Strange phenomena related to ordering problems in quantizations. (English) Zbl 1046.53057

This paper points out some questions of the theory of classical ordering. The basic model of the quantum picture is the Weyl algebra. To understand the “group” generated by the exponential functions of quadratic forms in the Weyl algebra, the authors introduce a geometrical object, which can be viewed as a generalization of the notion of covering space.
In this paper, the authors compute infinitesimal actions of quadratic forms in Weyl ordering and normal ordering, defining involutive distributions on the space of exponential functions.
They show that *-exponential functions of quadratic forms generate a groupe-like object, which looks like a non-trivial double cover of \(\text{SL}_\mathbb C(2)\). This object may be seen as a complexification of the metaplectic group \(Mp(2,\mathbb R)\).


53D50 Geometric quantization
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81S05 Commutation relations and statistics as related to quantum mechanics (general)
Full Text: EuDML