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Quantum compound Poisson processes and white noise analysis

Eugene Lytvynov

Published Jan 1, 1998• 1 citation
Publisher

Abstract

We consider a family B of self-adjoint commuting operators b ς = ∫ ς(t) dB t where B t is a quantum compound Poisson process in a Fock space. By using the projection spectral theorem, we construct the Fourier transform in generalized joint eigenvectors of the family B which is unitary between the Fock space and the L 2-space of compound Poisson white noise, (L 2CP). This construction gives the possibility of introducing spaces of test and generalized functions the dual pairing of which is determined by the inner product of (L 2 CP)

DOI
10.1007/978-3-0348-8789-2_11

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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.

数理解析研究所講究録•Aug 1, 2002•0 citations

Fock Space and the Poisson Process

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Using the Wiener–Poisson isomorphism, we show that if (Ft)0 ⩽ t ⩽ 1 is a real, bounded, predictable process adapted to the filtration of a compensated Poisson process (Xt)0 ⩽ t ⩽ 1, and if M ^ t is the operator corresponding to multiplication by M t = ∫ 0 t F s d X s , then for any regular self-adjoint quantum semimartingale J = ( J t ) 0 ⩽ t ⩽ 1 , the essentially self-adjoint quantum semimartingale ( M ^ t + J t ) 0 ⩽ t ⩽ 1 satisfies the quantum Ito formula. We also introduce a generalisation of the Poisson process to a measure space (M, M, μ) as an isometry I: L2 (M, M, μ) → L2(Ω, F, P) and give a new construction of the generalised Wiener–Poisson isomorphism WI: F+ (L2(M)) → L2 (Ω, F, P) using exponential vectors. Using C*-algebra theory, given any measure space we construct a canonical generalised Poisson process. Unlike other constructions, we make no a priori use of Poisson measures. 2000 Mathematics Subject Classification 60G20, 60G35, 46L53, 81S25.

Proceedings of the London Mathematical Society•Nov 1, 2004•0 citations

An Analytic Characterization of p , q -White Noise Functionals

Anis Riahi, Amine Ettaieb, Wathek Chammam +1 more

In this paper, a characterization theorem for the S -transform of infinite dimensional distributions of noncommutative white noise corresponding to the p , q -deformed quantum oscillator algebra is investigated. We derive a unitary operator U between the noncommutative L 2 -space and the p , q -Fock space which serves to give the construction of a white noise Gel’fand triple. Next, a general characterization theorem is proven for the space of p , q -Gaussian white noise distributions in terms of new spaces of p , q -entire functions with certain growth rates determined by Young functions and a suitable p , q -exponential map.

Journal of Mathematics•Dec 8, 2020•2 citations

Generalized Fock Spaces Based on the Chaotic Representation

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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 Mechanics on Lie Groups: I. Noncommutative Fourier Transforms

Mathieu Beauvillain, Blagoje Oblak, Marios Petropoulos

Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.

quant-pharXiv:2512.19840•Dec 24, 2025
We consider a family B of self-adjoint commuting operators b ς = ∫ ς(t) dB t where B t is a quantum compound Poisson process in a Fock space. By using the projection spectral theorem, we construct the Fourier transform in generalized joint eigenvectors of the family B which is unitary between the Fock space and the L 2-space of compound Poisson white noise, (L 2CP). This construction gives the possibility of introducing spaces of test and generalized functions the dual pairing of which is determined by the inner product of (L 2 CP)
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.
Using the Wiener–Poisson isomorphism, we show that if (Ft)0 ⩽ t ⩽ 1 is a real, bounded, predictable process adapted to the filtration of a compensated Poisson process (Xt)0 ⩽ t ⩽ 1, and if M ^ t is the operator corresponding to multiplication by M t = ∫ 0 t F s d X s , then for any regular self-adjoint quantum semimartingale J = ( J t ) 0 ⩽ t ⩽ 1 , the essentially self-adjoint quantum semimartingale ( M ^ t + J t ) 0 ⩽ t ⩽ 1 satisfies the quantum Ito formula. We also introduce a generalisation of the Poisson process to a measure space (M, M, μ) as an isometry I: L2 (M, M, μ) → L2(Ω, F, P) and give a new construction of the generalised Wiener–Poisson isomorphism WI: F+ (L2(M)) → L2 (Ω, F, P) using exponential vectors. Using C*-algebra theory, given any measure space we construct a canonical generalised Poisson process. Unlike other constructions, we make no a priori use of Poisson measures. 2000 Mathematics Subject Classification 60G20, 60G35, 46L53, 81S25.
In this paper, a characterization theorem for the S -transform of infinite dimensional distributions of noncommutative white noise corresponding to the p , q -deformed quantum oscillator algebra is investigated. We derive a unitary operator U between the noncommutative L 2 -space and the p , q -Fock space which serves to give the construction of a white noise Gel’fand triple. Next, a general characterization theorem is proven for the space of p , q -Gaussian white noise distributions in terms of new spaces of p , q -entire functions with certain growth rates determined by Young functions and a suitable p , q -exponential map.
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.
Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.