Abstract
The many-Hilbert-spaces theory of quantum measurements, which was originally proposed by S. Machida and the present author, is reviewed and developed. Dividing a typical quantum measurement in two successive steps, the first being responsible for spectral decomposition and the second for detection, we point out that the wave packet reduction by measurement takes place at the latter step, through interaction of an object system with one of the local systems of detectors. First we discuss the physics of the detection process, using numerical simulations for a simple detector model, and then formulate a general theory of quantum measurements to give the wave packet reduction in an explicit form as a sort of phase transition. The derivation is based on the macroscopic nature of the local system, to be represented in a continuous direct sum of many Hilbert spaces, and on the finite-size effect of the local system, to give phase shifts proportional to size parameters. We give a definite criterion for examining any instrument as to whether it works well as a detector or not. Finally, we compare the present theory with famous measurement theories and propose a possible experimental test to discriminate it from others. A few solvable detector models are also discussed.
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S. Machida and M. Namiki,Prog. Theor. Phys. 63, 1457–1473, 1833–1847 (1980);Proceedings of the International Symposium on Foundations of Quantum Mechanics—In the Light of New Technology (ISQM, Tokyo, 1983), S. Kamefuchi, H. Ezawa, Y. Murayama, M. Namiki, S. Nomura, Y. Ohnuki, and T. Yajima, eds. (Physical Society of Japan, Tokyo, 1984), pp. 127–135, 136–139;Fundamental Questions in Quantum Mechanics, L. M. Roth and A. Inomata, eds. (Gordon and Breach, New York, 1986), pp. 77–92; M. Namiki,Ann. N.Y. Acad. Sci. 480, 78–88 (1986);Microphysical Reality and Quantum Formalism, F. Selleri, G. Tarozzi, and Z. van der Merwe, eds. (Reidel, Dordrecht, 1987);Quantum Mechanics of Macroscopic Systems and Measurement Problem (in Japanese), Y. Otsuki, ed. (Kyoritsu, Tokyo, 1985), pp. 139–219.
H. Araki,Prog. Theor. Phys. 64, 719–730 (1980);Fundamental Aspects of Quantum Theory, V. Gorini and A. Frigerio, eds. (Plenum, New York, 1986), pp. 23–33.
See earlier papers of Machida and Namiki.
See, for example, K. M. McVoy,Ann. Phys. (New York) 54 552–565 (1969).
See Ref. 1 and Appendix.
Y. Murayama, H. Nakazato, M. Namiki, and I. Ohba, Movie entitled “Numerical Simulations of Reduction of Wave Packet in Quantum Measurement Processes” (Cinesll, Tokyo, 1986); Y. Murayama and M. Namiki,Proceedings of the International Conference on The Concept of Probability (Reidel, Dordrecht, 1988).
S. Nakajima, Private communication.
W. K. Wootters and W. H. Zurek,Phys. Rev. D 19, 443–456 (1979).
See, for example, many papers inProceedings of the Second International Symposium on Foundations of Quantum Mechanics (ISQM, Tokyo, 1986), M. Namiki, Y. Murayama, S. Nomura, and Y. Ohnuki, eds. (Physical Society of Japan, Tokyo, 1987).
S. Watanabe, Private communication. Does this remark mean that the naive realism can hardly work in quantum mechanics?
E. P. Wigner,Am. J. Phys. 31, 6–15 (1963).
See, for example, A. Daneri, A. Loinger, and G. M. Prosperi,Nucl. Phys. 3, 297–319 (1962).
See, for example,The Many-Worlds Interpretation of Quantum Mechanics, B. S. DeWitt and N. Graham, eds. (Princeton University Press, Princeton, 1973).
W. H. Zurek,Phys. Rev. D 26, 1862–1880 (1982). Many papers have been published on the environment theory, but our discussions focus on Zurek's theory here.
See Refs. 1 and 2. Note that we cannot define the inner product betweenu n (x; l) = sin(nπx/l) ∈ ℋ (l) andu m (x; l′) = sin(mπx/l′) ∈ ℋ (l′) ifl ≠l′
See, for example, H. Rauch, in 2nd Ref. 2, 277–288; in Ref. 9, pp. 3–17; M. Namiki, Y. Otake, and H. Soshi,Prog. Theor. Phys. 77, 508–513 (1987).
H. Araki and M. M. Yanase,Phys. Rev. 120, 622–626 (1960).
See Ref. 11 and the following papers: A. Fine,Phys. Rev. D 2, 2783–2787 (1970); A. Shimony,Phys. Rev. D 9, 2321–2323 (1974).
J. M. Jauch, E. P. Wigner, and M. M. Yanase,Nuovo Cimento 48, 144–151 (1967).
S. Machida and M. Namiki, in Ref. 9, pp. 355–359.
Y. Morikawa, M. Namiki, and Y. Otake,Prog. Theor. Phys. 78 (1987), in press.
B. d'Espagnat, LPTHE/45, November 1986.
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Namiki, M. Many-Hilbert-spaces theory of quantum measurements. Found Phys 18, 29–55 (1988). https://doi.org/10.1007/BF01882872
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DOI: https://doi.org/10.1007/BF01882872