Habib REBEI | IPEIN - Academia.edu (original) (raw)
Papers by Habib REBEI
Quantum Studies: Mathematics and Foundations, 2015
In this paper, we introduce the generalized Weyl operators canonically associated with the one-mo... more In this paper, we introduce the generalized Weyl operators canonically associated with the one-mode oscillator Lie algebra as unitary operators acting on the bosonic Fock space (C). Next, we establish the generalized Weyl relations and deduce a group structure on the manifold R 2 × [−π, π[×R generalizing the well-known Heisenberg one in a natural way.
Communications on Stochastic Analysis
We extend the Lie-algebra time shift technique, introduced in [2], from the usual Weyl algebra (a... more We extend the Lie-algebra time shift technique, introduced in [2], from the usual Weyl algebra (associated to the additive group of a Hilbert space H) to the generalized Weyl algebra (oscillator algebra), associated to the semi-direct product of the additive group H with the unitary group on H (the Euclidean group of H, in the terminology of [14]). While in the usual Weyl algebra the possible quantum extensions of the time shift are essentially reduced to isomorphic copies of the Wiener process, in the case of the oscillator algebra a larger class of Lévy process arises.Our main result is the proof of the fact that the generators of the quantum Markov semigroups, associated to these time shifts, share with the quantum extensions of the Laplacian the important property that the generalized Weyl operators are eigenoperators for them and the corresponding eigenvalues are explicitly computed in terms of the Lévy-Khintchin factor of the underlying classical Lévy process.
Probability and Mathematical Statistics
The theory of one-mode type Interacting Fock Space (IFS) allows us to construct the quantum decom... more The theory of one-mode type Interacting Fock Space (IFS) allows us to construct the quantum decomposition associated with stochastic processes on R with moments of any order. The problem to extend this result to processes without moments of any order is still open but the Araki-Woods- Parthasarathy-Schmidt characterization of Levy processes in terms of boson Fock spaces, canonically associated with the Levy-Khintchine functions of these processes, provides a quantum decomposition for them which is based on boson creations, annihilation and preservation operators rather than on their IFS counterparts. In order to compare the two quantum decompositions in their common domain of application (i.e., the Levy processes with moments of all orders) the first step is to give a precise formulation of the quantum decomposition for these processes and the analytical conditions of its validity. We show that these conditions distinguish three different notions of quantum decomposition of a Levy p...
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2010
ABSTRACT In this paper we introduce the quadratic Weyl operators canonically associated to the on... more ABSTRACT In this paper we introduce the quadratic Weyl operators canonically associated to the one mode renormalized square of white noise (RSWN) algebra as unitary operator acting on the one mode interacting Fock space {Γ, {ωn, n ∈ ℕ}, Φ} where {ωn, n ∈ ℕ} is the principal Jacobi sequence of the nonstandard (i.e. neither Gaussian nor Poisson) Meixner classes. We deduce the quadratic Weyl relations and construct the quadratic analogue of the Heisenberg Lie group with one degree of freedom. In particular, we determine the manifold structure of the group and introduce a local chart containing the identity on which the group law has a simple rational expression in the chart coordinates (see Theorem 6.3).
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2013
ABSTRACT The quantum decomposition of a classical random variable is one of the deep results of q... more ABSTRACT The quantum decomposition of a classical random variable is one of the deep results of quantum probability: it shows that any classical random variable or stochastic process has a built-in non-commutative structure which is intrinsic and canonical, and not artificially put by hands. Up to now the technique to deduce the quantum decomposition has been based on the theory of interacting Fock spaces and on Jacobi's tri-diagonal relation for orthogonal polynomials. Therefore it requires the existence of moments of any order and cannot be applied to random variables without this property. The problem to find an analogue of the quantum decomposition for random variables without finite moments of any order remained open for about fifteen years and nobody had any idea of how such a decomposition could look like. In the present paper we prove that any infinitely divisible random variable has a quantum decomposition canonically associated to its Lévy–Khintchin triple. The analytical formulation of this result is based on Kolmogorov representation of these triples in terms of 1–cocycles (helices) in Hilbert spaces and on the Araki–Woods–Parthasarathy–Schmidt characterization of these representation in terms of Fock spaces. It distinguishes three classes of random variables: (i) with finite second moment; (ii) with finite first moment only; (iii) without any moment. The third class involves a new type of renormalization based on the associated Lévy–Khinchin function.
It has been known for a long time that any infinitely divisible distribution (I.D.D) can be reali... more It has been known for a long time that any
infinitely divisible distribution (I.D.D) can be realized on a
symmetric Fock space with an appropriate noise space. This
realization led to a kind of correspondence between Lie
algebras and I.D.D. Namely, each I.D.D (or equivalently Lévy
process) leads to a such Lie algebra commutation relations.
In this context, it was shown (see [1]) that the the quantum
stochastic processes corresponding to the bounded form of
the oscillator algebra can not cover a large classes of Lévy
processes, in particular the non standard Meixner classes.
For this reason, we consider the unbounded form of the
oscillator algebra called the adapted oscillator algebra. Then,
we prove that its Fock representation can give rise to the
infinitely divisible processes such as the Gamma, Pascal and
the Meixner-Pollaczek.
In the present paper, we extend the notion of quantum time shift, and the related results obtaine... more In the present paper, we extend the notion of quantum time shift, and the related results obtained in , from representations of current algebras of the Heisenberg Lie algebra to representations of current algebras of the Oscillator Lie algebra. This produces quantum extensions of a class of classical Lévy processes much wider than the usual Brownian motion. In particular, this class of processes includes the Meixner processes and, by an approximation procedure, we construct quantum extensions of all classical Lévy processes with a Lévy measure with finite variance. Finally, we compute the explicit form of the action, on the Weyl operators of the initial space, of the generators of the quantum Markov processes canonically associated to the above class of Lévy processes. The emergence of the Meixner classes in connection with the renormalized second order white noise, is now well known. The fact that they also emerge from first order noise in a simple and canonical way comes somehow as a surprise.
Quantum Studies: Mathematics and Foundations, 2015
In this paper, we introduce the generalized Weyl operators canonically associated with the one-mo... more In this paper, we introduce the generalized Weyl operators canonically associated with the one-mode oscillator Lie algebra as unitary operators acting on the bosonic Fock space (C). Next, we establish the generalized Weyl relations and deduce a group structure on the manifold R 2 × [−π, π[×R generalizing the well-known Heisenberg one in a natural way.
Communications on Stochastic Analysis
We extend the Lie-algebra time shift technique, introduced in [2], from the usual Weyl algebra (a... more We extend the Lie-algebra time shift technique, introduced in [2], from the usual Weyl algebra (associated to the additive group of a Hilbert space H) to the generalized Weyl algebra (oscillator algebra), associated to the semi-direct product of the additive group H with the unitary group on H (the Euclidean group of H, in the terminology of [14]). While in the usual Weyl algebra the possible quantum extensions of the time shift are essentially reduced to isomorphic copies of the Wiener process, in the case of the oscillator algebra a larger class of Lévy process arises.Our main result is the proof of the fact that the generators of the quantum Markov semigroups, associated to these time shifts, share with the quantum extensions of the Laplacian the important property that the generalized Weyl operators are eigenoperators for them and the corresponding eigenvalues are explicitly computed in terms of the Lévy-Khintchin factor of the underlying classical Lévy process.
Probability and Mathematical Statistics
The theory of one-mode type Interacting Fock Space (IFS) allows us to construct the quantum decom... more The theory of one-mode type Interacting Fock Space (IFS) allows us to construct the quantum decomposition associated with stochastic processes on R with moments of any order. The problem to extend this result to processes without moments of any order is still open but the Araki-Woods- Parthasarathy-Schmidt characterization of Levy processes in terms of boson Fock spaces, canonically associated with the Levy-Khintchine functions of these processes, provides a quantum decomposition for them which is based on boson creations, annihilation and preservation operators rather than on their IFS counterparts. In order to compare the two quantum decompositions in their common domain of application (i.e., the Levy processes with moments of all orders) the first step is to give a precise formulation of the quantum decomposition for these processes and the analytical conditions of its validity. We show that these conditions distinguish three different notions of quantum decomposition of a Levy p...
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2010
ABSTRACT In this paper we introduce the quadratic Weyl operators canonically associated to the on... more ABSTRACT In this paper we introduce the quadratic Weyl operators canonically associated to the one mode renormalized square of white noise (RSWN) algebra as unitary operator acting on the one mode interacting Fock space {Γ, {ωn, n ∈ ℕ}, Φ} where {ωn, n ∈ ℕ} is the principal Jacobi sequence of the nonstandard (i.e. neither Gaussian nor Poisson) Meixner classes. We deduce the quadratic Weyl relations and construct the quadratic analogue of the Heisenberg Lie group with one degree of freedom. In particular, we determine the manifold structure of the group and introduce a local chart containing the identity on which the group law has a simple rational expression in the chart coordinates (see Theorem 6.3).
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2013
ABSTRACT The quantum decomposition of a classical random variable is one of the deep results of q... more ABSTRACT The quantum decomposition of a classical random variable is one of the deep results of quantum probability: it shows that any classical random variable or stochastic process has a built-in non-commutative structure which is intrinsic and canonical, and not artificially put by hands. Up to now the technique to deduce the quantum decomposition has been based on the theory of interacting Fock spaces and on Jacobi's tri-diagonal relation for orthogonal polynomials. Therefore it requires the existence of moments of any order and cannot be applied to random variables without this property. The problem to find an analogue of the quantum decomposition for random variables without finite moments of any order remained open for about fifteen years and nobody had any idea of how such a decomposition could look like. In the present paper we prove that any infinitely divisible random variable has a quantum decomposition canonically associated to its Lévy–Khintchin triple. The analytical formulation of this result is based on Kolmogorov representation of these triples in terms of 1–cocycles (helices) in Hilbert spaces and on the Araki–Woods–Parthasarathy–Schmidt characterization of these representation in terms of Fock spaces. It distinguishes three classes of random variables: (i) with finite second moment; (ii) with finite first moment only; (iii) without any moment. The third class involves a new type of renormalization based on the associated Lévy–Khinchin function.
It has been known for a long time that any infinitely divisible distribution (I.D.D) can be reali... more It has been known for a long time that any
infinitely divisible distribution (I.D.D) can be realized on a
symmetric Fock space with an appropriate noise space. This
realization led to a kind of correspondence between Lie
algebras and I.D.D. Namely, each I.D.D (or equivalently Lévy
process) leads to a such Lie algebra commutation relations.
In this context, it was shown (see [1]) that the the quantum
stochastic processes corresponding to the bounded form of
the oscillator algebra can not cover a large classes of Lévy
processes, in particular the non standard Meixner classes.
For this reason, we consider the unbounded form of the
oscillator algebra called the adapted oscillator algebra. Then,
we prove that its Fock representation can give rise to the
infinitely divisible processes such as the Gamma, Pascal and
the Meixner-Pollaczek.
In the present paper, we extend the notion of quantum time shift, and the related results obtaine... more In the present paper, we extend the notion of quantum time shift, and the related results obtained in , from representations of current algebras of the Heisenberg Lie algebra to representations of current algebras of the Oscillator Lie algebra. This produces quantum extensions of a class of classical Lévy processes much wider than the usual Brownian motion. In particular, this class of processes includes the Meixner processes and, by an approximation procedure, we construct quantum extensions of all classical Lévy processes with a Lévy measure with finite variance. Finally, we compute the explicit form of the action, on the Weyl operators of the initial space, of the generators of the quantum Markov processes canonically associated to the above class of Lévy processes. The emergence of the Meixner classes in connection with the renormalized second order white noise, is now well known. The fact that they also emerge from first order noise in a simple and canonical way comes somehow as a surprise.