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数理解析研究所講究録

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

Nobuhiro Asai

Published Aug 1, 2002• 0 citations
OpenAlex

Abstract

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.

Related papers

Stochastic analysis classical and quantum : perspectives of white noise theory

武幸 飛田

# Part 1: # White Noise Functional Approach to Polymer Entanglements (C C Bernido & M V Carpio-Bernido) # White Noise Analysis, Quantum Field Theory, and Topology (A Hahn) # A Topic on Noncanonical Representations of Gaussian Processes (Y Hibino) # Integral Representation of Hilbert-Schimdt Operators on Boson Fock Space (U C Ji) # The Dawn of White Noise Analysis (I Kubo) # White Noise Stochastic Integration (H-H Kuo) # Connes-Hida Calculus and Bismut-Quillen Superconnections (R Leandre & H Ouerdiane) # A Quantum Decomposition of Levy Processes (Y-J Lee & H-H Shih) # Generalized Entanglement and Its Classification (T Matsuoka) # A White Noise Approach to Fractional Brownian Motion (D Nualart) # Adaptive Dynamics in Quantum Information and Chaos (M Ohya) # Micro-Macro Duality in Quantum Physics (I Ojima) # White Noise Measures Associated to the Solutions of Stochastic Differential Equations (H Ouerdiane) # A Remark on Sets of Infinite Dimensional Spaces with Full or Zero Capacity (J Ren & M Rockner) # An Infinite Dimensional Laplacian in White Noise Theory (K Saito) # Invariance of Poisson Noise (Si Si et al.) # Nonequilibrium Steady States with Bose-Einstein Condensates (S Tasaki & T Matsui) # Multidimensional Skew Reflected Diffusions (G Trutnau) # On Quantum Mutual Type Entropies and Quantum Capacity (N Watanabe) # Part 2: # White Noise Calculus and Stochastic Calculus (L Accardi & A Boukas)

WORLD SCIENTIFIC eBooks•Jan 1, 2005•6 citations

The poisson process in quantum stochastic calculus

Shayanthan Pathmanathan

Given a compensated Poisson process $(X_t)_{t \geq 0}$ based on $(\Omega, \mathcal{F}, \mathbb{P})$, the Wiener-Poisson isomorphism $\mathcal{W} : \mathfrak{F}_+(L^2 (\mathbb{R}_+)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$ is constructed. We restrict the isomorphism to $\mathfrak{F}_+(L^2 [0,1])$ and prove some novel properties of the Poisson exponentials $\mathcal{E}(f) := \mathcal{W}(e(f))$. A new proof of the result $\Lambda_t + A_t + A^{\dagger}_t = \mathcal{W}^{-1}\widehat{X_t} \mathcal{W}$ is also given. The analogous results for $\mathfrak{F}_+(L^2 (\mathbb{R}_+))$ are briefly mentioned. The concept of a compensated Poisson process over $\mathbb{R}_+$ is generalised to any measure space $(M, \mathcal{M}, \mu)$ as an isometry $I : L^2(M, \mathcal{M}, \mu) \to L^2 (\Omega,\mathcal{F}, \mathbb{P})$ satisfying certain properties. For such a generalised Poisson process we recall the construction of the generalised Wiener-Poisson isomorphism, $\mathcal{W}_I : \mathfrak{F}_+(L^2(M)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$, using Charlier polynomials. Two alternative constructions of $\mathcal{W}_I$ are also provided, the first using exponential vectors and then deducing the connection with Charlier polynomials, and the second using the theory of reproducing kernel Hilbert spaces. Given any measure space $(M, \mathcal{M}, \mu)$, we construct a canonical generalised Poisson process $I : L^2 (M, \mathcal{M}, \mu) \to L^2(\Delta, \mathcal{B}, \mathbb{P})$, where $\Delta$ is the maximal ideal space, with $\mathcal{B}$ the completion of its Borel $\sigma$-field with respect to $\mathbb{P}$, of a $C^*$-algebra $\mathcal{A} \subseteq \mathfrak{B}(\mathfrak{F}_+(L^2(M)))$. The Gelfand transform $\mathcal{A} \to \mathfrak{B}(L^2(\Delta))$ is unitarily implemented by the Wiener-Poisson isomorphism $\mathcal{W}_I: \mathfrak{F}_+(L^2(M)) \to L^2(\Delta)$. This construction only uses operator algebra theory and makes no a priori use of Poisson measures. A new Fock space proof of the quantum Ito formula for $(\Lambda_t + A_t + A^{\dagger}_t)_{0 \leq t \leq 1}$ is given. If $(F_{\ \! \! t})_{0 \leq t \leq 1}$ is a real, bounded, predictable process with respect to a compensated Poisson process $(X_t)_{0 \leq t \leq 1}$, we show that if $M_t = \int_0^t F_s dX_s$, then on $\mathsf{E}_{\mathrm{lb}} := \mathrm{linsp} \{ e(f) : f \in L^2_{\mathrm{lb}}[0,1] \}$, $\mathcal{W}^{-1} \widehat{M_t} \mathcal{W} = \int_0^t \mathcal{W}^{-1} \widehat{F_s} \mathcal{W} (d\Lambda_s + dA_s + dA^{\dagger}_s),$ and that $(\mathcal{W}^{-1} \widehat{M_t} \mathcal{W})_{0 \leq t \leq 1}$ is an essentially self-adjoint quantum semimartingale. We prove, using the classical Ito formula, that if $(J_t)_{0 \leq t \leq 1}$ is a regular self-adjoint quantum semimartingale, then $(\mathcal{W} \widehat{M_t} \mathcal{W}^{-1} + J_t)_{0 \leq t \leq 1}$ is an essentially self-adjoint quantum semimartingale satisfying the quantum Duhamel formula, and hence the quantum Ito formula. The equivalent result for the sum of a Brownian and Poisson martingale, provided that the sum is essentially self-adjoint with core $\mathsf{E}_{\mathrm{lb}}$, is also proved.

•Jan 1, 2002•1 citation

Quantum Stochastic Calculus

Mou-Hsiung Chang

This chapter begins with introductions of symmetric Fock space Γ(ℍ) and symmetric Guichardet space Φ(ℍ) of a generic complex Hilbert space ℍ. It is shown that these 2 spaces are actually isomorphic and can and will be used interchangeably throughout the chapter. In particular, the simple and yet useful integral-summation formula (see Theorem (3.2.1)) for the Guichardet space is often employed to establish the essential results in quantum stochastic calculus. It has been shown that the symmetric Fock space (and hence the symmetric Guichadet space) provides a plausible mathematical tool for modeling phenomena in quantum optics or quantum electrodynamics. Many results in the quantum physics and quantum probability literature are actually established based on this concrete model space. When the generic complex Hilbert space ℍ = L 2 (ℝ + ; K) (where K is another complex Hilbert space), the class of exponential vectors along with 3 different types of quantum noise processes, namely, the creation, annihilation, and neutral (or guage) processes, can be introduced. It is shown that the subspace generated by the class of exponential vectors is dense in the Fock space. Therefore, it is convenient to verify the properties that hold for Fock space by verifying the same hold for the class of exponential vectors. Parallel to those of Itô integrals with respect to classical Brownian motion and/or Poisson process, the concepts of a quantum stochastic integral of a operator-valued process (as a member of the Fock space) with respect to each of the above mentioned quantum noise processes are constructed. The quantum stochastic calculus, parallel to those of classical Itô calculus, is developed. The theory of quantum stochastic calculus on the symmetric Fock space due originally to Hudson and Parthasarathy [HP84a] has been the topic of investigation by many researchers in the last 20 years. Much of he material presented in this chapter can be found in the books by Meyer [Mey95], Parthasarathy [Par92] and Attal [Att03].

Cambridge University Press eBooks•Feb 9, 2015•0 citations

Annihilation and creation operators in stochastic quantization

Toshio Fukumi

Annihilation and creation operators in quantum mechanics are constructed in the framework of Nelson's stochastic mechanics. Employing Ito's stochastic calculus, annihilation and creation operators were found to be written as (1/\ensuremath{\surd}2\ensuremath{\omega})[d/dt,\ensuremath{\partial}/\ensuremath{\partial}B\ifmmode \dot{}\else \.{}\fi{}(t)] and (1/\ensuremath{\surd}2\ensuremath{\omega})[d/dt,\ensuremath{\partial}/\ensuremath{\partial}B${\ifmmode \dot{}\else \.{}\fi{}}^{\mathrm{*}}$(t)], respectively, where B\ifmmode \dot{}\else \.{}\fi{}(t) is a Gaussian white noise and B${\ifmmode \dot{}\else \.{}\fi{}}^{\mathrm{*}}$(t) is a time-reversed Gaussian white noise. The operators were shown to have proper commutation relations and to provide a Hamiltonian which has an eigenvalue equation desired in the {N} representation as should exist in conventional quantum mechanics. Therefore, the present formulation provides a correspondence between stochastic mechanics and conventional quantum mechanics.

Physical Review A•Jul 1, 1989•3 citations

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

暢宏 浅井

数理解析研究所講究録•Aug 1, 2002•0 citations

Functional spaces and operators connected with some Lévy noises

E. Lytvynov

We review some recent developments in white noise analysis and quantum probability. We pay a special attention to spaces of test and generalized functionals of some Lévy white noises, as well as as to the structure of quantum white noise on these spaces.

arXiv:math/0608380•Aug 15, 2006•2 citations
IntroductionLet ata_{t}at​ and at∗a_{t}^{*}at∗​ be respectively annihilation and creation operators at apoint t∈Rt\in \mathrm{R}t∈R .In the theory of white noise operators [23][24], it is known that ata_{t}at​ is acontinuous linear operator from Γu(Ec)\Gamma_{u}(E\mathrm{c})Γu​(Ec) , the (Fock) space of test functions over acomplexified nuclear space EcE_{\mathrm{c}}Ec​ , into itself, and at is acontinuous linear operator from Γu(EC)∗\Gamma_{u}(E_{\mathbb{C}})^{*}Γu​(EC​)∗ , the dual space of Γu(EC)\Gamma_{u}(E_{\mathbb{C}})Γu​(EC​) , into itself.In paticular, at+at∗a_{t}+a_{t}^{*}at​+at∗​ and at+at∗+at∗at+Ia_{t}+a_{t}^{*}+a_{t}^{*}a_{t}+Iat​+at∗​+at∗​at​+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 JJJ -transform given by (2.4) and the holomorphy, but also through Sx-transform gllven by (3.3) depending on the exponential function ϕξX\phi_{\xi}^{X}ϕξX​ .The function ϕξX\phi_{\xi}^{X}ϕξX​ determines aunitary isomorphism between Boson Fock space and L2(E∗,μx)L^{2}(E^{*},\mu x)L2(E∗,μx) , X=GX=GX=G, PPP , where μG\mu_{G}μG​ and μP\mu_{P}μP​ are a Gaussian measure and aPoisson measure on E∗E^{*}E∗ , respectively.As is pointed out in Remark 4.3, the choice of ϕξX\phi_{\xi}^{X}ϕξX​ (kernel function of SXS_{X}SX​ -transform)is the essential part to represent the above different quantum noises acting on the same space Γu(Ec)∗\Gamma_{u}(E\mathrm{c})^{*}Γu​(Ec)∗ .In Appendix, some connections between [6][7][6][7][6][7] and [10] wiU\mathrm{w}\mathrm{i}\mathrm{U}wiU be noted.
# Part 1: # White Noise Functional Approach to Polymer Entanglements (C C Bernido & M V Carpio-Bernido) # White Noise Analysis, Quantum Field Theory, and Topology (A Hahn) # A Topic on Noncanonical Representations of Gaussian Processes (Y Hibino) # Integral Representation of Hilbert-Schimdt Operators on Boson Fock Space (U C Ji) # The Dawn of White Noise Analysis (I Kubo) # White Noise Stochastic Integration (H-H Kuo) # Connes-Hida Calculus and Bismut-Quillen Superconnections (R Leandre & H Ouerdiane) # A Quantum Decomposition of Levy Processes (Y-J Lee & H-H Shih) # Generalized Entanglement and Its Classification (T Matsuoka) # A White Noise Approach to Fractional Brownian Motion (D Nualart) # Adaptive Dynamics in Quantum Information and Chaos (M Ohya) # Micro-Macro Duality in Quantum Physics (I Ojima) # White Noise Measures Associated to the Solutions of Stochastic Differential Equations (H Ouerdiane) # A Remark on Sets of Infinite Dimensional Spaces with Full or Zero Capacity (J Ren & M Rockner) # An Infinite Dimensional Laplacian in White Noise Theory (K Saito) # Invariance of Poisson Noise (Si Si et al.) # Nonequilibrium Steady States with Bose-Einstein Condensates (S Tasaki & T Matsui) # Multidimensional Skew Reflected Diffusions (G Trutnau) # On Quantum Mutual Type Entropies and Quantum Capacity (N Watanabe) # Part 2: # White Noise Calculus and Stochastic Calculus (L Accardi & A Boukas)
Given a compensated Poisson process (Xt)t≥0(X_t)_{t \geq 0}(Xt​)t≥0​ based on (Ω,F,P)(\Omega, \mathcal{F}, \mathbb{P})(Ω,F,P), the Wiener-Poisson isomorphism W:F+(L2(R+))→L2(Ω,F,P)\mathcal{W} : \mathfrak{F}_+(L^2 (\mathbb{R}_+)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})W:F+​(L2(R+​))→L2(Ω,F,P) is constructed. We restrict the isomorphism to F+(L2[0,1])\mathfrak{F}_+(L^2 [0,1])F+​(L2[0,1]) and prove some novel properties of the Poisson exponentials E(f):=W(e(f))\mathcal{E}(f) := \mathcal{W}(e(f))E(f):=W(e(f)). A new proof of the result Λt+At+At†=W−1Xt^W\Lambda_t + A_t + A^{\dagger}_t = \mathcal{W}^{-1}\widehat{X_t} \mathcal{W}Λt​+At​+At†​=W−1Xt​​W is also given. The analogous results for F+(L2(R+))\mathfrak{F}_+(L^2 (\mathbb{R}_+))F+​(L2(R+​)) are briefly mentioned. The concept of a compensated Poisson process over R+\mathbb{R}_+R+​ is generalised to any measure space (M,M,μ)(M, \mathcal{M}, \mu)(M,M,μ) as an isometry I:L2(M,M,μ)→L2(Ω,F,P)I : L^2(M, \mathcal{M}, \mu) \to L^2 (\Omega,\mathcal{F}, \mathbb{P})I:L2(M,M,μ)→L2(Ω,F,P) satisfying certain properties. For such a generalised Poisson process we recall the construction of the generalised Wiener-Poisson isomorphism, WI:F+(L2(M))→L2(Ω,F,P)\mathcal{W}_I : \mathfrak{F}_+(L^2(M)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})WI​:F+​(L2(M))→L2(Ω,F,P), using Charlier polynomials. Two alternative constructions of WI\mathcal{W}_IWI​ are also provided, the first using exponential vectors and then deducing the connection with Charlier polynomials, and the second using the theory of reproducing kernel Hilbert spaces. Given any measure space (M,M,μ)(M, \mathcal{M}, \mu)(M,M,μ), we construct a canonical generalised Poisson process I:L2(M,M,μ)→L2(Δ,B,P)I : L^2 (M, \mathcal{M}, \mu) \to L^2(\Delta, \mathcal{B}, \mathbb{P})I:L2(M,M,μ)→L2(Δ,B,P), where Δ\DeltaΔ is the maximal ideal space, with B\mathcal{B}B the completion of its Borel σ\sigmaσ-field with respect to P\mathbb{P}P, of a C∗C^*C∗-algebra A⊆B(F+(L2(M)))\mathcal{A} \subseteq \mathfrak{B}(\mathfrak{F}_+(L^2(M)))A⊆B(F+​(L2(M))). The Gelfand transform A→B(L2(Δ))\mathcal{A} \to \mathfrak{B}(L^2(\Delta))A→B(L2(Δ)) is unitarily implemented by the Wiener-Poisson isomorphism WI:F+(L2(M))→L2(Δ)\mathcal{W}_I: \mathfrak{F}_+(L^2(M)) \to L^2(\Delta)WI​:F+​(L2(M))→L2(Δ). This construction only uses operator algebra theory and makes no a priori use of Poisson measures. A new Fock space proof of the quantum Ito formula for (Λt+At+At†)0≤t≤1(\Lambda_t + A_t + A^{\dagger}_t)_{0 \leq t \leq 1}(Λt​+At​+At†​)0≤t≤1​ is given. If (F  ⁣ ⁣t)0≤t≤1(F_{\ \! \! t})_{0 \leq t \leq 1}(F t​)0≤t≤1​ is a real, bounded, predictable process with respect to a compensated Poisson process (Xt)0≤t≤1(X_t)_{0 \leq t \leq 1}(Xt​)0≤t≤1​, we show that if Mt=∫0tFsdXsM_t = \int_0^t F_s dX_sMt​=∫0t​Fs​dXs​, then on Elb:=linsp{e(f):f∈Llb2[0,1]}\mathsf{E}_{\mathrm{lb}} := \mathrm{linsp} \{ e(f) : f \in L^2_{\mathrm{lb}}[0,1] \}Elb​:=linsp{e(f):f∈Llb2​[0,1]}, W−1Mt^W=∫0tW−1Fs^W(dΛs+dAs+dAs†),\mathcal{W}^{-1} \widehat{M_t} \mathcal{W} = \int_0^t \mathcal{W}^{-1} \widehat{F_s} \mathcal{W} (d\Lambda_s + dA_s + dA^{\dagger}_s),W−1Mt​​W=∫0t​W−1Fs​​W(dΛs​+dAs​+dAs†​), and that (W−1Mt^W)0≤t≤1(\mathcal{W}^{-1} \widehat{M_t} \mathcal{W})_{0 \leq t \leq 1}(W−1Mt​​W)0≤t≤1​ is an essentially self-adjoint quantum semimartingale. We prove, using the classical Ito formula, that if (Jt)0≤t≤1(J_t)_{0 \leq t \leq 1}(Jt​)0≤t≤1​ is a regular self-adjoint quantum semimartingale, then (WMt^W−1+Jt)0≤t≤1(\mathcal{W} \widehat{M_t} \mathcal{W}^{-1} + J_t)_{0 \leq t \leq 1}(WMt​​W−1+Jt​)0≤t≤1​ is an essentially self-adjoint quantum semimartingale satisfying the quantum Duhamel formula, and hence the quantum Ito formula. The equivalent result for the sum of a Brownian and Poisson martingale, provided that the sum is essentially self-adjoint with core Elb\mathsf{E}_{\mathrm{lb}}Elb​, is also proved.
This chapter begins with introductions of symmetric Fock space Γ(ℍ) and symmetric Guichardet space Φ(ℍ) of a generic complex Hilbert space ℍ. It is shown that these 2 spaces are actually isomorphic and can and will be used interchangeably throughout the chapter. In particular, the simple and yet useful integral-summation formula (see Theorem (3.2.1)) for the Guichardet space is often employed to establish the essential results in quantum stochastic calculus. It has been shown that the symmetric Fock space (and hence the symmetric Guichadet space) provides a plausible mathematical tool for modeling phenomena in quantum optics or quantum electrodynamics. Many results in the quantum physics and quantum probability literature are actually established based on this concrete model space. When the generic complex Hilbert space ℍ = L 2 (ℝ + ; K) (where K is another complex Hilbert space), the class of exponential vectors along with 3 different types of quantum noise processes, namely, the creation, annihilation, and neutral (or guage) processes, can be introduced. It is shown that the subspace generated by the class of exponential vectors is dense in the Fock space. Therefore, it is convenient to verify the properties that hold for Fock space by verifying the same hold for the class of exponential vectors. Parallel to those of Itô integrals with respect to classical Brownian motion and/or Poisson process, the concepts of a quantum stochastic integral of a operator-valued process (as a member of the Fock space) with respect to each of the above mentioned quantum noise processes are constructed. The quantum stochastic calculus, parallel to those of classical Itô calculus, is developed. The theory of quantum stochastic calculus on the symmetric Fock space due originally to Hudson and Parthasarathy [HP84a] has been the topic of investigation by many researchers in the last 20 years. Much of he material presented in this chapter can be found in the books by Meyer [Mey95], Parthasarathy [Par92] and Attal [Att03].
Annihilation and creation operators in quantum mechanics are constructed in the framework of Nelson's stochastic mechanics. Employing Ito's stochastic calculus, annihilation and creation operators were found to be written as (1/\ensuremath{\surd}2\ensuremath{\omega})[d/dt,\ensuremath{\partial}/\ensuremath{\partial}B\ifmmode \dot{}\else \.{}\fi{}(t)] and (1/\ensuremath{\surd}2\ensuremath{\omega})[d/dt,\ensuremath{\partial}/\ensuremath{\partial}B${\ifmmode \dot{}\else \.{}\fi{}}^{\mathrm{*}}$(t)], respectively, where B\ifmmode \dot{}\else \.{}\fi{}(t) is a Gaussian white noise and B${\ifmmode \dot{}\else \.{}\fi{}}^{\mathrm{*}}$(t) is a time-reversed Gaussian white noise. The operators were shown to have proper commutation relations and to provide a Hamiltonian which has an eigenvalue equation desired in the {N} representation as should exist in conventional quantum mechanics. Therefore, the present formulation provides a correspondence between stochastic mechanics and conventional quantum mechanics.