Summary
The quantum mechanical measurement process is analysed in detail, especially from the group theoretical point of view. The first three chapters review the results obtained before briefly. In §1, the classical theory of measurement, originated by von Neumann, is birefly described. The importance of considering more general schemes of measurements than the original one is, however, emphasized In §2, the role of conservation laws, in imposing limitations on the possibility of measurements, is discussed. Accurately measurable quantities are defined in this context. In §3, the role of conservation laws is further analysed. It is shown that the conserved quantity of the apparatus cannot be used for a “pointer reading.” The following two chapters discuss the nature of the exactly measurable quantities. In §4, exactly measurable quantities are explicitly determined. It is shown that the measurement of projection operators in the Hilbert space of the object which commute with the operators of the restricted Poincaré group give all the information on the states of the object that can be obtained by accurate measurements. In §5, the measurement on objects which are composed of more than one particle is discussed.
The last chapter, §6, deals with approximately measurable quantities, but no complete estimate of the unavoidable inaccuracy of the measurement of these is obtained. It also contains some concluding remarks.
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References
J. von Neumann, Mathematische Grundlagen der Quantrum-mechanik (Julius Springer, Berlin 1932 ). English translation by R.T. Beyer, University Press, Princeton, 1955.
The somewhat schematic and idealised nature of von Neumann’s measurement theory was surely widely recognised ever since his ideas were put forward.. For a discussion of this subject see, for instance, the first author’s summary at the London (Ontario) conference on the Epistenmology of Quantum Mechanics (March, 1971 ) to appear in Contemporary Research on the Foundations of Quantum Theory. C.A. Hooker, Editor (Reidel, 1973 ).
J. von Neumann, ref. 1. The fact that the states of the system on which the measuring apparatus acts in the prescribed fashion do not usually include all the states will be discussed elsewhere.
W. Pauli, Handbuch der Physik, 2nd Ed. (Springer Verlag, Berlin) Vol. 24, page 83. Reprinted by Edwards Bros., 1946. See particularly pages 143–154. D. Bohm, Phys. Rev. 85, 166, 180 (1952).
H. Margenau and R.N. Hill, Progr. Theor. Physics 26, 722, (1961). A rather complete summary and evaluation of the various views and attitudes is found in B. d’Espagnat’s Conceptual Foundations of Quantum Mechanics (W.A. Benjamin, Menlo Park, 1971 ).
This is, essentially, the attitude adopted by H.P. Stapp, Amer. Jour. of Phys. 40, 1098 (1972).
E.P. Wigner, Zeits. f. Phys. 133, 101 (1952); H. Araki and M.M. Yanase, Phys. Rev. 120, 622 (1960).
This point is most clearly stated in F. London and E. Bauer’s La Théorie de 1’ observation en Méchanique Quantique ( Hermann et Cie, Paris, 1939 ).
M.M. Yanase, Phys. Rev. 123, 666 (1961).
See e.g. E.P. Wigner, Amer. Jour. of Phys. 31, 6 (1963).
For a summary, see for instance Chapter 11 of A. Speiser’s Theorie der Gruppen von endlicher Ordnung (Julius Springer, Berlin 1927) or almost any book on the theory of group representations.
E.P. Wigner, Annals of Math. 40, 149 (1939); V. Bargmann and E.P. Wigner, Proc. Nat. Acad. Sc. 34, 211 (1948).
See reference 10 or any book on the application of the theory of group representations to quantum mechanics, such as the first author’s Group Theory (Academic Press, 1959) or M. Hamermesh’s Group Theory (Addison Wesley, 1962).
J. von Neumann, Annals of Math. 50, 401 (1949); F.I. Mautner, ibid. 51, 1 (1950); 52, 528 (1950); Proc. Amer. Math. Soc. 2 490 (1951).
F. Goldrich and E.P. Wigner, chapter in Magic without Magic, John R. Klauder, Editor (W.H. Freeman, 1972 ).
Y. Aharonov and L. Susskind, Phys. Rev. 155, 1428 (1967). M. Goldstone, personal communication.
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Wigner, E.P., Yanase, M.M. (1997). Analysis of the Quantum Mechanical Measurement Process. In: Wightman, A.S. (eds) Part I: Particles and Fields. Part II: Foundations of Quantum Mechanics. The Scientific Papers, vol A / 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-09203-3_59
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