Geometric quantization and no-go theorems

Viktor Ginzburg; Richard Montgomery

Banach Center Publications (2000)

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

Abstract

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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.

How to cite

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Ginzburg, 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 -

References

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