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
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Cattaneo, U., Comment. Math. Helvetici 54, 629 (1979).
Scutaru, H., Letters Math. Phys. 2, 101 (1979).
Castrigiano, D.P.L. and Heinrichs, R.W., Letters Math. Phys. 4, 169 (1980).
Mackey, G.W., Proc. Natl. Acad. Sci. U.S.A. 35, 537 (1949).
Davies, E.B. and Lewis, J.T., Commun. Math. Phys. 17, 239 (1969).
Neumann, H., Helv. Phys. Acta 45, 811 (1972).
Ali, S.T. and Emch, G.G., J. Math. Phys. 15, 176 (1974).
Ali, S.T. and Prugovečki, E., J. Math. Phys. 18, 219 (1977).
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.
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.
Naimark, M.A., C. R. (Doklady) Acad. Sci. URSS 41, 359 (1943).
Nachbin, L., Topology and Order, Van Nostrand Co. Inc., Princeton, N.J. (1965).
Dixmier, J., Les Algbrès d'operateurs dans l'espace Hilbertien, Gauthier-Villars, Paris (1957).
Kadison, R.V., Proc. Amer. Math. Soc. 12, 973 (1961).
Phelps, R.R., Lectures on Choquet's Theorem, Van Nostrand Co., Inc., Princeton, N.J. (1966).
Espelie, M.S., Pacific J. Math. 48, 57 (1973).
Berberian, S.K., Notes on Spectral Theory, Van Nostrand Co., Inc., Princeton, N.J. (1966).
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).
Takesaki, M., Acta Math. 119, 273 (1967).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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
