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Progress of Theoretical Physics

Implications of the Proposed Theory of Measurement

R. Fukuda

Published Jan 1, 1989• 7 citations
Publisher

Abstract

The proposed theory of measurement in quantum mechanics is further studied. Using a model, a relationship between the shift of the detector variable and the magnitude of the interference is derived and the approach to the macroscopic limit is discussed. Various implications our theory may have in other typical phenomena of the macroscopic body are discussed and the comparison with other theories of measurement is made.

DOI
10.1143/ptp.81.34

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The proposed theory of measurement in quantum mechanics is further studied. Using a model, a relationship between the shift of the detector variable and the magnitude of the interference is derived and the approach to the macroscopic limit is discussed. Various implications our theory may have in other typical phenomena of the macroscopic body are discussed and the comparison with other theories of measurement is made.
This paper is concerned with the description of the process of measurement within the context of a quantum theory of the physical world. It is noted that quantum mechanics permits a quasi-classical description (classical in the limited sense implied by the correspondence principle of Bohr) of those macroscopic phenomena in terms of which the observer forms his perceptions. Thus, the process of measurement in quantum mechanics can be understood on the quasi-classical level by transcribing from the strictly classical observables of Newtonian physics to their quasi-classical counterparts the known rules for the measurement of the former. The remaining physical problem is the delineation of the circumstances in which the correlation of a peculiarly quantum mechanical observable A with a classically measurable observable B can result in a significant measurement of A. This is undertaken within the context of quantum theory. The resulting clarification of the process of measurement has important implications relative to the philosophic interpretation of quantum mechanics.
The question as to whether quantum mechanics can be applied to a macroscopic system, which has recently been raised through discussions of the problem of how to interpret the so-called “reduction of a wave packet”, is discussed. It is shown by a detailed analysis of the destruction of interference which brings about the reduction, that a supplementary assumption on the preciseness of measurements must at least be introduced in order to interpret the reduction process as an objective process, and it does not, however, necessarily imply inapplicability of quantum mechanics to a macroscopic system.
The notion of the macroscopic in fundamental quantum theory is analyzed. After a brief summary of use of the term macroscopic, its use in quantum theory is compared with its previous use elsewhere. Next, the connections specifically to the Copenhagen interpretation of quantum mechanics and quantum measurement theory more generally, where this term first began to deviate from previous uses, are explained and exhibited through a number of examples. Then, recent attempts to define accurately and quantitatively the extent of being macroscopic, that is, macroscopicity are discussed and their implications considered. This is done most particularly in the realm of quantum optics, where it differs most from previous uses and has recently been of considerable interest. Finally, with the benefit of this analysis, recommendations are made regarding future use of the notion of the macroscopic in fundamental physics.
Macroscopic variables of the macroscopic system lose the quantum fluctuations in the macroscopic limit. A separate Hilbert space corresponds to each c-number value of these macroscopic variables. The measuring process in quantum mechanics consists of recording these c-number values and amounts to determination of the Hilbert space to which the system belongs. These observations provide a solution to the problem of measurement and lead to predictions concerning the c-number equations of motion of the macroscopic variables.
We present and discuss some models for the quantum measuring process. Our models indicate that the coherent state approach to classical limits in quantum theories becomes a powerful method for a unified description of various measuring processes. Each detector in the models has a parameter N , the number of its dynamical variables; the whole process of an interaction between an object system and the detector is calculable for any finite number N . It is shown that the detector states after the interaction are characterized by a macroscopic parameter, which is closely connected with the “reliability” of the detector. The reliability plays an essential role in eliminating the quantum interference between two states corresponding to two different positions of the pointer. Finally, the quantum mechanics of the detector reduces to classical mechanics on suitable “phase space” as N becomes large, as is required.