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Diffeomorphism Groups And Local Symmetries: Some Applications In Quantum Physics

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Symmetries in Science III

Abstract

This paper reviews results about unitary representations of diffeomorphism groups. Such groups occur naturally in a number of different physical contexts. Their areas of application include quantum mechanics and quantization methods, quantum field theory and local current algebra, general relativity and quantum gravity, and hydrodynamics and the quantization of fluids. In the case when the underlying manifold is a circle, they are related to Kac-Moody algebras, the Virasoro algebra, and the quantum mechanics of strings.

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References

  1. Y. Aharonov and D. Bohm (1959), Phys. Rev.115, 485.

    Google Scholar 

  2. B. Angermann, H.D. Doebner, and J. Tolar (1983), “Quantum Kinematics on Smooth Manifolds,” in S.I. Andersson and H.D. Doebner (eds.),Nonlinear Partial Differential Operators and Quantization Procedures, Springer Lecture Notes in Mathematics1037, 171 – 208.

    Google Scholar 

  3. L. Auslander and B. Kostant (1971), Inventiones Math.14., 255.

    Google Scholar 

  4. A. B. Borisov (1978), J. Phys. A: Math. Gen.11, 1957.

    Google Scholar 

  5. A. B Borisov (1979), J. Phys. A: Math. Gen.12, 1625.

    Google Scholar 

  6. R. Y. Chiao, A. Hansen, and A. A. Moulthrop (1985), Phys. Rev. Lett.54, 1339.

    Google Scholar 

  7. R. Y. Chiao, A. Hansen, and A. A. Moulthrop (1987), Phys. Rev. Lett.58, 175.

    Google Scholar 

  8. Y. Choquet-Bruhat, C. De Witt-Morette, and M. Dillard-Bleick (1977),Analysis, Manifolds and Physics( Amsterdam: North Holland).

    Google Scholar 

  9. R. F. Dashen and D. H. Sharp (1968), Phys. Rev.165, 1857.

    Google Scholar 

  10. H. D. Doebner and J. Tolar (1980), “On Global Properties of Quantum Systems,” in B. Gruber and R. S. Millman (eds.),Symposium on Symmetries in Science, Carbondale, Illinois 1979(New York: Plenum), 475 – 486.

    Google Scholar 

  11. J. L. Friedman and R. D. Sorkin (1980), Phys. Rev. Lett.44, 1100 – 1103.

    Google Scholar 

  12. A. Girard (1973), J. Math. Phys.14, 353 – 365.

    Google Scholar 

  13. P. Goddard and D. Olive (1986), Int’l. J. Mod. Phys. A1, 303.

    Google Scholar 

  14. G. A. Goldin (1969), “Current Algebras as Unitary Representations of Groups,” unpubl. Ph.D. thesis, Princeton University.

    Google Scholar 

  15. G. A. Goldin (1971), J. Math. Phys.12, 462 – 487.

    Google Scholar 

  16. G. A. Goldin (1984), “Diffeomorphism Groups, Semidirect Products, and Quantum Theory,” in J. E. Marsden (ed.),Fluids and Plasmas: Geometry and Dynamics, Contemp. Math.28(Am. Math. Soc.), 189 – 207.

    Google Scholar 

  17. G. A. Goldin (1984), “Representations of Semidirect Products of Diffeo—morphism Groups in Quantum Theory,” in W. W. Zachary (ed.),XIII International Colloquium on Group Theoretical Methods in Physics(Singapore: World Scientific), 261 – 264.

    Google Scholar 

  18. G. A. Goldin (1988), “Parastatistics, θ-Statistics, and Topological Quantum Mechanics from Unitary Representations of Dirreomorphism Groups,” in H. D. Doebner and J. D. Hennig (eds.), Procs. of the XV. Int’l. Conference on Differential Geometric Methods in Theoretical Physics (Teaneck, NJ: World Scientific), 197–207.

    Google Scholar 

  19. G. A. Goldin (in press), “On the Distinguishability of Particles Described by Unitary Representations of Diff(M).” Paper presented at the XVI. Colloquium on Group Theoretical Methods in Physics, Varna, Bulgaria, June 1987.

    Google Scholar 

  20. G. A. GoIdin, J. Grodnik, R. T. Powers, and D. H. Sharp (1974), J. Math. Phys.15, 88 – 100.

    Google Scholar 

  21. G. A. Goldin and R. Menikoff (1985). J. Math. Phys..26, 1880 – 1884.

    Google Scholar 

  22. G. A. Goldin, R. Menikoff, and D. H. Sharp (1980), J. Math. Phys.21, 650 – 664.

    Google Scholar 

  23. G. A. Goldin, R. Menikoff, and D. H. Sharp (1981), J. Math. Phys.22, 1664 – 1668.

    Google Scholar 

  24. G. A. Goldin, R. Menikoff, and D. H. Sharp (1983), Phys. Rev. Lett.51, 2246 – 2249.

    Google Scholar 

  25. G. A. Goldin, R. Menikoff, and D. H. Sharp (1983), Phys. Rev. Lett.51, 2246 – 2249.

    Google Scholar 

  26. G. A. Goldin, R. Menikoff, and D. H. Sharp (1983), J. Phys. A: Math. Gen.16, 1827 – 1833.

    Google Scholar 

  27. G. A. Goldin, R. Menikoff, and D. H. Sharp (1985), Phys. Rev. Lett.54, 603.

    Google Scholar 

  28. G. A. Goldin, R. Menikoff, and D. H. Sharp (1987), Phys. Rev. Lett.58, 174.

    Google Scholar 

  29. G. A. Goldin, R. Menikoff, and D. H. Sharp (1987), J. Math. Phys.28, 744 – 746.

    Google Scholar 

  30. G. A. Goldin, R. Menikoff, and D. H. Sharp (1987), Phys. Rev. Lett.58, 2162 – 2164.

    Google Scholar 

  31. G. A. Goldin, R. Menikoff, and D. H. Sharp (1987), “Diffeomorphism Groups, Coadjoint Orbits, and the Quantization of Classical Fluids,” and “Quantized Vortex Filaments in Incompressible Fluids,” in Y. S. Kim and W. W. Zachary (eds.),Procs. of the First Int’l. Conference on the Physics of Phase Space, Springer Lecture Notes in Physics 278, 360–362 and 363–365.

    Google Scholar 

  32. G. A. Goldin and D. H. Sharp (1970), “Lie Algebras of Local Currents and their Representations,” in V. Bargmann (ed.),Group Representations in Mathematics and Physics: Battelle Seattle 1969 Rencontres, Springer Lecture Notes in Physics6, 300 – 311.

    Google Scholar 

  33. G.A. Goldin and D. H. Sharp (1983), Commun. Math. Phys.92, 217–228,

    Google Scholar 

  34. G.A. Goldin and D. H. Sharp (1983), Phys. Rev. D28, 830 – 832.

    Google Scholar 

  35. J.Grodnik and D. H. Sharp (1970), Phys. Rev. D1, 1531.

    Google Scholar 

  36. J.Grodnik and D. H. Sharp (1970), Phys. Rev. D1, 1546.

    Google Scholar 

  37. A. Hansen, A. A. Moulthrop, and R. Y. Chiao (1985), Phys. Rev. Lett.55, 1431.

    Google Scholar 

  38. S. W. Hawking and G. F. R. Ellis (1973),The Large Scale Structure of Space-Time( London: Cambridge Univ. Press).

    Google Scholar 

  39. C. J. Isham (1981), Phys. Letts.106B, 188 – 192.

    Google Scholar 

  40. C. J. Isham (1983), “Quantum Field Theory and Spatial Topology,” in G. Denardo and H. D. Doebner (eds.),Conference on Differential Geometric Methods in Theoretical Physics: Trieste 1981( Singapore: World Scientific ), 171 – 185.

    Google Scholar 

  41. C.J. Isham (1984), “Topological and Global Aspects of Quantum Theory.” in B. DeWitt and R. Stora (eds.), Relatively, Groups and Topology II, Procs. 1983 Les Houches Summer School (Amsterdam: North Holland), 1059–1290.

    Google Scholar 

  42. I. M. Khalatnikov (1965),Introduction to the Theory of Superfluidity( New York: Benjamin).

    Google Scholar 

  43. A. A. Kirillov (1981), Ser. Math. Sov.1, 351.

    Google Scholar 

  44. M. G. G. Laidlaw and C. M. DeWitt (1971), Phys. Rev. D3, 1375.

    Google Scholar 

  45. S. Lang (1962),Introduction to Differentiable Manifolds( New York: Interscience).

    Google Scholar 

  46. G. W. Mackey (1976),The Theory of Unitary Group Representations( Chicago: Univ. of Chicago Press).

    Google Scholar 

  47. G. W. Mackey (1976), The Theory of Unitary Group Representations (Chicago: Univ. of Chicago Press).

    Google Scholar 

  48. J. E. Marsden and A. Weinstein (1983), “Coadjoint Orbits, Vortices, and Clebsch Variables for Incompressible Fluids,” in D. K. Campbell, H. A. Rose, and A. C. Scott (eds.),Procs. of the Los Alamos Conference “Order in Chaos”, Physica7D, 305.

    Google Scholar 

  49. R. Menikoff (1974), J. Math. Phys.15, 1138.

    Google Scholar 

  50. R. Menikoff (1974), J. Math. Phys.15, 1394.

    Google Scholar 

  51. R. Menikoff and D. H. Sharp (1977), J. Math. Phys.18, 471.

    Google Scholar 

  52. A. M. L. Messiah and O. W. Greenberg (1964), Phys. Rev.136, B248.

    Google Scholar 

  53. M. Rasetti and T. Regge (1975), Physica80A, 217.

    Google Scholar 

  54. M. Reed and B. Simon (1972),Methods of Modern Mathematical Physics, Vol. I: Functional Analysis (New York: Academic Press).

    Google Scholar 

  55. M. Reed and B. Simon (1975),Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self—Adjointness( New York: Academic Press).

    Google Scholar 

  56. R. H. Stolt and J. R. Taylor (1970), Nuclear Physics B19, 1.

    Google Scholar 

  57. H. Sugawara (1968), Phys. Rev.170, 1659.

    Google Scholar 

  58. W. Thirring (1978),Classical Dynamical Systems, transl. by E. Harrell ( New York: Springer).

    Google Scholar 

  59. A. M. Vershik, I. Gel ’fand, and M. I. Graev (1975), Usp. Mat. Nauk30, 3.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Depts. of Mathematics and Physics, Rutgers University, New Brunswick, 08903, N.J., USA

    Gerald A. Goldin

  2. Theoretical Division, Los Alamos National Laboratory, Los Alamos, 87545, N.M., USA

    David H. Sharp

Authors
  1. Gerald A. Goldin
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  2. David H. Sharp
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Editor information

Editors and Affiliations

  1. Southern Illinois University, Carbondale, Illinois, USA

    Bruno Gruber

  2. Yale University, New Haven, Connecticut, USA

    Francesco Iachello

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© 1989 Plenum Press, New York

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Goldin, G.A., Sharp, D.H. (1989). Diffeomorphism Groups And Local Symmetries: Some Applications In Quantum Physics. In: Gruber, B., Iachello, F. (eds) Symmetries in Science III. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0787-7_10

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  • DOI: https://doi.org/10.1007/978-1-4613-0787-7_10

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4612-8082-8

  • Online ISBN: 978-1-4613-0787-7

  • eBook Packages: Springer Book Archive

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