Skip to main content

Diffeomorphism Groups, Quasi-Invariant Measures, and Infinite Quantum Systems

  • Chapter
Symmetries in Science VIII

Abstract

This paper provides a brief introduction to how unitary representations of diffeomorphism groups can describe certain quantum systems having infinitely many degrees of freedom. It is a partial report of our joint work [1], based on the August 1994 talk by the first author at the Symmetries in Science VIII conference in Bregenz, Austria. We would like to express appreciation to the conference organizers, especially Professor Bruno Gruber, for the opportunity to present our results.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. A. Goldin and U. Moschella, Saclay preprint T94/029 (1994).

    Google Scholar 

  2. G. A. Goldin, “Current Algebras as Unitary Representations of Groups”, Ph. D. thesis, Princeton University (1969). G. A. Goldin and D. H. Sharp, in 1969 Battelle Rencontres: Group Representations, Lecture Notes in Physics 6, edited by V. Bargmann (Springer, Berlin, 1970), p. 300; G. A. Goldin, J. Math. Phys. 12, 462 (1971).

    Google Scholar 

  3. For more detailed reviews, see G. A, Goldin and D. H. Sharp, “Diffeomorphism Groups and Local Symmetries: Some Applications in Quantum Physics”, in Symmetries in Science III, edited by B. Gruber and F. lachello (New York: Plenum, 1989), p. 181; G. A. Goldin, “Predicting Anyons: The Origins of Fractional Statistics in Two-Dimensional Space”, in Symmetries in Science V, edited by B. Gruber, L. C. Biedenharn, and H.-D. Doebner (New York: Plenum, 1991), p. 259; G. A. Goldin and D. H. Sharp, Int. J. Mod. Phys. B5, 2625 (1991); and G. A. Goldin, Int. J. Mod. Phys. B6, 1905 (1992).

    Google Scholar 

  4. R. Dashen and D. H. Sharp, Phys. Rev. 165 (1968), 1867.

    Article  ADS  Google Scholar 

  5. H.-D. Doebner and J. Tolar, in Symposium on Symmetries in Science, Carbondale, Illinois 1979, edited by B. Gruber and R. S. Millman (Plenum, New York, 1980), p. 475; B. Angermann, H.-D. Doebner, and J. Tolar, in Nonlinear Partial Differential Operators and Quantization Procedures, Lecture Notes in Mathematics 1037, edited by S. I. Andersson and H.-D. Doebner (Springer, Berlin, 1983), p. 171; H.-D. Doebner, H. J. Elmers, and W. Heidenreich, J. Math. Phys. 30 (1989), 1053.

    Google Scholar 

  6. G. A. Goldin, R. Menikoff, and D. H. Sharp, J. Math. Phys. 21, 650 (1980); J. Math, Phys. 22, 1664 (1981); Phys. Rev. Lett. 51, 2246 (1983); G. A. Goldin and D. H. Sharp, Phys. Rev. D28, 830 (1983).

    Google Scholar 

  7. J. M. Leinaas and J. Myrheim, Nuovo Cimento 37B, 1 (1977); J. M. Leinaas, Nuovo Cimento 4A, 19 (1978); Fort. Phys. 28, 579 (1980).

    Google Scholar 

  8. F. Wilczek, Phys. Rev. Lett. 48, 1144 (1982); Phys. Rev. Lett. 49, 957 (1982).

    Google Scholar 

  9. H.-D. Doebner and G. A. Goldin, Phys. Letts. A162, 397 (1992); G. A. Goldin, Int. J. Mod. Phys. B6, 1905 (1992); H.-D. Doebner and G. A. Goldin, J. Phys. A: Math. Gen. 27, 1771 (1994).

    Google Scholar 

  10. G. A. Goldin, J. Grodnik, R. T. Powers, and D.H. Sharp, J. Math. Phys. 15, 88 (1974); R. Menikoff, J. Math. Phys. 15 (1974), 1138 and 1394; A. M. Vershik, I. M. Gelfand, and M. I. Graev, Dokl. Akad. Nauk, SSSR 232, 745 (1977).

    Google Scholar 

  11. I. M. Gelfand and N. Ya Vilenkin, Generalized Functions, Vol. 4 (New York: Academic Press, 1964).

    Google Scholar 

  12. M. Rasetti and T. Regge, Physica 80A, 217 (1975); J. Marsden and A. Weinstein, Physica 7D, 305 (1983); G. A. Goldin, R. Menikoff, and D. H. Sharp, Phys. Rev. Lett. 58, 2162 (1987); Phys. Rev. Lett. 67, 3499 (1991). For recent reviews of related results see G. A. Goldin and D. H. Sharp, Int. J. Mod. Phys. B5, 2625 (1991), and G. A. Goldin, Int. J. Mod. Phys. B6, 1905 (1992).

    Google Scholar 

  13. S. W. Hawking, “The Path-Integral Approach to Quantum Gravity”, in General Relativity: An Einstein Centenary Survey , edited by S. W. Hawking and W. Israel, Cambridge University Press (1979).

    Google Scholar 

  14. A. Ashtekar and J. Lewandowski, “Representation Theory of Analytic Holonomy C*-algebras”, in Knots and Quantum Gravity, edited by J. Baez, Oxford University Press (1994, in press); A. Ashtekar, D. Marolf, and J. Mourao, “Integration on the Space of Connections Modulo Gauge Transformations” (March 1994 preprint).

    Google Scholar 

  15. W. Feller An Introduction to Probability Theory and Its Applications, Vol. II Second Edition, Wiley, New York (1971).

    MATH  Google Scholar 

  16. G. Baker, Phys. Rev. 126, 2071 (1962); M. Kac, G. Uhlenbeck, and P. Hemmer, J. Math. Phys. 4, 216 (1963); F. Dyson, Commun. Math. Phys. 12, 1961 (1969).

    Google Scholar 

  17. E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. 16, 407 (1961), Rev. Mod. Phys. 36, 856 (1964). See also J. M. Luck, J. Stat. Phys. 72, 417 (1993) and references therein.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer Science+Business Media New York

About this chapter

Cite this chapter

Goldin, G.A., Moschella, U. (1995). Diffeomorphism Groups, Quasi-Invariant Measures, and Infinite Quantum Systems. In: Gruber, B. (eds) Symmetries in Science VIII. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-1915-7_13

Download citation

  • DOI: https://doi.org/10.1007/978-1-4615-1915-7_13

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-5783-4

  • Online ISBN: 978-1-4615-1915-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics