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The reconstruction problem at prequantic level. (English) Zbl 0789.58042
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 629-636 (1992).
The author extends the results from J. Marsden, R. Montgomery and T. Raţiu [Reduction, symmetry and Berry’s phase in mechanics (to appear)] concerning the reconstruction of the dynamics on the phase space $$M$$ with a symmetry group from that of the reduced phase space, at the prequantic level. The main result is
Theorem 4.1. Let $$H$$ be a $$G$$-invariant smooth function on $$T^*M$$ and $$H_ \mu$$ its restriction to $$(T^*M)_ \mu$$. Then the prequantum operator $$Q_ H$$ on the extended phase space $$T^*M$$ can be determined from the prequantum operator $$Q_{H_ \mu}$$ on the reduced phase space $$(T^*M)_ \mu$$ via the connection 1-form $$\alpha^ \mu$$, $$\widetilde\gamma$$ and $$\gamma$$.
For the entire collection see [Zbl 0764.00002].
Reviewer: V.Oproiu (Iaşi)
##### MSC:
 53D50 Geometric quantization 70G10 Generalized coordinates; event, impulse-energy, configuration, state, or phase space for problems in mechanics 81S10 Geometry and quantization, symplectic methods
##### Keywords:
reconstruction; dynamics; phase space; symmetry