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Communications in Mathematical Physics

, Volume 165, Issue 2, pp 281–296 | Cite as

Toeplitz quantization of Kähler manifolds anggl(N), N→∞ limits

  • Martin Bordemann
  • Eckhard Meinrenken
  • Martin Schlichenmaier
Article

Abstract

For general compact Kähler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes earlier results of teh authors and Klimek and Lesniewski obtained for the tours and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finitedimensional matrix algebrasgl(N), N→∞.

Keywords

Neural Network Manifold Statistical Physic Complex System Nonlinear Dynamics 
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References

  1. 1.
    Arnol'd, V.I.: Mathematical methods of classical mechanics. Berin, Heidelberg, New York: Springer 1980Google Scholar
  2. 2.
    Bayen, F., Flato, M., Fronsdal, C., Liochnerowicz, A., Sternheimer, D.: Deformation theory and quantization. Ann. Phys.111, 61–110 (part I), 111–151 (part II) (1978)Google Scholar
  3. 3.
    Berezin, F.A.: Quantizatiion. Math. USSR Izv.8, no. 5, 1109–1165 (1974); Quantization in complex symmetric spaces. Math. USSR Izv.9, no. 2, 341–379 (1975); General concept of Quantization. Commun. Math. Phys.40, 153–174 (1975)Google Scholar
  4. 4.
    Bordemann, M., Hoppe, J.: The dynamics oof relativistic membranes. I: Reduction to 2-dimensional fluid dynamics. To appear in Phys. Lett. BGoogle Scholar
  5. 5.
    Bordemann, M., Hoppe, J., Schaller, P., Sichlichenmaier, M.:gl(∞) and geometric quantization. Commun. Math. Phys.138, 209–244 (1991).Google Scholar
  6. 6.
    Bordemann, M., Meinrenken, E., Römer, H.: Total space quantization of Kähler manifolds. Reprint Appril 1993, Freiburg THEP 93/5Google Scholar
  7. 7.
    Boutet de Monvel, L., Guillemin, V.: The spectral theory of Toeplitz operators. Ann. Math. Studies, Nr. 99, Princeton, NJ: Princeton Universeity Press 1981Google Scholar
  8. 8.
    Boutet de Monvel, Sjöstrand, J.: Sur la singularité des noyaux de Bergma et de Szegö. Astérisque34–34, 123–164 (1976)Google Scholar
  9. 9.
    Cahen, M., Gutt, S., Rawnsley, J.H.: Quantization of Kähler manifolds I: Geometric interpretation of Berezin's quantization. JGP7, no. 1, 45–65 (1990); Quantization of Kähler manifolds II. Preprint (1991)Google Scholar
  10. 10.
    Calabi, E.: Isometric imbedding of compelx manifolds. Ann. Math.58, 1–23 (1953)Google Scholar
  11. 11.
    DeWilde, M., Lecomte, P.B.A.: Existence of star products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds. Lett. Math. Phys.7, 487–496 (1983)Google Scholar
  12. 12.
    Dixmier, J.C.:C *-Aalgebras. Amsterdam, New York, Oxford: North-Holland 1977Google Scholar
  13. 13.
    Ebin, D.G., Mardsen, J.: Group of diffeomorphisms and the motion of an incompressible fluid. Ann. Math.92, 102–163 (1970)Google Scholar
  14. 14.
    Fairlie, D., Fletscher, P., Zachos, C.N.: Trigonometric structure constants for new infinite algebras. Phys. Lett. B218, 203 (1989)Google Scholar
  15. 15.
    Farkas, H.M., Kra, I.: Rieman surfaces. Berlin, Heidelberg, New York: Springer 1980Google Scholar
  16. 16.
    Gilkey, P.: Invariance theory, the heat equation, and the Atiyah-Singer index theorem. Wilmington: Publish or Perish 1984.Google Scholar
  17. 17.
    Griffiths, Ph., Harris, J.: Principles of algebraic geometry. New York: John Wiley 1978Google Scholar
  18. 18.
    Guillemin, V.: Some classical theorems in spectral theory revisited. Seminars on singularities of solutions of linear partial differential equations. Ann. Math. Studies, Nr. 91, Hörmader, L., (ed.) Princeton, NJ: Princeton University Press 1979, pp. 219–259Google Scholar
  19. 19.
    Guillemin, V.: Symplectic spinors and partial differential equations. Géométrie symplectic et physique mathématique (Aix en Provence (1974), Coll. Int. CNRS 237, pp. 217–252Google Scholar
  20. 20.
    Guillemin, V., Uribe, A.: Circular symmetry and the trace formula. Invent. Math.96, 385–423 (1989)Google Scholar
  21. 21.
    Hörmander, L.: Fourier integral operators I. Acta Math.127, 79–183 (1971)Google Scholar
  22. 22.
    Hörmander, L.: The analysis of linear partial differential operators. Vol. I–IV, Berlin, Heidelberg, New York: Springer 1985Google Scholar
  23. 23.
    Hoppe, J.: Quantum theory of a relativistic surface ... MIT PhD Thesis 1982, Elem. Part. Res. J. (Kyoto)83 (1989/1990), no. 3Google Scholar
  24. 24.
    Klimek, S., Kesniewski A.: Quantum Rieman surfaces: I. Teh unit disc. Commun. Math. Phys.146, 103–122 (1992); Quantum Riemann Surfaces: II. The discrete series. Lett. Math. Phys.24, 125–139 (1992)Google Scholar
  25. 25.
    Meinrenken, E.: Semiklassische Nähreungen und mikrolokale Analysis. Dissertation (in preparation), Fak. für Physik, University of FreiburgGoogle Scholar
  26. 26.
    Mumford, D.: Algebraic geometry I. Complex projective varieties. Berlin, Heidelberg, New York: Springer 1976Google Scholar
  27. 27.
    Miumford, D.: Curves and their Jacobians. Ann. Arbor: The University of Michigan Press 1976Google Scholar
  28. 28.
    Rawnsley, J.H.: Coherent states and Kähleer manifolds. Quant. J. Math. Oxford28, no. 2, 403–415 (1977)Google Scholar
  29. 29.
    Schlichemaier, M.: An introduction to Rieman surfaces, algebraic cures and moduli spaces. Lecture Notes in Physics322. Berlin, Heidelberg, New York: Springer 1989Google Scholar
  30. 30.
    Simon, B.: The classical limit of quantum partition functions. Commun. Math. Phys.71, 247–276 (1980)Google Scholar
  31. 31.
    Tuynman, G.M.: Generalized Bergman kernels and geometric quantization. J. Math. Phys.28, 573–583 (1987)Google Scholar
  32. 32.
    Tuynmann, G.M.: Quantization: Towards a comparison between methods. J. Math. Phys.28, 2829–2840 (1987)Google Scholar
  33. 33.
    Wells, R.O.: Differential analysis on compelx manifolds. Berlin, Heidelberg, New York: Springer 1980Google Scholar
  34. 34.
    Woodhouse, N.: Geometric quantization. Oxford: Clarendon Press 1980Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Martin Bordemann
    • 1
  • Eckhard Meinrenken
    • 2
  • Martin Schlichenmaier
    • 3
  1. 1.Department of PhysicsUniversity of FreiburgFreiburgGermany
  2. 2.Department of MathematicsM.I.T.CambridgeUSA
  3. 3.Department of Mathematics and computer ScienceUniverseity of MannheimMannheimGermany

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