Quantization in complex symmetric spaces

By means of the method described in the author's paper “Quantization” (Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1116–1174), we construct the quantization of a classical mechanics whose phase space is a classical complex symmetric space. We establish the important qualitative differences between the quantization of such mechanics and the quantization of ordinary mechanics with plane phase spaces: for all the spaces considered, except for the sphere, Planck's constant is bounded above. Moreover, in the compact case Planck's constant takes on only discrete values.
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