Summary
We illustrate the rôle played by the concept of metastability of a macrosystem in the quantum theory of measurement, by discussing the two significant cases of the scintillation counter and of the spark chamber.
Riassunto
Si illustra l’ufficio della nozione di metastabilità (nel senso della meccanica statistica) di un sistema macroscopico nella teoria quantistica della misurazione, discutendo i due casi significativi del contatore a scintillazione e della camera a scintilla.
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References
A. Daneri, A. Loinger andG. M. Prosperi:Nucl. Phys.,33, 297 (1962). For a qualitative discussion, seeP. Caldirola:Scientia,58 (Nov. 1964).
J. von Neumann:Mathematical Foundations of Quantum Mechanics, Chapts. V and VI (Princeton, 1955).
L. Rosenfeld:Suppl. Progr. Theor. Phys., extra n., (1965), p. 222.
E. P. Wigner:Am. Journ. Phys.,31, 6 (1963). This article is essentially devoted to a discussion of the merits and defects of von Neumann’s approach.
A. Shimony:Am. Journ. Phys.,31, 755 (1963). In this review article, the Author raises the following objections to our approach: i) The existence of large-scale quantal behaviours, as,e.g., the spin echoes and the Mössbauer effect, in which unexpected coherent contributions are obtained from the parts of macrosystems, is a caution against underestimating the ingenuity of the experimenters regarding phase relations in macrosystems; ii) from the fact that two macro-observers use the same mixture to describe the system composed of the micro-object and the measuring (macro-) apparatus prior to their respective readings, one can infer only that they make the same statistical predictions correctly characterizing an ensemble of similar situations. However, the agreement of the two observers in aspecific reading of the apparatus would be a mere coincidence. Our answers are as follows:Adv. i). — The states involved in large-scale quantal phenomena have nothing to do with themacroscopic states, which serve as final states of a measuring (macro-) apparatus. See, in particular, the examples discussed byA. Peres andN. Rosen:Phys. Rev.,135, B 1486 (1964).Adv. ii). — We think—withBohr, Jordan andLudwig (cf.e.g. Ludwig’s bookDie Grundlagen der Quantenmechanik (Berlin, 1954))—that quantum mechanics describes the microsystems only through their interactions with the macrosystems, which are the only systems interesting the common experience of the daily life. In our opinion, all the statements concerning the macroscopic state of a large body have a meaning independently of its observation. Weassume that two persons who look at a macroscopic body, or who perform, in particular, a reading of an apparatus, do always «see the same thing». Accordingly, the problem is only to prove that the actual reading of an apparatus may be consistently interpreted as resulting simply in an increase of information (transition from the mixture to one of the composing elements). (Shimony’s objections have also been criticized byM. N. Hack:Am. Journ. Phys.,32, 890 (1964)).
P. A. Moldauer:Am. Journ. Phys.,32, 172 (1964).
M. M. Yanase:Am. Journ. Phys.,32, 208 (1964).
J. M. Jauch:Helv. Phys. Acta,37, 193 (1964). This Author claims to have dissolved, by means of a suitable analysis, the entire problem of the measuring process into a pseudoproblem. In reality, he has missed the point. As it has been pointed out byRosenfeld (private communication to one of us), «he has unfortunately dismissed as unimportant the behaviour of the amplifier, which … is just the key to the understanding of the reduction [of the wave packet]. If one reads Jauch’s text carefully, one notices the logical gap in his argument just at this point; in fact, he does not give any physical justification for considering the probability operator of the microsystem II [the microscopic part of the measuring device], which gives him the reduction.» In Jauch’s opinion, any attempt, like ours, of finding in the macroscopic nature of the measuring instruments the reasons for the occurrence of a mixture at the end of the measuring process would be doomed to failure. He claims that this conclusion follows from Wigner’s article quoted in (4), in which it would be proved that all the above attempts must be necessarily inconsistent with the linear time-evolution laws of quantum theory. However, a careful analysis of Wigner’s paper makes clear that this Author has in reality only reminded us, quite correctly and quite obviously, that the time-evolved of a pure state, as given by the Schrödinger equation, cannot be a mixture. Now, we haveproved in our paper, making an essential use of the quantal ergodic theorem, that, for all practical purposes, a macro-observer may describe the behaviour of the global system micro-object plus macro-apparatus, at the end of the process of measurement, by means of a given mixture. (Obviously, this is true only for the macroscopic observables of the apparatus, and not for any mathematically conceivable dynamical variable!) We remark further that, in our approach, «the reduction of the initial state of the atomic system has nothing to do with the interaction between this system and the measuring apparatus: in fact, it is related to a process taking place in the latter apparatus after all interaction with the atomic system has ceased.» (Rosenfeld,loc. cit. in (3)). Consequently, Jauch’s statement is not correct. Another argument against this statement is the following. IfJauch were right, the linear character of the Schrödinger equation ought to be inconsistent with any kind of macroscopic description of the world; in particular, inconsistent with the master equation (which, on the contrary, has been rigorously proved, at least in a certain number of cases) and with the thermodynamical description of the macroscopic processes by means of equations connecting only macroscopic quantities. Actually, all the macroscopic descriptions entail that the «interference» terms, in so far as the macroscopic variables are concerned, may be neglected.
In our theory, the measuring apparatus is schematized as a macrosystem, which has, besides the energy, at least another macroscopic constant of the motion. The value of this constant characterizes an invariant manifold of the Hilbert state space of the system, which we have called «channel».
See the itemQuantum Theory of Measurement byG. M. Prosperi, in theDictionary of Physics (Oxford, 1964).
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Daneri, A., Loinger, A. & Prosperi, G.M. Further remarks on the relations between statistical mechanics and quantum theory of measurement. Nuovo Cimento B (1965-1970) 44, 119–128 (1966). https://doi.org/10.1007/BF02710429
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DOI: https://doi.org/10.1007/BF02710429