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\(C^*\)-Quadratic Quantization

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Abstract

In the first part of the paper we introduce a new parametrization for the manifold underlying quadratic analogue of the usual Heisenberg group introduced in Accardi et al. (Infin Dimens Anal Quantum Probab Relat Top 13:551–587, 2010) which makes the composition law much more transparent. In the second part of the paper the new coordinates are used to construct an inductive system of \(*\)-algebras each of which is isomorphic to a finite tensor product of copies of the one-mode quadratic Weyl algebra. We prove that the inductive limit \(*\)-algebra is factorizable and has a natural localization given by a family of \(*\)-sub-algebras each of which is localized on a bounded Borel subset of \(\mathbb {R}\). Moreover, we prove that the family of quadratic analogues of the Fock states, defined on the inductive family of \(*\)-algebras, is projective hence it defines a unique state on the limit \(*\)-algebra. Finally we complete this \(*\)-algebra under the (minimal regular) \(C^*\)-norm thus obtaining a \(C^*\)-algebra.

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Acknowledgements

The authors express their deep gratitude to Gerardo Morsella for precious suggestions that allowed to considerably improve the present paper. L. Accardi acknowledges support by the Russian Science Foundation N. RSF 14-11-00687, Steklov Mathematical Institute. A. Dhahri acknowledges support by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant 2016R1C1B1010008) and by the intramural research Grant of Chungbuk National University in 2015. H. Rebei gratefully acknowledges Qassim University, represented by the Deanship of Scientific Research, on the support for this research.

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Authors and Affiliations

  1. Volterra Center, University of Tor Vergata Roma.II, Via Columbia 2, 00133, Roma, Italy

    Luigi Accardi

  2. Department of Mathematics, Chungbuk National University, 1 Chungdae-ro, Seowon-gu, Cheongju, 362-763, Korea

    Ameur Dhahri

  3. Department of Mathematics, College of Science, Qassim University, P.O.Box 5155, Buraydah, 51422, Kingdom of Saudi Arabia

    Habib Rebei

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  1. Luigi Accardi
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  2. Ameur Dhahri
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  3. Habib Rebei
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Correspondence to Habib Rebei.

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Accardi, L., Dhahri, A. & Rebei, H. \(C^*\)-Quadratic Quantization. J Stat Phys 172, 1187–1209 (2018). https://doi.org/10.1007/s10955-018-2085-y

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