Comments on Comments on Comments

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Comments on Comments on Comments: A Reply to Margenau and Wigner Hilary Putnam Philosophy of Science, Vol. 31, No. 1. (Jan., 1964), pp. 1-6. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28196401%2931%3A1%3C1%3ACOCOCA%3E2.0.CO%3B2-2 Philosophy of Science is currently published by The University of Chicago Press.

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Philosophy of Science

VOL.

January,

31

I 964

NO. I

DISCUSSION : COMMENTS ON COMMENTS ON COMMENTS A REPLY TO MAWGENAU AND WIGNER* HILARY P U T N A M Massachusetts Institute of Technology

T h e Margenau and Wigner "Comments" [2] on my "Comments on the Paper of David Sharp", [3, 41 is a strange document. First the authors say, in effect, "had anything been wrong (with the fundamentals of quantum mechanics) we should certainly have heard". Then they issue various obiter dicta (e.g., the "cut between observer and object" is unavoidable in quantum mechanics; the-highly subjectivistic -London-Bauer treatment of quantum mechanics is described, along with von Neumann's book, as "the most compact and explicit formulation of the conceptual structure of quantum mechanics"). My assumption 2 (that the whole universe is a system) is described as "not supportable", because "the measurement is an interaction between the object and the observer". T h e "object" (the closed system) cannot include the observer. The issues involved in this discussion are fundamental ones. I believe that the conceptual structure of quantum mechanics today is as unhealthy as the conceptual structure of the calculus was at the time Berkeley's famous criticism was issued. For this reason-as much to emphasize the seriousness of the present situatioil in the foundations of quantum mechanics as to remove confusions that may be left in the mind of the general reader upon reading the Margenau and Wigner "Commentsv-I intend to restate the main points of my previous "Comments", and to show in detail why the Margenau and Wigner remarks fail completely to meet them. 1. The main point. Let S be a system which is "isolated" (as well as possible) during an interval to < t < t,, and whose state at to is known, let M be a measuring system which interacts with S so as to measure an observable 0 at tl, and let T be the "rest of the universe". In quantum mechanics, a physical situation is described by giving two things: a Hamiltonian and a state function. T h e usual way of obtaining an approximate description of the situation of the system S is simply to set the interaction of M T with S equal to zero for the interval to < t < t,. This, of course, is only an approximation-rigorously, the interaction between S and M T never completely vanishes, as Sharp and I both pointed out in our papers. What then is the rigorous description of the system S ? The answer, surprisingly, is that usual quantum mechanics provides no rigorous,

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* Received

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October, 1962.

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HILARY PUTNAM

contradiction-free account at all ! (The parallel with the 18th century situation in the foundations of the calculus is surprisingly close: setting dx = 0 after one has divided by dx "works". But mathematically this procedure is wholly unjustified, and it took the work of Weierstrauss and the development of the concept of a limit to provide a rigorous, thoroughly justifiable procedure.) In fact, if we take account of the fact that S is not strictly isolated (i.e., Hamiltonian (interaction between M T and S ) # 0), then, by an elementary calculation1, S cannot be assigned any state function. Also, since M T generates a field (however weak) which would have to be exactly known to describe the situation of S by means of a Hamiltonian, and by quantum mechanics itself, one cannot exactly know this field, since one cannot know the simultaneous positions and momenta of its sources, S cannot be assigned a Hamiltonian either. So the "approximation" made in quantum mechanics-setting Hamiltonian (interaction M T and S) = 0--is like the "approximation" setting dx = 0, and not like the legitimate approximations in classical mechanics, which can always in principle be dispensed with. I t is an algorithm which "worlis", but which has not, to date, been grounded in a consistent and, in principle, mathematically rigorous theory.

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2. T h e Margenau-Wigner reply. Margenau and Wigner reply: "Overall consistency of all parts of quantum mechanics, especially when that theory is forced to make reference to 'the entire universe' has never been proven or claimed." This is the only reference to the main point of my "Comments", and it gives the erroneous impression that the point we have just reviewed depends on treating "the entire universe" (S M T) as a system with a $-function of its own.

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3. Cosmological p r o b l e m s n o t relevant. Margenau and Wigner's phraseology"especially when that theory is forced to make reference to 'the entire universe' " (italics mine)-suggests that by 'the entire universe' I must have meant the cosmological universe and that I sought to embroil quantum mechanics in the problems of cosmology. Nothing could be wider of the mark. Footnote 1 of my paper made it clear that the question is whether quantum mechanics can consistently treat measurement as an interaction taking place within a single closed system (containing the observer). There is no objection to "idealizing" by setting Hamiltonian (T, M S ) = 0. After all, it is purely contingent that T is not just empty space. But it is not purely contingent that M is not just empty space: empty space cannot make measurements. If we do attempt to treat all measurements-that is to say, all the measurements we are interested in-as taking place within one closed system (as we would in classical physics), then we must imagine that the "rest of the universe", T, is just empty space, or at least that no measurements are carried out by observers in T upon M S. Otherwise, (1) the main point (see above) is not taken care of at all, and (2) we are not imagining that all measurements relevant in the context take place in one closed system (which is the question at issue). Margenau and Wigner write, "In fact, if one wants to ascertain the result of the measurement, one has to observe the measuring apparatus, i.e., carry out a measurement on it." As an argument against the "one closed system" view this is worthless, since it presupposes that the observer is not part of M.

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4. It is n o t t r u e t h a t "the object c a n n o t b e t h e whole tiniverse". Margenau and Wigner also state that Von Neumann's axioms for quantum mechanics are in-

' Cf. 141, p. 227, equation (4), and p. 230 ff.

COMMENTS ON COMMENTS ON COMMENTS

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compatible with the assumption that a closed system which contains the observer (the "entire universe") is a system in the sense of quantum mechanics. I t is true that if we make the assumption that "measurement" involves the interaction of the system under consideration with an outside system, then we cannot also assume that "the entire universe" is a system. Must we make this assumption ? In my "Comments", I suggested that it might be possible to give it up, but I did not give details. Since this is the point (the "cut" between observer and object) that Margenau and Wigner say is central to all of quantum mechanics, I will now be more explicit on this point. Let M and S be as before, and let T be empty (so that the "entire universe" consists of M S for present purposes). Von Neumann postulates that when M measures an observable 0 in S , then S is thrown into a new state, an eigenstate of the observable 0. Which eigenstate of 0 S is in is determined by M. According to Bohr, this is done in a wholly classical manner-that is, the process by which some macro-observable in M (say a pointer reading) comes to register the value corresponding to the 0-state S is in can be explained by classical physics. In particular, M can be treated using classical physics a l o n e ~ n l yS has to be described quantum mechanically. Of course, the "cut" can be shifted-that is, a proper part of M (always including the observer) can be taken as the measuring system M', while the rest of M can be adjoined to S to make the new observed system St.But, however we make the "cut", the measured system S is thought of as obligingly "jumping" into an eigenstate of 0 so that a classical system M can measure 0 in a purely classical way. This is not only implausible on the face of it, but inconsistent since S cannot, strictly speaking, have states of its own, as has already been pointed out. What is consistent and what also seems to avoid the whole difficulty, is to say that the interaction between M and S causes the entire system M S to go into an eigenstate of 0. In other words, assume: 0-measurement causes the entire universe to go into an eigenstate of 0. This assumption is consistent with the mathematical formalism of quantum mechanics-in fact, more consistent than the assumption that S alone jumps into an eigenstate of 0, as we have seen-and expresses the view that the measuring system is a part of the total system under consideration, and not an "outside" system.

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5. Quantum mechanics and classical physics. In the preceding section, I referred to a well-known peculiarity of the received interpretation of quantum mechanics (the so-called "Copenhagen Interpretation")-namely, S is ascribed a #function, and treated according to the laws of quantum mechanics, while the measuring system or "observer", M, is treated as a classical object. Thus quantum mechanics