Comments on the Paper of David Sharp

Comments on the Paper of David Sharp Hilary Putnam Philosophy of Science, Vol. 28, No. 3. (Jul., 1961), pp. 234-237. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28196107%2928%3A3%3C234%3ACOTPOD%3E2.0.CO%3B2-6 Philosophy of Science is currently published by The University of Chicago Press.

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DISCUSSION

COMMENTS ON TME PAPER OF DAVID SHARP HILARY P U T N A M

Mr. Sharp's resolution of the Einstein-Podolsky-Rosen paradox is, in my opinion, the right one. However, the resolution does not in my opinion show that "God's in his heaven; all's right with the world (quantum mechanics)." This is in no way a criticism of Sharp ; rather, it is inevitable that any resolution of this paradox, dealing as it does with the quantum mechanical concept of << measurement", must tangle with some of the real difficulties which are lurking in the notion. The purpose of these comments is to use Mr. Sharp's remarks as a basis for bringing out one of the serious conceptual difficulties connected with measurement. First a couple of words on the status of elementary, non-relativistic quantum mechanics: I t is widely believed that this theory is (a) mathematically rigorous; and (b) fits nature as well as can be expected of a non-relativistic theory; that is, provided distances are not "too small" or velocities "too great"' predictions may be expected to be accurate to several decimal places. Also, (c) it is widely believed that Von Neumann's axiomatization gave the theory, including << measurement" a logically rigorous formulation. I am not a physicist. Hence I will not comment on point (6). However, it is my opinion that claims (a) and (c) are simply false. I maintain that a formalization of quantum mechanics, in the strict sense of the term "formalization", which is physically acceptable and also has the minimal virtue that can be expected of a formalization-simple consistency-is not today in sight. In some cases, of course, formalization is of little philosophical or scientific significance. In the case of quantum mechanics, however, the difficulties that obstruct formalization are important for the reason that they also obstruct any attempt at a reascnable interpretation of the theory as a whole. Some of the difficulties have to do with the very "formation rules" of the calculus: What should count as a "well formed formula" of quantum mechanics (under a suitable formalization) ? Here, however, let us consider a difficulty which arises when we attempt to decide what the "assumptions" of the theory are; or, rather, to what the theory commits us. Consider the following statements:

(1) A "measurement" on a system S requires the interaction of S with some outside system T.

* Received

October, 1960.

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(2) T h e "whole universe" is a system in the sense of quantum mechanics (possesses a state function).l First of all, I shall show that no physically acceptable version of present quantum mechanics can incorporate both of these statements. In particular, the interpretations (of elementary non-relativistic quantum mechanics) suggested by Bohm2 and Ludwig3 fall to the ground on just this point. do this, observe that from (1) and (2) we can immediately obtain: (3) T h e whole universe never undergoes measurement. But according to quantum mechanics there are only two ways in with a system can change its state: "motion" and "measurement". And "motion" is simply a continuous change of state according to the Schrodinger equation. So we have: (4) 'The whole universe is a system whose time development obeys the Schrodinger equation at all times (i.e., no "reduction of the wave paclset"). But from (4) a host of empirically false consequences follow! For example, let "system 1" be a single electron, and let "observable A" be the position of system 1. Then rl (as Sharp points out) is also an "observaSle" (with a spectluni of eigenfunctions and eigenvalues) in the larger system (the "whole universe"). And it can be rigorously proved from (4) that the state function of the larger system cannot be an eigenfunction of the observable A at two different times (unless "measurement" takes place, contrary to (3) and (4)). But this implies that the observable A is never measured (even in the sub-systems), which could easily contradict experience! Since (1) and (2) are physically incompatible, let us next consider the "move" of rejecting (2) while retaining (1). In this case there are still difficulties with theories like those of Bohnl and Ludwig, (which attempt to reduce ic: Guuodlagcn der Quantenmechanik", (Springer Verlag, Berlin, 1954). 4 Strictly speaking, the eigenfunctions of position are not functions but rather distributions in the sense of Schwa1.t~.

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versions (including Von Neumann's) of elementary, non-relativistic quantum mechanics that any system which is not closed should have a state function of its own! Thus, if one rejects (2) we obtain: (5) There are no "systems" in the sense of quantum mechanics-i.e., physical systems whose states can be represented by vectors in Hilbert space ("state functions"). (!) I t may be argued that, although (5) does follow from quantum mechanics (if (2) is rejected); nonetheless there are systems whose interaction with "the rest of the universe" is so weak that they may be treated as isolated systems as a justi$able approximation. Here I agree with Sharp. An approximation is justifiable if the same results (within the specified limits) would have been obtained (perhaps not as easily, however) without the approximation. Let us ask: without the approximation (treating systems which are not isolated as if they were isolated), would the same results (expectation values) follow from quantum mechanics? T h e answer is no! For unless we pass to a state-function of a closed system we cannot even ideally, even "in principle", get any expectation values out of quantum mechanics whatsoever, in the light of (5). Thus treating non-isolated systems as if they were isolated at one stage in a computation and then later recognizing that a "measurement" has taken place (which implies that the system is not and could never have been strictly isolated), is not an approximation in the usual sense (a mere mathematical convenience) but a part of the theory. T o put it another way, what is used in practice is an inconsistent theory, which can nevertheless serve as a useful prediction-algorithm, provided one acquires a certain artfulness in its employment. T h e difficulty (in rigourously assigning a state-function to a system which is not closed) is that the influence of the "outside" must be represented by a potential which enters into the particular form of the operator H in the Schriidinger equation. But this "influence" is a field with outside sources. These sources are themselves subject to quantum-mechanical uncertainties; hence the "potential" would itself be a new kind of object, if one attempted to represent it rigourously, namely an uncertain potentiaL5 One might attempt to mcdify quantum mechanics in this direction (incorporating "uncertain potentials"); but this would be a major mathematical undertaking, and no mere formalization of the present "theory". What I want to emphasize is that the present theory does something which is essentially untenable: namely, it treats the universe as consisting of two kinds of objects, "classical" objects not subject to uncertainties, and micro-objects with the former measuring the latter. That this account cannotbe true follows from quantum mechanics itself. The last alternative is to give up (I). I n this case (2) might or might not be retained. This alternative is the one that is most appealing to me; however, little attention appears to have been given it by physicists. Here too, however, This was suggested by B. Kayser, a fellow-student of Sharp's at Princeton.

COMMENTS ON THE PAPER OF DAVID SHARP

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the problems go beyond mere formalization. Namely it becomes necessary to elaborate a concept according to which at least macro-observables retain sharp values without being "measured" from outside. T o discuss this would go beyond the bounds of these present comments. T o conclude, there is major "unfinished business" in the very foundations of quantum mechanics, and in particular in the elaboration of a consistent and physically acceptable theory of measurement. Only when such a theory has been elaborated will a Jinal resolution of the Einstein-Podolsky-Rosen Paradox be possible.