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Weyl algebra and a realization of the unitary symmetry. (English) Zbl 0992.16019
Slovák, Jan (ed.) et al., Proceedings of the 17th winter school “Geometry and physics”, Srní, Czech Republic, January 11-18, 1997. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 54, 101-111 (1998).
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 \(\text{Sp}(d,\mathbb{C})\)-invariant model to the \(\text{GL}(d,\mathbb{C})\)- and, at last, \(\text{U}(d)\)-invariant models are performed.
For the entire collection see [Zbl 0904.00040].
Reviewer: A.A.Bogush (Minsk)
16S32 Rings of differential operators (associative algebraic aspects)
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
22E70 Applications of Lie groups to the sciences; explicit representations
22E60 Lie algebras of Lie groups
17B45 Lie algebras of linear algebraic groups