Bulletin of the Korean Mathematical Society

CONVOLUTIONS OF WHITE NOISE OPERATORS

Un-Cig Ji, Young-Yi Kim

Published Sep 30, 2011• 4 citations

Abstract

Motivated by the convolution product of white noise functionals, we introduce a new notion of convolution products of white noise operators. Then we study several interesting relations between the convolution products and the quantum generalized Fourier-Mehler transforms, and study a quantum-classical correspondence.

DOI: 10.4134/bkms.2011.48.5.1003

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Bulletin of the Korean Mathematical Society

CONVOLUTIONS OF WHITE NOISE OPERATORS

Un-Cig Ji, Young-Yi Kim

Published Sep 30, 2011• 4 citations

Publisher

Abstract

DOI: 10.4134/bkms.2011.48.5.1003

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数理解析研究所講究録Sep 1, 19950 citations

Multiplication Operators by White Noise Delta Functions and Associated Differential Equations

Luigi Accardi, Un Cig Ji, Kimiaki Sait�

Journal of Stochastic AnalysisJun 22, 20230 citations

QUANTUM WHITE NOISE DERIVATIVES AND ASSOCIATED DIFFERENTIAL EQUATIONS FOR WHITE NOISE OPERATORS

Un Cig Ji, Nobuaki Obata

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Higher Powers of Quantum White Noises in Terms of Integral Kernel Operators

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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.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.

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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.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.

数理解析研究所講究録Aug 1, 20020 citations

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.