Skip to main content
SpringerLink
Log in

Geometric Quantization

  • Chapter
Dynamical Systems IV

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 4))

Abstract

The word “quantization” is used both in physical and in mathematical works in many different senses. In recent times this has come to be reflected explicitly in the terminology: the terms “asymptotic”, “deformational”, “geometric” quantization, etc., have emerged.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M.F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond., A 308 (1983), 523–615.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Alexeev and S. Shatashvili, Path integral quantization of the coadjoint orbits of the Virasoro group and 1-D gravity, Nuclear Phys. B323 (1989), 719–733.

    Article  MathSciNet  Google Scholar 

  3. B] J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization,In: Progress in Mathematics, Vol. 107, Birkhäuser (1993) 1–300.

    Google Scholar 

  4. N. Berline and M. Vergne, The equivariant index and Kirillov’s character formula, Amer. J. Math. 107 (1985), 1159–1190.

    Article  MathSciNet  MATH  Google Scholar 

  5. J.J. Duistermaat and G.J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Inventiones Mathematicae 69 (1982), 259–268.

    Article  MathSciNet  MATH  Google Scholar 

  6. V. Guillemin, Moment maps and combinatorial invariants of Hamiltonian T^ -spaces, Progress in Mathematics 122, Birkhäuser (1994) 1 — I50

    Google Scholar 

  7. V. Guillemin and S. Sternberg, Homogeneous quantization and multiplicities of group representations, J. Funct. Anal. 47 (1982), 344–380.

    Article  MathSciNet  MATH  Google Scholar 

  8. G.J. Heckman, Projections orbits and asymptotic behavior of multiplicities for compact connected Lie group, Inventiones Mathematicae 67 (1982), 333–356.

    Article  MathSciNet  MATH  Google Scholar 

  9. L. Jeffrey and F. Kirwan, Localization for non abelian group actions, Topology 34, No. 2 (1995) 291–327.

    Article  MathSciNet  MATH  Google Scholar 

  10. A.A. Kirillov, Kähler structures on K-orbits of the group of diffeomorphisms of the circle,Funkts. Anal. Prilozh. 21, No. 2 (1987), 42–45. Enlish transi.: Funct. Anal. Appt. 21, No. 2 (1987), 122125.

    Google Scholar 

  11. A.A. Kirillov, The orbit method, I: Geometric quantization, II: Infinite-dimensional Lie groups and Lie algebras, Contemporary Mathematics 145 (1993), 1–63.

    Google Scholar 

  12. A.A. Kirillov, Introduction to the Theory of Representations and Noncommutative Harmonic Analysis, Itogi Nauki Tekh., Ser. Sovrem. Probt. Mat., Fundam. Napravleniya 22 (1988), 5162. English transl. in: Representation Theory and Noncommutative Harmonic Analysis I, Encyclopaedia of Mathematical Sciences 22, Springer, 1994, 1–156.

    Google Scholar 

  13. Y. Karshon and S. Tolman, The moment map and line bundles over presymplectic toric manifolds, J. Diff. Geom. 38 (1993), 465–484.

    MathSciNet  MATH  Google Scholar 

  14. A.A. Kirillov and D.V. Yuriev, Kähler geometry of the infinite-dimensional space M = Dijj(S 1 )/ Rot(SI), Funct. Anal. Appt. 21, No. 4 (1987), 284–294.

    Google Scholar 

  15. R. Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations, Annals of Math. II. Ser. 141, No. 1 (1995), 87–129.

    Google Scholar 

  16. M. Vergne, Quantification géométrique et multiplicités,C. R. Acad. Sci. Paris 319 (1994), 327332.

    Google Scholar 

  17. E. Witten, Two dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), 303–368.

    Google Scholar 

  18. Adler, M.: On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-Devries type equations. Invent. Math. 50, 219–248 (1979). Zbl. 393. 35058

    Google Scholar 

  19. Arnol’d, V.I.: Mathematical Methods of Classical Mechanics. Nauka, Moscow 1974, 431 p. (English translation: Graduate Texts in Mathematics 60, Springer-Verlag, New YorkHeidelberg-Berlin 1978, 462 p.). Zbl. 386. 70001

    Google Scholar 

  20. Arnol’d, V.I., Givental’, A.B.: Symplectic geometry, in the collection: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundamental’nye Napravleniya (Contemporary Problems of Mathematics, Fundamental Directions) 4. VINITI, Moscow 1985, pp. 5–139 (English translation: see present volume, pp. 1–136). Zbl. 592. 58030

    Google Scholar 

  21. Atiyah, M.F., Schmid, W.: A geometric construction of the discrete series for semi-simple Lie groups. Invent. Math. 42, 1–62 (1977). Zbl. 373. 22001

    Google Scholar 

  22. Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform. Part I. Commun. Pure Appl. Math. /4, 187–214 (1961). Zbl. 107, 91

    Google Scholar 

  23. Berezin, F.A.: Quantization. Izv. Akad. Nauk SSSR, Ser. Mat. 38, 1116–1175 (1974) (English translation: Math. USSR, Izv. 8, 1109–1165 (1974)). Zbl. 312. 53049

    Google Scholar 

  24. Berezin, F.A.: Introduction to Algebra and Analysis with Anticommuting Variables. Moscow State University, Moscow 1983, 208 p. (English translation: Part I of: Introduction to Superanalysis. Mathematical Physics and Applied Mathematics 9. D. Reidel, DordrechtBoston-Tokyo 1987, 424 p.). Zbl. 527. 15020

    Google Scholar 

  25. Berezin, F.A., Lejtes, D.A.: Supermanifolds. Dokl. Akad. Nauk SSSR 224, 505–508 (1975) (English translation: Soy. Math., Dokl. /6, 1218–1222 (1975)). Zbl. 331. 58005

    Google Scholar 

  26. Blattner, R.J.: The metalinear geometry of non-real polarizations, in: Differential Geometrical Methods in Mathematical Physics. Proceedings of the Symposium Held at the University of Bonn, July 1–4, 1975. Lect. Notes Math. 570, Springer-Verlag, Berlin—Heidelberg—New York 1977, pp. 11–45. Zbl. 346. 53027

    Google Scholar 

  27. Duflo, M.: Théorie de Mackey pour les groupes de Lie algébriques. Acta Math. 149, 153–213 (1982). Zbl. 529. 22011

    Google Scholar 

  28. Duflo, M.: Construction de représentations unitaires d’un groupe de Lie, in: Harmonic Analysis and Group Representations. Cours d’été du CIME, Cortona, 1980. Liguori Editore, Naples 1982, pp. 129–221

    Google Scholar 

  29. Faddeev, L.D., Takhtadzhyan, L.A.: Simple connection between the geometric and the Hamiltonian representations of integrable nonlinear equations. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 115, 264–273 (1982) (English translation: J. Sov. Math. 28, 800–806 (1985)). Zbl. 505. 35082

    Google Scholar 

  30. Faddeev, L.D., Yakubovskij, O.A.: Lectures on Quantum Mechanics for Students of Mathematics (in Russian). Leningrad State University, Leningrad 1980, 200 p.

    Google Scholar 

  31. Fomenko, A.T., Mishchenko, A.S.: Integrability of Euler equations on semisimple Lie algebras. Tr. Semin. Vektorn. Tenzorn. Anal. Prilozh. Geom. Mekh. Fiz. 19, 3–94 (1979) (English translation: Sel. Math. Sov. 2, 207–291 (1982)). Zbl. 452. 58015

    Google Scholar 

  32. Fomenko, A.T., Trofimov, V.V.: Liouville integrability of Hamiltonian systems on Lie algebras. Usp. Mat. Nauk 39, No. 2, 3–56 (1984) (English translation: Russ. Math. Surv. 39, No. 2, 1–67 (1984)). Zbl. 549. 58024

    Google Scholar 

  33. Guillemin, V., Sternberg, S.: Geometric Asymptotics. Mathematical Surveys 14. American Mathematical Society, Providence, R.I. 1977, 474 p. Zbl. 364. 53011

    Google Scholar 

  34. Hörmander, L.: An Introduction to Complex Analysis in Several Variables. Van Nostrand, Princeton 1966, 208 p. Zbl. 138, 62

    Google Scholar 

  35. Hurt, N.: Geometric Quantization in Action. Mathematics and Its Applications 8. D. Reidel, Dordrecht—Boston—London 1983, 336 p. Zbl. 505. 22007

    Google Scholar 

  36. Joseph, A.: The algebraic method in representation theory (enveloping algebras), in: Group Theoretical Methods in Physics. Fourth International Colloquium, Nijmegen 1975. Lect. Notes Phys. 50. Springer-Verlag, Berlin—Heidelberg—New York 1976, pp. 95–109. Zbl. 374. 17003

    Google Scholar 

  37. Kirillov, A.A.: Unitary representations of nilpotent Lie groups. Usp. Mat. Nauk 17, No. 4, 57–110 (1962) (English translation: Russ. Math. Surv. /7, No. 4, 53–104 (1962)). Zbl. 106, 250

    Google Scholar 

  38. Kirillov, A.A.: Construction of unitary irreducible representations of Lie groups. Vestn. Mosk. Univ., Ser. 125, No. 2, 41–51 (1970) (English translation: Mosc. Univ. Math. Bull. 25 (1970), Nr. 1–2, 69–76 (1972)). Zbl. 249. 22011

    Google Scholar 

  39. Kirillov, A.A.: Lectures on the Theory of Group Representations. Part IV. Group Representations and Mechanics (in Russian). Moscow State University, Moscow 1971

    Google Scholar 

  40. Kirillov, A.A.: Elements of the Theory of Representations (Second edition). Nauka, Moscow 1978, 343 p. (English translation (of the first edition): Grundlehren der mathematischen Wissenschaften 220, Springer-Verlag, Berlin—Heidelberg—New York 1976, 315 p.). Zbl. 342. 22001

    Google Scholar 

  41. Kirillov, A.A.: Invariant operators on geometric quantities, in the collection: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. (Contemporary Problems of Mathematics) 16. VINITI, Moscow 1980, p. 3–29 (English translation: J. Soy. Math. 18, 1–21 (1982)). Zbl. 453. 53008

    Google Scholar 

  42. Koopman, B.O.: Hamiltonian systems and transformations in Hilbert space. Proc. Natl. Acad. Sci. USA 17, 315–318 (1931) Zbl. 2, 57

    Google Scholar 

  43. Kostant, B.: Quantization and unitary representations. Part I: Prequantization, in: Lectures in Modern Analysis and Applications III. Lect. Notes Math. 170. Springer-Verlag, BerlinHeidelberg—New York 1970, pp. 87–208. Zbl. 223. 53028

    Google Scholar 

  44. Kostant, B.: Symplectic spinors, in:. Symp. Math. 14, Istituto Naz. di Alt. Mat. Roma. Academic Press, London—New York 1974, pp. 139–152. Zbl. 321. 58015

    Google Scholar 

  45. Kostant, B.: Graded manifolds, graded Lie theory, and prequantization, in: Differential Geometrical Methods in Mathematical Physics. Proceedings of the Symposium Held at the University of Bonn, July 1–4, 1975. Lect. Notes Math. 570. Springer-Verlag, Berlin-Heidelberg-New York 1977, pp. 177–306. Zbl. 358. 53024

    Google Scholar 

  46. Leray, J.: Analyse lagrangienne et mécanique quantique. Séminaire sur les Equations aux Dérivées Partielles (1976–1977). I, Exp. No. 1, Collège de France, Paris 1977, 303 p.; and Recherche coopérative sur programme CNRS No. 25, IRMA de Strasbourg 1978 (revised and translated into English: Lagrangian Analysis and Quantum Mechanics. The MIT Press, Cambridge, Mass.-London 1981, 271 p.). Zbl. 483. 35002

    Google Scholar 

  47. Lion, G., Vergne, M.: The Weil Representation, Maslov Index and Theta Series. Prog. Math., Boston 6. Birkhäuser, Boston-Basel-Stuttgart 1980, 337 p. Zbl. 444. 22005

    Google Scholar 

  48. Maslov, V.P.: Théorie des perturbations et méthodes asymptotiques. Moscow State University, Moscow 1965 (French translation, with two additional articles by V.I. Arnol’d and V.C. Bouslaev [Buslaev]: Dunod Gauthier-Villars, Paris 1972, 384 p.). Zbl. 247. 47010

    Google Scholar 

  49. Moser, J.: On the volume elements on a manifold. Trans. Am. Math. Soc. 120, 286–294 (1965). Zbl 141. 194

    Google Scholar 

  50. Newlander, A., Nirenberg, L.: Complex analytic coordinates in almost complex manifolds. Ann. Math., II. Ser. 65, 391–404 (1957). Zbl. 79. 161

    Google Scholar 

  51. Ol’shanetskij, M.A., Perelomov, A.M.: Explicit solutions of some completely integrable Hamiltonian systems. Funkts. Anal. Prilozh. 11, No. 1, 75–76 (1977) (English translation: Funct. Anal. Appl. 11, 66–68 (1977)). Zbl. 351. 58001

    Google Scholar 

  52. Onofri, E.: Dynamical quantization of the Kepler manifold. J. Math. Phys. 17, 401–408 (1976)

    Article  MathSciNet  Google Scholar 

  53. Rejman, A.G., Semenov-Tyan-Shanskij, M.A.: Current algebras and nonlinear partial differential equations. Dokl. Akad. Nauk SSSR 251, 1310–1314 (1980) (English translation: Soy. Math., Dokl. 21, 630–634 (1980)). Zbl. 501. 58018

    Google Scholar 

  54. Segal, I.E.: Quantization of nonlinear systems. J. Math. Phys. /, 468–488 (1960). Zbl. 99, 224

    Google Scholar 

  55. Simms, D.J., Woodhouse, N.M.J.: Lectures on Geometric Quantization. Lect. Notes Phys. 53. Springer-Verlag, Berlin-Heidelberg-New York 1976, 166 p. Zbl. 343. 53023

    Google Scholar 

  56. niatycki, J.: On cohomology groups appearing in geometric quantization, in: Differential Geometrical Methods in Mathematical Physics. Proceedings of the Symposium Held at the University of Bonn, July 1–4, 1975. Lect. Notes Math. 570. Springer-Verlag, Berlin-Heidelberg-New York 1977, pp. 46–66. Zbl. 353. 53019

    Google Scholar 

  57. niatycki, J.: Geometric Quantization and Quantum Mechanics. Applied Mathematical Sciences 30. Springer-Verlag, New York-Heidelberg-Berlin 1980, 230 p. Zbl. 429. 58007

    Google Scholar 

  58. Souriau, J.-M.: Quantification géométrique. Commun. Math. Phys. 1, 374–398 (1966)

    MathSciNet  MATH  Google Scholar 

  59. Souriau, J.-M.: Structure des systèmes dynamiques. Maîtrises de mathématiques. Dunod, Paris 1970, 414 p. Zbl. 186, 580

    Google Scholar 

  60. Souriau, J.-M.: Thermodynamique et géométrie, in: Differential Geometrical Methods in Mathematical Physics II. Proceedings, University of Bonn, July 13–16, 1977. Lect. Notes Math. 676. Springer-Verlag, Berlin-Heidelberg-New York 1978, pp. 369–397. Zbl. 397. 53048

    Google Scholar 

  61. Souriau, J.-M.: Groupes différentiels, in: Differential Geometrical Methods in Mathematical Physics. Proceedings of the Conferences Held at Aix-en-Provence, September 3–7, 1979 and Salamanca, September 10–14, 1979. Lect. Notes Math. 836. Springer-Verlag, Berlin-Heidelberg-New York 1980, pp. 91–128. Zbl. 501. 58010

    Google Scholar 

  62. Van Hove, L.: Sur certaines représentations unitaires d’un groupe infini de transformations. Mem. Cl. Sci., Collect. Octavo, Acad. R. Belg. 26, no. 6 (1951), 102 p. Zbl. 45, 387

    Google Scholar 

  63. Weinstein, A.: Symplectic geometry. Bull. Am. Math. Soc., New Ser. 5, 1–13 (1981). Zbl. 465. 58013

    Google Scholar 

  64. Woodhouse, N.M.J.: Twistor theory and geometric quantization, in: Group Theoretical Methods in Physics. Fourth International Colloquium, Nijmegen 1975. Lect. Notes Phys. 50. Springer-Verlag, Berlin-Heidelberg-New York 1976, pp. 149–163. Zbl. 364. 53015

    Google Scholar 

Download references

Authors
  1. A. A. Kirillov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Steklov Mathematical Institute, ul. Gubkina 8, 117966, Moscow, Russia

    V. I. Arnold

  2. CEREMADE, Université Paris 9 — Dauphine, Place du Marechal de Lattre de Tassigny, 75775, Paris Cedex 16-e, France

    V. I. Arnold

  3. Institute of Physical Sciences and Technology, University of Maryland, 20742-2431, College Park, MD, USA

    S. P. Novikov

  4. L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, 2 Kosygina str., 117940, Moscow, Russsia

    S. P. Novikov

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Kirillov, A.A. (2001). Geometric Quantization. In: Arnold, V.I., Novikov, S.P. (eds) Dynamical Systems IV. Encyclopaedia of Mathematical Sciences, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06791-8_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-06791-8_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08297-9

  • Online ISBN: 978-3-662-06791-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation