White Noise Operators Research Articles

Quantum continuity equations and quantum evolution systems for white noise operators

Quantum continuity equations and quantum evolution systems for white noise operators

Quantum analogues of white noise delta functions

In this paper, we construct a quantum analogue of a white noise delta function. For our main purpose, we first discuss several white noise delta functions, specially, an infinite-dimensional analogue of Donsker’s delta function. Then as a general setting, we construct white noise delta functions on white noise test functionals and we derive a Cauchy problem whose solution is given by the white noise delta function. Also, we prove that the white noise delta function is an eigenvector of the exotic Laplacian. By applying the canonical topological isomorphisms between the spaces of white noise operators and the spaces of white noise functionals, we formulate a quantum analogue of the white noise delta function which satisfies a quantum white noise differential equation associated with the quantum Volterra Laplacian. Finally, we prove that quantum analogue of the white noise delta function is an eigenoperator of the quantum exotic Laplacian.

Wick Ordering Associated with the (p, q)-Gaussian White Noise Process

In this paper the quantum decomposition associated with the noncommutative analog of Gaussian white noise processes for the relation of the Chakrabarti–Jagannathan deformed quantum oscillator algebra is investigated. The constructed decomposition with the (p, q)-deformed commutation relations of the pointwise creation and annihilation operators is used to solve the problem of normal wick ordering for white noise operators on the (p, q)-deformed Fock space.

Characterization Theorems for the Quantum White Noise Gross Laplacian and Applications

This paper reports on the characterization of the quantum white noise (QWN) Gross Laplacian based on nuclear algebra of white noise operators acting on spaces of entire functions with \(\theta \)-exponential growth of minimal type. First, we use extended techniques of rotation invariance operators, the commutation relations with respect to the QWN-derivatives and the QWN-conservation operator. Second, we employ the new concept of QWN-convolution operators. As application, we study and characterize the powers of the QWN-Gross Laplacian. As for their associated Cauchy problem it is solved using a QWN-convolution and Wick calculus.

Quantum white noise Gaussian kernel operators

The quantum white noise (QWN) Gaussian kernel operators (with operators parameters) acting on nuclear algebra of white noise operators is introduced by means of QWN- symbol calculus. The quantum-classical correspondence is studied. An integral representation in terms of the QWN- derivatives and their adjoints is obtained. Under some conditions on the operators parameters, we show that the composition of the QWN-Gaussian kernel operators is a QWN-Gaussian kernel operator with other parameters. Finally, we characterize the QWN-Gaussian kernel operators using important connection with the QWN- derivatives and their adjoints.

Euler’s Theorem for Homogeneous White Noise Operators

In this paper we introduce a new notion of λ −order homogeneous operators on the nuclear algebra of white noise operators. Then, we give their Fock expansion in terms of quantum white noise (QWN) fields \(\{a_{t},\: a^{*}_{t}\, ; \; t\in \mathbb {R}\}\). The quantum extension of the scaling transform enables us to prove Euler’s theorem in quantum white noise setting.

An implementation problem for boson fields and quantum Girsanov transform

We study an implementation problem for quadratic functions of annihilation and creation operators on a boson field in terms of quantum white noise calculus. The implementation problem is shown to be equivalent to a linear differential equation for white noise operators containing quantum white noise derivatives. The solution is explicitly obtained and turns out to form a class of white noise operators including generalized Fourier–Gauss and Fourier–Mehler transforms, Bogoliubov transform, and a quantum extension of the Girsanov transform.

Characterization of the QWN-conservation operator and applications

Based on the finding that the quantum white noise (QWN) conservation operator is a Wick derivation operator acting on white noise operators, we characterize the aforementioned operator by using an extended techniques of rotation invariance operators in a first place. In a second place, we use a new idea of commutation relations with respect to the QWN-derivatives. Eventually, we use the action on the number operator. As applications, we invest these results to study three types of Wick differential equations.

Factorization property of convolutions of white noise operators

We first study a general type of convolutions, parameterized by operators, on white noise functionals and study a relation between the convolution and the generalized Fourier-Mehler transform. Secondly, we extend the convolution of generalized white noise functionals to a convolution of white noise operators. Then, as factorization properties of the convolutions, we study a relation between the convolution and quantum generalized Fourier-Mehler transform.

Operator theory: quantum white noise approach

we develop an operator theory on a nuclear algebra of white noise operators in terms of the quantum white noise (QWN) derivatives and their dual adjoints. Using an adequate definition of a QWN-symbol transformation, we discuss QWN-integral-sum kernel operators which give the Fock expansion of the QWN-operators (i.e. the linear operators acting on nuclear algebra of white noise operators). As application, we characterize all rotation invariant QWN-operators by means of the QWN-conservation operator, the QWN-Gross Laplacians. These topics are expected to open a new area in QWN infinite-dimensional analysis.

Quantum [formula omitted]-potentials associated to quantum Ornstein–Uhlenbeck semigroups

Using the quantum Ornstein–Uhlenbeck (O–U) semigroups (introduced in Rguigui [21]) and based on nuclear infinite dimensional algebra of entire functions with a certain exponential growth condition with two variables, the quantum λ-potential and the generalised quantum (λ1,λ2)-potential appear naturally for λ,λ1,λ2∈(0,∞). We prove that the solution of Poisson equations associated with the suitable quantum number operators can be expressed in terms of these potentials. Using a useful criterion for the positivity of generalised operators, we demonstrate that the solutions of the Cauchy problems associated to the quantum number operators are positive operators if the initial condition is also positive. In this case, we show that these solutions, the quantum λ-potential and the generalised quantum (λ1,λ2)-potential have integral representations given by positive Borel measures. Based on a new notion of positivity of white noise operators, the aforementioned potentials are shown to be Markovian operators whenever λ∈[1,∞) and λ1λ2⩾λ1+λ2.

Quantum Ornstein–Uhlenbeck semigroups

Based on nuclear infinite-dimensional algebra of entire functions with a certain exponential growth condition with two variables, we define a class of operators which gives in particular three semigroups acting on continuous linear operators, called the quantum Ornstein–Uhlenbeck (O–U) semigroup, the left quantum O–U semigroup and the right quantum O–U semigroup. Then, we prove that the solution of the Cauchy problem associated with the quantum number operator, the left quantum number operator and the right quantum number operator, respectively, can be expressed in terms of such semigroups. Moreover, probabilistic representations of these solutions are given. Eventually, using a new notion of positive white noise operators, we show that the aforementioned semigroups are Markovian.

QWN-EULER OPERATOR AND ASSOCIATED CAUCHY PROBLEM

In this paper the quantum white noise (QWN)-Euler operator [Formula: see text] is defined as the sum [Formula: see text], where [Formula: see text] and NQ stand for appropriate QWN counterparts of the Gross Laplacian and the conservation operator, respectively. It is shown that [Formula: see text] has an integral representation in terms of the QWN-derivatives [Formula: see text] as a kind of functional integral acting on nuclear algebra of white noise operators. The solution of the Cauchy problem associated to the QWN-Euler operator is worked out in the basis of the QWN coordinate system.

CONVOLUTIONS OF WHITE NOISE OPERATORS

Motivated by the convolution product of white noise functionals, we introduce a new notion of convolution products of white noise operators. Then we study several interesting relations between the convolution products and the quantum generalized Fourier-Mehler transforms, and study a quantum-classical correspondence.

Nuclear Realization of Virasoro–Zamolodchikov-w∞ ⋆-Lie Algebras Through the Renormalized Higher Powers of Quantum Meixner White Noise

By using an appropriate one-mode type interacting Fock spaces, [Formula: see text], introduced in [1], we define a nuclear triple [Formula: see text] of test and generalized functions, with θ being a suitable Young function. Moreover, we prove general characterization theorems for the fundamental nuclear spaces. For the applications, we introduce new renormalized products for the generators of the renormalized higher powers of white noise ⋆-Lie algebra and the Virasoro-Zamolodchikov-w∞ ⋆-Lie algebra. Then we show that these new renormalized products lead to nuclear realizations of these Lie algebras in terms of quantum Meixner white noise operators.

Pascal white noise calculus

In this paper white noise analysis with respect to the Lévy process with negative binomial distributed marginals is investigated. An appropriate space of distributions, ℰ ′, is used to describe the structure of the Hilbert space of quadratic integrable functionals with respect to the Pascal white noise measure ΛNB. The constructed decomposition is used to define a nuclear triple of test and generalized functions, where θ is a Young function satisfying some suitable conditions. By using the 𝒮-transform and the symbol transform σNB, a general characterization theorems are proven for Pascal white noise distributions, white noise test functions and white noise operators in terms of analytical functions with growth condition of exponential type. As application, some quantum stochastic differential equations are solved with special emphasis on Wick calculus.

Annihilation-Derivative, Creation-Derivative and Representation of Quantum Martingales

On the basis of the quantum white noise theory we introduce the notion of creation- and annihilation-derivatives of Fock space operators and study the differentiability of white noise operators. We define the Hitsuda–Skorohod quantum stochastic integrals by the adjoint actions of quantum stochastic gradients and show explicit formulas for their creation- and annihilation-derivatives. As an application, we derive direct formulas for the integrands in the quantum stochastic integral representation of a regular quantum martingale.

UNITARY REPRESENTATIONS OF THE WITT AND sl(2, ℝ)-ALGEBRAS THROUGH RENORMALIZED POWERS OF THE QUANTUM PASCAL WHITE NOISE

By using an appropriate space of distributions, [Formula: see text], we derive the chaos decomposition property of the Hilbert space of quadratic integrable functionals with respect to the Pascal white noise measure ΛNB. The constructed decomposition is used to define a nuclear triple [Formula: see text] of test and generalized functions, where θ is a Young function satisfying some suitable conditions. A general characterization theorems are proven for the Pascal white noise distributions, white noise test functions and white noise operators in terms of analytical functions with growth condition of exponential type. By using appropriate renormalization procedure, we obtain the representation of the square of white noise obtained by Accardi–Franz–Skeide in Ref. 5. Finally, we investigate the main aim of this paper which is to give unitary equivalent representations of the Witt algebra in the basis of Pascal white noise theory.

Quantum Lévy Laplacian and associated heat equation

As a non-commutative extension of the Lévy Laplacian for entire functions on a nuclear space, we define the quantum Lévy Laplacian acting on white noise operators. We solve a heat type equation associated with the quantum Lévy Laplacian and study its relation to the classical Lévy heat equation. The solution to the quantum Lévy heat equation is obtained also from a normal-ordered white noise differential equation involving the quadratic quantum white noise.

The multitime correlation functions, free white noise, and the generalized Poisson statistics in the low density limit

In the present paper the low density limit of the nonchronological multitime correlation functions of boson number type operators is investigated. We prove that the limiting truncated nonchronological correlation functions can be computed using only a subclass of diagrams associated to noncrossing pair partitions and thus coincide with nontruncated correlation functions of suitable free number operators. The independent in the limit subalgebras are found and the limiting statistics is investigated. In particular, it is found that the cumulants of certain elements coincide in the limit with the cumulants of the Poisson distribution. An explicit representation of the limiting correlation functions and thus of the limiting algebra is constructed in a special case through suitably defined quantum white noise operators.