Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and Theoretical

Theq-Meixner distributions associated with aq-deformed symmetric Fock space

Hiroaki Yoshida

Published Mar 28, 2011• 1 citation
Publisher

Abstract

In this paper, we consider the self-adjoint operator of the linear sum of the double q -creation, the double q -annihilation, the q -number, and the scalar operators on a q -deformed symmetric Fock space and investigate its distribution. By the self-adjointness of the operator, the theory of orthogonal polynomials can help us to determine the distributions well. It will be found that the distributions are in the types of the q -Meixner family.

DOI
10.1088/1751-8113/44/16/165306

Related papers

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.

Journal of Mathematical Physics•Aug 1, 2000•17 citations

Radial Bargmann representation for the Fock space of type B

Nobuhiro Asai, Marek Bożejko, Takahiro Hasebe

Let να,q be the probability and orthogonality measure for the q-Meixner-Pollaczek orthogonal polynomials, which has appeared in the work of Bożejko, Ejsmont, and Hasebe [J. Funct. Anal. 269, 1769–1795 (2015)] as the distribution of the (α, q)-Gaussian process (the Gaussian process of type B) over the (α, q)-Fock space (the Fock space of type B). The main purpose of this paper is to find the radial Bargmann representation of να,q. Our main results cover not only the representation of q-Gaussian distribution by van Leeuwen and Maassen [J. Math. Phys. 36, 4743–4756 (1995)] but also of q2-Gaussian and symmetric free Meixner distributions on ℝ. In addition, non-trivial commutation relations satisfied by (α, q)-operators are presented.

arXiv:1512.08862•Journal of Mathematical Physics•Jan 13, 2016•5 citations

On $(alpha, q)$-Poisson Operators and Their Distributions (Mathematical aspects of quantum fields and related topics)

Nobuhiro Asai

We propose an (α, q)-analogue of the Poisson operator on the Fock space of Bożejko-Ejsmont-Hasebe [6] and discuss a probability law of the operator. We show that the probability law is expressed by the q-Meixner distribution. Our results contain not only symmetric distributions as in [6], but also non-symmetric ones such as free Poisson, q and q²- deformations of Poisson, Pascal, Gamma, and Meixner distributions. This is the summary of the paper [4].

•Jun 1, 2021•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

Quantum Decomposition Associated with the q-Deformed Lévy–Meixner White Noise

Anis Riahi, Amine Ettaieb

In this paper we start with a new detailed construction of the one-mode type q-Lévy-Meixner Fock space [Formula: see text] which serves to obtain the quantum decomposition associated with the q-deformed Lévy-Meixner white noise processes. More precisely, based on the notion of quantum decomposition and the orthogonalization of polynomials of noncommutative q-Lévy-Meixner white noise [Formula: see text], we study the chaos property of the noncommutative L2-space with respect to the vacuum expectation τ. Next, we determine the distribution of the q-Lévy-Meixner operator J(χD) = ⟨ω, χD⟩ and as a consequence we give some useful properties of the q-Lévy-Meixner white noise process.

Open Systems & Information Dynamics•Jun 1, 2020•1 citation

Spectra of Observables in the q-Oscillator and q-Analogue of the Fourier Transform

Anatoliy U. Klimyk

Spectra of the position and momentum operators of the Biedenharn-Macfarlane q-oscillator (with the main relation aa + -qa + a = 1) are studied when q > 1.These operators are symmetric but not self-adjoint.They have a one-parameter family of selfadjoint extensions.These extensions are derived explicitly.Their spectra and eigenfunctions are given.Spectra of different extensions do not intersect.The results show that the creation and annihilation operators a + and a of the q-oscillator for q > 1 cannot determine a physical system without further more precise definition.In order to determine a physical system we have to choose appropriate self-adjoint extensions of the position and momentum operators.

arXiv:math-ph/0508032•Symmetry Integrability and Geometry Methods and Applications•Oct 21, 2005•5 citations
In this paper, we consider the self-adjoint operator of the linear sum of the double q -creation, the double q -annihilation, the q -number, and the scalar operators on a q -deformed symmetric Fock space and investigate its distribution. By the self-adjointness of the operator, the theory of orthogonal polynomials can help us to determine the distributions well. It will be found that the distributions are in the types of the q -Meixner family.
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.
Let να,q be the probability and orthogonality measure for the q-Meixner-Pollaczek orthogonal polynomials, which has appeared in the work of Bożejko, Ejsmont, and Hasebe [J. Funct. Anal. 269, 1769–1795 (2015)] as the distribution of the (α, q)-Gaussian process (the Gaussian process of type B) over the (α, q)-Fock space (the Fock space of type B). The main purpose of this paper is to find the radial Bargmann representation of να,q. Our main results cover not only the representation of q-Gaussian distribution by van Leeuwen and Maassen [J. Math. Phys. 36, 4743–4756 (1995)] but also of q2-Gaussian and symmetric free Meixner distributions on ℝ. In addition, non-trivial commutation relations satisfied by (α, q)-operators are presented.
On (alpha,q)(alpha, q)(alpha,q)-Poisson Operators and Their Distributions (Mathematical aspects of quantum fields and related topics)
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.
In this paper we start with a new detailed construction of the one-mode type q-Lévy-Meixner Fock space [Formula: see text] which serves to obtain the quantum decomposition associated with the q-deformed Lévy-Meixner white noise processes. More precisely, based on the notion of quantum decomposition and the orthogonalization of polynomials of noncommutative q-Lévy-Meixner white noise [Formula: see text], we study the chaos property of the noncommutative L2-space with respect to the vacuum expectation τ. Next, we determine the distribution of the q-Lévy-Meixner operator J(χD) = ⟨ω, χD⟩ and as a consequence we give some useful properties of the q-Lévy-Meixner white noise process.
Spectra of the position and momentum operators of the Biedenharn-Macfarlane q-oscillator (with the main relation aa + -qa + a = 1) are studied when q > 1.These operators are symmetric but not self-adjoint.They have a one-parameter family of selfadjoint extensions.These extensions are derived explicitly.Their spectra and eigenfunctions are given.Spectra of different extensions do not intersect.The results show that the creation and annihilation operators a + and a of the q-oscillator for q > 1 cannot determine a physical system without further more precise definition.In order to determine a physical system we have to choose appropriate self-adjoint extensions of the position and momentum operators.