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Obstructions to Quantization

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Mechanics: From Theory to Computation

Summary

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.

This paper is dedicated to the memory of Juan-Carlos Simo

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References

  1. Abraham, R. & Marsden, J.E. [1978] Foundations of Mechanics. Second Ed. (Benjamin-Cummings, Reading, MA).

    MATH  Google Scholar 

  2. 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.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. 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.

    Chapter  Google Scholar 

  4. Arens, R. & Babbit, D. [1965] Algebraic difficulties of preserving dynamical relations when forming quantum-mechanical operators J. Math. Phys. 6, 1071–1075.

    Article  ADS  Google Scholar 

  5. Ashtekar, A. [1980] On the relation between classical and quantum variables. Commun. Math. Phys. 71, 59–64.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Atkin, C.J. [1984] A note on the algebra of Poisson brackets. Math. Proc. Camb. Phil. Soc. 96, 45–60.

    Article  MathSciNet  MATH  Google Scholar 

  7. 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.

    MathSciNet  MATH  Google Scholar 

  8. 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. 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. Barut, A.O. & Raczka, R. [1986] Theory of Group Representations and Applications. Second Ed. (World Scientific, Singapore).

    MATH  Google Scholar 

  11. 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. 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.

    Chapter  Google Scholar 

  13. 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. Bratteli, O. & Robinson, D.W. [1979] Operator Algebras and Quantum Statistical Mechanics I. (Springer, New York).

    Google Scholar 

  15. Chernoff, P.R. [1981] Mathematical obstructions to quantization. Hodronic J.. 4, 879898.

    MathSciNet  MATH  Google Scholar 

  16. Chernoff, P.R. [1988] Seminar on representations of diffeomorphism groups. Unpublished notes.

    Google Scholar 

  17. Chernoff, P.R. [1995] Irreducible representations of infinite dimensional transformation groups and Lie algebras I. J. Funct. Anal. 130, 255–282.

    Article  MathSciNet  MATH  Google Scholar 

  18. Cohen, L. [1966] Generalized phase-space distribution functions. J. Math. Phvs. 7, 781 - 786.

    Article  ADS  Google Scholar 

  19. Dirac, P.A.M. [1967] The Principles of Quantum Mechanics. Revised Fourth Ed. (Oxford Univ. Press. Oxford).

    Google Scholar 

  20. Doebner, H.D. and Melsheimer, O. [1968] Limitable dynamical groups in quantum mechanics I. J. Math. Phys. 9, 1638 - 1656.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. Emch, G.G. [1972] Algebraic Methods in Statistical Mechanics and Quantum Field Theory. (Wiley, New York).

    MATH  Google Scholar 

  22. 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. 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. Folland, G.B. [1989] Harmonic Analysis in Phase Space. Ann. Math. Ser. 122(Princeton University Press, Princeton).

    MATH  Google Scholar 

  25. Fronsdal, C. [1978] Some ideas about quantization. Rep. Math. Phvs. 15, 111 - 145.

    Article  ADS  Google Scholar 

  26. Ginzburg, V.L. & Montgomery, R. [1997] Geometric quantization and no-go theorems. Preprint dg-ga/9703010.

    Google Scholar 

  27. Glimm, J. & Jaffe, A. [1981] Quantum Physics. A Functional Integral Point of View. (Springer Verlag, New York).

    MATH  Google Scholar 

  28. Gotay, M.J. [1980] Functorial geometric quantization and Van Hove’s theorem. Int. J. Theor: Phys. 19, 139 - 161.

    Article  MathSciNet  MATH  Google Scholar 

  29. Gotay, M.J. [1987] A class of non-polarizable symplectic manifolds. Mh. Math. 103, 27 - 30.

    Article  MathSciNet  MATH  Google Scholar 

  30. 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. Gotay, M.J. [1999] On the Groenewold-Van Hove problem for R 2n. J. Math. Phvs. 40, 2107 - 2116.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Gotay, M.J. & Grabowski, J. [1999] On quantizing nilpotent and solvable basic algebras. Preprint math-ph/9902012.

    Google Scholar 

  33. Gotay, M.J. & Grabowski, J. [2000] On quantizing semisimple basic algebras. In preparation.

    Google Scholar 

  34. Gotay, M.J., Grabowski, J., & Grundling, H.B. [2000] An obstruction to quantizing compact symplectic manifolds. Proc. Amer: Math. Soc. 128, 237 - 243.

    Article  MathSciNet  MATH  Google Scholar 

  35. Gotay, M.J. & Grundling, H.B. [1997] On quantizing T*S 1.Rep. Math. Phys. 40, 107–123.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. 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. Gotay, M.J., Grundling, H., &Hurst, C.A. [1996] A Groenewold-Van Hove theorem for S 2. Trans. Amer: Math. Soc. 348 1579–1597.

    Article  MathSciNet  MATH  Google Scholar 

  38. Gotay, M.J., Grundling, H., & Tuynman, G.T. [1996] Obstruction results in quantization theory. J. Nonlinear Sci. 6, 469 - 498.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. Grabowski, J. [1978] Isomorphisms and ideals of the Lie algebras of vector fields. Invent. Math. 50, 13 - 33.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. Grabowski, J. [1985] The Lie structure of C*and Poisson algebras. Studia Math. 81, 259–270.

    MathSciNet  MATH  Google Scholar 

  41. Groenewold, H.J. [1946] On the principles of elementary quantum mechanics. Physica 12, 405 - 460.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. Guillemin, V. & Sternberg, S. [1984] Symplectic Techniques in Physics. (Cambridge Univ. Press, Cambridge).

    MATH  Google Scholar 

  43. Helton, J.W. & Miller, R.L. [1994] NC Algebra: A Mathematica Package for Doing Non Commuting Algebra. v0.2 ncalg@ucsd.edu.(USCD, La Jolla).

    Google Scholar 

  44. Hennings, M.A. [1986] Fronsdal 5-quantization and Fell inducing. Math. Proc. Carnb. Phil. Soc. 99, 179 - 188.

    Article  MathSciNet  MATH  Google Scholar 

  45. 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. Joseph, A. [1970] Derivations of Lie brackets and canonical quantization. Commun. Math. Phys. 17, 210 - 232.

    Article  ADS  MATH  Google Scholar 

  47. Kerner, E.H. & Sutcliffe, W.G. [1970] Unique Hamiltonian operators via Feynman path integrals. J. Math. Phys. 11, 391–393.

    Article  ADS  Google Scholar 

  48. 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. 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. 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. Mackey, G.W. [1976] The Theory of Unitary Group Representations(University of Chicago Press, Chicago).

    MATH  Google Scholar 

  52. Margenau, H. & Cohen, L. [1967] Probabilities in quantum mechanics. In: Quantum Theory and Reality. Bunge, M., Ed. (Springer-Verlag, New York), 71–89.

    Chapter  Google Scholar 

  53. Marsden, J.E. & Ratiu, T.S. [1994] Introduction to Mechanics and Symmetry.(Springer-Verlag, New York).

    Book  MATH  Google Scholar 

  54. Mnatsakanova, M., Morchio, G., Strocchi, F., & Vernov, Yu. [1998] Irreducible representations of the Heisenberg algebra in Krein spaces. J. Math. Phys. 39, 2969–2982.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  55. Onishchik, A.L. [1994] Topology of Transitive Transformation Groups. (Johann Ambrosius Barth, Leipzig).

    MATH  Google Scholar 

  56. Reed, M. & Simon, B. [1972] Functional Analysis I. (Academic Press, New York).

    Google Scholar 

  57. Rieffel, M.A. [1989] Deformation quantization of Heisenberg manifolds. Commun. Math. Phys. 122, 531–562.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  58. Rieffel, M.A. [1990] Deformation quantization and operator algebras. Proc. Sym. Pure Math. 45, 411–423.

    MathSciNet  Google Scholar 

  59. 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. Rieffel, M.A. [1998] Questions on quantization. In: Operator Algebras and Operator Theory. Contemp. Math. 228, 315–326.

    Article  MathSciNet  Google Scholar 

  61. 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).

    Book  Google Scholar 

  62. Souriau, J.-M. [1997] Structure of Dynamical Systems.(Birkhäuser, Boston).

    Book  MATH  Google Scholar 

  63. Tuynman, G.M. [1998] Prequantization is irreducible. Indag. Mathem. 9, 607–618.

    Article  MathSciNet  MATH  Google Scholar 

  64. Urwin, R.W. [1983] The prequantization representations of the Poisson Lie algebra. Adv. Math. 50, 126–154.

    Article  MathSciNet  MATH  Google Scholar 

  65. van Hove, L. [1951] Sur certaines représentations unitaires d’un groupe infini de transformations. Proc. Roy. Acad. Sci. Belgium26, 1–102.

    Google Scholar 

  66. 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. Varadarajan, V.S. [1984] Lie Groups, Lie Algebras and Their Representations. (Springer-Verlag, New York).

    MATH  Google Scholar 

  68. Velhinho, J. [1998] Some remarks on a full quantization of the torus. Int. J. Mod. Phys.A13, 3905–3914.

    MathSciNet  ADS  Google Scholar 

  69. von Neumann, J. [1955] Mathematical Foundations of Quantum Mechanics. (Princeton. Univ. Press, Princeton).

    MATH  Google Scholar 

  70. Weinstein, A. [1989] Cohomology of symplectomorphism groups and critical values of hamiltonians. Math. Z. 201, 75–82.

    Article  MathSciNet  MATH  Google Scholar 

  71. Wildberger, N. [1983] Quantization and harmonic analysis on Lie groups. Dissertation, Yale University.216 M. J. Gotay.

    Google Scholar 

  72. Woodhouse, N.M.J. [1992] Geometric quantization. Second Ed. (Clarendon Press, Oxford).

    MATH  Google Scholar 

  73. Ziegler, F. [1996] Quantum representations and the orbit method. Thesis, Université de Provence.

    Google Scholar 

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  1. Department of Mathematics, University of Hawaii, 2565 The Mall, Honolulu, HI, 96822, USA

    M. J. Gotay

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Gotay, M.J. (2000). Obstructions to Quantization. In: Mechanics: From Theory to Computation. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1246-1_7

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  • DOI: https://doi.org/10.1007/978-1-4612-1246-1_7

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