A second quantization procedure is formulated in the framework of Nelson’s stochastic mechanics. It is shown that the annihilation operator is given by an integration over a Gaussian white noise Ḃ(s) with the kernel exp(iω(t−s)) and the creation operator is given by an integration over a time reversed Gaussian white noise Ḃ*(s) with the kernel exp(−iω(t−s)). The eigenfunctions on which the operators act are shown to be Gaussian random variables whose collection is dense in a Hilbert space. Employing the above operators, we can construct {N}-representation in the stochastic mechanics.A second quantization procedure is formulated in the framework of Nelson’s stochastic mechanics. It is shown that the annihilation operator is given by an integration over a Gaussian white noise Ḃ(s) with the kernel exp(iω(t−s)) and the creation operator is given by an integration over a time reversed Gaussian white noise Ḃ*(s) with the kernel exp(−iω(t−s)). The eigenfunctions on which the operators act are shown to be Gaussian random variables whose collection is dense in a Hilbert space. Employing the above operators, we can construct {N}-representation in the stochastic mechanics.

Journal of Mathematical PhysicsNov 1, 19920 citations

On Quantum White Noises and Related Transformations (Trends in Infinite Dimensional Analysis and Quantum Probability)

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.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

QUANTUM WHITE NOISE ANALYSIS AND QUANTUM STOCHASTIC EQUATIONS

V. P. Belavkin, Jaeseong Heo, Un Cig Ji

Jan 1, 20111 citation

Gaussian quantum quadratic stochastic operators

Nasir Ganikhodjaev

Aug 22, 20160 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.