Obstructions to Quantization

  • M. J. Gotay


Quantization is not a straightforward proposition, as demonstrated by Groenewold’s and Van Hove’s discovery, more than fifty years ago, of an “obstruction” to quantization. Their “no-go theorems” assert that it is in principle impossible to consistently quantize every classical polynomial observable on the phase space R 2n in a physically meaningful way. Similar obstructions have been recently found for S 2 and T*S 1, buttressing the common belief that no-go theorems should hold in some generality. Surprisingly, this is not so—it has just been proven that there are no obstructions to quantizing either T 2 or T*R +.

In this paper we work towards delineating the circumstances under which such obstructions will appear, and understanding the mechanisms which produce them. Our objectives are to conjecture—and in some cases prove—generalized Groenewold-Van Hove theorems, and to determine the maximal Lie subalgebras of observables which can be consistently quantized. This requires a study of the structure of Poisson algebras of symplectic manifolds and their representations. To these ends we include an exposition of both prequantization (in an extended sense) and quantization theory, here formulated in terms of “basic algebras of observables” We then review in detail the known results for R 2n , S 2, T*S 1, T 2, and T*R +, as well as recent theoretical work. Our discussion is independent of any particular method of quantization; we concentrate on the structural aspects of quantization theory which are common to all Hilbert space-based quantization techniques.


Poisson Bracket Symplectic Manifold Basic Algebra Poisson Algebra Coadjoint Orbit 
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  1. [AM]
    Abraham, R. & Marsden, J.E. [1978] Foundations of Mechanics. Second Ed. (Benjamin-Cummings, Reading, MA).zbMATHGoogle Scholar
  2. [AA]
    Aldaya, V. & Azcárraga, J.A. [1982] Quantization as a consequence of the symmetry group: An approach to geometric quantization. J. Math. Phys. 23, 1297–1305.MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. [ADT]
    Angermann, B., Doebner, H.-D. & Tolar, J. [1983] Quantum kinematics on smooth manifolds. In: Nonlinear Partial Differential Operators and Quantization Procedures. Andersson, S.I. & Doebner, H.-D., Eds. Lecture Notes in Math. 1087, 171–208.CrossRefGoogle Scholar
  4. [AB]
    Arens, R. & Babbit, D. [1965] Algebraic difficulties of preserving dynamical relations when forming quantum-mechanical operators J. Math. Phys. 6, 1071–1075.ADSCrossRefGoogle Scholar
  5. [AS]
    Ashtekar, A. [1980] On the relation between classical and quantum variables. Commun. Math. Phys. 71, 59–64.MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. [At]
    Atkin, C.J. [1984] A note on the algebra of Poisson brackets. Math. Proc. Camb. Phil. Soc. 96, 45–60.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [Av1]
    Avez, A. [1974] Représentation de l’algèbre de Lie des symplectomorphismes par des opérateurs bornés. C.R. Acad. Sc. Paris Sér: A. 279, 785–787.MathSciNetzbMATHGoogle Scholar
  8. [Av2]
    Avez, A. [1974–1975] Remarques sur les automorphismes infinitésimaux des variétés symplectiques compactes. Rend. Sem. Mat. Univers. Politecn. Torino, 33, 5–12.Google Scholar
  9. [Av3]
    Avez, A. [1980] Symplectic group, quantum mechanics and Anosov’s systems. In: Dynamical Systems and Microphysics. Blaquiere, A. et al., Eds. (Springer, New York) 301–324.Google Scholar
  10. [BaRa]
    Barut, A.O. & Raczka, R. [1986] Theory of Group Representations and Applications. Second Ed. (World Scientific, Singapore).zbMATHGoogle Scholar
  11. [BFFLS]
    Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., & Stemheimer, D. [1978] De-formation theory and quantization I, II. Ann. Phys. 110, 61–110, 111–151.Google Scholar
  12. [B11]
    Blattner, R.J. [1983] On geometric quantization. In: Non-Linear Partial Differential Operators and Quantization Procedures. Andersson, S.I. & Doebner, H.-D., Eds. Lecture Notes in Math. 1087, 209–241.CrossRefGoogle Scholar
  13. [B12]
    Blattner, R.J. [1991] Some remarks on quantization. In: Symplectic Geometry and Mathematical Physics. Donato. P. et al., Eds. Progress in Math. 99 (Birkhäuser, Boston) 37–47.Google Scholar
  14. [BrRo]
    Bratteli, O. & Robinson, D.W. [1979] Operator Algebras and Quantum Statistical Mechanics I. (Springer, New York).Google Scholar
  15. [Ch1]
    Chernoff, P.R. [1981] Mathematical obstructions to quantization. Hodronic J.. 4, 879898.MathSciNetzbMATHGoogle Scholar
  16. [Ch2]
    Chernoff, P.R. [1988] Seminar on representations of diffeomorphism groups. Unpublished notes.Google Scholar
  17. [Ch3]
    Chernoff, P.R. [1995] Irreducible representations of infinite dimensional transformation groups and Lie algebras I. J. Funct. Anal. 130, 255–282.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [Co]
    Cohen, L. [1966] Generalized phase-space distribution functions. J. Math. Phvs. 7, 781 - 786.ADSCrossRefGoogle Scholar
  19. [Di]
    Dirac, P.A.M. [1967] The Principles of Quantum Mechanics. Revised Fourth Ed. (Oxford Univ. Press. Oxford).Google Scholar
  20. [DM]
    Doebner, H.D. and Melsheimer, O. [1968] Limitable dynamical groups in quantum mechanics I. J. Math. Phys. 9, 1638 - 1656.MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. [Em]
    Emch, G.G. [1972] Algebraic Methods in Statistical Mechanics and Quantum Field Theory. (Wiley, New York).zbMATHGoogle Scholar
  22. [Fi]
    Filippini, R.J. [1995] The symplectic geometry of the theorems of Borel-Weil and Peter-Weyl. Thesis, University of California at Berkeley.Google Scholar
  23. [F1]
    Flato, M. [1976] Theory of analytic vectors and applications. In: Mathematical Physics and Physical Mathematics. Maurin, K. & Riczka, R.. Eds. (Reidel, Dordrecht) 231–250.Google Scholar
  24. [Fo]
    Folland, G.B. [1989] Harmonic Analysis in Phase Space. Ann. Math. Ser. 122(Princeton University Press, Princeton).zbMATHGoogle Scholar
  25. [Fr]
    Fronsdal, C. [1978] Some ideas about quantization. Rep. Math. Phvs. 15, 111 - 145.ADSCrossRefGoogle Scholar
  26. [GM]
    Ginzburg, V.L. & Montgomery, R. [1997] Geometric quantization and no-go theorems. Preprint dg-ga/9703010.Google Scholar
  27. [GJ]
    Glimm, J. & Jaffe, A. [1981] Quantum Physics. A Functional Integral Point of View. (Springer Verlag, New York).zbMATHGoogle Scholar
  28. [Go1]
    Gotay, M.J. [1980] Functorial geometric quantization and Van Hove’s theorem. Int. J. Theor: Phys. 19, 139 - 161.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [Go2]
    Gotay, M.J. [1987] A class of non-polarizable symplectic manifolds. Mh. Math. 103, 27 - 30.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [Go3]
    Gotay, M.J. [1995] On a full quantization of the torus. In: Quantization, Coherent States and Complex Structures, Antoine, J.-P. et al., Eds. (Plenum, New York) 55–62.Google Scholar
  31. [Go4]
    Gotay, M.J. [1999] On the Groenewold-Van Hove problem for R 2n. J. Math. Phvs. 40, 2107 - 2116.MathSciNetADSzbMATHCrossRefGoogle Scholar
  32. [GGra1]
    Gotay, M.J. & Grabowski, J. [1999] On quantizing nilpotent and solvable basic algebras. Preprint math-ph/9902012.Google Scholar
  33. [GGra2]
    Gotay, M.J. & Grabowski, J. [2000] On quantizing semisimple basic algebras. In preparation.Google Scholar
  34. [GGG]
    Gotay, M.J., Grabowski, J., & Grundling, H.B. [2000] An obstruction to quantizing compact symplectic manifolds. Proc. Amer: Math. Soc. 128, 237 - 243.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [GGru1]
    Gotay, M.J. & Grundling, H.B. [1997] On quantizing T*S 1.Rep. Math. Phys. 40, 107–123.MathSciNetADSzbMATHCrossRefGoogle Scholar
  36. [GGru2]
    Gotay, M.J. & Grundling, H. [1999] Nonexistence of finite-dimensional quantizations of a noncompact symplectic manifold. In: Differential Geonnetry and Applications, Koldr. I. et al., Eds. (Masaryk Univ., Brno) 593–596.Google Scholar
  37. [GGH]
    Gotay, M.J., Grundling, H., &Hurst, C.A. [1996] A Groenewold-Van Hove theorem for S 2. Trans. Amer: Math. Soc. 348 1579–1597.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [GGT]
    Gotay, M.J., Grundling, H., & Tuynman, G.T. [1996] Obstruction results in quantization theory. J. Nonlinear Sci. 6, 469 - 498.MathSciNetADSzbMATHCrossRefGoogle Scholar
  39. [Gra1]
    Grabowski, J. [1978] Isomorphisms and ideals of the Lie algebras of vector fields. Invent. Math. 50, 13 - 33.MathSciNetADSzbMATHCrossRefGoogle Scholar
  40. [Gra2]
    Grabowski, J. [1985] The Lie structure of C*and Poisson algebras. Studia Math. 81, 259–270.MathSciNetzbMATHGoogle Scholar
  41. [Gro]
    Groenewold, H.J. [1946] On the principles of elementary quantum mechanics. Physica 12, 405 - 460.MathSciNetADSzbMATHCrossRefGoogle Scholar
  42. [GS]
    Guillemin, V. & Sternberg, S. [1984] Symplectic Techniques in Physics. (Cambridge Univ. Press, Cambridge).zbMATHGoogle Scholar
  43. [HM]
    Helton, J.W. & Miller, R.L. [1994] NC Algebra: A Mathematica Package for Doing Non Commuting Algebra. v0.2, La Jolla).Google Scholar
  44. [He]
    Hennings, M.A. [1986] Fronsdal 5-quantization and Fell inducing. Math. Proc. Carnb. Phil. Soc. 99, 179 - 188.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [Is]
    Isham, C.J. [1984] Topological and global aspects of quantum theory. In: Relativity, Groups and Topology II. DeWitt, B.S. & Stora, R., Eds. (North-Holland, Amsterdam) 1059–1290.Google Scholar
  46. [Jo]
    Joseph, A. [1970] Derivations of Lie brackets and canonical quantization. Commun. Math. Phys. 17, 210 - 232.ADSzbMATHCrossRefGoogle Scholar
  47. [KS]
    Kerner, E.H. & Sutcliffe, W.G. [1970] Unique Hamiltonian operators via Feynman path integrals. J. Math. Phys. 11, 391–393.ADSCrossRefGoogle Scholar
  48. [Ki]
    Kirillov, A.A. [1990] Geometric quantization. In: Dynamical Systems IV: Symplectic Geometry and Its Applications. Arnol’d, V.I. and Novikov, S.P., Eds. Encyclopaedia Math. Sci. IV. (Springer, New York) 137–172.Google Scholar
  49. [Ku]
    Kuryshkin, V.V. [1972] La mécanique quantique avec une fonction non-négative de distribution dans l’espace des phases. Ann. Inst. H. Poincaré 17, 81 - 95.Google Scholar
  50. [KLZ]
    Kuryshkin, V.V., Lyabis, I.A., & Zaparovanny, Y.I. [1978] Sur le problème de la regle de correspondence en théorie quantique. Ann. Fond. L. de Broglie. 3, 45 - 61.Google Scholar
  51. [Ma]
    Mackey, G.W. [1976] The Theory of Unitary Group Representations(University of Chicago Press, Chicago).zbMATHGoogle Scholar
  52. [MC]
    Margenau, H. & Cohen, L. [1967] Probabilities in quantum mechanics. In: Quantum Theory and Reality. Bunge, M., Ed. (Springer-Verlag, New York), 71–89.CrossRefGoogle Scholar
  53. [MR]
    Marsden, J.E. & Ratiu, T.S. [1994] Introduction to Mechanics and Symmetry.(Springer-Verlag, New York).zbMATHCrossRefGoogle Scholar
  54. [MMSV]
    Mnatsakanova, M., Morchio, G., Strocchi, F., & Vernov, Yu. [1998] Irreducible representations of the Heisenberg algebra in Krein spaces. J. Math. Phys. 39, 2969–2982.MathSciNetADSzbMATHCrossRefGoogle Scholar
  55. [On]
    Onishchik, A.L. [1994] Topology of Transitive Transformation Groups. (Johann Ambrosius Barth, Leipzig).zbMATHGoogle Scholar
  56. [ReSi]
    Reed, M. & Simon, B. [1972] Functional Analysis I. (Academic Press, New York).Google Scholar
  57. [Ri1]
    Rieffel, M.A. [1989] Deformation quantization of Heisenberg manifolds. Commun. Math. Phys. 122, 531–562.MathSciNetADSzbMATHCrossRefGoogle Scholar
  58. [Ri2]
    Rieffel, M.A. [1990] Deformation quantization and operator algebras. Proc. Sym. Pure Math. 45, 411–423.MathSciNetGoogle Scholar
  59. [Ri3]
    Rieffel, M.A. [1993] Quantization and C* -algebras. In: C* -Algebras: 1943–1993, A Fifty Year Celebration. Doran, R.S., Ed. Contemp. Math. 167, 67–97.Google Scholar
  60. [Ri4]
    Rieffel, M.A. [1998] Questions on quantization. In: Operator Algebras and Operator Theory. Contemp. Math. 228, 315–326.MathSciNetCrossRefGoogle Scholar
  61. [Ro]
    Robert, A. [1983] Introduction to the Representation Theory of Compact and Locally Compact Groups. London Math. Soc. Lect. Note Ser. 80 (Cambridge U. P., Cambridge).CrossRefGoogle Scholar
  62. [So]
    Souriau, J.-M. [1997] Structure of Dynamical Systems.(Birkhäuser, Boston).zbMATHCrossRefGoogle Scholar
  63. [Tu]
    Tuynman, G.M. [1998] Prequantization is irreducible. Indag. Mathem. 9, 607–618.MathSciNetzbMATHCrossRefGoogle Scholar
  64. [Ur]
    Urwin, R.W. [1983] The prequantization representations of the Poisson Lie algebra. Adv. Math. 50, 126–154.MathSciNetzbMATHCrossRefGoogle Scholar
  65. [VH1]
    van Hove, L. [1951] Sur certaines représentations unitaires d’un groupe infini de transformations. Proc. Roy. Acad. Sci. Belgium26, 1–102.Google Scholar
  66. [VH2]
    van Hove, L. [1951] Sur le problème des relations entre les transformations unitaires de la mécanique quantique et les transformations canoniques de la mécanique classique. Acad. Roy. Belgique Bull. Cl. Sci. (5) 37, 610 - 620.Google Scholar
  67. [Va]
    Varadarajan, V.S. [1984] Lie Groups, Lie Algebras and Their Representations. (Springer-Verlag, New York).zbMATHGoogle Scholar
  68. [Ve]
    Velhinho, J. [1998] Some remarks on a full quantization of the torus. Int. J. Mod. Phys.A13, 3905–3914.MathSciNetADSGoogle Scholar
  69. [VN]
    von Neumann, J. [1955] Mathematical Foundations of Quantum Mechanics. (Princeton. Univ. Press, Princeton).zbMATHGoogle Scholar
  70. [We]
    Weinstein, A. [1989] Cohomology of symplectomorphism groups and critical values of hamiltonians. Math. Z. 201, 75–82.MathSciNetzbMATHCrossRefGoogle Scholar
  71. [Wi]
    Wildberger, N. [1983] Quantization and harmonic analysis on Lie groups. Dissertation, Yale University.216 M. J. Gotay.Google Scholar
  72. [Wo]
    Woodhouse, N.M.J. [1992] Geometric quantization. Second Ed. (Clarendon Press, Oxford).zbMATHGoogle Scholar
  73. [zi]
    Ziegler, F. [1996] Quantum representations and the orbit method. Thesis, Université de Provence.Google Scholar

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© Springer Science+Business Media New York 2000

Authors and Affiliations

  • M. J. Gotay
    • 1
  1. 1.Department of MathematicsUniversity of HawaiiHonoluluUSA

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