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数理解析研究所講究録

Graphical representations of the $q$-creation and the $q$-annihilation operators and set partition statistics (Infinite Dimensional Analysis and Quantum Probability Theory)

Hiroaki Yoshida

Published Aug 1, 2001• 0 citations
OpenAlex

Abstract

In this note, we introduce acertain graphical representations of the $q$-creation the $q$-annihilation, the $q$-numb er, and the scalar operators on the $q$-Fock space.These are given as the cards on which some flow lines are drawn and increasing and decreasing of the number of lines correspond to creation and annihilation, respectively.By the cards arrangement technique with these cards, of which machinery is the same as of the partition function for the lattice spins model in statistical me- chanics, we write the moments of the $q$-Poisson random variable using the number of arc crossings, which is the same set partition statistic as restricted crossings investigated by P. Biane.This result suggests another $q$-deformed moments-cumulants relations, which interpolates between the usual and the free cases exactly.This is the joint work with Naoko Saitoh at Ochanomizu University.1The $q$ -Fock space For aHibert space 7{ and $q\in(-1,1)$ , the $q$-Fock space $\mathcal{F}_{q}(\mathcal{H})$ can be defined as follows: (see, for instance, [BS1], [BKS]) Let $\mathcal{F}^{fin}(\mathcal{H})$ be the linear span of vectors of the form $\xi_{1}\otimes\cdots\otimes\xi_{n}\in \mathcal{H}^{\otimes n}$ , where $n$ varies in $\mathrm{z}_{\geq 0}$ and we put $\mathcal{H}^{\otimes 0}\cong \mathrm{C}\Omega$ for some distinguished vector $\Omega$ called vacuum.We consider the sesquilinear form $\langle\cdot|\cdot\rangle_{q}$ on $\mathcal{F}^{J^{jn}}(\mathcal{H})$ given by the sesquilinear extension of $\langle\xi_{1}\otimes\cdots\otimes\xi_{n}|\eta_{1}\otimes\cdots\otimes\eta_{m}\rangle_{q}=\delta_{n,m}\sum_{\pi\in S_{n}}q^{i(\pi)}\langle\xi_{1}|\eta_{\pi(1)}\rangle\cdots\langle\xi_{n}|\eta_{\pi(n)}\rangle$ ,

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Graphical representations of the $q$-creation and the $q$-annihilation operators and set partition statistics (Infinite Dimensional Analysis and Quantum Probability Theory)

裕亮 吉田

数理解析研究所講究録•Aug 1, 2001•0 citations

Fock representations of Q-deformed commutation relations

Marek Bożejko, Eugene Lytvynov, Janusz Wysoczański

We consider Fock representations of the Q-deformed commutation relations ∂s∂t†=Q(s,t)∂t†∂s+δ(s,t) for s,t∈T. Here T:=Rd (or more generally T is a locally compact Polish space), the function Q:T2→C satisfies |Q(s,t)|≤1 and Q(s,t)=Q(t,s)¯, and ∫T2h(s)g(t)δ(s,t) σ(ds)σ(dt):=∫Th(t)g(t) σ(dt), σ being a fixed reference measure on T. In the case, where |Q(s,t)|≡1, the Q-deformed commutation relations describe a generalized statistics studied by Liguori and Mintchev. These generalized statistics contain anyon statistics as a special case (with T=R2 and a special choice of the function Q). The related Q-deformed Fock space F(H) over H:=L2(T→C,σ) is constructed. An explicit form of the orthogonal projection of H⊗n onto the n-particle space Fn(H) is derived. A scalar product in Fn(H) is given by an operator Pn≥0 in H⊗n which is strictly positive on Fn(H). We realize the smeared operators ∂t† and ∂t as creation and annihilation operators in F(H), respectively. Additional Q-commutation relations are obtained between the creation operators and between the annihilation operators. They are of the form ∂s†∂t†=Q(t,s)∂t†∂s†, ∂s∂t=Q(t,s)∂t∂s, valid for those s,t∈T for which |Q(s, t)| = 1.

arXiv:1603.03075•Journal of Mathematical Physics•Jul 1, 2017•11 citations

q -deformed Poisson random variables on q-Fock space

Naoko Saitoh, Hiroaki Yoshida

The q-deformed Poisson distribution has been introduced as the orthogonalizing probability measure for a certain q-deformation of Charlier polynomials, which is reduced to the usual Poisson distribution in the limit q→1 and takes the free Poisson distribution in case of q=0. We give the operator on the q-Fock space, which has the q-deformed Poisson distribution with respect to the vacuum state. It is a linear combination of a q-number operator, a q-Gaussian random variable, and a scalar operator. This representation is of the same form as of Hudson and Parthasarathy on the symmetric Fock space.

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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

Introduction to the "second quantization" formalism for non-relativistic quantum mechanics: A possible substitution for Sections 6.7 and 6.8 of Feynman's "Statistical Mechanics"

Hal Tasaki

This is a self-contained and hopefully readable account on the method of creation and annihilation operators (also known as the Fock space representation or the "second quantization" formalism) for non-relativistic quantum mechanics of many particles. Assuming knowledge only on conventional quantum mechanics in the wave function formalism, we define the creation and annihilation operators, discuss their properties, and introduce corresponding representations of states and operators of many-particle systems. As the title of the note suggests, we cover most topics treated in sections 6.7 and 6.8 of Feynman's "Statistical Mechanics: A Set of Lectures". As a preliminary, we also carefully discuss the symmetry of wave functions describing indistinguishable particles. We note that all the contents of the present note are completely standard, and the definitions and the derivations presented here have been known to many. Although the style of the present note may be slightly more mathematical than standard physics literatures, we do not try to achieve full mathematical rigor.(Note to experts: In particular we here DERIVE the (anti)commutation relations of the creation and annihilation operators, rather than simply declaring them. In this sense our approach is quite close to that of Feynman's. But we here focus on the action of creation/annihilation operators on general $N$ body wave functions, while Feynman makes a heavy use of Slater-determinant-type states from the beginning. We hope that our presentation provides a better perspective on the formalism.)

arXiv:1812.10732•Dec 27, 2018•1 citation

Quantum Fields and Processes

John Edward Gough, Joachim Kupsch

Wick ordering of creation and annihilation operators is of fundamental importance for computing averages and correlations in quantum field theory and, by extension, in the Hudson–Parthasarathy theory of quantum stochastic processes, quantum mechanics, stochastic processes, and probability. This book develops the unified combinatorial framework behind these examples, starting with the simplest mathematically, and working up to the Fock space setting for quantum fields. Emphasizing ideas from combinatorics such as the role of lattice of partitions for multiple stochastic integrals by Wallstrom–Rota and combinatorial species by Joyal, it presents insights coming from quantum probability. It also introduces a 'field calculus' which acts as a succinct alternative to standard Feynman diagrams and formulates quantum field theory (cumulant moments, Dyson–Schwinger equation, tree expansions, 1-particle irreducibility) in this language. Featuring many worked examples, the book is aimed at mathematical physicists, quantum field theorists, and probabilists, including graduate and advanced undergraduate students.

Cambridge University Press eBooks•Mar 23, 2018•2 citations
Graphical representations of the qqq-creation and the qqq-annihilation operators and set partition statistics (Infinite Dimensional Analysis and Quantum Probability Theory)
In this note, we introduce acertain graphical representations of the qqq-creation the qqq-annihilation, the qqq-numb er, and the scalar operators on the qqq-Fock space.These are given as the cards on which some flow lines are drawn and increasing and decreasing of the number of lines correspond to creation and annihilation, respectively.By the cards arrangement technique with these cards, of which machinery is the same as of the partition function for the lattice spins model in statistical me- chanics, we write the moments of the qqq-Poisson random variable using the number of arc crossings, which is the same set partition statistic as restricted crossings investigated by P. Biane.This result suggests another qqq-deformed moments-cumulants relations, which interpolates between the usual and the free cases exactly.This is the joint work with Naoko Saitoh at Ochanomizu University.1The qqq -Fock space For aHibert space 7{ and q∈(−1,1)q\in(-1,1)q∈(−1,1) , the qqq-Fock space Fq(H)\mathcal{F}_{q}(\mathcal{H})Fq​(H) can be defined as follows: (see, for instance, [BS1], [BKS]) Let Ffin(H)\mathcal{F}^{fin}(\mathcal{H})Ffin(H) be the linear span of vectors of the form ξ1⊗⋯⊗ξn∈H⊗n\xi_{1}\otimes\cdots\otimes\xi_{n}\in \mathcal{H}^{\otimes n}ξ1​⊗⋯⊗ξn​∈H⊗n , where nnn varies in z≥0\mathrm{z}_{\geq 0}z≥0​ and we put H⊗0≅CΩ\mathcal{H}^{\otimes 0}\cong \mathrm{C}\OmegaH⊗0≅CΩ for some distinguished vector Ω\OmegaΩ called vacuum.We consider the sesquilinear form ⟨⋅∣⋅⟩q\langle\cdot|\cdot\rangle_{q}⟨⋅∣⋅⟩q​ on FJjn(H)\mathcal{F}^{J^{jn}}(\mathcal{H})FJjn(H) given by the sesquilinear extension of ⟨ξ1⊗⋯⊗ξn∣η1⊗⋯⊗ηm⟩q=δn,m∑π∈Snqi(π)⟨ξ1∣ηπ(1)⟩⋯⟨ξn∣ηπ(n)⟩\langle\xi_{1}\otimes\cdots\otimes\xi_{n}|\eta_{1}\otimes\cdots\otimes\eta_{m}\rangle_{q}=\delta_{n,m}\sum_{\pi\in S_{n}}q^{i(\pi)}\langle\xi_{1}|\eta_{\pi(1)}\rangle\cdots\langle\xi_{n}|\eta_{\pi(n)}\rangle⟨ξ1​⊗⋯⊗ξn​∣η1​⊗⋯⊗ηm​⟩q​=δn,m​∑π∈Sn​​qi(π)⟨ξ1​∣ηπ(1)​⟩⋯⟨ξn​∣ηπ(n)​⟩ ,
Graphical representations of the qqq-creation and the qqq-annihilation operators and set partition statistics (Infinite Dimensional Analysis and Quantum Probability Theory)
We consider Fock representations of the Q-deformed commutation relations ∂s∂t†=Q(s,t)∂t†∂s+δ(s,t) for s,t∈T. Here T:=Rd (or more generally T is a locally compact Polish space), the function Q:T2→C satisfies |Q(s,t)|≤1 and Q(s,t)=Q(t,s)¯, and ∫T2h(s)g(t)δ(s,t) σ(ds)σ(dt):=∫Th(t)g(t) σ(dt), σ being a fixed reference measure on T. In the case, where |Q(s,t)|≡1, the Q-deformed commutation relations describe a generalized statistics studied by Liguori and Mintchev. These generalized statistics contain anyon statistics as a special case (with T=R2 and a special choice of the function Q). The related Q-deformed Fock space F(H) over H:=L2(T→C,σ) is constructed. An explicit form of the orthogonal projection of H⊗n onto the n-particle space Fn(H) is derived. A scalar product in Fn(H) is given by an operator Pn≥0 in H⊗n which is strictly positive on Fn(H). We realize the smeared operators ∂t† and ∂t as creation and annihilation operators in F(H), respectively. Additional Q-commutation relations are obtained between the creation operators and between the annihilation operators. They are of the form ∂s†∂t†=Q(t,s)∂t†∂s†, ∂s∂t=Q(t,s)∂t∂s, valid for those s,t∈T for which |Q(s, t)| = 1.
The q-deformed Poisson distribution has been introduced as the orthogonalizing probability measure for a certain q-deformation of Charlier polynomials, which is reduced to the usual Poisson distribution in the limit q→1 and takes the free Poisson distribution in case of q=0. We give the operator on the q-Fock space, which has the q-deformed Poisson distribution with respect to the vacuum state. It is a linear combination of a q-number operator, a q-Gaussian random variable, and a scalar operator. This representation is of the same form as of Hudson and Parthasarathy on the symmetric Fock space.
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.
This is a self-contained and hopefully readable account on the method of creation and annihilation operators (also known as the Fock space representation or the "second quantization" formalism) for non-relativistic quantum mechanics of many particles. Assuming knowledge only on conventional quantum mechanics in the wave function formalism, we define the creation and annihilation operators, discuss their properties, and introduce corresponding representations of states and operators of many-particle systems. As the title of the note suggests, we cover most topics treated in sections 6.7 and 6.8 of Feynman's "Statistical Mechanics: A Set of Lectures". As a preliminary, we also carefully discuss the symmetry of wave functions describing indistinguishable particles. We note that all the contents of the present note are completely standard, and the definitions and the derivations presented here have been known to many. Although the style of the present note may be slightly more mathematical than standard physics literatures, we do not try to achieve full mathematical rigor.(Note to experts: In particular we here DERIVE the (anti)commutation relations of the creation and annihilation operators, rather than simply declaring them. In this sense our approach is quite close to that of Feynman's. But we here focus on the action of creation/annihilation operators on general NNN body wave functions, while Feynman makes a heavy use of Slater-determinant-type states from the beginning. We hope that our presentation provides a better perspective on the formalism.)
Wick ordering of creation and annihilation operators is of fundamental importance for computing averages and correlations in quantum field theory and, by extension, in the Hudson–Parthasarathy theory of quantum stochastic processes, quantum mechanics, stochastic processes, and probability. This book develops the unified combinatorial framework behind these examples, starting with the simplest mathematically, and working up to the Fock space setting for quantum fields. Emphasizing ideas from combinatorics such as the role of lattice of partitions for multiple stochastic integrals by Wallstrom–Rota and combinatorial species by Joyal, it presents insights coming from quantum probability. It also introduces a 'field calculus' which acts as a succinct alternative to standard Feynman diagrams and formulates quantum field theory (cumulant moments, Dyson–Schwinger equation, tree expansions, 1-particle irreducibility) in this language. Featuring many worked examples, the book is aimed at mathematical physicists, quantum field theorists, and probabilists, including graduate and advanced undergraduate students.