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].

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 |

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\textit{M. Puta}, in: 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; Budapest: János Bolyai Mathematical Society. 629--636 (1992; Zbl 0789.58042)