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Infinite Dimensional Analysis Quantum Probability and Related Topics

Infinite degrees of freedom Weyl representation: Characterization and application

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

Published Mar 1, 2018• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0219025718500029

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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.
IntroductionLet ata_{t}at​ and at∗a_{t}^{*}at∗​ be respectively annihilation and creation operators at apoint t∈Rt\in \mathrm{R}t∈R .In the theory of white noise operators [23][24], it is known that ata_{t}at​ is acontinuous linear operator from Γu(Ec)\Gamma_{u}(E\mathrm{c})Γu​(Ec) , the (Fock) space of test functions over acomplexified nuclear space EcE_{\mathrm{c}}Ec​ , into itself, and at is acontinuous linear operator from Γu(EC)∗\Gamma_{u}(E_{\mathbb{C}})^{*}Γu​(EC​)∗ , the dual space of Γu(EC)\Gamma_{u}(E_{\mathbb{C}})Γu​(EC​) , into itself.In paticular, at+at∗a_{t}+a_{t}^{*}at​+at∗​ and at+at∗+at∗at+Ia_{t}+a_{t}^{*}+a_{t}^{*}a_{t}+Iat​+at∗​+at∗​at​+I are cffied the quantum Gaussian white noise and the quantum Poisson white noise, respectively (cf.[11] [24]).These are known to be elemental generalzed qunatum stochastic processes.The main purpose of this work is to examine relationships between classical and quantum white noises from the point of the classical white noise theory of Gaussian [19][20] and Poisson types [13].Then it will be clarified that quantum Gaussian and Poisson white noises are the Fock space realzations of classical Gaussian and Poisson white noises, respectively.We shall discuss them in Section 4. As amatter of fact, one can see that such facts can be proved not only through JJJ -transform given by (2.4) and the holomorphy, but also through Sx-transform gllven by (3.3) depending on the exponential function ϕξX\phi_{\xi}^{X}ϕξX​ .The function ϕξX\phi_{\xi}^{X}ϕξX​ determines aunitary isomorphism between Boson Fock space and L2(E∗,μx)L^{2}(E^{*},\mu x)L2(E∗,μx) , X=GX=GX=G, PPP , where μG\mu_{G}μG​ and μP\mu_{P}μP​ are a Gaussian measure and aPoisson measure on E∗E^{*}E∗ , respectively.As is pointed out in Remark 4.3, the choice of ϕξX\phi_{\xi}^{X}ϕξX​ (kernel function of SXS_{X}SX​ -transform)is the essential part to represent the above different quantum noises acting on the same space Γu(Ec)∗\Gamma_{u}(E\mathrm{c})^{*}Γu​(Ec)∗ .In Appendix, some connections between [6][7][6][7][6][7] and [10] wiU\mathrm{w}\mathrm{i}\mathrm{U}wiU be noted.
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 construct a new class of representations of the canonical commutation relations, which generalizes previously known classes. We perturb the infinitesimal generator of the initial Fock representation (in other words, the free quantum field) by a function of the field which is square-integrable with respect to the associated Gaussian measure. We characterize perturbations which lead to representations of the canonical commutation relations. We provide conditions entailing the irreducibility of such representations, show explicitly that our class of representations subsumes previously studied classes, and give necessary and sufficient conditions for our representations to be unitarily equivalent, or just quasi-equivalent, with Fock, coherent or quasifree representations. 1991 Mathematics Subject Classification: 81S05, 46L60, 81T05.