Geometric and deformation quantization

Details Geometric and deformation quantization Geometric and deformation quantization

2009-03-30

arxiv.org/abs/0903.5336v2

Physics

Mathematical Physics

28 pages; sections 4.3 & 5.1 corrected + minor corrections

Scientific paper

We present a simple geometric construction linking geometric to deformation quantization. Both theories depend on some apparently arbitrary parameters, most importantly a polarization and a symplectic connection, and for real polarizations we find a compatibility condition restricting the set of admissible connections. In the special case when phase space is a cotangent bundle this compatibility condition has many solutions, and the resulting quantum theory not only reproduces the well-known geometric quantization scheme, but also allows to quantize all interesting observables. For K\"ahler manifolds there is no compatibility condition, but a canonical choice for the parameters. The explicit form of the observables however remains undetermined.

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