Normal-Ordered Forms in Fock Space by Quantum White Noise Derivatives

Un Cig Ji, Nobuaki Obata

Published Jan 1, 2013• 5 citations
Publisher

Abstract

The normal-ordered form of the composition of the exponential of a quadratic annihilation operator and its adjoint is derived by means of a new method based on quantum white noise calculus. Our method consists of two steps: (i) to derive a system of differential equations consisting of quantum white noise derivatives and Wick products; and (ii) to solve explicitly the system of differential equations.

DOI
10.4036/iis.2013.201

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The normal-ordered form of the composition of the exponential of a quadratic annihilation operator and its adjoint is derived by means of a new method based on quantum white noise calculus. Our method consists of two steps: (i) to derive a system of differential equations consisting of quantum white noise derivatives and Wick products; and (ii) to solve explicitly the system of differential equations.
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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.