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Naive Realism about Operators

  • Martin Daumer
  • Detlef Dürr
  • Sheldon Goldstein
  • Nino Zanghì

Abstract

A source of much difficulty and confusion in the interpretation of quantum mechanics is a “naive realism about operators.” By this we refer to various ways of taking too seriously the notion of operator-as-observable, and in particular to the all too casual talk about “measuring operators” that occurs when the subject is quantum mechanics. Without a specification of what should be meant by “measuring” a quantum observable, such an expression can have no clear meaning. A definite specification is provided by Bohmian mechanics, a theory that emerges from Schrödinger’s equation for a system of particles when we merely insist that “particles” means particles. Bohmian mechanics clarifies the status and the role of operators as observables in quantum mechanics by providing the operational details absent from standard quantum mechanics. It thereby allows us to readily dismiss all the radical claims traditionally enveloping the transition from the classical to the quantum realm- for example, that we must abandon classical logic or classical probability. The moral is rather simple: Beware naive realism, especially about operators!

Keywords

Wave Function Quantum Mechanic Quantum Theory Hide Variable Quantum Formalism 
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.

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References

  1. Albert, D. Z.: 1992, Quantum Mechanics and Experience, Harvard University Press, Cambridge, MA.Google Scholar
  2. Bell, J. S.: 1964, ‘Onthe Einstein-Podolski-Rosen Paradox’, Physics 1, 195–200. Reprinted in Bell 1987.Google Scholar
  3. Bell, J. S.: 1966, ‘On the problem of hidden variables in quantum mechanics’, Review of Modern Physics 38, 447–452. Reprinted in Bell 1987.zbMATHCrossRefGoogle Scholar
  4. Bell, J. S.: 1981, ‘Quantum mechanics for cosmologists’, in Quantum Gravity 2, C. Isham, R. Penrose, and D. Sciama (eds.), Oxford University Press, New York, pp. 611–637. Reprinted in Bell 1987.Google Scholar
  5. Bell, J. S.: 1982, ‘On the impossible pilot wave’, Foundations of Physics 12, 989–999. Reprinted in Bell, 1987.MathSciNetCrossRefGoogle Scholar
  6. Bell, J. S.: 1987, Speakable and unspeakable in quantum mechanics, Cambridge University Press, Cambridge.zbMATHGoogle Scholar
  7. Bell, J. S.: 1990, ‘Against “measurement”’, Physics World 3, 33–40. [Also appears in’ sixty-two Years of Uncertainty: Historical, Philosophical, and Physical Inquiries into the Foundations of Quantum Mechanics’, Plenum Press, New York, pp. 17–31.]Google Scholar
  8. Beltramerti, E. G. and Cassinelli, G.: 1981, The Logic of Quantum Mechanics, Reading, Mass.Google Scholar
  9. Berndl, K., Dürr, D., Goldstein, S., and Zanghi, N.: 1996, ‘EPR-Bell Nonlocality, Lorentz Invariance, and Bohmian Quantum Theory’, quant-ph/9510027, preprint. (To appear in Physical Review A, April 1996.)Google Scholar
  10. Berndl, K., Dürr, D., Goldstein, S., Peruzzi G., and Zanghi, N.: 1995, ‘On the Global Existence of Bohmian Mechanics’, Communications in Mathematical Physics 173, 647–673.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Birkhoff, G. and von Neumann, J.: 1936, ‘The logic of Quantum Mechanics’, Ann. Math. 37.Google Scholar
  12. Böhm, D.: 1952, ‘A Suggested Interpretation of the Quantum Theory in Terms of ‘Hidden’ Variables, I and II,’ Physical Review 85, 166–193. Reprinted in Wheeler and Zurek 1983, pp. 369–396.CrossRefGoogle Scholar
  13. Bohm, D. and Hiley, B. J.: 1993, The Undivided Universe: An Ontological Interpretation of Quantum Theory, Routledge & Kegan Paul, London.Google Scholar
  14. Daumer, M., Dürr, D., Goldstein, S., and Zanghì, N.: 1996, ‘On the role of operators in quantum theory’, in preparation.Google Scholar
  15. Davies, E. B.: 1976, Quantum Theory of Open Systems, Academic Press, London.zbMATHGoogle Scholar
  16. De Witt, B. S. and Graham, N. (eds.): 1973, The Many-Worlds interpretation of Quantum Mechanics, Princeton, N.J..Google Scholar
  17. Dieks, D.: 1991, ‘On some alleged difficulties in interpretation of quantum mechanics’, Synthese 86, 77–86.MathSciNetCrossRefGoogle Scholar
  18. Dürr, D., Goldstein, S., and Zanghì, N.: 1992, ‘Quantum Equilibrium and the Origin of Absolute Uncertainty’, Journal of Statistical Physics 67, 843–907; “Quantum Mechanics, Randomness, and Deterministic Reality,” Physics Letters A 172, 6–12.MathSciNetzbMATHCrossRefGoogle Scholar
  19. Dürr, D., Goldstein, S., and Zanghì, N.: 1996, ‘Bohmian Mechanics as the Foundation of Quantum Mechanics’, in Bohmian Mechanics and Quantum Theory: An Appraisal, J. Cushing, A. Fine and S. Goldstein (eds.), Kluwer Academic Press.Google Scholar
  20. Everett, H.: 1957, ‘Relative state formulation of quantum mechanics’, Review of Modern Physics 29, 454–462. Reprinted in De Witt et al. 1973, and Wheeler et al. 1983.MathSciNetCrossRefGoogle Scholar
  21. Gell-Mann, M. and Hartle, J. B.: 1993, ‘Classical Equations for Quantum Systems’, Physical Review D 47, 3345–3382.MathSciNetCrossRefGoogle Scholar
  22. Ghirardi, G. C., Rimini, A., and Weber, T.: 1986, ‘Unified Dynamics for Microscopic and Macroscopic Systems’, Physical Review D 34, 470–491.MathSciNetzbMATHCrossRefGoogle Scholar
  23. Ghirardi, G. C., Pearle, P., and Rimini, A.: 1990, ‘Markov Processes in Hilbert Space and Continuous Spontaneous Localization of Systems of Identical Particles’, Physical Review A 42, 78–89.MathSciNetCrossRefGoogle Scholar
  24. Ghirardi, G. C., Grassi, R., and Benatti, F.: 1995, ‘Describing the Macroscopic World: Closing the Circle within the Dynamical Reduction Program’, Foundations of Physics 23, 341–364.MathSciNetCrossRefGoogle Scholar
  25. Gleason, A. M: 1957, ‘Measures on the Closed Subspaces of a Hilbert Space’, Journal of Mathematics and Mechanics 6, 885–893.MathSciNetzbMATHGoogle Scholar
  26. Griffiths, R. B: 1984, ‘Consistent Histories and the Interpretation of Quantum Mechanics’ Journal of Statistical Physics 36, 219–272.MathSciNetzbMATHCrossRefGoogle Scholar
  27. Goldstein, S.: 1987, ‘Stochastic Mechanics and Quantum Theory’, Journal of Statistical Physics 47, 645–667.MathSciNetCrossRefGoogle Scholar
  28. Jauch, J. M. and Piron, C.: 1963, ‘Can Hidden Variables be Excluded in Quantum Mechanics?’, Helvetica Phisica Acta 36, 827–837.MathSciNetzbMATHGoogle Scholar
  29. Jauch, J. M.: 1968, Foundations of Quantum Mechanics, Addison-Wesley, Reading, Mass.zbMATHGoogle Scholar
  30. Holland, P. R.: 1993, The Quantum Theory of Motion, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
  31. Kochen, S. and Specker, E. P.: 1967, ‘The Problem of Hidden Variables in Quantum Mechanics’, Journal of Mathematics and Mechanics 17, 59–87.MathSciNetzbMATHGoogle Scholar
  32. Kochen, S.: ‘A New Interpretation of Quantum Mechanics’, in P. Lathi and P. Mittelstaedt (eds.), Symposium on the Foundations of Modern Physics, World Scientific, Singapore.Google Scholar
  33. Mott, N. F: 1929, Proceedings of the Royal Society A 124, 440.Google Scholar
  34. Nelson, E.: 1966 ‘Derivation of the Schrödinger Equation from Newtonian Mechanics’, Physical Review 150, 1079–1085.CrossRefGoogle Scholar
  35. Nelson, E.: 1985 Quantum Fluctuations, Princeton University Press, Princeton.zbMATHGoogle Scholar
  36. Omnès, R.: 1988, ‘Logical Reformulation of Quantum Mechanics I’, Journal of Statistical Physics 53, 893–932.MathSciNetzbMATHCrossRefGoogle Scholar
  37. von Neumann, J.: 1932, Mathematische Grundlagen der Quantenmechanik Springer Verlag, Berlin. English translation: 955, Princeton University Press, Princeton.zbMATHGoogle Scholar
  38. Schrödinger, E.: 1935, ‘Die gegenwärtige Situation in der Quantenmechanik,’ Die Naturwissenschaften 23, 807–812, 824–828, 844–849. [Also appears in translation as “The Present Situation in Quantum Mechanics,” in Wheeler and Zurek 1983, pp. 152–167.]CrossRefGoogle Scholar
  39. van Fraassen, B.: 1991, Quantum Mechanics, an Empiricist View, Oxford University Press, Oxford.Google Scholar
  40. Wheeler, J. A. and Zurek, W. H. (eds.): 1983, Quantum Theory and Measurement, Princeton University Press, Princeton.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1997

Authors and Affiliations

  • Martin Daumer
    • 1
  • Detlef Dürr
    • 1
  • Sheldon Goldstein
    • 2
  • Nino Zanghì
    • 3
  1. 1.Mathematisches Institut der Universität MünchenMünchenGermany
  2. 2.Department of MathematicsRutgers UniversityNew BrunswickUSA
  3. 3.INFNIstituto di Fisica dell’Università di GenovaGenovaItaly

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