Quantization on curved manifolds. (English) Zbl 1009.53061

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 3rd international conference on geometry, integrability and quantization, Varna, Bulgaria, June 14-23, 2001. Sofia: Coral Press Scientific Publishing. 64-104 (2002).
Since the early days of quantum mechanics, several methods have been developed in order to deal with manifolds supporting non-trivial topology. Among them, two techniques have received a great attention because they are most geometrical in nature. These are the Kostant-Souriau geometric quantization scheme and the so-called constrained quantum mechanics. The present paper is an excellent survey of these two methods and both approaches are illustrated in full detail. The presented examples include the \(n\)-dimensional oscillator and the Kepler problem which are treated within geometric quantization scheme using the well-known Marsden-Weinstein reduction and even a combination of both methods is applied in the study of geodesic flow on axisymmetric ellipsoids.
For the entire collection see [Zbl 0980.00035].


53D50 Geometric quantization
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53D20 Momentum maps; symplectic reduction
81S10 Geometry and quantization, symplectic methods