Geometric Quantization and No Go Theorems

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A geometric quantization of a Kähler manifold, viewed as a symplectic manifold, depends on the complex structure compatible with the symplectic form. The quantizations form a vector bundle over the space of such complex structures. Having a canonical quantization would amount to finding a natural (projectively) flat connection on this vector bundle. We prove that for a broad class of manifolds, including symplectic homogeneous spaces (e.g., the sphere), such connection does not exist. This is a consequence of a ``no go'' theorem claiming that the entire Lie algebra of smooth functions on a compact symplectic manifold cannot be quantized, i.e., it has no essentially nontrivial finite-dimensional representations.
AMS-LaTeX, 10 pages

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Differential Geometry, 53F05 (Primary) 58F06, 81S10 (Secondary)

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http://salesiana.dossiersoluciones.com/handle/123456789/152220

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/dg-ga/9703010
 http://arxiv.org/abs/dg-ga/9703010

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arXiv