Deformation quantization

Alan Weinstein

Séminaire Bourbaki (1993-1994)

  • Volume: 36, page 389-409
  • ISSN: 0303-1179

How to cite


Weinstein, Alan. "Deformation quantization." Séminaire Bourbaki 36 (1993-1994): 389-409. <>.

author = {Weinstein, Alan},
journal = {Séminaire Bourbaki},
keywords = {deformation quantization},
language = {eng},
pages = {389-409},
publisher = {Société Mathématique de France},
title = {Deformation quantization},
url = {},
volume = {36},
year = {1993-1994},

AU - Weinstein, Alan
TI - Deformation quantization
JO - Séminaire Bourbaki
PY - 1993-1994
PB - Société Mathématique de France
VL - 36
SP - 389
EP - 409
LA - eng
KW - deformation quantization
UR -
ER -


  1. [BFFLS] Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz A., and Sternheimer, D., Deformation theory and quantization, I and II, Ann. Phys.111, (1977), 61-151. Zbl0377.53025MR496157
  2. [Be1] Berezin, F.A., Some remarks about the associated envelope of a Lie algebra, Funct. Anal. Appl.1, (1967), 91-102. Zbl0227.22020
  3. [Be2] Berezin, F.A., Quantization, Math USSR Izv.8 (1974), 1109-1165. Zbl0312.53049
  4. [BoG] Boutet de Monvel, L., and Guillemin, V., The spectral theory of Toeplitz operators, Annals. of Math. Studies99, Princeton University Press, Princeton, 1981. Zbl0469.47021MR620794
  5. [CaGR] Cahen, M., Gutt, S., and Rawnsley, J., Quantization of Kähler Manifolds. II, Trans. Amer. Math. Soc.337 (1993), 73-98. Zbl0788.53062MR1179394
  6. [Co] Connes, A., Non-commutative differential geometry, Publ. Math. IHES62 (1986), 41-144. Zbl0592.46056
  7. [CoFS] Connes, A., Flato, M., and Sternheimer, D., Closed star-products and cyclic cohomology, Lett. Math. Phys.24 (1992), 1-12. Zbl0767.55005MR1162894
  8. [Cz] Czyz, J.,On geometric quantization and its connections with the Maslov theory, Rep. Math. Phys.15 (1979), 57-97. Zbl0419.58009MR551131
  9. [DaP] Dazord, P., and Patissier, G., La première classe de Chern comme obstruction à la quantification asymptotique, Symplectic geometry, groupoids, and integrable systems, Séminaire sud-Rhodanien de géométrie à Berkeley (1989), P. Dazord and A. Weinstein, eds., Springer-MSRI Series (1991), 73-97. Zbl0732.58020MR1104920
  10. [De] Deligne, P., Unpublished letters and lectures at IAS, Princeton, 1993. 
  11. [DeL1] De Wilde, M., and Lecomte, P., Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds, Lett. Math. Phys.7 (1983), 487-496. Zbl0526.58023MR728644
  12. [DeL2] De Wilde, M., and Lecomte, P., Formal deformations of the Poisson Lie algebra of a symplectic manifold and star-products. Existence, equivalence, derivations, in M. Hazewinkel and M. Gerstenhaber, eds., Deformation Theory of Algebras and Structures and Applications, Kluwer Acad. Pub., Dordrecht (1988), 897-960. Zbl0685.58039MR981635
  13. [DeL3] De Wilde, M., and Lecomte, P., Existence of star-products revisited, Note di Matematica10, Suppl. 1 (1990), 205-216. Zbl0776.53023MR1193524
  14. [Di] Dirac, P.A.M., The principles of quantum mechanics, Clarenden Press, Oxford, 1930. Zbl0012.18104MR23198
  15. [Do] Donin, J., On the quantization of Poisson brackets, Advances in Math. (to appear). Zbl0931.53040MR1445363
  16. [EW] Emmrich, C., and Weinstein, A., The differential geometry of Fedosov's quantization, Lie Theory and Geometry, in Honor of B. Kostant, J.L. Brylinski, R. Brylinski, V. Guillemin, and V. Kac, eds., Progress in Mathematics, Birkhäuser, New York (to appear). Zbl0846.58031MR1327527
  17. [Fe1] Fedosov, B.V., Formal quantization, Some Topics of Modern Mathematics and their Applications to Problems of Mathematical Physics (in Russian), Moscow (1985), 129-136. MR933154
  18. [Fe2] Fedosov, B.V., Index theorem in the algebra of quantum observables, Sov. Phys. Dokl.34 (1989), 318-321. MR998039
  19. [Fe3] Fedosov, B.V., A simple geometrical construction of deformation quantization, J. Diff. Geom. (to appear). Zbl0812.53034MR1293654
  20. [Fe4] Fedosov, B.V., Proof of the index theorem for deformation quantization, Advances in Partial Differential Equations, Akademie Verlag, Berlin (to appear). MR1389013
  21. [Fe5] Fedosov, B.V., Reduction and eigenstates in deformation quantization, Advances in Partial Differential Equations, Akademie Verlag, Berlin (to appear). Zbl0809.58012MR1287670
  22. [Fe6] Fedosov, B.V., A trace density in deformation quantization, preprint, Moscow Institute of Physics and Technology, 1994. MR1389014
  23. [FS] Flato, M., and Sternheimer, D., Closedness of star products and cohomologies, Lie Theory and Geometry, in Honor of B. Kostant, J.L. Brylinski, R. Brylinski, V. Guillemin, and V. Kac, eds., Progress in Mathematics, Birkhäuser, New York (to appear). Zbl0853.58058MR1327536
  24. [Ge] Gerstenhaber, M., On the deformation of rings and algebras, Annals of Math., 79 (1964), 59-103. Zbl0123.03101MR171807
  25. [Gu] Gutt, S., Equivalence of deformations of twisted products on a symplectic manifold, Lett. Math. Phys.3 (1979), 495-502. MR555333
  26. [KM1] Karasev M.V., and Maslov, V.P., Pseudodifferential operators and a canonical operator in general symplectic manifolds, Math. USSR Izvestia23 (1984), 277-305. Zbl0554.58048
  27. [KM2] Karasev, M.V. and Maslov, V.P., Nonlinear Poisson brackets: geometry and quantization, Translations of mathematical monographs, v. 119, Amer. Math. Soc., Providence, 1993. Zbl0776.58003MR1214142
  28. [L] Lie, S., Theorie der Transformationsgruppen, (Zweiter Abschnitt, unter Mitwirkung von Prof. Dr. Friedrich Engel), Leipzig, Teubner, 1890. Zbl20.0368.01JFM23.0364.01
  29. [LtF] Littlejohn, R.G., and Flynn, W.G., Geometric phases in the asymptotic theory of coupled wave equations, Phys. Rev. A44 (1991), 5239-5256. MR1133847
  30. [Mi-Ru] Min-Oo, and Ruh, E., Comparison theorems for compact symmetric spaces, Ann. Sci. École Norm. Sup.12 (1979), 335-353. Zbl0424.53024MR559346
  31. [Ms] Moser, J., On the volume elements on a manifold, Trans. Amer. Math. Soc.120 (1965), 280-296. Zbl0141.19407MR182927
  32. [My] Moyal, J., Quantum mechanics as a statistical theory, Proc. Camb. Phil. Soc.45 (1949), 99-124. Zbl0031.33601MR29330
  33. [NT1] Nest, R., and Tsygan, B., Algebraic index theorem, Comm. Math. Phys., (to appear). Zbl0887.58050MR1350407
  34. [NT2] Nest, R., and Tsygan, B., Algebraic index theorem for families, Advances in Math. (to appear). Zbl0837.58029MR1337107
  35. [OMY1] Omori, H., Maeda, Y., and Yoshioka, A., Weyl manifolds and deformation quantization, Advances in Math.85 (1991), 224-255. Zbl0734.58011MR1093007
  36. [OMY2] Omori, H., Maeda, Y., and Yoshioka, A., Existence of a closed star-product, Lett. Math. Phys.26 (1992), 285-294. Zbl0771.58017MR1205289
  37. [Ri] Rieffel, M.A., Deformation quantization and operator algebras, Proc. Symp. Pure Math.51 (1990), 411-423. Zbl0715.46038MR1077400
  38. [Ru] Ruh, E., Cartan connections, Proc.Symp.Pure Math.54 (1993), 505-519. Zbl0796.53037MR1216642
  39. [S] Sternberg, S., Celestial Mechanics Part II, W.A. Benjamin, New York, 1969. Zbl0194.56702
  40. [Ta] Tamarkin, D.E., Topological invariants of connections on symplectic manifolds, Funct. Anal. Appl. (to appear). Zbl0878.58002MR1375540
  41. [Ts] Tsujishita, T., On variation bicomplexes associated to differential equations, Osaka J. Math19 (1982), 311-363. Zbl0524.58041MR667492
  42. [V] Vey, J., Déformation du crochet de Poisson sur une variété symplectique, Comment. Math. Helv.50 (1975), 421-454. Zbl0351.53029MR420753
  43. [Wi1] Weinstein, A., Fourier integral operators, quantization, and the spectrum of a Riemannian manifold. Colloque Internationale du Centre National de la Recherche Scientifique No. 237. Géométrie Symplectique et Physique Mathématique (1976), 289-298. Zbl0327.58013MR650990
  44. [Wi2] Weinstein, A., Noncommutative geometry and geometric quantization, Symplectic Geometry and Mathematical Physics: actes du colloque en l'honneur de Jean-Marie Souriau, P. Donato et al eds., Birkhäuser (1991), 446-461. Zbl0756.58022MR1156554
  45. [Wy] Weyl, H., The theory of groups and quantum mechanics, Dover, New York, 1931. Zbl0041.56804

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.