# Article

 Title: Weyl algebra and a realization of the unitary symmetry (English) Author: Strasburger, Aleksander Language: English Journal: Proceedings of the 17th Winter School "Geometry and Physics" Volume: Issue: 1997 Year: Pages: [101]-111 . Category: math . Summary: In the paper the origins of the intrinsic unitary symmetry encountered in the study of bosonic systems with finite degrees of freedom and its relations with the Weyl algebra (1979, Jacobson) generated by the quantum canonical commutation relations are presented. An analytical representation of the Weyl algebra formulated in terms of partial differential operators with polynomial coefficients is studied in detail. As a basic example, the symmetry properties of the $d$-dimensional quantum harmonic oscillator are considered. In this case the usual set of the creation and annihilation operators is used as basis of the Weyl algebra under consideration. From this model are deduced the underlying symplectic structure and some of its consequences. In particular, the transitions from the initial $\text{Sp}(d,\bbfC)$-invariant model to the $\text{GL}(d,\bbfC)$- and, at last, $\text{U}(d)$-invariant models are performed. (English) MSC: 16S32 MSC: 17B45 MSC: 22E60 MSC: 22E70 MSC: 33C80 MSC: 81R05 idZBL: Zbl 0992.16019 idMR: MR1662731 . Date available: 2009-07-13T21:40:08Z Last updated: 2012-09-18 Stable URL: http://hdl.handle.net/10338.dmlcz/701620 .

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