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 Mathematics

An Analytic Characterization of p , q -White Noise Functionals

Anis Riahi, Amine Ettaieb, Wathek Chammam, Ziyad A. Alhussain

Published Dec 8, 2020• 2 citations
Publisher

Abstract

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.

DOI
10.1155/2020/6319138

Related papers

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

Infinite Dimensional Ito Algebras of Quantum White Noise

V. P. Belavkin

A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Ito B*-algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Ito algebras of quantum Brownian and quantum Levy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Ito algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin theorem in the classical case.

arXiv:math/0512288•Dec 13, 2005•4 citations

GENERALIZED WHITE NOISE OPERATOR FIELDS AND QUANTUM WHITE NOISE DERIVATIVES

Un Cig Ji, Nobuaki Obata

Regarding a Fock space operator as a function of quantum white noise = ( at;a t ; t2 T ), we introduce its quantum white noise derivatives (qwn- derivatives) as a kind of functional derivatives with respect to at and a t . We prove that every white noise operator is dierentiable and the qwn-derivatives form a gen- eralized white noise operator eld on T. We show a relation between qwn-derivatives and generalized quantum stochastic integrals. We obtain a condition under which qwn-derivatives are dened pointwisely.

•Jan 1, 2007•12 citations

Infinite degrees of freedom Weyl representation: Characterization and application

Abdessatar Barhoumi, Bilel Kacem Ben Ammou, Hafedh M. S. Rguigui

By means of infinite-dimensional nuclear spaces, we generalize important results on the representation of the Weyl commutation relations. For this purpose, we construct a new nuclear Lie group generalizing the groups introduced by Parthasarathy [An Introduction to Quantum Stochastic Calculus (Birkhäuser, 1992)] and Gelfand–Vilenkin [Generalized Functions (Academic Press, 1964)] (see Ref. 15). Then we give an explicit construction of Weyl representations generated from a non-Fock representation. Moreover, we characterize all these Weyl representations in quantum white noise setting.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Mar 1, 2018•0 citations

The q-Gamma White Noise

Hakeem A. Othman

Abstract For 0 < q < 1 and 0 < α < 1, we construct the infinite dimensional q-Gamma white noise measure γα,q by using the Bochner-Minlos theorem. Then we give the chaos decomposition of an L2 space with respect to the measure γα,q via an isomorphism with the 1-mode type interacting Fock space associated to the q-Gamma measure.

Tatra Mountains Mathematical Publications•Jun 1, 2016•4 citations

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
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.
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.
A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Ito B*-algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Ito algebras of quantum Brownian and quantum Levy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Ito algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin theorem in the classical case.
Regarding a Fock space operator as a function of quantum white noise = ( at;a t ; t2 T ), we introduce its quantum white noise derivatives (qwn- derivatives) as a kind of functional derivatives with respect to at and a t . We prove that every white noise operator is dierentiable and the qwn-derivatives form a gen- eralized white noise operator eld on T. We show a relation between qwn-derivatives and generalized quantum stochastic integrals. We obtain a condition under which qwn-derivatives are dened pointwisely.
By means of infinite-dimensional nuclear spaces, we generalize important results on the representation of the Weyl commutation relations. For this purpose, we construct a new nuclear Lie group generalizing the groups introduced by Parthasarathy [An Introduction to Quantum Stochastic Calculus (Birkhäuser, 1992)] and Gelfand–Vilenkin [Generalized Functions (Academic Press, 1964)] (see Ref. 15). Then we give an explicit construction of Weyl representations generated from a non-Fock representation. Moreover, we characterize all these Weyl representations in quantum white noise setting.
# 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)