Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2000 | 51 | 1 | 69-77
Tytuł artykułu

Geometric quantization and no-go theorems

Treść / Zawartość
Warianty tytułu
Języki publikacji
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.
Słowa kluczowe
Opis fizyczny
  • Department of Mathematics, University of California at Santa Cruz, Santa Cruz, CA 95064, U.S.A.
  • Department of Mathematics, University of California at Santa Cruz, Santa Cruz, CA 95064, U.S.A.
  • [Ati] M. F. Atiyah, phGeometry and physics of knots, Cambridge University Press, Cambridge, 1990.
  • [Atk] C. J. Atkin, phA note on the algebra of Poisson brackets, Math. Proc. Cambridge Philos. Soc. 96 (1984), 45-60.
  • [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.
  • [Av2] A. Avez, phRemarques sur les automorphismes infinitésimaux des variétés symplectiques compactes, Rend. Sem. Mat. Univ. Politec. Torino 33 (1974-75), 5-12.
  • [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.
  • [ADPW] S. Axelrod, S. Della Pietra and E. Witten, phGeometric quantization of Chern-Simons gauge theories, J. Differential Geom. 33 (1991), 787-902.
  • [Ba] A. Banyaga, phSur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv. 53 (1978), 174-227.
  • [BU] D. Borthwick and A. Uribe, phAlmost complex structures and geometric quantization, Math. Res. Lett. 3 (1996), 845-861.
  • [Du] J. J. Duistermaat, phThe heat kernel Lefschetz fixed point formula for the $Spin^ℂ$ Dirac operator, Birkhäuser, Boston, 1996.
  • [Fr] D. S. Freed, phReview of 'The heat kernel Lefschetz fixed point formula for the $Spin^ℂ$ Dirac operator' by J. J. Duistermaat, Bull. Amer. Math. Soc. 34 (1997), 73-78.
  • [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.
  • [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.
  • [GGT] M. J. Gotay, H. B. Grundling and G. M. Tuyman, phObstruction results in quantization theory, J. Nonlinear Sci. 6 (1996), 469-498.
  • [Gr] J. Grabowski, phIsomorphisms and ideals of the Lie algebras of vector fields, Invent. Math. 50 (1978), 13-33.
  • [Gu] V. Guillemin, phStar products on compact pre-quantizable symplectic manifolds, Lett. Math. Phys. 35 (1995), 85-89.
  • [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.
  • [Hi] N. J. Hitchin, phFlat connections and geometric quantization, Comm. Math. Phys. 131 (1990), 347-380.
  • [LV] G. Lion and M. Vergne, phThe Weil representation, Maslov index and theta series, Birkhäuser, Boston, 1980.
  • [Om] H. Omori, phInfinite dimensional Lie transformation groups, Lect. Notes in Math., no. 427, Springer-Verlag, New York, 1974.
  • [SP] M. E. Shanks and L. E. Pursel, phThe Lie algebra of smooth manifolds, Proc. Amer. Math. Soc. 5 (1954), 468-472.
  • [We] A. Weinstein, phDeformation quantization, Séminaire Bourbaki, Vol. 1993/94. Astérisque No. 227 (1995), Exp. No. 789, 5, 389-409.
  • [Wo] N. M. J. Woodhouse, phGeometric quantization, Oxford University Press, New York, 1992.
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.