Transforms in Quantum White Noise

Takeyuki Hida

Un Cig Ji, Nobuaki Obata

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.

Nobuhiro Asai

IntroductionLet $a_{t}$ and $a_{t}^{*}$ be respectively annihilation and creation operators at apoint $t\in \mathrm{R}$ .In the theory of white noise operators [23][24], it is known that $a_{t}$ is acontinuous linear operator from $\Gamma_{u}(E\mathrm{c})$ , the (Fock) space of test functions over acomplexified nuclear space $E_{\mathrm{c}}$ , into itself, and at is acontinuous linear operator from $\Gamma_{u}(E_{\mathbb{C}})^{*}$ , the dual space of $\Gamma_{u}(E_{\mathbb{C}})$ , into itself.In paticular, $a_{t}+a_{t}^{*}$ and $a_{t}+a_{t}^{*}+a_{t}^{*}a_{t}+I$ are cffied the quantum Gaussian white noise and the quantum Poisson white noise, respectively (cf.[11] [24]).These are known to be elemental generalzed qunatum stochastic processes.The main purpose of this work is to examine relationships between classical and quantum white noises from the point of the classical white noise theory of Gaussian [19][20] and Poisson types [13].Then it will be clarified that quantum Gaussian and Poisson white noises are the Fock space realzations of classical Gaussian and Poisson white noises, respectively.We shall discuss them in Section 4. As amatter of fact, one can see that such facts can be proved not only through $J$ -transform given by (2.4) and the holomorphy, but also through Sx-transform gllven by (3.3) depending on the exponential function $\phi_{\xi}^{X}$ .The function $\phi_{\xi}^{X}$ determines aunitary isomorphism between Boson Fock space and $L^{2}(E^{*},\mu x)$ , $X=G$, $P$ , where $\mu_{G}$ and $\mu_{P}$ are a Gaussian measure and aPoisson measure on $E^{*}$ , respectively.As is pointed out in Remark 4.3, the choice of $\phi_{\xi}^{X}$ (kernel function of $S_{X}$ -transform)is the essential part to represent the above different quantum noises acting on the same space $\Gamma_{u}(E\mathrm{c})^{*}$ .In Appendix, some connections between $[6][7]$ and [10] $\mathrm{w}\mathrm{i}\mathrm{U}$ be noted.

Un Cig Ji, Nobuaki Obata

Luigi Accardi, Un Cig Ji, Kimiaki Sait�

We establish explicit forms of the multiplication operators induced by white noise delta functions, which are closely related to the Bogoliubov transformation and a quantum analogue of Girsanov transform.Then we study the differential equations for operators associated with the multiplication operators by the white noise delta functions.

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