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DOE OSTI · 22787954

C{sup ∗}-Quadratic Quantization

Accardi, Luigi·Dhahri, Ameur·Rebei, Habib
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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 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{sup ∗}-norm thus obtaining a C{sup ∗}-algebra.

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71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICSALGEBRACOORDINATESHEISENBERG PICTUREMATHEMATICAL MANIFOLDSQUANTIZATIONRENORMALIZATIONTENSORS

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Accardi, Luigi, Dhahri, Ameur, Rebei, Habib. 2018-09-15. C{sup ∗}-Quadratic Quantization. https://doi.org/10.1007/s10955-018-2085-y

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