La Rivista del Nuovo Cimento (1978-1999)

, Volume 18, Issue 4, pp 1–27 | Cite as

The complementarity of quantum observables: Theory and experiments

  • P. Busch
  • P. J. Lahti


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  1. [1]
    According to Wolfgang Pauli, the new quantum theory could have been called the theory of complementarity. (Pauli W.,General Principles of Quantum Mechanics (Springer, Berlin) 1980, original German edition: 1933). This is one example showing the central importance of the notion of complementarity in the discussions on the foundations of quantum mechanics.CrossRefGoogle Scholar
  2. [2]
    Grangier Ph., Roger G. andAspect A.,Europhys. Lett.,1 (1986) 173;Aspect A. andGrangier Ph.,Hyperfine Interactions,37 (1987) 3; in:Sixty-Two Years of Uncertainty: Historical, Philosophical and Physical Inquiries into the Foundations of Quantum Mechanics, edited byA. I. Miller (Plenum Press, New York, N.Y.) 1990, p. 45.ADSCrossRefGoogle Scholar
  3. [3]
    This approach is discussed in a greter detail, for instance, in the monograph byBusch P., Grabowski M. andLahti P. J.,Operational Quantum Physics, Lect. Notes Phys., m.31 (Springer, Berlin) 1995.Google Scholar
  4. [4]
    For a more detailed presentation of measurement theory the reader may wish to consult the monograph byBusch P., Lahti P. J. andMittelstaedt P.,The Quantum Theory of Measurement, Lect. Notes Phys., m.2 (Springer, Berlin) 1991, second revised edition, 1995 (in press).CrossRefGoogle Scholar
  5. [5]
    von Neumann J.,Mathematische Grundlagen der Quantenmechanik (Springer, Berlin) 1981, original edition: 1932.zbMATHGoogle Scholar
  6. [6]
    Ludwig G.,Foundations of Quantum Mechanics (Springer, Berlin) 1983.CrossRefzbMATHGoogle Scholar
  7. [7]
    Davies E. B. andLewis J. T.,Commun. Math. Phys.,17 (1970) 239. See alsoLahti P. J., inRecent Development in Quantum Logic, edited byP. Mittelstaedt andE.-W. Stachow (Bibliographisches Institut, Mannheim) 1985, p. 61.MathSciNetADSCrossRefzbMATHGoogle Scholar
  8. [8]
    Bohr N.,Nature (1928) 580;Phys. Rev.,48 (1935) 696.Google Scholar
  9. [9]
    We recall that for two bounded self-adjoint operatorsA andB the orderingA ≤ B is defined as 〈φ|Aφ〉 〈φ|Bφ〉 for all φ ∃ 27–01.Google Scholar
  10. [10]
    For two bounded positive operatorsA, B the greatest lower bound may or may not exist among the positive operators. If it does exist it is denotedAB. Google Scholar
  11. [11]
    Kraus K.,Phys. Rev. D,10 (1987) 3070.MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    For pairs of state transformations Φ1, Φ2 the ordering Φ1 Φ2 is defined as Φ1(T)≤ ≤ Φ2(T) for all statesT. If for two state transformations Φ1, Φ2 the greatest lower bound exists it is denoted Φ1 ∧ Φ2 Google Scholar
  13. [13]
    Busch P. andLahti P. J.,Phys. Rev. D,29 (1984) 1634.MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    Garrison J. C. andWong J.,J. Math. Phys.,11 (1970) 2242.MathSciNetADSCrossRefzbMATHGoogle Scholar
  15. [15]
    Lenard A.,J. Funct. Anal.,10 (1972) 410.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Lahti P. J. andYlinen K.,J. Math. Phys.,28 (1987) 2614.MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    Busch P. andLahti P. J.,Phys. Lett. A,115 (1986) 259.MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    Ylinen K.,J. Math. Anal. Appl.,137 (1989) 185.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Heisenberg W.,Z. Phys.,43 (1927) 172.ADSCrossRefzbMATHGoogle Scholar
  20. [20]
    Busch P.,Int. J. Theor. Phys.,24 (1985) 63.MathSciNetCrossRefGoogle Scholar
  21. [21]
    Davies E. B.,Quantum Theory of Open Systems (Academic Press, New York, N.Y.) 1976;Prugovečki E.,Stochastic Quantum Mechanics and Quantum Spacetime (Reidel, Dordrecht) 1986.zbMATHGoogle Scholar
  22. [22]
    Busch P.,Phys. Rev. D,33 (1986) 2253.MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    Mlak W.,Hilbert Spaces and Operator Theory (Kluwer, Dordrecht) 1991.zbMATHGoogle Scholar
  24. [24]
    Busch P.,J. Math. Phys.,25 (1984) 1794.MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    Busch P., Grabowski M. andLahti P. J.,Ann. Phys. (N.Y.),237 (1995) 1.MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    Prasad S., Scully M. O. andMartienssen W.,Opt. Commun.,62 (1987) 139;Campos R. A., Saleh B. E. A. andTeich M. C.,Phys. Rev. A,40 (1989) 1371.ADSCrossRefGoogle Scholar
  27. [27]
    Perelomov A. M.,Generalized Coherent States and their Applications (Springer, Berlin) 1986.CrossRefzbMATHGoogle Scholar
  28. [28]
    Mittelstaedt P., Prieur A. andSchieder R.,Found. Phys.,17 (1987) 893.ADSCrossRefGoogle Scholar
  29. [29]
    Sanders B. C. andMilburn G. J.,Phys. Rev. A,39 (1989) 694.ADSCrossRefGoogle Scholar
  30. [30]
    Busch P.,Found. Phys.,17 (1987) 905.MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica 1995

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of HullHullUK
  2. 2.Department of PhysicsUniversity of TurkuTurkuFinland

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