Star product on the Euclidean motion group in the plane. (English) Zbl 1471.81058

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 22nd international conference on geometry, integrability and quantization, Varna, Bulgaria, June 8–13, 2020. Sofia: Bulgarian Academy of Sciences, Institute of Biophysics and Biomedical Engineering. Geom. Integrability Quantization 22, 209-218 (2021).
Summary: In this work, we perform exact and concrete computations of star-product of functions on the Euclidean motion group in the plane, and list its \(C\)-star-algebra properties. The star-product of phase space functions is one of the main ingredients in phase space quantum mechanics, which includes Weyl quantization and the Wigner transform, and their generalizations. These methods have also found extensive use in signal and image analysis. Thus, the computations we provide here should prove very useful for phase space models where the Euclidean motion groups play the crucial role, for instance, in quantum optics.
For the entire collection see [Zbl 1468.53002].


81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
22E70 Applications of Lie groups to the sciences; explicit representations
53D55 Deformation quantization, star products
46L05 General theory of \(C^*\)-algebras
47N70 Applications of operator theory in systems, signals, circuits, and control theory
68U10 Computing methodologies for image processing
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[1] < +∞.
[2] Here, the operator W f W l is bounded, since f, l ∈ S(E(2)) and F ∈ L 2 (S 1 ). The property W f W l < +∞ translates to the property f s l ∈ S(E(2)).
[3] If W f , W h , and W l are Weyl quantizations associated with f , h and l, re-spectively, then [(W f W h )W l F ](x) = [W f (W h W l )F ](x).
[4] The composition of Weyl transforms is associative and so is s on E(2).
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