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On the general quantum theory of measurement

  • Arthur I. Fine (a1)

Extract

We use the term ‘measurement’ to refer to the interaction between an object and an apparatus on the basis of which information concerning the initial state of the object may be obtained from information on the resulting state of the apparatus. The quantum theory of measurement is a quantum theoretic investigation of such interactions in order to analyse the correlations between object and apparatus that measurement must establish. Although there is a sizeable literature on quantum measurements there appear to be just two sorts of interactions that have been employed. There are the ‘disturbing’ interactions consistent with the analysis of Landau and Peierls (8) as developed by Pauli (11) and by Landau and Lifshitz (7), and there are the ‘non-disturbing’ interactions explicitly set out by von Neumann ((10), chs. 5, 6), and that dominate the literature. In this paper we shall investigate the most general types of interactions that could possibly constitute measurements and provide a precise mathematical characterization (section 2). We shall then examine an interesting subclass, corresponding to Landau's ideas, that contains both of the above sorts of measurements (section 3). Finally, we shall discuss von Neumann measurements explicitly and explore the purported limitations suggested by Wigner(12) and Araki and Yanase (2). We hope, in this way, to provide a comprehensive basis for discussions of quantum measurements.

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References

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(1)Albertson, J.Quantum-mechanical measurement operator. Phys. Rev. 129 (1963), 940943.
(2)Araki, H. and Yanase, M.Measurement of quantum mechanical operators. Phys. Rev. 120 (1961), 622626.
(3)Durand, L.On the theory of measurement in quantum mechanical systems. Philos. Sci. 27 (1960), 115133.
(4)Fine, A.Dissertation (unpublished) University of Chicago (1963); Realism in quantum measurements, Methodology and science (forthcoming).
(5)Jauch, J. M.The problem of measurement in quantum mechanics. Helv. Phys. Acta 37 (1964), 293316.
(6)Jauch, J. M., Wigner, E. P. and Yanase, M. M.Some comments concerning measurements in quantum mechanics. Nuovo Cimento 48 (1967), 144151.
(7)Landau, L. and Lifshitz, E.Quantum mechanics pp. 2124. (Pergamon; London, 1958).
(8)Landau, L. and Peierls, . Erweiterung des Unbestimmtheitsprinzips für die relativistische Quantentheorie. Z. Physik 69 (1931), 5669.
(9)Margenau, H.Measurements and quantum states, I and II. Philos. Sci. 30 (1963), 116, 138–157.
(10)von Neumann, J.Mathematical Foundations of Quantum Mechanics (Princeton Univ.; Princeton, 1955).
(11)Pauli, W. Prinzipien der Quantentheorie, 5, Part 1, 1–168. In Handbuch der Physik ed. Flügge, S. (Springer-Verlag; Berlin, 1958).
(12)Wigner, E. P.Die Messung quantenmechanischer Operatoren. Z. Physik 133 (1952), 101108.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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