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When there is a family of complex structures on the phase space, parametrized by a set S, the prequantum Hilbert spaces produced by geometric quantization, using the half-form correction, also depends on these parameters. This way we obtain a field of Hilbert spaces p : H → S. We show that this field can have natural inequivalent smooth Hilbert bundle structures.

Smooth structures on the field of prequantum Hilbert spaces