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 $\text{Sp}(d,ℂ)$-invariant model to the $\text{GL}(d,ℂ)$- and, at last, $\text{U}\left(d\right)$-invariant models are performed.

EUDML-ID : urn:eudml:doc:220853
Mots clés:

@article{701620,
     title = {Weyl algebra and a realization of the unitary symmetry},
     booktitle = {Proceedings of the 17th Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {1998},
     pages = {[101]-111},
     mrnumber = {MR1662731},
     zbl = {0992.16019},
     url = {http://dml.mathdoc.fr/item/701620}
}

Strasburger, Aleksander. Weyl algebra and a realization of the unitary symmetry, dans Proceedings of the 17th Winter School "Geometry and Physics", GDML_Books,  (1998), pp. [101]-111. http://gdmltest.u-ga.fr/item/701620/