Geometric Quantization and No Go Theorems
Abstract
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 finitedimensional representations.
 Publication:

eprint arXiv:dgga/9703010
 Pub Date:
 March 1997
 DOI:
 10.48550/arXiv.dgga/9703010
 arXiv:
 arXiv:dgga/9703010
 Bibcode:
 1997dg.ga.....3010G
 Keywords:

 Mathematics  Differential Geometry;
 53F05 (Primary) 58F06;
 81S10 (Secondary)
 EPrint:
 AMSLaTeX, 10 pages