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Bargmann-Fock realization of the noncommutative torus. (English) Zbl 1319.22005

Kielanowski, Piotr (ed.) et al., Geometric methods in physics. XXXII workshop, Białowieża, Poland, June 30 – July 6, 2013. Selected papers. Cham: Birkhäuser/Springer (ISBN 978-3-319-06247-1/hbk; 978-3-319-06248-8/ebook). Trends in Mathematics, 39-48 (2014).
First, the authors present an interpretation of the Bargmann transform as a correspondence between state spaces that is similar to commonly considered intertwiners in the representation theory of finite groups. Then, the non-commutative torus is nothing else than the range of the star-exponential for the Heisenberg group within the Kirillov orbit method context. A realization is obtained of the non-commutative torus as acting on a Fock space of entire functions. A relation of the classical approach to the non-commutative torus in the context of Weyl quantization to its realization in terms of the canonical quantization is obtained. After certain remarks on the geometric quantization of co-adjoint orbits, the authors describe real and complex polarizations. Next they present some properties of the Heisenberg group and discuss the representations from real and complex polarizations. Then the intertwiners and the Bargmann transform are presented. In the last section of the paper, the authors discuss star-exponentials and non-commutative tori. They present some aspects of the Weyl calculus, the star-exponentials and an approach to the non-commutative torus. Finally they obtain the Bargmann-Fock realization of the non-commutative torus.
For the entire collection see [Zbl 1296.00056].

MSC:

22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
53D35 Global theory of symplectic and contact manifolds
58B34 Noncommutative geometry (à la Connes)
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