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.
@article{bwmeta1.element.bwnjournal-article-bcpv51z1p69bwm,
author = {Ginzburg, Viktor and Montgomery, Richard},
title = {Geometric quantization and no-go theorems},
journal = {Banach Center Publications},
volume = {51},
year = {2000},
pages = {69-77},
zbl = {0995.53056},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv51z1p69bwm}
}
Ginzburg, Viktor; Montgomery, Richard. Geometric quantization and no-go theorems. Banach Center Publications, Tome 51 (2000) pp. 69-77. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv51z1p69bwm/
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