Foundations of Physics Letters, Vol. 5, No. 2, 1992

RESOLVING THE MEASUREMENT PROBLEM: REPLY TO ALLEN STAIRS Arthur Fine Department of Philosophy Northwestern University Evanston, Illinois 60208 Received March 13, 1992

We defend an account of quantum measurement as a species of selective interactions, an account that gets beyond the insolubility theorem to resolve the quantum measurement problem. In the course of the defense we propose a novel analysis of inexact measurements and discuss the problem of how to treat the states and selective interactions of composite systems. Key words: approximate measurement, composite systems, insolubility theorem, quantum measurement problem. 1. I N T R O D U C T I O N In the foregoing piece [1], Allen Stairs takes a critical look at a proposal I have sketched for how to deal with the quantum measurement problem [2,3]. His examination turns up both virtues and vices, providing an occasion to test the resources of the proposal. I am grateful to Stairs for this opportunity. The measurement problem rests on a general insolubility theorem [4-6] that is usually thought to derive from a conflict between linear dynamics and the rule of silence. Linearity entangles the object and apparatus states in a non-factorizable superposition, and the rule of silence applied to such superposed states forbids us from attributing a definite apparatus result to the interaction. In the papers that Stairs cites [2,3] I show that there is a significant third player in the genesis of the problem; namely, the application of the interaction formalism itself. Revising the rules for using that formalism, I suggest, provides a way out of the conflict, a way that respects the usual dynamics and the usual interpretive practices. 125 0894-9875/92/04004)125506.50/0 © 1992 Plenum Publishing Corporation

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My way of resolving the measurement problem involves the idea that some interactions (not necessarily measurements alone) are selective. They are sensitive only to certain aspects or features of a system, in the sense that the result of the interaction depends on those features only and not on the whole thing. 1 This nonholistic conception calls for a way of treating interactions with only an "aspect" (or a "feature") of a system. Where the feature corresponds to the probability distribution for an observable, I suggest we represent it by a mixed state whose density operator yields the probability distribution in question. This gives an objective, nonignorance interpretation to certain mixed states for a single system. They represent "features" or "aspects" of the system. As a necessary condition on a measurement interaction, I adopt the common requirement that the interaction with the apparatus transfer the probability distribution from the object observable to the apparatus observable. The basic proposal, then, is to regard the measurement of an observable E on a system in state ~t as a measurement interaction that selects the aspect of the system corresponding to the probability distribution for E that is determined by state ~. Given the right deployment of the interaction formalism, it is then trivial to show that measurements produce results. I would emphasize that this is a purely physical conception that regards measurements as a species of selective interactions. This conception involves no conscious "observers." Neither does it require the triggering of amplification devices (and thus can readily accommodate negative-result measurements). When the interacting parts are represented by mixed states, this account of measurement uses only the language of elementary quantum theory (without recourse to infinite limits), and it can be given a general and precise mathematical treatment. I illustrate a typical case below. Suppose that the object o has initial state ~g = Y~an(~nwhere the d~n are the eigenstates of the observable E which is to be measured. [I will write the corresponding density operator as Pbg], sometimes calling it the initial (pure) state.] Let the initial "tuned" state of the measuring apparatus a be given by pure state ~ (density operator P[{]), where the "pointer positions" are eigenvalues belonging to the eigenstates ~n. If U is the unitary operator corresponding to the measurement interaction, I will suppose that U correlates initial eigenstates of the object observable E with final eigenstates of the apparatus pointer position observable. Thus

U(¢n®~) = *fn®~n where ~)fnmay or may not be the same as q~n, depending on whether the measurement is disturbing. My proposal is that the measurement only interacts with the E-aspect of the object o. This aspect of o is represented

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by the mixture with density operator ~lanl21~n].

The measurement

interaction then looks like this

Y, lanlZl~¢n]®P[~]

U

> 2 lanl2p [¢fn®~a]

(1)

where the final object-apparatus state is a mixture over pointer positions, as required. Stairs raises a series of questions and problems with regard to this account. Let me take them, roughly, in order. 2. I N E X A C T M E A S U R E M E N T S First off, Stairs suggests that my proposal is vulnerable to an observation that Albert and Loewer [7] make in criticism of several other proposals; namely, that they are unstable with regard to approximate measurements. Thus Stairs suggests that in an approximate measurement, according to my account, the final object-apparatus state will not be a mixture over pointer positions. Unlike the usual problem of measurement, the problem over inexact measurements arises even in the special situation where the initial object state is an eigenstate q~nof E, not a superposition of eigenstates. In this situation my proposal coincides with the orthodox application of the interaction formalism, and the unitary interaction looks like (*n®~)

U

> (Ofn®~')

(2)

where ~' = ~bn~n, and one of the Ibn12is distinguished by being very close to, but still different from, 1. (This way of treating approximate measurements is discussed in [6]; another way is in [8].) Here we see (already) that the final object-apparatus state ~' is not an eigenstate (or a mixture of eigenstates) of the pointer positions, but rather a superposition. On my view, however, in the transformation (2) above, no measurement of E has actually taken place. That is, the interaction represented by U does not in fact transfer the probability for the nth value of E (which is 1) to the probability for some pointer position. Thus my proposal does not fail to produce "results" for a certain class of measurements (namely, app.roxlmate or inexact measurements). Rather, I would suggest that just as m buying a car, where we recognize that "almost-new" is not really new at all; so too we should recognize that "approximate-measurements" (if

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treated via (2), as above) are not really measurements at all. Still, a problem remains. For, regardless of whether it is a "measurement," the transformation (2) does not leave the apparatus with a definite pointer position, and one may easily find that circumstance unsettling. Think of SchrSdinger's cat. Indeed, it may be instructive to run this puzzle with the cat. In the usual formulation, the cat being dead or alive is made to indicate whether an atom in a radioactive pile has, respectively, decayed or not within an hour. The interaction between the atom-cat systems, while mediated by the cruel contraption involving the prussic acid, hammer and all, is nevertheless strict. It transfers the probability for an atomic decay (which is 50:50) onto the probability for the cat to be alive, which thus functions as a proper indicator variable for measuring the decay. According to my treatment of the interaction, after an hour the cat would be either alive or dead (with equal probability). Now if we loosen the link between the atom and the cat (e.g., by making the acid solution weak enough not to kill the cat, although very likely to do so) then this interaction may count as an approximate measurement of the decay. If the cat is found dead, then very likely the atom has decayed -- although there is a small chance of an error here. Still, before we look to see whether the cat is alive or not, we would like it definitely to be one or the other. In the case of a transformation like that above, however, the quantum description of the cat, before observation, has it in a state that is superposed over live-cat and dead-cat states° The rule of silence says that here we should not ask the very question we want answered. Suppose, however, that we could get over this difficulty for the special situation where initially the atomic pile is in an eigenstate (to decay in an hour, or not to decay in an hour). Then if we follow my proposal and treat the cat as interacting only with the decay aspect of the pile, we would solve the problem (i.e., leave the cat either alive or dead) not only in this special starting state but also in the general case of an object state initially superposed over these eigenstates. As I see it, the difficulty derives from two sources. First, if we treat the apparatus (here the cat) as initially in a specially tuned (or "neutral") eigenstate, we may already have given up the game before any interaction at all. For example, if the initial cat state is a superposition over live-cat and dead-cat states, then we are in trouble regardless of any measurements. Thus I think one should treat the initial cat (apparatus) state as a mixture over an appropriate set of determinate variables. This is a well-known strategy for representing macroscopic observables more generally, although on the usual conception it is not sufficient to get around the measurement problem. The second source of the problem with inexact measurements comes from the scheme that represents them by way of interactions, as in (2), that do not strictly correlate initial decay eigenstates with evolved cat variables relating to its health. Thus I propose to treat approximate measurements differently.

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I will treat inexact measurements as exact measurements of an inexact variable, like a sharp photograph of a cloud or bank of fog. This will allow for a selective application of the interaction formalism. Suppose we let each eigenstate {n correspond to a "state of health" of the cat; that is, to a particular assignment of definite values to all the relevant variables that enter into the determination of the cat's health (blood pressure, pulse rate, EKG pattern, etc.) Let Pn be the probability that the cat is alive in state {nMy picture is that "being alive" probabilistically supervenes in these other variables, and may not itself correspond to any proper quantum mechanical observable. (This is like the treatment of macroscopic variables in [9], only without the infinite direct sums.) We can suppose that the initial cat state is a mixture ~lcnl21~n] and we assume that at the start of the cruel experiment the cat is alive, so that ~1 Icnl2pn -~ 1. For definiteness, consider the state of the object (the radioactive pile) initially to be the eigenstate I~Dwhere it is certain to decay in an hour. We will suppose the measurement interaction to be represented by a unitary U satisfying, for each n, (q~D®~n) f

U > (~f®2f "'~D "~n )

(3)

•

where the evolved cat state ~n' which U correlates to the initial decay state qbD, is among the eigenstates {~m}- Then in the actual measurement we have U I~(~D]®Z[Cnl2P[~n] ) V[qbfD] ®Z [Cn[2P[~fn]. (4) The probability that the cat is alive at the end of an hour is 2 Icnl2ptn, where ptn is the probability for being alive in state ~ .

We can

suppose that Y~ ICnl2pfn~ 0. So, if there is an atomic decay, the cat will probably register that by its death -- although there may be some error. Thus the preceding scheme represents an approximate measurement of the decay. At the end of the interaction (but before we look) the cat is either alive or dead, probably dead. It depends on the ptn, which is to say, on the effect the decay has on the disposition of the life-determining variables.

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Moreover all these variables, on which the cat's life depends, are themselves definite. When we run the experiment from an initial radioactive state that is superposed over decay and not-decay states, my proposal is that we regard the interaction with the cat (plus contraption) as responding only to the decay feature of the pile. That means we replace the superposed state by the corresponding mixture and apply the interaction formalism to the mixture. Because of linearity, the result will be a mixture of states like that just calculated (the right side of (4)). These final states will have the same determinate character. In them the cat will be alive or dead, according to the state of the health variables, and these variables themselves will be definite. Moreover, these nice features do not depend on the accuracy of the measurement. Even an inaccurate measurement, treated as an exact determination of a probability, will produce definite results with respect to the state of the cat's health. Finally, if the cat should find itself in other interactions that are not measurements, one can try these same techniques to preserve its integrity as a determinate creature. It simply depends on being justified in using the interaction formalism selectively, and on being careful to represent initial cat states as the right sort of mixtures. As for the puzzling transformation (2) with which we began, we see that it is not necessary to suppose that it occurs in inexact measurements. Indeed, there is no reason to suppose that it occurs in nature at all.

3. EPR PROBLEMS Stairs thinks that the EPR situation (say, the measurement of the left-hand component on a pair of spatially separated electrons in the singlet state) tests the idea of a selective interaction. When we perform a measurement of the spin Sx in the x direction on the left-hand system (LHS), Stairs thinks that "intuitions about locality combine with further intuitions about selective interaction to suggest that the apparatus should interact only with the left-hand system." In discussing the idea of a selective interaction I have not explicitly considered interactions with composite systems. Stairs raises the issue of just how to proceed. Before doing so we need to answer a preliminary question. The interaction is supposed to be with the probabilistic aspect of the object system that corresponds to the observable being measured. The question is this. When we measure the x-spin on the left component of a composite system, what observable is being measured and on what system.'? Two possible answers suggest themselves: (i) The observable is Sx defined on the space of the LHS. (ii) The observable is (Sx ® I) defined on the tensor product space of the composite system. The first answer is not adequate, because there are two probabilistic aspects to spin on the LHS. There are the simple probabilities for the spin values. But there are also the correlations between those values and values of observables on the RHS. These correlations come from joint probabilities

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for pairs of observables each one defined on different factors of the product space. Thus the aspect we are after is an aspect of the composite system itself, and we must look to (ii). At this point, Stairs treatment for how to proceed is compelling. I would only insist that the issue is not actually resolved (as he suggests that it is) simply by moving to the algebra of operators that commute with the left-hand x-spin (or, as Stairs nicely points out, to the Ltiders' rule). For we have first to determine on what space these operators are defined: on the space of the LHS or on the whole tensor product. If we took the first option, then treating the spin aspect as involving all the observables commuting with Sx on the LHS would still not get at the correlations. On the other hand, once we have settled on (Sx ® I) in order to get the correlations, it is clear that we need to capture the joint distributions for any observable on the RHS. The most elegant way to do that is to regard the (Sx ® I) aspect as involving the subalgebra of observables on the product space that commute with (Sx ® I), as Stairs suggests. Then the density operator we want would be the canonical operator for representing the singlet state, if (Sx ® I) were a superselection operator. There is a less elegant, but transparent way of getting the same result. To find the mixture to represent the x-spin aspect, begin with the singlet state function, in the fight basis, and then take the mixture that carries the probability distribution, as illustrated in the Introduction. Using that same notation, we can represent the singlet state V as = 2an ~n

(5)

L R where each ~n = ~)n®~n , and where the components are the oppositevalued (i.e., I+, - ) and I-,+) ) eigenstates of the x-spin on the LHS and the RHS, respectively. Then the mixture that represents the aspect of the composite system that enters into the measurement has density operator 2 lanl21~n].

(6)

This is exactly the operator that Stairs arrives at for the proper starting state. He is fight. The moral is that the aspect involved in the measurement of one component of a composite system is an aspect that depends on the state of the whole composite system, and not just on the measured component. (For a discussion of when composites have states, see Sec. 5 below.) Stairs seems to think that this moral involves an unacceptable nonlocality. In the first instance, I think that is because Stairs runs together two different ideas: the idea of interacting with only an aspect or a feature of a system and that of interacting with a localized part. If we think of color or

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shape or crystal structure, for instance, there is no reason to regard an aspect or feature as localized at all. Indeed, aspects --like patterns-- are frequently global. So there is no reason to suppose, as Stairs suggests, that an aspect for a measurement on one component of a composite system only pertains to that component. In the second place, Stairs also suggests that, by implicating operators like I®Sx in the x-spin measurement on the LHS, one must understand that LHS measurement to involve some kind of interaction with the RHS. As he points out, however, in a different context I have denied any such cross-component interaction. I still do. In my view the object we interact with is a system with two spatially separated components, so a non-local object. The feature of that object that takes part in the interaction is global; it is an algebra (or family of probability distributions) that depends on both separated components. However, this is a functional or algebraic "dependence" and does not necessarily have to do with any physical interaction between the components. (Remember the difference between correlation and causality.) So my picture is that measuring the x-spin on the LHS constitutes a local interaction with a global feature of the whole composite system. What I mean to deny, in calling the interaction local, is that it involves any physical disturbance that bridges the two components. Thus I continue to resist the different picture that Stairs calls up when he says that measuring the LHS has a "very robust consequence" for the RHS. That language suggests causal connection, physical determination, or some similar influence from one component system to the other. I do not think anything of the sort goes on in EPR. Indeed, if we consider alternative space-time descriptions the "consequence" that Stairs refers to on the RHS can be made to disappear. For, suppose we redescribe the measurement on the LHS in a new frame F' moving with a very tiny velocity with respect to the frame F of Stairs' "robust consequence." And suppose the measurement R on the RHS is spacelike separated from the measurement L on the LHS, where R comes after L in the F frame. As judged in F, the probability at R for x-spin up will be either 1 or 0, depending on the LHS outcome. This is the consequence Stairs points to. But, as judged in F' the probability at R for x-spin up may be the same (whichever of 0 or 1 corresponds to the LHS 1 depending on whether L, as judged in F', comes outcome) or it may be 2' (respectively) before or after R. Thus the consequence depends on the reference frame, and is not robust at all. Of course Stairs knows this problem well. I take it that his challenge, then, is for me to resolve the apparent and frame-relative nonlocality. Frankly, I do not see why I should take up the challenge here. There are special and serious nonlocality problems that arise in some attempts to deal with the measurement problem. For example, Gisin has shown that nonlinear collapse theories permit superluminal signalling [10]. Stairs' concerns are not like this. His concerns are addressed by getting free of the "intuition" that aspects should not be global, and distinguishing

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between a non-local interaction and a local interaction with a global feature. Once we do so, my account of measurement does not raise questions regarding locality other than those already present in standard quantum theory. Elsewhere I have emphasized that quantum entanglement presents us with correlations between the outcomes of random experiments and suggested the idea of "random choices in harmony" to help train one's intuitions away from the idea that these correlations involve some mysterious sort of nonlocal influence [10, 12, 13]. But understanding entanglement is not the problem addressed in the papers on measurement theory. Instead, I have tried to deal with different problems differently. In not treating all problems at once my procedure is nonholistic, which brings me to the next topic. 4. Q U A N T U M M E T A P H Y S I C S Stairs has some other problems on his agenda. He is concerned that when, as above, we take the measurement of local observables to involve an aspect of the whole compound system we embrace a holism or nonseparability that I have elsewhere been reluctant to acknowledge. Of course I have not been reluctant to acknowledge that we need to use the non-factorizable composite state function to get the right joint probabilities or that in the singlet state the individual components do not have their own state functions. This constitutes what Stairs refers to, respectively, as holism (i.e., non-particularism) and non-separability. If that is all there is to it, then I have always been a holist and non-separatist. I suspect, however, that there is more. I suspect that in using these particular titles one is supposed to be indicating allegiance to a quite general and dramatically non-classical worldview, or the like, and gaining something from this association. I have no such allegiance and I do not acknowledge (because it is not true) that conferring metaphysical titles on features relating to non-factorizable quantum state functions helps us understand the phenomena involved in EPR. Indeed, I have argued strenuously that this kind of word-mongering contributes to a metaphysical sideshow that takes us away from understanding things. Stairs illustrates the danger when he wonders whether we can recover the initial state by somehow adding up its aspects. Here he is clearly embarked on exploring the suitability of a parts-and-whole ontology, an enterprise into which he may have been tempted by my references to aspects as "parts" (see Note 1). In fact, Stairs sees quite clearly what the relation is of an aspect to the whole state. As he says, the aspect represents a probability distribution over a restricted set of events, one of many distributions that the whole state function prescribes. But Stairs makes this clarification in the context of a different metaphysical picture. This time it is probability that needs a metaphysical back-up. Apparently probability distributions are not enough, Stairs thinks they need (somehow) to be grounded by "propensities." Frankly, I am not sure that

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much is gained by this except to enshrine a realist vocabulary. Moreover, Stairs' suggested construal of propensities is not exactly in accord with my own understanding of the quantum probabilities. I always took the Born rule literally. When it says that the quantum probabilities are probabilities for measurement outcomes, I took that to refer to the outcome a typical measurement interaction has on the measuring apparatus. This is not the same as a "tendency" to realize or display some latent property of the object, as Stairs suggests. So, my own picture, which I regard as pretty standard quantum theory, is a bit more pragmatic and less realist than Stairs'. This section may be somewhat of a digression. For the issues here do not concern my measurement scheme so much as a philosophically "correct" language in which to embed it. Stairs does have a final and serious worry, however, to which I now turn.

5. MEASUREMENT AND PROJECTION In order to track the change undergone by the joint object-apparatus (o + a) system, Stairs raises a series of questions concerning what transition occurs during a measurement. The joint system seems to move from an initial pure state to a final mixture, and Stairs associates any such motion with a projection. I have some trouble with this line of inquiry. Recall the simplified situation sketched in the Introduction, where the object o has a pure initial state ~t = 3".an qbn (with (~n eigenstates of the observable E to be measured) and the measuring apparatus a starts out in a pure state ~ (where the "pointer positions" are eigenvalues belonging to the eigenstates {n). Where U is the unitary operator corresponding to the measurement interaction, we assume that

f®

U((~n®~)= On ~n. The measurement interaction, then, is given by eq. (1): lanl21~On]®P[~] ~

U

~ lanlZP[~fn®~]

and the final object-apparatus state is a mixture over pointer positions. Stairs notes that in the above interaction the starting state for the joint (o + a) system is the mixture on the left side of the arrow. He asks whether this is "really" the state of (o + a) as the interaction begins. On my behalf he answers "no," for otherwise he thinks that I will simply have posited a projection at the beginning of the measurement. Implicit in this line of

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inquiry is Stairs' assumption that there "really is" a state of (o + a) just prior to measurement; namely, the pure state ~®~. This assumption is part of the usual application of the product formalism for non-interacting systems. Applying this assumption, we would say that the joint system begins in a pure state and winds up in a mixture. But then, as Stairs remarks, under this interpretation the measurement itself amounts to a kind of projection of the joint (o + a) system. My reservations concern the "really is" assumption embodied in the usual deployment of the interaction formalism. Just because there is a state for each of two independent systems (prior to any interaction between them), why must the "composite" of these systems be thought of as an additional physical system with its own, state? It was yon Neumann, I believe, who first introduced the formalism for composite systems. In his classic discussion [14, Chap. VI, Sec. 2] he makes it plain that the state description for a composite system is defined by the requirement that it yield all the probabilistic information pertaining to the component systems. (Actually von Neumann talks of expectation values rather then probability, but it amounts to the same thing.) In the case of non-interacting systems nothing is gained by employing the composite state formalization, for there are no probability distributions derivable from the composite state ~t®~ that are not already available from the states of the individual systems, plus the assumption of their (stochastic) independence. Recall that my whole effort has been to get at the measurement problem by questioning the way the interaction formalism is used, suggesting a different and more contextdependent usage. Thus I do not think that we need to regard the noninteracting object and apparatus systems as forming some new "joint system" with its own specially assigned "state." On this understanding, the "joint system" does not undergo a projection, or any other kind of transition, from an initial pure state ~®~ to any other. What we do need is to provide a way of determining what happens to systems as the result of an interaction. That, I take it, is what the interaction formalism is actually designed to do. So, instead of thinking in terms of transitions on composite systems, in dealing with a measurement interaction, I have been thinking of the effects of the object on the apparatus. What I suggest is that if o has state Wo and a has state Wa at the start of an E selective interaction, then the actual effect of the interaction is obtained by starting the interaction in the composite state Wo(E)®Wa, where Wo(E) is the density operator that represents the E aspect of a. Accordingly, it is U(Wo(E)®Wa)U -t that shows the effect of the interaction on a. For example, as a result of the transition represented by (1), the apparatus winds up in some state ~n. Thus the apparatus goes from pure state ~ to pure state ~n, with probability tan12. Since generally I(~nl~)l2 ~ lan[2, this probabilistic transition is not a projection (in the usual sense).

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Using the interaction formalism in this fashion represents a hypothesis; namely, that the way the object system participates in the interaction is adequately represented only by considering its E-aspect alone. It is a consequence of this hypothesis that in tracking the effect of the object on the apparatus, the interaction formalism also keeps track of what happens to the object in the course of the measurement. Thus, the way the object shows up in the final state U(Wo(E)®Wa)U -1 represents how the interaction has affected it. In the transition represented by (1), the object goes from the initial pure state V = Y.anqbn to some final

state

Ofn, with

probability lan12. If ~fn = ~n, then this transition is a projection, otherwise not. It depends on the details of the actual interaction. (This account of the object transitions raises the question of how to treat, say, a Stern-Gerlach experiment with the beams recombined, or a delayed choice experiment. These, and other "paradoxes" are discussed in [3]. A similar difficulty for the apparatus transitions may be raised by the possibility of exhibiting macroscopic quantum coherence in an rf SQUID [15]. I believe this possibility can be treated similarly.) What then of the (o+a) system? As emphasized above, it does not have its own initial "state." But it does wind up in the mixture represented by U(Wo(E)®Wa)U-t; e.g., the right side of the arrow in eq. (1). Indeed, the whole idea of a selective interaction is to find a "substate" for the subalgebra that represents the E aspect and to let the interaction formalism tell us what happens in a selective interaction by applying it to the "things" that are actually interacting; namely, the E aspect of o and the apparatus. Accordingly, the end product of this interaction is the state that applies to the joint (o+a) system at the conclusion of the measurement. In interactions like (1), the object starts out in pure state V and winds up in qb~, with probability lan12. Similarly, the apparatus starts out in pure state ~ and winds up in pure state ~n, with probability lan12. (Moreover, these final states are linked.) Thus, whether or not we call it a "projection," the effect of the measurement on the object and apparatus systems is a probabilistic transition. This transition is a consequence of two separate conditions: (a) that the interaction between object and apparatus is selective for the observable being measured and (b) that the evolution operator U governing the interaction transfers probabilities from the object observable to the pointer positions. Taken together, these two conditions imply the above probabilistic transitions (although neither one does if taken alone). Without laying too much stress on the concept of "explanation," I do think this account of the object/apparatus transitions has explanatory value. That is, I do think this treatment of measurements helps us understand why measurements have results.

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As every investigator knows, however, demands for explanation tend to outstrip whatever level of understanding one may have achieved. So here one may wish to go further and ask whether there is an algorithm, or a deeper physical analysis, that will tell us in general what aspects of an object are interacted with by what interactions. That is, having shown that we can understand measurement results as the dynamic effects of certain selective interactions, we could go on to ask what makes an interaction selective. I have been adopting a pragmatic attitude to this question, suggesting that we learn which interactions are selective by seeing their results, and that the major task is to provide a reasonably general and precise way to codify selectivity by incorporating it into the quantum theory. (Although the work here is certainly on a totally different plane, still my inspiration for what constitutes acceptable theory construction is drawn from Newton's attitude toward the "causes" of gravity and from Einstein's on the velocity of light as a limiting velocity -- more generally, on the need he saw for "principle" theories.) Perhaps there are deeper and more informative answers to the question of why the results of a particular interaction depend only on interacting with a subalgebra of the whole system, or how that selectivity comes about. Alternatively, we may come to see this fact itself as not further reducible. We could come to regard selectivity as a starting point for scientific understanding, a basic and primitive concept on which investigations in the quantum domain turn. It is too soon to tell. I would only urge that in evaluating a proposal we not insist that it be made to provide answers, or even a heuristic, for every (apparently) sensible question. No proposal in the history of science would pass such a test. Stairs concludes his critique with a question about Geiger counters. He pictures the way the counter works like this: a decay occurs and it then induces an interaction with the Geiger counter to produce a click. Well, maybe; although that is not what quantum mechanics says. Quantum mechanics is less committal. Bracketing the measurement problem, quantum mechanics just says that the "click" emerges from an interaction with the radioactive substance. It does not say that the decay comes first, occurring independently of the click. My treatment of the cat example, above, is a formal sketch that corresponds to this non-committal account, a n d it is a sketch where we do not have to "bracket" the measurement problem. But suppose Stairs is right. Suppose there are spontaneous reductions of the wave packet, independent of any interactions at all. In that case quantum mechanics is wrong. The unitary dynamics of the quantum theory, then, will need to be replaced by some on-going stochastic process and the rule of silence will also have to be broken. (Concerning the latter see, e.g., the discussion of "tails" in [16].) No compelling account of this sort now exists. Neither do we know of phenomena which require it, at least not if my proposals about selective interactions and the measurement problem are sound.

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In any case, I have pursued my investigation of the measurement problem under two strong constraints: tolerance for the established interpretive practices regarding superpositions, and the assumption that the quantum dynamics are correct. This is not to say that I frown upon more visionary projects. Certainly not, for we can all use some vision. Let me close by thanking Allen Stairs, again, for his constructive critique. I hope that others will profit from our exchange as much as I have from considering the points he raises. I would also thank the Editor for facilitating this discussion. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

A. Stairs, Found. Phys. Lett. 5 (2), (1992), preceding paper. A. Fine, in R. Kargon and P. Achinstein, eds., Kelvin's Baltimore Lectures and Modern Theoretical Physics (M.I.T. Press, Cambridge, 1987), p. 491. A. Fine, in S. French, ed., Festschrift for Heinz Post (Kluwer, Dordrecht, forthcoming). E.P. Wigner, Am. J. Phys. 31 6 (1963). A. Fine, Phys. Rev. D2 2783 (1970). A. Shimony, Phys. Rev. D9 2321 (1974). D. Albert and B. Loewer, in A. Fine et al., eds., PSA 1990, Vol.1 (Philosophy of Science Association, East Lansing, 1990), p. 277. M.H. Fehrs and A. Shimony, Phys. Rev. D9 2317 (1974). S. Machida and M. Namiki, Prog. Theor. Phys. 63 1457, 1833 (1980). N. Gisin, Helv. Phys. Acta 62 363 (1989). A. Fine, in P. Asquith and R. Giere, eds., PSA 1980, Vol.2 (Philosophy of Science Association, East Lansing, 1981), p. 535. A. Fine, J. Phil. 79 733 (1982). A. Fine, in J. Cushing and E. McMullin, eds., Philosophical Consequences of Quantum Theory (University of Notre Dame Press, Notre Dame, 1989), p. 175. J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955). A. Leggett, in B. J. Hiley and D. Peat, eds., Quantum Implications (Routledge & Kegan Paul, London, 1987). G. Ghirardi and P. Pearle, in A. Fine et al., eds., PSA 1990, Vol.2 (Philosophy of Science Association, East Lansing, 1991), p. 19.

NOTE . In carlier expositions I have also used the word "part," and spokcn of interacting with only part of a system. Stairs' critique has made me aware that the languages of "parts" lends itself to compositional (i.e.,

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part-whole) metaphors that can lead one astray. Thus I shall try to avoid that language here.