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Communications on Stochastic Analysis

Renormalized square of white noise quantum time shift

Luigi Accardi, Habib Ouerdiane, Habib Rebei

Published Jun 1, 2012• 6 citations
Publisher

Abstract

Quantum time shifts are a very special class of continuous time random walk type additive processes on Lie algebras whose exponentiation gives rise to quantum Markov processes whose infinitesimal generators possess a total set of unitary eigen-operators: such a class of quantum Markov generators was not known in the literature.This property generalizes the known fact that trigonometric exponentials are eigenvectors of the usual classical Laplacian and was first noticed in connection with the quantum Laplacian in [5].In that case the eigen-operators turn out to be the Weyl operators, which are known to be non-commutative extensions of the trigonometric exponentials.A generalization of these results, from the Weyl algebra to the harmonic oscillator algebra was obtained in [6].In the present paper we extend the above results to the renormalized square of white noise algebra.This provides among other things, a quantum extension of the Markov generators of the Meixner processes.To our knowledge such an extension was not previously known.

DOI
10.31390/cosa.6.2.01

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Quantum time shifts are a very special class of continuous time random walk type additive processes on Lie algebras whose exponentiation gives rise to quantum Markov processes whose infinitesimal generators possess a total set of unitary eigen-operators: such a class of quantum Markov generators was not known in the literature.This property generalizes the known fact that trigonometric exponentials are eigenvectors of the usual classical Laplacian and was first noticed in connection with the quantum Laplacian in [5].In that case the eigen-operators turn out to be the Weyl operators, which are known to be non-commutative extensions of the trigonometric exponentials.A generalization of these results, from the Weyl algebra to the harmonic oscillator algebra was obtained in [6].In the present paper we extend the above results to the renormalized square of white noise algebra.This provides among other things, a quantum extension of the Markov generators of the Meixner processes.To our knowledge such an extension was not previously known.
Abstract In the present paper, we extend the notion of quantum time shift, and the related results obtained in [Accardi, Barhoumi and Ouerdiane, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9: 215–248, 2006], 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.
We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators in the space of functions on a Hilbert space which are square integrable with respect to a shift-invariant measure. We study unitary groups of shift operators in the phase space and averaging of such shifts by Gaussian vectors, which form semigroups of self-adjoint contractions: we find conditions for their strong continuity and establish properties of their generators. Significant differences in their properties allow us to show the absence of the Fourier transform as a unitary transformation that implements the unitary equivalence of these compressive semigroups. Next, we prove the Taylor formula for a certain special subset of smooth functions for shifting to a non-finite vector. It allows us to prove convergence of quantum random walks in the coordinate representation to the evolution of a diffusion process, as well as convergence of quantum random walks in both coordinate and momentum representations to the evolution semigroup of a quantum oscillator in an infinite-dimensional phase space. We find the special essential common domain of generators of semigroups arising in averaging of random shift operators both in position and momentum representations. The invariance of this common domain with respect to both semigroups allows to establish properties of a convex combination of both generators. That convex combination are Hamiltonians of infinite-dimentional quantum oscillators. Thus, we obtain that a Weyl representation of a random walk in an infinite dimensional phase space describes the semigroup of self-adjoint contractions whose generator is the Hamiltonian of an infinite dimensional harmonic oscillator.
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).
We study the fundamental properties of classical and quantum Markov processes generated by q-Bessel operators and their extension to the algebra of all bounded operators on the Hilbert space [Formula: see text]. In particular, we find a suitable generalized Gorini–Kossakowski–Sudarshan–Lindblad representation for the infinitesimal generator of q-Bessel operator and show that both the classical and quantum Markov processes are transient for α > 0 and recurrent for α = 0. We also show that they do not admit invariant states and, moreover that the support projection of any initial state instantaneously fills the full space.
Abstract Random walks in a real separable Hilbert space correspond to a composition of random unitary operators in the space of functions square integrable with respect to a translation-invariant measure, as well as to a composition of random unitary channels in the space of quantum states. Conditions on the distribution of random walks that are necessary and sufficient for the continuity of the mathematical expectations of their representations in the Hilbert space and the space of quantum states are obtained. The dynamics of the set of quantum states obtained by averaging random shifts over the semigroup (with respect to the convolution operation) of centered Gaussian measures on the Hilbert space is studied. It is shown that, depending on the property of the covariance operator of the Gaussian measure to have a nuclear square root, such dynamics is either a strongly continuous quantum dynamical semigroup or instantaneously destroys the set of normal states. In the latter case, the dynamics of singular quantum states is described in terms of the evolution of mean values on an operator algebra that implements a representation of canonical commutation relations.