Skip to main content
SpringerLink
Log in

Unitarity in “Quantization Commutes with Reduction”

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. In this paper, we study the geometric quantization of the symplectic quotient M // G. Guillemin and Sternberg [Invent. Math. 67, 515–538 (1982)] have shown, under suitable regularity assumptions, that there is a natural invertible map between the quantum Hilbert space over M //G and the G-invariant subspace of the quantum Hilbert space over M.

Reproducing other recent results in the literature, we prove that in general the natural map of Guillemin and Sternberg is not unitary, even to leading order in Planck’s constant. We then modify the quantization procedure by the “metaplectic correction” and show that in this setting there is still a natural invertible map between the Hilbert space over M // G and the G-invariant subspace of the Hilbert space over M. We then prove that this modified Guillemin–Sternberg map is asymptotically unitary to leading order in Planck’s constant. The analysis also shows a good asymptotic relationship between Toeplitz operators on M and on M // G.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Axelrod S., Della Pietra S. and Witten E. (1991). Geometric quantization of Chern-Simons gauge theory. J. Diff. Geom. 33: 787–902

    MATH  Google Scholar 

  2. Baez J.C., Segal I.E. and Zhou Z.-F. (1992). Introduction to algebraic and constructive quantum field theory. Princeton Series in Physics. Princeton University Press, Princeton, NJ

    Google Scholar 

  3. Banyaga A. and Hurtubise D.E. (2004). A proof of the Morse–Bott lemma. Exp. Math. 22(4): 365–373

    MATH  Google Scholar 

  4. Bleistein N. and Handelsman R.A. (1975). Asymptotic Expansions of Integrals. Dover Inc., New York

    MATH  Google Scholar 

  5. Borthwick D., Paul T. and Uribe A. (1995). Legendrian distributions with applications to relative Poincaré series. Invent. Math. 122(2): 359–402

    Article  MATH  ADS  Google Scholar 

  6. Borthwick D. and Uribe A. (1996). Almost complex strucutres and geometric quantization. Math. Res. Lett. 3(6): 845–861

    MATH  Google Scholar 

  7. Borthwick D. and Uribe A. (2000). Nearly Kählerian embeddings of symplectic manifolds. Asian J. Math. 4(3): 599–620

    MATH  Google Scholar 

  8. Boutetde Monvel L. and Guillemin V. (1981). The Spectral Theory of Toeplitz Operators, Vol. 99 Ann. of Math. Studies. Princeton University Press, Princeton, NJ

    Google Scholar 

  9. Charles L. (2007). Semi-classical properties of geometric quantization with metaplectic correction. Comm. Math. Phys. 270(2): 445–480

    Article  MATH  ADS  Google Scholar 

  10. Charles L. (2006). Toeplitz operators and Hamiltonian torus actions. J. Func. Anal. 236(1): 299–350

    Article  MATH  Google Scholar 

  11. Chavel I. (2006). Riemannian Geometry: A Modern Introduction, Second Edition, Vol. 98 Cambridge studies in advanced mathematics. Cambridge University Press, New York

    Google Scholar 

  12. Czyz J. (1978). On some approach to geometric quantization. Diff. Geom. Methods in Math. Phys. 676: 315–328

    Article  Google Scholar 

  13. Dirac P.A.M. (1964). Lectures on Quantum Mechanics. Yeshiva University, New York

    Google Scholar 

  14. Donaldson, S.K.: Remarks on gauge theory, complex geometry and 4-manifold topology. In: S.M. Atiyah, D. Iagolnitzer, editors, Fields Medallists’ Lectures, 2nd Edition, no. 9 in World Scientific Series in 20th Century Mathematics, 2004

  15. Driver B.K. and Hall B.C. (1999). Yang–Mills theory and the Segal–Bargmann transform. Commun. Math. Phys. 201: 249–290

    Article  MATH  ADS  Google Scholar 

  16. Driver, B.K., Hall, B.C.: The energy representation has no non-zero fixed vectors. In: Stochastic processes, physics and geometry: new interplays, II, number 29 in Conference Proceedings, Providence, RI: Amer. Math. Soc. 2000

  17. Duistermaat J. and Kolk J. (2000). Lie Groups. Springer-Verlag, Berlin

    MATH  Google Scholar 

  18. Flensted-Jensen M. (1978). Spherical functions of a real semisimple Lie group. A method of reduction to the complex case. J. Func. Anal. 30(1): 106–146

    Article  MATH  Google Scholar 

  19. Florentino C., Matias P., Mourão J. and Nunes J.P. (2003). Coherent state transforms and vector bundles on elliptic curves. J. Func. Anal. 204(2): 355–398

    Article  MATH  Google Scholar 

  20. Flude, J.P.M.: Geometric asymptotics of spin. Thesis, U. Nottingham, UK, 1998

  21. Gilkey P.B., Leahy J.V. and Park J. (1999). Spectral geometry, Riemannian submersions, and the Gromov–Lawson conjecture. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton, FL

    Google Scholar 

  22. Gotay M.J. (1986). Constraints, reduction and quantization. J. Math. Phys. 27(8): 2051–2066

    Article  MATH  ADS  Google Scholar 

  23. Griffiths P. and Harris J. (1978). Principles of Algebraic Geometry. John Wiley & Sons, New York

    MATH  Google Scholar 

  24. Guillemin V. and Sternberg S. (1982). Geometric Quantization and Multiplicities of Group Representations. Invent. Math. 67: 515–538

    Article  MATH  ADS  Google Scholar 

  25. Hall B.C. (2001). Coherent states and the quantization of (1 + 1)-dimensional Yang–Mills theory. Rev. Math. Phys. 13(10): 1281–1305

    Article  MATH  Google Scholar 

  26. Hall B.C. (2002). Geometric quantization and the generalized segal–bargmann transform for lie groups of compact type. Commun. Math. Phys. 226: 233–268

    Article  MATH  ADS  Google Scholar 

  27. Heinzner P., Huckleberry A. and Loose F. (1994). Kählerian extensions of the symplectic reduction. J. reine angew. Math. 455: 123–140

    MATH  Google Scholar 

  28. Huebschmann J. (2006). Kähler quantization and reduction. J. reine angew. Math. 591: 75–109

    MATH  Google Scholar 

  29. Jeffrey L.C. and Kirwan F.C. (1997). Localization and the quantization conjecture. Topology 36(3): 647–693

    Article  MATH  Google Scholar 

  30. Kempf, G., Ness, L.: The length of vectors in representation spaces. In: Algebraic Geometry (Proceedings of the Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), no. 732 Lecture Notes in Math., Berlin: Springer, 1979, pp 233–243

  31. Knapp A. (2002). Lie Groups: Beyond an Introduction, 2nd Edition, Vol. 140 Progress in Mathematics. Basel-Boston, Birkhäuser

    Google Scholar 

  32. Landsman N. (1995). Rieffel induction as generalized quantum Marsden–Weinstein reduction. J. Geom. Phys. 15(4): 285–319

    Article  MATH  Google Scholar 

  33. Landsman N. and Wren K. (1997). Constrained quantization and θ-angles. Nuc. Phys. B 502(3): 537–560

    Article  MATH  ADS  Google Scholar 

  34. Ma X. and Marinescu G. (2004). Generalized Bergman kernels on symplectic manifolds. C. R. Acad. Sci. Paris, Ser. I 339(7): 493–498, Full Version: http://arxiv.org/list/math.DG/0411559, 2004

    MATH  Google Scholar 

  35. Ma X. and Zhang W. (2005). Bergman kernels and symplectic reduction. C. R. Acad. Sci. Paris, Ser. I 341: 297–302

    MATH  Google Scholar 

  36. Ma, X., Zhang, W.: Bergman kernels and symplectic reduction. http://arxiv.org/list/math.DG/0607605, 2006

  37. Marsden J. and Weinstein A. (1974). Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5(1): 121–130

    Article  MATH  Google Scholar 

  38. Meinrenken E. (1998). Symplectic Surgery and the Spinc-Dirac Operator. Adv. Math. 134(2): 240–277

    Article  MATH  Google Scholar 

  39. Mumford D., Fogarty J. and Kirwan F. (1994). Geometric Invariant Theory, Third Edition, Volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Springer-Verlag, Berlin

    Google Scholar 

  40. Paoletti R. (2005). The Szëgo Kernel of a symplectic quotient. Adv. Math. 197(2): 523–553

    Article  MATH  Google Scholar 

  41. Sjamaar R. (1995). Holomorphic slices, symplectic reduction and multiplicities of representations. Ann. Math. (2) 141(1): 87–129

    Article  MATH  Google Scholar 

  42. Sjamaar R. (1996). Symplectic reduction and Riemann–Roch formulas for multiplicities. Bull. Amer. Math. Soc. (N.S.) 33(3): 327–388

    Article  MATH  Google Scholar 

  43. Stenzel M. (1999). The Segal–Bargmann transform on a symmetric space of compact type. J. Func. Anal. 165: 44–58

    Article  MATH  Google Scholar 

  44. Tian Y. and Zhang W. (1998). An analytic proof of the geometric quantization conjecture of Guillemin–Sternberg. Invent. Math. 132: 229–259

    Article  MATH  Google Scholar 

  45. Woodhouse N.M.J. (1991). Geometric Quantization. Oxford University Press, Inc., New York

    Google Scholar 

  46. Wren K. (1998). Constrained quantization and θ-angles. II. Nuc. Phys. B 521(3): 471–502

    Article  MATH  ADS  Google Scholar 

  47. Zheng, F.: Complex Differential Geometry, Volume 18 of Studies in Advanced Mathematics. Providence, RI: Amer. Math. Soc./ Int. Press, 2000

Download references

Author information

Authors and Affiliations

  1. University of Notre Dame, Notre Dame, IN, 46556-4618, USA

    Brian C. Hall & William D. Kirwin

Authors
  1. Brian C. Hall
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. William D. Kirwin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Brian C. Hall.

Additional information

Communicated by J.Z. Imbrie

Supported in part by NSF Grant DMS-02000649.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hall, B.C., Kirwin, W.D. Unitarity in “Quantization Commutes with Reduction”. Commun. Math. Phys. 275, 401–442 (2007). https://doi.org/10.1007/s00220-007-0303-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-007-0303-6

Keywords

Search

Navigation