Skip to main content

A geometrical property of POV-measures and systems of covariance

V. Quantization Methods

Part of the Lecture Notes in Mathematics book series (LNM,volume 905)

Abstract

A Choquet type of an integral representation is found for a class of normalized positive operator valued (POV) measures on a Hilbert space. An arbitrary POV-measure within this class is thereby represented uniquely as an integral over projection valued (PV) measures. As an application, the case of a commutative system of covariance (representing a generalization of the imprimitivity theorem of Mackey) is discussed. The relevance of these results to the theory of quantum mechanical observables, admitting stochastic value spaces, is pointed out.

Keywords

  • Extreme Point
  • Closed Subgroup
  • Borel Function
  • Weak Closure
  • Bounded Borel Function

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Cattaneo, U., Comment. Math. Helvetici 54, 629 (1979).

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Scutaru, H., Letters Math. Phys. 2, 101 (1979).

    CrossRef  MathSciNet  Google Scholar 

  3. Castrigiano, D.P.L. and Heinrichs, R.W., Letters Math. Phys. 4, 169 (1980).

    CrossRef  MATH  Google Scholar 

  4. Mackey, G.W., Proc. Natl. Acad. Sci. U.S.A. 35, 537 (1949).

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Davies, E.B. and Lewis, J.T., Commun. Math. Phys. 17, 239 (1969).

    CrossRef  MathSciNet  Google Scholar 

  6. Neumann, H., Helv. Phys. Acta 45, 811 (1972).

    MathSciNet  Google Scholar 

  7. Ali, S.T. and Emch, G.G., J. Math. Phys. 15, 176 (1974).

    CrossRef  MathSciNet  Google Scholar 

  8. Ali, S.T. and Prugovečki, E., J. Math. Phys. 18, 219 (1977).

    CrossRef  MATH  Google Scholar 

  9. Ali, S.T. and Prugovečki, E., ‘Consistent models of spin o and 1/2 extended particles scattering in external fields', in Mathematical Methods and Applications of Scattering Theory, Eds., J.A. De Santo, A. W. Saenz and W.W. Zachary; Series: Lecture Notes in Physics; Springer-Verlag, Berlin-Heidelberg-New York, Vol. 130 (1980), p. 197.

    Google Scholar 

  10. Ali, S.T., ‘Aspects of relativistic quantum mechanics on phase space', in Differential Geometric Methods in Mathematical Physics, Ed., H.D. Doebner, Series: Lecture Notes in Physics; Springer-Verlag, Berlin-Heidelberg-New York, Vol. 139 (1981), p. 49.

    CrossRef  Google Scholar 

  11. Naimark, M.A., C. R. (Doklady) Acad. Sci. URSS 41, 359 (1943).

    MathSciNet  MATH  Google Scholar 

  12. Nachbin, L., Topology and Order, Van Nostrand Co. Inc., Princeton, N.J. (1965).

    MATH  Google Scholar 

  13. Dixmier, J., Les Algbrès d'operateurs dans l'espace Hilbertien, Gauthier-Villars, Paris (1957).

    Google Scholar 

  14. Kadison, R.V., Proc. Amer. Math. Soc. 12, 973 (1961).

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Phelps, R.R., Lectures on Choquet's Theorem, Van Nostrand Co., Inc., Princeton, N.J. (1966).

    MATH  Google Scholar 

  16. Espelie, M.S., Pacific J. Math. 48, 57 (1973).

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. Berberian, S.K., Notes on Spectral Theory, Van Nostrand Co., Inc., Princeton, N.J. (1966).

    MATH  Google Scholar 

  18. See, for example, B. SZ-Nagy, Extensions of Linear Transformations in Hilbert Space which Extend Beyond this Space, Appendix to F. Riesz and B. SZ-Nagy, Functional Analysis, Frederick Ungar, New York (1960).

    Google Scholar 

  19. Takesaki, M., Acta Math. 119, 273 (1967).

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1982 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Twareque Ali, S. (1982). A geometrical property of POV-measures and systems of covariance. In: Doebner, HD., Andersson, S.I., Petry, H.R. (eds) Differential Geometric Methods in Mathematical Physics. Lecture Notes in Mathematics, vol 905. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092439

Download citation

  • DOI: https://doi.org/10.1007/BFb0092439

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11197-9

  • Online ISBN: 978-3-540-39002-2

  • eBook Packages: Springer Book Archive