# Geometric quantization and no-go theorems

Viktor Ginzburg; Richard Montgomery

Banach Center Publications (2000)

- Volume: 51, Issue: 1, page 69-77
- ISSN: 0137-6934

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topGinzburg, Viktor, and Montgomery, Richard. "Geometric quantization and no-go theorems." Banach Center Publications 51.1 (2000): 69-77. <http://eudml.org/doc/209045>.

@article{Ginzburg2000,

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.},

author = {Ginzburg, Viktor, Montgomery, Richard},

journal = {Banach Center Publications},

keywords = {geometric quantization; no-go theorems; connections on vector bundles},

language = {eng},

number = {1},

pages = {69-77},

title = {Geometric quantization and no-go theorems},

url = {http://eudml.org/doc/209045},

volume = {51},

year = {2000},

}

TY - JOUR

AU - Ginzburg, Viktor

AU - Montgomery, Richard

TI - Geometric quantization and no-go theorems

JO - Banach Center Publications

PY - 2000

VL - 51

IS - 1

SP - 69

EP - 77

AB - 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.

LA - eng

KW - geometric quantization; no-go theorems; connections on vector bundles

UR - http://eudml.org/doc/209045

ER -

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