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

top
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

top

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

top
  1. [Ati] M. F. Atiyah, phGeometry and physics of knots, Cambridge University Press, Cambridge, 1990. 
  2. [Atk] C. J. Atkin, phA note on the algebra of Poisson brackets, Math. Proc. Cambridge Philos. Soc. 96 (1984), 45-60. 
  3. [Av1] A. Avez, phReprésentation de l'algèbre de Lie des symplectomorphismes par des opérateurs bornés, C. R. Acad. Sci. Paris Sér. A 279 (1974), 785-787. Zbl0297.53019
  4. [Av2] A. Avez, phRemarques sur les automorphismes infinitésimaux des variétés symplectiques compactes, Rend. Sem. Mat. Univ. Politec. Torino 33 (1974-75), 5-12. 
  5. [ADL] A. Avez, A. Diaz-Miranda and A. Lichnerowicz, phSur l'algèbre des automorphismes infinitésimaux d'une variété symplectique, J. Differential Geom. 9 (1974), 1-40. 
  6. [ADPW] S. Axelrod, S. Della Pietra and E. Witten, phGeometric quantization of Chern-Simons gauge theories, J. Differential Geom. 33 (1991), 787-902. Zbl0697.53061
  7. [Ba] A. Banyaga, phSur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv. 53 (1978), 174-227. Zbl0393.58007
  8. [BU] D. Borthwick and A. Uribe, phAlmost complex structures and geometric quantization, Math. Res. Lett. 3 (1996), 845-861. Zbl0872.58030
  9. [Du] J. J. Duistermaat, phThe heat kernel Lefschetz fixed point formula for the S p i n ℂ Dirac operator, Birkhäuser, Boston, 1996. 
  10. [Fr] D. S. Freed, phReview of ’The heat kernel Lefschetz fixed point formula for the S p i n ℂ Dirac operator’ by J. J. Duistermaat, Bull. Amer. Math. Soc. 34 (1997), 73-78. 
  11. [GGG] M. J. Gotay, J. Grabowski and H. B. Grundling, phAn obstruction to quantizing compact symplectic manifolds, Proc. Amer. Math. Soc. 128 (2000), 237-243. Zbl1097.53502
  12. [GGH] M. J. Gotay, H. B. Grundling and A. Hurst, phA Groenewold-Van Hove theorem for S 2 , Trans. Amer. Math. Soc. 348 (1996), 1579-1597. 
  13. [GGT] M. J. Gotay, H. B. Grundling and G. M. Tuyman, phObstruction results in quantization theory, J. Nonlinear Sci. 6 (1996), 469-498. Zbl0863.58030
  14. [Gr] J. Grabowski, phIsomorphisms and ideals of the Lie algebras of vector fields, Invent. Math. 50 (1978), 13-33. 
  15. [Gu] V. Guillemin, phStar products on compact pre-quantizable symplectic manifolds, Lett. Math. Phys. 35 (1995), 85-89. Zbl0842.58041
  16. [GU] V. Guillemin and A. Uribe, phThe Laplace operator on the nth tensor power of a line bundle: eigenvalues which are uniformly bounded in n, Asymptotic Anal. 1 (1988), 105-113. Zbl0649.53026
  17. [Hi] N. J. Hitchin, phFlat connections and geometric quantization, Comm. Math. Phys. 131 (1990), 347-380. 
  18. [LV] G. Lion and M. Vergne, phThe Weil representation, Maslov index and theta series, Birkhäuser, Boston, 1980. Zbl0444.22005
  19. [Om] H. Omori, phInfinite dimensional Lie transformation groups, Lect. Notes in Math., no. 427, Springer-Verlag, New York, 1974. 
  20. [SP] M. E. Shanks and L. E. Pursel, phThe Lie algebra of smooth manifolds, Proc. Amer. Math. Soc. 5 (1954), 468-472. 
  21. [We] A. Weinstein, phDeformation quantization, Séminaire Bourbaki, Vol. 1993/94. Astérisque No. 227 (1995), Exp. No. 789, 5, 389-409. 
  22. [Wo] N. M. J. Woodhouse, phGeometric quantization, Oxford University Press, New York, 1992. 

NotesEmbed ?

top

You must be logged in to post comments.