Skip to main content

Connections on symplectic manifolds and geometric quantization

Part I Proceedings Of The International Colloquium Of The C.N.R.S. Held At Aix-en-Provence, September 3–7, 1979 Edited By J.M. Souriau

Part of the Lecture Notes in Mathematics book series (LNM,volume 836)

Resume

Etant donné une variété symplectique munie des sous-fibrés Lagrangiens F,G ⊂ TℂM complémentaires, nous considérons des connexions symplectiques distinguées. Ces connexions ∇ sont utilisées pour quantifier les germes de fonctions dans les faisceaux C kF par des opérateurs différentiels dans un fibré quantique en lignes Q, qui est muni d'une connexion ∇Q plate en direction de F. Aux propres choix de ∇, ∇Q et d'une suite c de nombres réels, nous définissons des lois de quantification, qui généralizent les quantifications de Kostant-Souriau, de Weyl ou à l'ordre (anti-)normale.

Keywords

  • Vector Bundle
  • Symplectic Manifold
  • Geometric Quantization
  • Hermitian Manifold
  • Order Differential Operator

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.00
Price excludes VAT (Canada)
  • Compact, lightweight 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G.S. Agarwal, E. Wolf Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics I Phys.Rev. D 2 (1970), 2161–2186

    MathSciNet  MATH  Google Scholar 

  2. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer Deformation theory and quantization I Ann.Phys. 111 (1978), 61–110

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. R.Bott in: Lectures on algebraic and differential topology Springer Lect.Notes in Math., vol.279

    Google Scholar 

  4. H. Daughaday, B.P. Nigam Function in quantum mechanics which corresponds to a given function in classical mechanics Phys.Rev. 139 B (1965), 1436–1442

    CrossRef  MathSciNet  Google Scholar 

  5. M. Flato, A. Lichnerowicz, D. Sternheimer Crochet de Moyal-Vey et quantification C.R.Acad.Sc.Paris, sér.A A 283 (1976) 19–24

    MathSciNet  MATH  Google Scholar 

  6. K. Gawędzki Fourier-like kernels in geometric quantization Diss.Math. 128 (1976), 1–83

    MathSciNet  MATH  Google Scholar 

  7. H.Hess On a geometric quantization scheme generalizing those of Kostant-Souriau and Czyż, to app. in: Proc.Inform.Meet. on Diff.Geom.Meth. in Physics, Clausthal 1978

    Google Scholar 

  8. H.Hess forthcoming thesis

    Google Scholar 

  9. L. Hörmander The Frobenius-Nirenberg theorem Arkiv för Matematik 5 (1965), 425–432

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. F.W.Kamber, P.Tondeur Foliated bundles and characteristic classes Springer Lect.Notes in Math., vol.493 (1975)

    Google Scholar 

  11. B.Kostant On the definition of quantization in: Coll.Int. du CNRS, Géométrie symplectique et physique mathématique, Aix-en-Provence 1974, ed. CNRS (1976)

    Google Scholar 

  12. B.Kostant Symplectic spinors in: Conv. di geom. simplett. e fisica matem., INDAM Rome 1973, Sympos.Math. XIV, Academic Press N.Y. (1974)

    Google Scholar 

  13. J.H. Rawnsley On the cohomology groups of a polarisation and diagonal quantisation Trans.Am.Math.Soc. 230 (1977), 235–255

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. J.H. Rawnsley Flat partial connections and holomorphic structures in C vector bundles Proc.Am.Math.Soc. 73 (1979), 391–397

    MathSciNet  MATH  Google Scholar 

  15. J. Underhill Quantization on a manifold with connection J.Math.Phys. 19 (1978), 1932–1935

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. J. Underhill, S.Taraviras Weyl quantization on a sphere in: Springer Lect.Notes in Phys., vol.50 (1976), 210–216

    CrossRef  MathSciNet  Google Scholar 

  17. I. Vaisman Cohomology and differential forms Dekker, N.Y. (1973)

    MATH  Google Scholar 

  18. J. Vey Déformation du crochet de Poisson sur une variété symplectique Comment.Math.Helv. 50 (1975), 421–454

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. A. Weinstein Symplectic manifolds and their Lagrangian submanifolds Adv.Math. 6 (1971), 329–346

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1980 Springer-Verlag

About this paper

Cite this paper

Hess, H. (1980). Connections on symplectic manifolds and geometric quantization. In: García, P.L., Pérez-Rendón, A., Souriau, J.M. (eds) Differential Geometrical Methods in Mathematical Physics. Lecture Notes in Mathematics, vol 836. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0089731

Download citation

  • DOI: https://doi.org/10.1007/BFb0089731

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-10275-5

  • Online ISBN: 978-3-540-38405-2

  • eBook Packages: Springer Book Archive