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 finite-dimensional representations.
- Publication:
-
eprint arXiv:dg-ga/9703010
- Pub Date:
- March 1997
- DOI:
- 10.48550/arXiv.dg-ga/9703010
- arXiv:
- arXiv:dg-ga/9703010
- Bibcode:
- 1997dg.ga.....3010G
- Keywords:
-
- Mathematics - Differential Geometry;
- 53F05 (Primary) 58F06;
- 81S10 (Secondary)
- E-Print:
- AMS-LaTeX, 10 pages