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Finding your marbles in wavefunction collapse theories
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Studies in History and Philosophy of Modern Physics 34 (2003) 607–620
Finding your marbles in wavefunction collapse theories Daniel Parker Department of Philosophy, University of Maryland, College Park, MD 20742, USA
Abstract Lewis (Br. J. Philos. Sci. 48 (1997) 313) has recently presented an argument claiming that, under the Ghirardi–Rimini–Weber (GRW) theory of quantum mechanics, arithmetic does not apply to ordinary macroscopic objects such as marbles (known as the Counting Anomaly). In this paper, I disentangle two different lines of Lewis’s argument, one devoted to what I call the standard GRW interpretation and the other to the mass density interpretation (MDI). I present both strains of Lewis’s argument, and move on to criticise Lewis’s position, focusing on his argument with respect to MDI. I analyse the structure of his argument, and follow this with a novel refutation of Lewis’s argument, drawing on the original presentation of MDI as developed by Ghirardi et al. (Found. Phys. 25 (1995) 5). I briefly consider the debate that ensued between Bassi and Ghirardi and Clifton and Monton and interpret it within the context of my analysis. I conclude that Lewis’s Counting Anomaly fails to generate a genuine problem. r 2003 Elsevier Ltd. All rights reserved. Keywords: Counting Anomaly; Ghirardi–Rimini–Weber (GRW); Wave packet reduction; Mass density interpretation; Bassi; Ghirardi; Clifton; Monton
1. Introduction In 1997, Peter Lewis published an essay identifying a potential problem with the Ghirardi–Rimini–Weber (GRW) theory of wave packet reduction in quantum mechanics. Lewis contended that a consequence of the GRW theory is that arithmetic fails for ordinary macroscopic objects. He argued that the GRW theory entails that if each of n non-interacting marbles is in a box, we are not justified in claiming that all the marbles are in the box, for large values of n. Following Clifton E-mail address: [email protected] (D. Parker). 1355-2198/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S1355-2198(03)00065-0
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and Monton (1999), we shall call this the ‘‘Counting Anomaly.’’ Ghirardi and Bassi (1999) later countered Lewis’s arguments, claiming that Lewis had misunderstood the GRW theory and that the Counting Anomaly could never occur. Here I argue that Ghirardi and Bassi’s conclusions are basically correct, although their arguments are misleading. Ghirardi and Bassi’s poor reasoning led Clifton and Monton (1999) to develop a different counterargument to Lewis, claiming that his conclusions were correct but that they could never be empirically confirmed. Clifton and Monton pointed out the numerous flaws in Ghirardi and Bassi’s paper, which elicited a rather sharp retort from Bassi and Ghirardi and led them to reformulate their objections to Lewis (Bassi & Ghirardi, 1999) based on the proposed mass density interpretation (hereafter MDI) as formulated by Ghirardi, Grassi, and Benatti (1995). Clifton and Monton (2000) have defended their position against Bassi and Ghirardi’s attack and reconstructed their arguments to account for MDI. More recently, Bassi and Ghirardi (2001) have clarified their claims in response to Clifton and Monton’s arguments against MDI. The current state of this debate leaves Lewis’s original arguments far behind. Indeed, the present positions of Ghirardi and Bassi, and Clifton and Monton, are hardly faithful to Lewis. While they still argue Lewis’s Counting Anomaly, the chronology and structure of the debate have led some issues, which I argue are peripheral and irrelevant, to take on lives of their own. Therefore, I propose a ‘‘Back to Lewis’’ movement. In the following pages, I will point out a flaw in Lewis’s position that has hitherto been unnoticed, or perhaps only hinted at. To this end, I will discuss how the debate has missed the crucial element of Lewis’s presentation of the Counting Anomaly in MDI by focusing on a separate issue, namely what it means for the mass of an object (microscopic or macroscopic) to be ‘‘accessible,’’ or ‘‘objectively localised.’’ While the precise meaning of ‘‘accessibility’’ in MDI and its consequences may be interesting (and problematic) in its own right, this paper argues that the original formalism of MDI (Ghirardi et al., 1995) is sufficient to refute Lewis’s Counting Anomaly. It is the central claim of this paper that Lewis’s argument fails, though MDI may prove untenable on other grounds. Specifically, MDI may still retain the ‘‘tails problem’’ of the original GRW theory, though it provides an interpretation that claims to sweep it under the rug. The problems of developing a satisfactory realist interpretation of MDI are well beyond the intended scope of this essay. Instead, I argue that the debate over the significance and appropriateness of the MDI criterion for the ‘‘objective localisability’’ or ‘‘accessibility’’ of mass, discussed at great length by Bassi and Ghirardi, and Clifton and Monton, should be divorced from talk of the Counting Anomaly. This paper will proceed as follows. In Section 2, I briefly summarise the GRW theory and MDI, which develops a criterion for determining when a mass distribution may be treated as a classical object and a second criterion to compare two mass distributions. Section 3 presents an outline of Lewis’s paper, followed in Section 4 by a critical evaluation of Lewis’s arithmetical conundrum. Finally, in Section 5, I trace the history of the debate that ensued, speculating as to the reasons why it has reached the unnecessary level of complexity that it has, and I reply to the
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arguments raised by Bassi and Ghirardi, and by Clifton and Monton, that are relevant to the analysis.
2. The GRW theory and MDI 2.1. The GRW theory According to the standard GRW dynamics, any given particle follows the usual . Schrodinger dynamics most of the time.1 On rare occasions, however, a particle suffers a ‘‘hit,’’ where its wavefunction, C; is multiplied by a normalised Gaussian centred on a position selected with probability density jCj2 by the Born probability rule. The width of the Gaussian and the frequency of a ‘‘hit’’ for a given particle are based on dimensional analysis and assigned approximate values of 105 cm and 107 s1, respectively. The width of the localisation distribution is chosen so that the particle is well localised without creating an unphysically large conjugate momentum distribution. More generally, the choice of frequency parameter is justified by . allowing single particles to follow Schrodinger evolution virtually all the time, while large systems (i.e., macroscopic objects) undergo frequent spontaneous collapses so as to conform to our everyday experiences (Ghirardi, Rimini, & Weber, 1986). A much-discussed aspect of this theory, the one that Lewis exploits, is the ‘‘tails’’ problem. Due to the Gaussian nature of the ‘‘hitting’’ function, the resultant wavefunction does not have compact support, and it cannot be localised in a bounded region. Alternatively phrased, the wavefunction has diminishing but nonzero tails going off to infinity in every direction: even after a collapse, a particle’s wavefunction is at once non-zero everywhere and has a non-zero probability of undergoing a further collapse anywhere. However, this seemingly undesirable trait is both unavoidable (Albert & Loewer, 1996) and a necessary part of the theory (Ghirardi, Grassi, & Pearle, 1990). Another way to approach the tails problem is to see that the GRW theory must deny the eigenvalue–eigenstate link. In the canonical formulation of quantum mechanics, mutually exclusive states of affairs are formalised through the use of orthogonal vectors in Hilbert space. Upon measurement, a vector representing a superposition of two orthogonal states instantaneously collapses on to one of these states and yields the value associated with that state. However, this does not happen in the GRW theory; a particle is never exactly in any particular localised region. Since the collapse state cannot, even in principle, be perfectly localised, two particles in different locations always have overlapping wavefunctions and therefore are only approximately orthogonal. Both the standard GRW interpretation and MDI share these underlying dynamics. In what I call ‘‘the standard GRW interpretation,’’ two states, jCS and jFS; are said to be orthogonal if /CjFSod; where d is some small parameter (Lewis, 1997; 1
This basic description is also appropriate to more sophisticated spontaneous collapse theories like the CSL model of Pearle (1989).
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Ghirardi et al., 1995). The distinctive feature of the standard GRW interpretation is the use of the inner product rule for assigning properties to the physical systems, to be contrasted with the rules associated with MDI, as described below.2 2.2. The MDI Ghirardi et al. (1995) introduced MDI, which utilises two devices for ascribing properties to states. The first, the R measure, identifies a criterion for determining when a mass distribution may be treated as an objectively localised classical object. That is, it replaces the notion of a position eigenstate as a means of determining when a mass may be taken as having a definite position. The second measure, the D function, replaces the inner product as a way of comparing two mass distributions. Each of these will be developed below. Ghirardi et al. defined a mass density operator for the mass density in a localised cell j X Mj ¼ mk NjðkÞ ; ð1Þ k
where mk is the mass of a particle of type k and NjðkÞ is the number density of such particles in a given cell j: They went on to define the average mass density, Mj ; in a cell in a natural fashion Mj ¼ /CjMj jCS
ð2Þ
and the variance Vj ¼ /Cj½Mj /CjMj jCS2 jCS: R2j
ð3Þ
Vj =M2j ;
¼ is much smaller than one, the mass in a In the case where the ratio, cell j is said to be ‘‘objectively localised’’ or ‘‘accessible’’ (Ghirardi et al., 1995, p. 18). Let us pause here to evaluate the implications of this accessibility criterion. Ghirardi et al. (1995) noted that if a mass distribution is significantly superposed between two or more cells, one cannot discuss it in a classical manner; i.e., it is not ‘‘objectively localised’’ or ‘‘accessible.’’ Thus, Ghirardi et al. required some method to distinguish between such cases. To do this, Ghirardi et al. chose the R measure as a means of demarcating between accessible and inaccessible mass distributions. When the R measure obtains, it implies that the mass in a cell j behaves as if the mass in the cell was the total mass of the object, since the mass density of the tails is negligible and experimentally undetectable. To demonstrate the appropriateness of the R criterion, Bassi and Ghirardi (2001) considered a classical analogue: a table. If a hypothetical table were to lose a single nucleon it would still make sense to refer to it as the same table, since the masses of the original table and the table that had lost a nucleon would be observationally indistinguishable. Similarly, according to Bassi and Ghirardi, one can safely ignore the contribution of the tails of an objectively 2 The standard GRW interpretation includes formulations equivalent to the inner product rule, including the fuzzy link (PosR) rule described by Albert and Loewer (1996) and used by Clifton and Monton (1999).
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localised mass distribution since it would not contribute significantly to the total mass of the distribution. To further motivate this criterion, let us look briefly at the following example, presented by Ghirardi et al. (1995). Consider the two states pffiffiffi B " A B jC" S ¼ 1= 2ðjCA ð4Þ N S þ jCN SÞ and jC SXjFN=2 S#jFN=2 S where jC# S refers to N masses superposed between the regions A and B; while jC# S is the state describing the composite system where half the mass is in region A and half in region B: The two states have the same mass expectation value under (2), but only the second state is to be considered objectively localised under the R measure (the reader is referred to the original paper for the details of the example). To introduce the second measure, the ‘‘mass distance’’ or D function, Ghirardi et al. (1995) noted that the Hilbert space formalism is inadequate to compare the similarity of two macroscopic mass distributions. As an example, they considered three macroscopic states, jCA S; jCB S and jCA0 S; where jCA S is a state with centre of mass localised at point A, jCB S; a state with centre of mass at a point B far away fron A, and jCA0 S; a state with centre of mass at A but differing from jCA S by the position of one microscopic particle and, therefore, nearly orthogonal to jCA S: Clearly, according to Ghirardi et al., jCA S and jCB S must be characterised as macroscopically different states, but jCA S and jCA0 S should be characterised as identical. However, in the Hilbert space topology, both jCB S and jCA0 S are nearly orthogonal to jCA S; and, therefore, the pairs are treated as equally dissimilar. To distinguish such cases, they proposed a function to determine the macroscopic equivalence of two mass distributions qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X DðjcS; jfSÞ ¼ ðMj Nj Þ2 ; ð5Þ j where Mj and Nj are the average mass densities of cell j for the two states jcS and jfS; respectively. Ghirardi et al. defined two states as macroscopically equivalent if DðjcS; jfSÞpe
ð6Þ
where e is taken to be 1018 m0 and m0 is the mass of a nucleon. However, not any two states can be reliably assessed by the D function as a measure of macroscopic equivalence. Two states can share the same value of M for all j and yet not be equivalent, as the states described in (4) demonstrate. Without some means of restricting the arguments of the ‘‘mass distance’’ function D; it is be easy to gerrymander states that the function would classify as equivalent when in fact they were not. Hence, the domain of the D function is limited to the ‘‘proper subset AðSÞ of UðSÞ of those states which are allowed by the [GRW] dynamics’’ (Ghirardi et al., 1995, p. 26). UðSÞ referred to the unit sphere of the Hilbert space associated with the system of interest S: The content of the subset AðSÞ was left rather vague in Ghirardi et al., as the authors readily admitted. However, they indicated that the properties of the states in AðSÞ could be made sufficiently precise if necessary and that they certainly wished to exclude the comparison of states like those of (4), which the D function would otherwise classify as equivalent.
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3. Lewis’s Counting Anomaly 3.1. The anomaly under the standard GRW interpretation Lewis’s argument looked at both the standard GRW interpretation and the MDI, and he drew the same conclusion in each case, viz., that arithmetic fails. What Ghirardi and Bassi, and Clifton and Monton failed to do is distinguish between the two distinct lines of reasoning in Lewis’s paper. Consequently, much of the debate between the two groups hinges on their inability to consider each of Lewis’s arguments in turn. So let us try. Lewis (1997) considered a marble, which may be inside or outside a box. According to the GRW theory, if the marble is inside the box, then its wavefunction is jcS ¼ ajinS þ bjoutS 2
2
ð7Þ 2
2
and 1 > jaj bjbj > 0 with the further condition that jaj þ jbj ¼ 1: This is the best one can do in the GRW theory. There is never a situation where the marble is wholly inside or outside the box; that is, the marble is never in one of the eigenstates jinS or joutS: However, for a sufficiently close to 1, one is permitted to infer that the marble is indeed within the confines of the box (Ghirardi et al., 1995). Lewis accepted this stipulation of the GRW theory, and considered the case of n marbles in a box. For n non-interacting marbles in a box (achieved by making the box extremely large), the vector in the composite Hilbert space is jcSall ¼ ðajinS þ bjoutSÞ1 #ðajinS þ bjoutSÞ2 #?#ðajinS þ bjoutSÞn
ð8Þ
and the pure state where all the marbles are in the box is jinSall ¼ jinS1 #jinS2 #?#jinSn :
ð9Þ
Taking the inner product between the two states, the probability of finding all the marbles in the box is j/CjinSj2all ¼ jaj2n :
ð10Þ
Since jaj2 is smaller than 1, for large n the inner product between the states jcSall and jinSall becomes arbitrarily small. Hence, even though each marble is individually in the box, all the marbles are not in the box. So arithmetic fails. 3.2. The anomaly under the MDI It was Lewis’s contention that some unacceptable consequence such as the one described above must arise for any proposed interpretation of the GRW theory. He claimed that the argumenty does not presuppose any particular kind of rule for determining, given the physical state of the system, how many marbles are in the boxy Whatever rule we might consider must tell us something about how many marbles are in the box when the state of the system is given by [(8)]y [I]f the argument is a
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good one, then any rule one might propose for the interpretation of the GRW theory is doomed to failure (1997, p. 319). As a concrete example, Lewis looked at MDI and recovered the same anomaly, but the argument took on a slightly different form. He first considered whether the mass density for an n marble system is accessible inside and outside the box. For such a system with many discrete cells, divided into regions A (marble sized and in the box) and B (marble sized and outside the box), the relevant mass density parameters for the state (8) are MA ¼ njaj2 m
VA ¼ n2 jaj2 jbj2 m2
R2A ¼ jbj2 =jaj2 51;
MB ¼ njbj2 m
VB ¼ n2 jaj2 jbj2 m2
R2B ¼ jaj2 =jbj2 b1;
ð11Þ
where m is the mass of a marble (Lewis, 1997). This shows that the mass of the n marbles is accessible or objectively localised inside the box and not so outside the box, according to the proposed definition of accessibility. Therefore, the n marbles can be understood to be inside the box, as one would hope. Thus, Lewis conceded that according to the R measure, all the marbles are claimed to be in the box. However, he argued that this is unacceptable, for (10) implies that the chance of finding all n marbles in the box is very small (Lewis, 1997, p. 320). Thus, according to Lewis, MDI fails to prevent the anomaly. Furthermore, Lewis found a second problem with MDI’s claim that all the marbles can be understood to be in the box. In this second case, he considered the ‘‘mass distance’’ function, D (Lewis, 1997, p. 328). For a one marble system in a box, the ‘‘distance’’ between its wavefunction and the eigenstate jinS is ffi pffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X X 2 2 2 2 DðjinS; jcSÞ ¼ ðm jaj mÞ þ ðjbj mÞ ð12Þ ¼ 2K jbj2 m; jAA jAB where K is the number of cells in a marble-sized region of space. For a small enough choice of b; one may claim that the ‘‘distance’’ between the two states satisfies condition (6). However, for an n marble system, the distance is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Xn X X 2 2 2 2 DðjinSall ; jcSall Þ ¼ ðm jaj mÞ þ ðjbj mÞ i¼1 jAAi jABi pffiffiffiffiffiffiffiffiffi ¼ 2nK jbj2 m: ð13Þ Lewis now could assert that, no matter how small b is chosen, one can increase the number of marbles n to a large enough value to ensure that DðjinSall0 jcSall Þ > e: His conclusion in this case was Examining the single-marble state [(7)] according to this rule, we find that for values of a and b typically produced by the GRW dynamics, [(7)] is indeed macroscopically equivalent to the eigenstate jinS of a marble being in the boxy However, if we examine the n-marble state [(8)] in the same way, we find that as n becomes large, [(8)] is not macroscopically equivalent to the eigenstate jinSall of all n marbles being in the box. But this cannot be the case, since the same rule tells us that each marble taken individually is in the box, and this is precisely what it means for all n marbles to be in the box (Lewis, 1997, p. 320; emphasis added).
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The important point here is that the Counting Anomaly (or some other unacceptable consequence) is manifest in two different forms under MDI: 1. The R measure’s claim that all the marbles are in the box contradicts the GRW theory, which, by (10), shows that ‘‘the chance of finding all n marbles in the box on observation is very small’’ (1997, p. 320). 2. While the R measure claims that all the marbles are in the box, the D measure states otherwise, and they are thus inconsistent.
4. The Counting Anomaly in the MDI The strategy of this section will be to refute Lewis’s Counting Anomaly by establishing the contrapositive to Lewis’s condition that ‘‘if the argument is a good one, then any rule one might propose for the interpretation of the GRW theory is doomed to failure,’’ and showing that some rule is not so doomed. Specifically, I show that neither of Lewis’s arguments with respect to MDI hold upon closer scrutiny so that MDI is not doomed to failure, at least not for the reasons Lewis gives. As a consequence, I conclude that Lewis’s argument is not a good one. Lewis’s first argument, that MDI claims that (8) is a state where all the marbles are in the box while (10) shows otherwise, falls flat. MDI rejects the use of the inner product rule, used in (10), to ascribe properties to states. Alternatively stated, the argument of Section 3.1 does presuppose a rule for determining how many marbles are in the box, namely the inner product rule. But the R and D measures are intended to replace the inner product, not complement it. This point was clearly made by Ghirardi et al. (1995) and echoed with particular reference to Lewis in Bassi and Ghirardi’s papers (see Ghirardi & Bassi, 1999, p. 53; Bassi & Ghirardi, 1999, p. 722). The inconsistency between the properties assigned by the standard GRW interpretation and MDI is to be expected and is entirely unproblematic: they are two different interpretational systems. Lewis’s second argument against MDI, that the R and D measures are inconsistent, is somewhat more complicated. To refute this line of reasoning, it will suffice to look at the D function for a single marble in a box, jCS; though the following argument obviously holds for the n marble system in a box. When introducing MDI, Ghirardi et al. asserted that ‘‘if one does not restrict the set of all possible states of the Hilbert space of ‘our universe,’ one unavoidably meets situations which cannot be consistently described in terms of the function M’’ (1995 p. 14). As we shall see presently, this condition was ignored in Lewis’s analysis. The GRW theory, as well as its extension to MDI, does restrict the possible states of the Hilbert space in ‘‘our universe.’’ An enumeration of the possible and impossible states in ‘‘our universe’’ is not necessary, for the relevant set of such states is already at hand. We have already noted that the state jcS of (7) is permitted by the theory while the state jinS is not. If this were not the case, there would be no Counting Anomaly. So it should come as no surprise (to Ghirardi or anyone else) that Lewis
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found an inconsistency in the GRW theory when he employed states that do not exist in ‘‘our universe!’’. Let us flesh this out a bit. When Lewis utilised the D function to compare the mass distributions of the states jinSall and jcSall ; he used one state that is not realisable under the GRW theory and one that is. Implicit in the definition of the D function, as discussed at the end of Section 2, is the stipulation that the mass distributions being compared are possible! Recall that the domain of the D measure was restricted to a particular subset of the states realisable by the GRW dynamics. It is therefore an illegal move to calculate the D function with one argument being jCS and the other as a state corresponding to a mass m inside the box (jinS), since such a mass distribution cannot be realised under the GRW dynamics. If one wanted to put a mass m inside the box, this could only be accomplished by placing a marble of mass m=jaj2 in the box. But such a mass distribution would have tails outside the box, and the D measure would fail to characterise the mass densities as identical, as one would expect. One can now see why Ghirardi and Bassi’s original refutation of Lewis’s argument failed. When they argued, ‘‘Thus the difficulty which arises, according to Lewis, in connection with a state like [___] does not present itself for the simple reason that such a state never occurs, its existence being forbidden by the GRW theory itself’’ (1999 p. 52). Ghirardi and Bassi filled the blank with a state equivalent to (8), rather than one such as (9). Furthermore, Ghirardi et al. (1995, pp. 27–28) exacerbated the problem by directly comparing, in their earlier paper, the two states jcS and jinS exactly as Lewis did. Their motivation here, however, was to point out that the D function succeeds, in the same way as the inner product /cjinS does, in describing an objective M state as ‘‘almost an eigenstate’’ of the mass being entirely within the box, not to authorise the use of non-existent mass distributions as arguments for the D function. It is unfortunate that such a move tacitly endorses the consideration of such states. It is important to emphasise that nothing has been presented here that moves beyond the original presentation of MDI developed by Ghirardi et al. (1995). The Counting Anomaly can be refuted entirely on MDI’s own terms. As a semantic rule, the R criterion provides a framework for mapping the properties of MDI states onto meaningful classical properties such as ‘‘in a box’’; that is, it determines when states can be discussed in a classical manner. The D measure allows MDI states to be compared directly with one another. The motivations for each of these criteria have already been discussed. When Ghirardi et al. used terms such as ‘‘macroscopically equivalent,’’ one should understand that this equivalence can hold only between potentially realisable MDI states, since no other type of state can arise given the dynamics of the theory. Now, one can debate whether MDI succeeds in developing a correspondence between its states and macroscopic properties, but this is a separate matter from describing or comparing the properties of the MDI states themselves. Lewis’s mistake was to conflate how the MDI criteria are used. He tried to derive a conclusion, namely that the property ‘‘all the marbles are in the box’’ does not obtain, from the D measure, whose use is restricted to comparing potentially
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realisable MDI states. Ghirardi et al.’s disregard for their own restrictions notwithstanding, we can see why Lewis’s argument fails to produce a genuine anomaly: the mass of the marbles is accessible; that is, it can be described as objectively localised. Both strains of Lewis’s argument against MDI fail to hold. Lewis’s first argument, that the standard GRW interpretation conflicts with MDI, is neither damning nor compelling. His second argument, that the R and D measures conflict, is simply incorrect. I conclude that MDI is not doomed to failure for any reason that Lewis provided, and therefore that his argument is not a good one.
5. The Ghirardi–Bassi and Clifton–Monton debate Lewis’s argument against MDI fails. Yet Lewis could be, and has been, read as having two separate arguments: one for the standard GRW interpretation (as developed in Section 3.1) and one for MDI (Section 3.2). Perhaps examining Lewis’s argument with respect to the standard GRW interpretation is otiose, as MDI is now taken to be the physically more relevant context in which to discuss the GRW theory and is ‘‘now universally accepted’’ by dynamical reduction theorists (Bassi & Ghirardi, 1999, p. 723), but this was not apparent early in the debate over the Counting Anomaly. In the remaining pages, the debate that ensued between the Ghirardi/Bassi and Clifton/Monton teams will be traced, with an eye towards sorting out how the conflation of Lewis’s two lines of reasoning has sustained it. Let us begin with Ghirardi and Bassi’s (1999) original reply to Lewis. They began by denying that Lewis correctly characterised the state that describes all the marbles as being in the box. They claimed that for a two-particle system, the state jcS1;2 ¼ ðajinS þ bjoutSÞ1 #ðajinS þ bjoutSÞ2
ð14Þ
is ‘‘non-sensical,’’ and the equivalent state jcS1;2 ¼ a2 jinS1 #jinS2 þ abjoutS1 #jinS2 þ abjinS1 #joutS2 þ b2 joutS1 #joutS2
ð15Þ
is required to make sense of Lewis’s argument. They further posited that in a small amount of time, the state will collapse on to one of the terms in (15). Surely, as Clifton and Monton (1999, p. 705) pointed out, this argument is faulty. Even if the system did undergo such a collapse, one would again be left in a state like (14), and we would be right back where we started. Ghirardi and Bassi went on to say that for the Counting Anomaly to be manifest, it would require more marbles than there would be even if the entire mass of the universe were to consist of marbles. These are hardly convincing reasons to reject Lewis’s conclusions. However, there are some informative aspects to Ghirardi and Bassi’s paper. First, they specifically pointed to Ghirardi et al.’s original discussion of the D function, pointing out that ‘‘the GRW theory, while claiming that a marble in the state jcS can be asserted ‘to be within the box,’ due to its basic dynamical features has precise physical implications which Lewis seems to ignore concerning hypothetical
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situations in which more than one marble is in such a state’’ (1999, p. 53). This statement meshes well with the preceding analysis of Lewis’s discussion of the D function. Clearly, it supports the claim that Lewis has missed the point of the D measure by conflating ‘‘precise physical implications’’ and ‘‘hypothetical situations.’’ As argued above, an accessible mass distribution is intended to indicate a classical state of affairs, one where it is appropriate to suppress the contribution of the tails. In addition, the analysis of such distributions does not admit the comparison of accessible states with ‘‘hypothetical’’ ones that are impossible under the GRW theory. On the other hand, it is a mystery how having more than one marble in a state like jcS poses any problem whatsoever. Since Lewis’s argument assumed that the marbles do not interact with one another, the composite n marble system should be considered as a collection of individual marbles. One could just well consider a collection of n spatially separated boxes, each containing a marble in a state like jcS; and Lewis’s argument would go through just as well. Ghirardi and Bassi further claimed that their arguments, based on the standard GRW interpretation, are equally applicable under MDI, and this sowed the seed for later confusion. To their credit however, they took the time to discuss the fact that no real wavefunction, even in the canonical formulation of quantum mechanics, can have compact support. They also outlined a method, analogous to the MDI, to define the ‘‘objective’’ position of a wavefunction. However, they formulated this method to deal with measurements in the usual quantum mechanics, and did not apply it directly to the GRW theory (Ghirardi & Bassi, 1999, pp. 56–60). Thus, we see that many elements of the preceding analysis were already present in Ghirardi and Bassi’s paper. Unfortunately, these elements were not clearly emphasised or stated in their work. It is small wonder, then, that Clifton and Monton re-examined Lewis’s original conclusions and contested Ghirardi and Bassi’s analysis. In essence, Clifton and Monton (1999) argued that Lewis’s critique of the GRW theory is correct, but that in principle, it could never become manifest. In short, they argued that in order to actually count the n marbles, one would have to operationalise the procedure by entangling the individual marbles with some sort of pointer, which would always yield a definite result concerning the number of marbles in the box. Their analysis of Lewis’s arguments excluded MDI, and focused solely on the standard GRW interpretation. Given Ghirardi and Bassi’s claims that their arguments directed toward the standard GRW interpretation were equally applicable to the MDI, one can hardly fault Clifton and Monton for concentrating on the standard GRW interpretation. Yet Bassi and Ghirardi were highly critical in their response. In their retort, Bassi and Ghirardi (1999) now claimed that the Counting Anomaly is only analysable through MDI. They call Clifton and Monton’s analysis ‘‘not appropriate and superfluous’’ (p. 719). If the MDI is the only viable method of treating Lewis’s arguments, then this is the case. But this was not previously assumed. In fact, one can show that Clifton and Monton’s elaborate line of reasoning is superfluous (but not inappropriate) since they consider the standard
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GRW interpretation in their first paper (1999). Consider their argument, abridged and adapted to MDI: if one can only talk of the total mass in the box, then the mass of the individual marbles is entangled with a pointer variable, viz., the mass of the entire system. The Counting Anomaly cannot arise since weighing the box will always yield a definite result. In order to operationalise the procedure, one measures the total weight of the box, which is naturally correlated under the MDI with the weight of the individual marbles. Under MDI, Clifton and Monton’s argument takes on a particularly simple form. But the essential disagreement remains: Clifton and Monton claim that the result of such a measurement would not record n marbles in the box, while Bassi and Ghirardi claim that it would. Bassi and Ghirardi’s (1999) new constructive argument against the Counting Anomaly added nothing to the debate. They simply reproduced the R measure for determining the accessibility of a mass distribution, and claimed that, given the ontology of the interpretation, ‘‘the marbles can therefore be claimed to be all within the box, and the total mass within the box is actually the one corresponding to all of them being in the boxythe very structure of the theory guarantees that this is the case’’ (p. 725). As I have argued above, this is essentially right, but doesn’t address where Lewis went wrong. Lewis plainly acknowledged the fact that the R measure claims that all the marbles are in the box, but argued that this cannot be so because the chances of finding them all there on observation is very small, and the D measure states otherwise. Obviously, Bassi and Ghirardi have done little to further the argument, aside from shifting the focus to the MDI. This leads to the next chapter in the debate. Clifton and Monton (2000) looked at MDI in terms of their previous arguments and concluded, ‘‘either the Counting Anomaly arises, or some other (equally surprising) anomaly will arise’’ (p. 158). Now it may be the case that some other problems arise from MDI, but the Counting Anomaly will not be one of them. Let us try to make this clear. First, Clifton and Monton posited that ‘‘one can treat the semantics for dynamical reduction theories as not adding anything of ontological import to them’’ (2000, p. 155) and question the meaning of the terms ‘‘accessibility’’ and ‘‘objectivity’’ of a mass density when the condition R51 obtains. Bassi and Ghirardi were clear on what is meant by these terms, as the quotation cited earlier from p. 725 in their 1999 paper shows. This seems to be the essential disagreement between the two sides of the debate: Bassi and Ghirardi maintained that a marble is ‘‘in the box’’ just in case it is accessible or objectively localised within the box, for this is precisely what it means for a marble to be ‘‘in the box.’’ Conversely, Clifton and Monton argued that the marble is not truly ‘‘in the box’’ when the R measure obtains, but only works as a semantic rule for property assignment. This is surely an issue that requires resolution, but it has absolutely nothing to do with the Counting Anomaly (since the problem can be stated equally well for a one marble system as it can for an n marble system). The argument put forward in this paper assumes the validity of MDI in order to show that the Counting Anomaly fails under it. Again, the Counting Anomaly poses no problems for MDI, and the virtues and faults of the interpretation should be discussed without reference to Lewis’s arguments.
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Second, Clifton and Monton (2000) assumed that if the total mass inside the box is given by jaj2 nm under MDI, then n can be chosen such that the mass in the box is not greater than mðn xÞ; for some arbitrary natural number x: In this situation, Clifton and Monton tried to place the hypothetical mass mðn xÞ (one not allowed by MDI) inside the box. This manner of comparing hypothetical masses and potentially realisable masses merely reformulates Lewis’s discussion of the D measure, where an unrealisable mass distribution is compared to a realisable one, and this objection is thus met by my earlier argument that it is invalid to compare possible mass distributions within a box to impossible ones under the D function. The hypothetical mass does not exist in our world, and cannot denote a real state of affairs.
6. Conclusion I have argued in this paper that Lewis’s Counting Anomaly does not obtain under the MDI. The structure of MDI clearly prevents the anomaly from arising. It is unfortunate that the chronology of the debate between Bassi and Ghirardi, and Clifton and Monton masked some crucial points related to the differences between structures of the standard GRW interpretation and MDI. Furthermore, no one had before bothered to critically evaluate Lewis’s arguments against MDI, choosing instead to focus on the notions of accessibility and objective localizability. I conclude that MDI is sufficient, on its own terms, to block the Counting Anomaly.
Acknowledgements I would like to thank Joseph Berkovitz, Jeffrey Bub, Roman Frigg and several anonymous referees for their insightful comments on earlier drafts of this paper.
References . Albert, D., & Loewer, B. (1996). Tails of Schrodinger’s cat. In R. Clifton (ed.), Perspectives on Quantum Reality (pp. 81–91). Dordrecht: Kluwer. Bassi, A., & Ghirardi, G. C. (1999). More about dynamical reduction and the enumeration principle. British Journal for Philosophy of Science, 50, 719–734. Bassi, A., & Ghirardi, G. C. (2001). Counting marbles: Reply to Clifton and Monton. British Journal for Philosophy of Science, 52, 125–130. Clifton, R., & Monton, B. (1999). Losing your marbles in wave function collapse theories. British Journal for Philosophy of Science, 50, 697–717. Clifton, R., & Monton, B. (2000). Counting marbles with accessible mass density: A reply to Bassi and Ghirardi. British Journal for Philosophy of Science, 51, 155–164. Ghirardi, G. C., & Bassi, A. (1999). Do dynamical reduction models imply that arithmetic does not apply to ordinary macroscopic objects? British Journal for Philosophy of Science, 50, 49–64. Ghirardi, G. C., Grassi, R., & Benatti, F. (1995). Describing the macroscopic world: Closing the circle within the dynamical reduction program. Foundations of Physics, 25, 5–38.
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Ghirardi, G. C., Grassi, R., & Pearle, P. (1990). Relativistic dynamical reduction models: General framework and examples. Foundations of Physics, 20, 1271–1316. Ghirardi, G. C., Rimini, A., & Weber, T. (1986). Unified dynamics for macroscopic and microscopic systems. Physical Review D, 34, 470–491. Lewis, P. (1997). Quantum mechanics, orthogonality, and counting. British Journal for Philosophy of Science, 48, 313–328. Pearle, P. (1989). Combining stochastic state-vector reduction with spontaneous localization. Physical Review A, 39, 2277–2289.
Studies in History and Philosophy of Modern Physics 34 (2003) 607–620
Finding your marbles in wavefunction collapse theories Daniel Parker Department of Philosophy, University of Maryland, College Park, MD 20742, USA
Abstract Lewis (Br. J. Philos. Sci. 48 (1997) 313) has recently presented an argument claiming that, under the Ghirardi–Rimini–Weber (GRW) theory of quantum mechanics, arithmetic does not apply to ordinary macroscopic objects such as marbles (known as the Counting Anomaly). In this paper, I disentangle two different lines of Lewis’s argument, one devoted to what I call the standard GRW interpretation and the other to the mass density interpretation (MDI). I present both strains of Lewis’s argument, and move on to criticise Lewis’s position, focusing on his argument with respect to MDI. I analyse the structure of his argument, and follow this with a novel refutation of Lewis’s argument, drawing on the original presentation of MDI as developed by Ghirardi et al. (Found. Phys. 25 (1995) 5). I briefly consider the debate that ensued between Bassi and Ghirardi and Clifton and Monton and interpret it within the context of my analysis. I conclude that Lewis’s Counting Anomaly fails to generate a genuine problem. r 2003 Elsevier Ltd. All rights reserved. Keywords: Counting Anomaly; Ghirardi–Rimini–Weber (GRW); Wave packet reduction; Mass density interpretation; Bassi; Ghirardi; Clifton; Monton
1. Introduction In 1997, Peter Lewis published an essay identifying a potential problem with the Ghirardi–Rimini–Weber (GRW) theory of wave packet reduction in quantum mechanics. Lewis contended that a consequence of the GRW theory is that arithmetic fails for ordinary macroscopic objects. He argued that the GRW theory entails that if each of n non-interacting marbles is in a box, we are not justified in claiming that all the marbles are in the box, for large values of n. Following Clifton E-mail address: [email protected] (D. Parker). 1355-2198/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S1355-2198(03)00065-0
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and Monton (1999), we shall call this the ‘‘Counting Anomaly.’’ Ghirardi and Bassi (1999) later countered Lewis’s arguments, claiming that Lewis had misunderstood the GRW theory and that the Counting Anomaly could never occur. Here I argue that Ghirardi and Bassi’s conclusions are basically correct, although their arguments are misleading. Ghirardi and Bassi’s poor reasoning led Clifton and Monton (1999) to develop a different counterargument to Lewis, claiming that his conclusions were correct but that they could never be empirically confirmed. Clifton and Monton pointed out the numerous flaws in Ghirardi and Bassi’s paper, which elicited a rather sharp retort from Bassi and Ghirardi and led them to reformulate their objections to Lewis (Bassi & Ghirardi, 1999) based on the proposed mass density interpretation (hereafter MDI) as formulated by Ghirardi, Grassi, and Benatti (1995). Clifton and Monton (2000) have defended their position against Bassi and Ghirardi’s attack and reconstructed their arguments to account for MDI. More recently, Bassi and Ghirardi (2001) have clarified their claims in response to Clifton and Monton’s arguments against MDI. The current state of this debate leaves Lewis’s original arguments far behind. Indeed, the present positions of Ghirardi and Bassi, and Clifton and Monton, are hardly faithful to Lewis. While they still argue Lewis’s Counting Anomaly, the chronology and structure of the debate have led some issues, which I argue are peripheral and irrelevant, to take on lives of their own. Therefore, I propose a ‘‘Back to Lewis’’ movement. In the following pages, I will point out a flaw in Lewis’s position that has hitherto been unnoticed, or perhaps only hinted at. To this end, I will discuss how the debate has missed the crucial element of Lewis’s presentation of the Counting Anomaly in MDI by focusing on a separate issue, namely what it means for the mass of an object (microscopic or macroscopic) to be ‘‘accessible,’’ or ‘‘objectively localised.’’ While the precise meaning of ‘‘accessibility’’ in MDI and its consequences may be interesting (and problematic) in its own right, this paper argues that the original formalism of MDI (Ghirardi et al., 1995) is sufficient to refute Lewis’s Counting Anomaly. It is the central claim of this paper that Lewis’s argument fails, though MDI may prove untenable on other grounds. Specifically, MDI may still retain the ‘‘tails problem’’ of the original GRW theory, though it provides an interpretation that claims to sweep it under the rug. The problems of developing a satisfactory realist interpretation of MDI are well beyond the intended scope of this essay. Instead, I argue that the debate over the significance and appropriateness of the MDI criterion for the ‘‘objective localisability’’ or ‘‘accessibility’’ of mass, discussed at great length by Bassi and Ghirardi, and Clifton and Monton, should be divorced from talk of the Counting Anomaly. This paper will proceed as follows. In Section 2, I briefly summarise the GRW theory and MDI, which develops a criterion for determining when a mass distribution may be treated as a classical object and a second criterion to compare two mass distributions. Section 3 presents an outline of Lewis’s paper, followed in Section 4 by a critical evaluation of Lewis’s arithmetical conundrum. Finally, in Section 5, I trace the history of the debate that ensued, speculating as to the reasons why it has reached the unnecessary level of complexity that it has, and I reply to the
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arguments raised by Bassi and Ghirardi, and by Clifton and Monton, that are relevant to the analysis.
2. The GRW theory and MDI 2.1. The GRW theory According to the standard GRW dynamics, any given particle follows the usual . Schrodinger dynamics most of the time.1 On rare occasions, however, a particle suffers a ‘‘hit,’’ where its wavefunction, C; is multiplied by a normalised Gaussian centred on a position selected with probability density jCj2 by the Born probability rule. The width of the Gaussian and the frequency of a ‘‘hit’’ for a given particle are based on dimensional analysis and assigned approximate values of 105 cm and 107 s1, respectively. The width of the localisation distribution is chosen so that the particle is well localised without creating an unphysically large conjugate momentum distribution. More generally, the choice of frequency parameter is justified by . allowing single particles to follow Schrodinger evolution virtually all the time, while large systems (i.e., macroscopic objects) undergo frequent spontaneous collapses so as to conform to our everyday experiences (Ghirardi, Rimini, & Weber, 1986). A much-discussed aspect of this theory, the one that Lewis exploits, is the ‘‘tails’’ problem. Due to the Gaussian nature of the ‘‘hitting’’ function, the resultant wavefunction does not have compact support, and it cannot be localised in a bounded region. Alternatively phrased, the wavefunction has diminishing but nonzero tails going off to infinity in every direction: even after a collapse, a particle’s wavefunction is at once non-zero everywhere and has a non-zero probability of undergoing a further collapse anywhere. However, this seemingly undesirable trait is both unavoidable (Albert & Loewer, 1996) and a necessary part of the theory (Ghirardi, Grassi, & Pearle, 1990). Another way to approach the tails problem is to see that the GRW theory must deny the eigenvalue–eigenstate link. In the canonical formulation of quantum mechanics, mutually exclusive states of affairs are formalised through the use of orthogonal vectors in Hilbert space. Upon measurement, a vector representing a superposition of two orthogonal states instantaneously collapses on to one of these states and yields the value associated with that state. However, this does not happen in the GRW theory; a particle is never exactly in any particular localised region. Since the collapse state cannot, even in principle, be perfectly localised, two particles in different locations always have overlapping wavefunctions and therefore are only approximately orthogonal. Both the standard GRW interpretation and MDI share these underlying dynamics. In what I call ‘‘the standard GRW interpretation,’’ two states, jCS and jFS; are said to be orthogonal if /CjFSod; where d is some small parameter (Lewis, 1997; 1
This basic description is also appropriate to more sophisticated spontaneous collapse theories like the CSL model of Pearle (1989).
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Ghirardi et al., 1995). The distinctive feature of the standard GRW interpretation is the use of the inner product rule for assigning properties to the physical systems, to be contrasted with the rules associated with MDI, as described below.2 2.2. The MDI Ghirardi et al. (1995) introduced MDI, which utilises two devices for ascribing properties to states. The first, the R measure, identifies a criterion for determining when a mass distribution may be treated as an objectively localised classical object. That is, it replaces the notion of a position eigenstate as a means of determining when a mass may be taken as having a definite position. The second measure, the D function, replaces the inner product as a way of comparing two mass distributions. Each of these will be developed below. Ghirardi et al. defined a mass density operator for the mass density in a localised cell j X Mj ¼ mk NjðkÞ ; ð1Þ k
where mk is the mass of a particle of type k and NjðkÞ is the number density of such particles in a given cell j: They went on to define the average mass density, Mj ; in a cell in a natural fashion Mj ¼ /CjMj jCS
ð2Þ
and the variance Vj ¼ /Cj½Mj /CjMj jCS2 jCS: R2j
ð3Þ
Vj =M2j ;
¼ is much smaller than one, the mass in a In the case where the ratio, cell j is said to be ‘‘objectively localised’’ or ‘‘accessible’’ (Ghirardi et al., 1995, p. 18). Let us pause here to evaluate the implications of this accessibility criterion. Ghirardi et al. (1995) noted that if a mass distribution is significantly superposed between two or more cells, one cannot discuss it in a classical manner; i.e., it is not ‘‘objectively localised’’ or ‘‘accessible.’’ Thus, Ghirardi et al. required some method to distinguish between such cases. To do this, Ghirardi et al. chose the R measure as a means of demarcating between accessible and inaccessible mass distributions. When the R measure obtains, it implies that the mass in a cell j behaves as if the mass in the cell was the total mass of the object, since the mass density of the tails is negligible and experimentally undetectable. To demonstrate the appropriateness of the R criterion, Bassi and Ghirardi (2001) considered a classical analogue: a table. If a hypothetical table were to lose a single nucleon it would still make sense to refer to it as the same table, since the masses of the original table and the table that had lost a nucleon would be observationally indistinguishable. Similarly, according to Bassi and Ghirardi, one can safely ignore the contribution of the tails of an objectively 2 The standard GRW interpretation includes formulations equivalent to the inner product rule, including the fuzzy link (PosR) rule described by Albert and Loewer (1996) and used by Clifton and Monton (1999).
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localised mass distribution since it would not contribute significantly to the total mass of the distribution. To further motivate this criterion, let us look briefly at the following example, presented by Ghirardi et al. (1995). Consider the two states pffiffiffi B " A B jC" S ¼ 1= 2ðjCA ð4Þ N S þ jCN SÞ and jC SXjFN=2 S#jFN=2 S where jC# S refers to N masses superposed between the regions A and B; while jC# S is the state describing the composite system where half the mass is in region A and half in region B: The two states have the same mass expectation value under (2), but only the second state is to be considered objectively localised under the R measure (the reader is referred to the original paper for the details of the example). To introduce the second measure, the ‘‘mass distance’’ or D function, Ghirardi et al. (1995) noted that the Hilbert space formalism is inadequate to compare the similarity of two macroscopic mass distributions. As an example, they considered three macroscopic states, jCA S; jCB S and jCA0 S; where jCA S is a state with centre of mass localised at point A, jCB S; a state with centre of mass at a point B far away fron A, and jCA0 S; a state with centre of mass at A but differing from jCA S by the position of one microscopic particle and, therefore, nearly orthogonal to jCA S: Clearly, according to Ghirardi et al., jCA S and jCB S must be characterised as macroscopically different states, but jCA S and jCA0 S should be characterised as identical. However, in the Hilbert space topology, both jCB S and jCA0 S are nearly orthogonal to jCA S; and, therefore, the pairs are treated as equally dissimilar. To distinguish such cases, they proposed a function to determine the macroscopic equivalence of two mass distributions qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X DðjcS; jfSÞ ¼ ðMj Nj Þ2 ; ð5Þ j where Mj and Nj are the average mass densities of cell j for the two states jcS and jfS; respectively. Ghirardi et al. defined two states as macroscopically equivalent if DðjcS; jfSÞpe
ð6Þ
where e is taken to be 1018 m0 and m0 is the mass of a nucleon. However, not any two states can be reliably assessed by the D function as a measure of macroscopic equivalence. Two states can share the same value of M for all j and yet not be equivalent, as the states described in (4) demonstrate. Without some means of restricting the arguments of the ‘‘mass distance’’ function D; it is be easy to gerrymander states that the function would classify as equivalent when in fact they were not. Hence, the domain of the D function is limited to the ‘‘proper subset AðSÞ of UðSÞ of those states which are allowed by the [GRW] dynamics’’ (Ghirardi et al., 1995, p. 26). UðSÞ referred to the unit sphere of the Hilbert space associated with the system of interest S: The content of the subset AðSÞ was left rather vague in Ghirardi et al., as the authors readily admitted. However, they indicated that the properties of the states in AðSÞ could be made sufficiently precise if necessary and that they certainly wished to exclude the comparison of states like those of (4), which the D function would otherwise classify as equivalent.
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3. Lewis’s Counting Anomaly 3.1. The anomaly under the standard GRW interpretation Lewis’s argument looked at both the standard GRW interpretation and the MDI, and he drew the same conclusion in each case, viz., that arithmetic fails. What Ghirardi and Bassi, and Clifton and Monton failed to do is distinguish between the two distinct lines of reasoning in Lewis’s paper. Consequently, much of the debate between the two groups hinges on their inability to consider each of Lewis’s arguments in turn. So let us try. Lewis (1997) considered a marble, which may be inside or outside a box. According to the GRW theory, if the marble is inside the box, then its wavefunction is jcS ¼ ajinS þ bjoutS 2
2
ð7Þ 2
2
and 1 > jaj bjbj > 0 with the further condition that jaj þ jbj ¼ 1: This is the best one can do in the GRW theory. There is never a situation where the marble is wholly inside or outside the box; that is, the marble is never in one of the eigenstates jinS or joutS: However, for a sufficiently close to 1, one is permitted to infer that the marble is indeed within the confines of the box (Ghirardi et al., 1995). Lewis accepted this stipulation of the GRW theory, and considered the case of n marbles in a box. For n non-interacting marbles in a box (achieved by making the box extremely large), the vector in the composite Hilbert space is jcSall ¼ ðajinS þ bjoutSÞ1 #ðajinS þ bjoutSÞ2 #?#ðajinS þ bjoutSÞn
ð8Þ
and the pure state where all the marbles are in the box is jinSall ¼ jinS1 #jinS2 #?#jinSn :
ð9Þ
Taking the inner product between the two states, the probability of finding all the marbles in the box is j/CjinSj2all ¼ jaj2n :
ð10Þ
Since jaj2 is smaller than 1, for large n the inner product between the states jcSall and jinSall becomes arbitrarily small. Hence, even though each marble is individually in the box, all the marbles are not in the box. So arithmetic fails. 3.2. The anomaly under the MDI It was Lewis’s contention that some unacceptable consequence such as the one described above must arise for any proposed interpretation of the GRW theory. He claimed that the argumenty does not presuppose any particular kind of rule for determining, given the physical state of the system, how many marbles are in the boxy Whatever rule we might consider must tell us something about how many marbles are in the box when the state of the system is given by [(8)]y [I]f the argument is a
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good one, then any rule one might propose for the interpretation of the GRW theory is doomed to failure (1997, p. 319). As a concrete example, Lewis looked at MDI and recovered the same anomaly, but the argument took on a slightly different form. He first considered whether the mass density for an n marble system is accessible inside and outside the box. For such a system with many discrete cells, divided into regions A (marble sized and in the box) and B (marble sized and outside the box), the relevant mass density parameters for the state (8) are MA ¼ njaj2 m
VA ¼ n2 jaj2 jbj2 m2
R2A ¼ jbj2 =jaj2 51;
MB ¼ njbj2 m
VB ¼ n2 jaj2 jbj2 m2
R2B ¼ jaj2 =jbj2 b1;
ð11Þ
where m is the mass of a marble (Lewis, 1997). This shows that the mass of the n marbles is accessible or objectively localised inside the box and not so outside the box, according to the proposed definition of accessibility. Therefore, the n marbles can be understood to be inside the box, as one would hope. Thus, Lewis conceded that according to the R measure, all the marbles are claimed to be in the box. However, he argued that this is unacceptable, for (10) implies that the chance of finding all n marbles in the box is very small (Lewis, 1997, p. 320). Thus, according to Lewis, MDI fails to prevent the anomaly. Furthermore, Lewis found a second problem with MDI’s claim that all the marbles can be understood to be in the box. In this second case, he considered the ‘‘mass distance’’ function, D (Lewis, 1997, p. 328). For a one marble system in a box, the ‘‘distance’’ between its wavefunction and the eigenstate jinS is ffi pffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X X 2 2 2 2 DðjinS; jcSÞ ¼ ðm jaj mÞ þ ðjbj mÞ ð12Þ ¼ 2K jbj2 m; jAA jAB where K is the number of cells in a marble-sized region of space. For a small enough choice of b; one may claim that the ‘‘distance’’ between the two states satisfies condition (6). However, for an n marble system, the distance is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Xn X X 2 2 2 2 DðjinSall ; jcSall Þ ¼ ðm jaj mÞ þ ðjbj mÞ i¼1 jAAi jABi pffiffiffiffiffiffiffiffiffi ¼ 2nK jbj2 m: ð13Þ Lewis now could assert that, no matter how small b is chosen, one can increase the number of marbles n to a large enough value to ensure that DðjinSall0 jcSall Þ > e: His conclusion in this case was Examining the single-marble state [(7)] according to this rule, we find that for values of a and b typically produced by the GRW dynamics, [(7)] is indeed macroscopically equivalent to the eigenstate jinS of a marble being in the boxy However, if we examine the n-marble state [(8)] in the same way, we find that as n becomes large, [(8)] is not macroscopically equivalent to the eigenstate jinSall of all n marbles being in the box. But this cannot be the case, since the same rule tells us that each marble taken individually is in the box, and this is precisely what it means for all n marbles to be in the box (Lewis, 1997, p. 320; emphasis added).
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The important point here is that the Counting Anomaly (or some other unacceptable consequence) is manifest in two different forms under MDI: 1. The R measure’s claim that all the marbles are in the box contradicts the GRW theory, which, by (10), shows that ‘‘the chance of finding all n marbles in the box on observation is very small’’ (1997, p. 320). 2. While the R measure claims that all the marbles are in the box, the D measure states otherwise, and they are thus inconsistent.
4. The Counting Anomaly in the MDI The strategy of this section will be to refute Lewis’s Counting Anomaly by establishing the contrapositive to Lewis’s condition that ‘‘if the argument is a good one, then any rule one might propose for the interpretation of the GRW theory is doomed to failure,’’ and showing that some rule is not so doomed. Specifically, I show that neither of Lewis’s arguments with respect to MDI hold upon closer scrutiny so that MDI is not doomed to failure, at least not for the reasons Lewis gives. As a consequence, I conclude that Lewis’s argument is not a good one. Lewis’s first argument, that MDI claims that (8) is a state where all the marbles are in the box while (10) shows otherwise, falls flat. MDI rejects the use of the inner product rule, used in (10), to ascribe properties to states. Alternatively stated, the argument of Section 3.1 does presuppose a rule for determining how many marbles are in the box, namely the inner product rule. But the R and D measures are intended to replace the inner product, not complement it. This point was clearly made by Ghirardi et al. (1995) and echoed with particular reference to Lewis in Bassi and Ghirardi’s papers (see Ghirardi & Bassi, 1999, p. 53; Bassi & Ghirardi, 1999, p. 722). The inconsistency between the properties assigned by the standard GRW interpretation and MDI is to be expected and is entirely unproblematic: they are two different interpretational systems. Lewis’s second argument against MDI, that the R and D measures are inconsistent, is somewhat more complicated. To refute this line of reasoning, it will suffice to look at the D function for a single marble in a box, jCS; though the following argument obviously holds for the n marble system in a box. When introducing MDI, Ghirardi et al. asserted that ‘‘if one does not restrict the set of all possible states of the Hilbert space of ‘our universe,’ one unavoidably meets situations which cannot be consistently described in terms of the function M’’ (1995 p. 14). As we shall see presently, this condition was ignored in Lewis’s analysis. The GRW theory, as well as its extension to MDI, does restrict the possible states of the Hilbert space in ‘‘our universe.’’ An enumeration of the possible and impossible states in ‘‘our universe’’ is not necessary, for the relevant set of such states is already at hand. We have already noted that the state jcS of (7) is permitted by the theory while the state jinS is not. If this were not the case, there would be no Counting Anomaly. So it should come as no surprise (to Ghirardi or anyone else) that Lewis
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found an inconsistency in the GRW theory when he employed states that do not exist in ‘‘our universe!’’. Let us flesh this out a bit. When Lewis utilised the D function to compare the mass distributions of the states jinSall and jcSall ; he used one state that is not realisable under the GRW theory and one that is. Implicit in the definition of the D function, as discussed at the end of Section 2, is the stipulation that the mass distributions being compared are possible! Recall that the domain of the D measure was restricted to a particular subset of the states realisable by the GRW dynamics. It is therefore an illegal move to calculate the D function with one argument being jCS and the other as a state corresponding to a mass m inside the box (jinS), since such a mass distribution cannot be realised under the GRW dynamics. If one wanted to put a mass m inside the box, this could only be accomplished by placing a marble of mass m=jaj2 in the box. But such a mass distribution would have tails outside the box, and the D measure would fail to characterise the mass densities as identical, as one would expect. One can now see why Ghirardi and Bassi’s original refutation of Lewis’s argument failed. When they argued, ‘‘Thus the difficulty which arises, according to Lewis, in connection with a state like [___] does not present itself for the simple reason that such a state never occurs, its existence being forbidden by the GRW theory itself’’ (1999 p. 52). Ghirardi and Bassi filled the blank with a state equivalent to (8), rather than one such as (9). Furthermore, Ghirardi et al. (1995, pp. 27–28) exacerbated the problem by directly comparing, in their earlier paper, the two states jcS and jinS exactly as Lewis did. Their motivation here, however, was to point out that the D function succeeds, in the same way as the inner product /cjinS does, in describing an objective M state as ‘‘almost an eigenstate’’ of the mass being entirely within the box, not to authorise the use of non-existent mass distributions as arguments for the D function. It is unfortunate that such a move tacitly endorses the consideration of such states. It is important to emphasise that nothing has been presented here that moves beyond the original presentation of MDI developed by Ghirardi et al. (1995). The Counting Anomaly can be refuted entirely on MDI’s own terms. As a semantic rule, the R criterion provides a framework for mapping the properties of MDI states onto meaningful classical properties such as ‘‘in a box’’; that is, it determines when states can be discussed in a classical manner. The D measure allows MDI states to be compared directly with one another. The motivations for each of these criteria have already been discussed. When Ghirardi et al. used terms such as ‘‘macroscopically equivalent,’’ one should understand that this equivalence can hold only between potentially realisable MDI states, since no other type of state can arise given the dynamics of the theory. Now, one can debate whether MDI succeeds in developing a correspondence between its states and macroscopic properties, but this is a separate matter from describing or comparing the properties of the MDI states themselves. Lewis’s mistake was to conflate how the MDI criteria are used. He tried to derive a conclusion, namely that the property ‘‘all the marbles are in the box’’ does not obtain, from the D measure, whose use is restricted to comparing potentially
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realisable MDI states. Ghirardi et al.’s disregard for their own restrictions notwithstanding, we can see why Lewis’s argument fails to produce a genuine anomaly: the mass of the marbles is accessible; that is, it can be described as objectively localised. Both strains of Lewis’s argument against MDI fail to hold. Lewis’s first argument, that the standard GRW interpretation conflicts with MDI, is neither damning nor compelling. His second argument, that the R and D measures conflict, is simply incorrect. I conclude that MDI is not doomed to failure for any reason that Lewis provided, and therefore that his argument is not a good one.
5. The Ghirardi–Bassi and Clifton–Monton debate Lewis’s argument against MDI fails. Yet Lewis could be, and has been, read as having two separate arguments: one for the standard GRW interpretation (as developed in Section 3.1) and one for MDI (Section 3.2). Perhaps examining Lewis’s argument with respect to the standard GRW interpretation is otiose, as MDI is now taken to be the physically more relevant context in which to discuss the GRW theory and is ‘‘now universally accepted’’ by dynamical reduction theorists (Bassi & Ghirardi, 1999, p. 723), but this was not apparent early in the debate over the Counting Anomaly. In the remaining pages, the debate that ensued between the Ghirardi/Bassi and Clifton/Monton teams will be traced, with an eye towards sorting out how the conflation of Lewis’s two lines of reasoning has sustained it. Let us begin with Ghirardi and Bassi’s (1999) original reply to Lewis. They began by denying that Lewis correctly characterised the state that describes all the marbles as being in the box. They claimed that for a two-particle system, the state jcS1;2 ¼ ðajinS þ bjoutSÞ1 #ðajinS þ bjoutSÞ2
ð14Þ
is ‘‘non-sensical,’’ and the equivalent state jcS1;2 ¼ a2 jinS1 #jinS2 þ abjoutS1 #jinS2 þ abjinS1 #joutS2 þ b2 joutS1 #joutS2
ð15Þ
is required to make sense of Lewis’s argument. They further posited that in a small amount of time, the state will collapse on to one of the terms in (15). Surely, as Clifton and Monton (1999, p. 705) pointed out, this argument is faulty. Even if the system did undergo such a collapse, one would again be left in a state like (14), and we would be right back where we started. Ghirardi and Bassi went on to say that for the Counting Anomaly to be manifest, it would require more marbles than there would be even if the entire mass of the universe were to consist of marbles. These are hardly convincing reasons to reject Lewis’s conclusions. However, there are some informative aspects to Ghirardi and Bassi’s paper. First, they specifically pointed to Ghirardi et al.’s original discussion of the D function, pointing out that ‘‘the GRW theory, while claiming that a marble in the state jcS can be asserted ‘to be within the box,’ due to its basic dynamical features has precise physical implications which Lewis seems to ignore concerning hypothetical
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situations in which more than one marble is in such a state’’ (1999, p. 53). This statement meshes well with the preceding analysis of Lewis’s discussion of the D function. Clearly, it supports the claim that Lewis has missed the point of the D measure by conflating ‘‘precise physical implications’’ and ‘‘hypothetical situations.’’ As argued above, an accessible mass distribution is intended to indicate a classical state of affairs, one where it is appropriate to suppress the contribution of the tails. In addition, the analysis of such distributions does not admit the comparison of accessible states with ‘‘hypothetical’’ ones that are impossible under the GRW theory. On the other hand, it is a mystery how having more than one marble in a state like jcS poses any problem whatsoever. Since Lewis’s argument assumed that the marbles do not interact with one another, the composite n marble system should be considered as a collection of individual marbles. One could just well consider a collection of n spatially separated boxes, each containing a marble in a state like jcS; and Lewis’s argument would go through just as well. Ghirardi and Bassi further claimed that their arguments, based on the standard GRW interpretation, are equally applicable under MDI, and this sowed the seed for later confusion. To their credit however, they took the time to discuss the fact that no real wavefunction, even in the canonical formulation of quantum mechanics, can have compact support. They also outlined a method, analogous to the MDI, to define the ‘‘objective’’ position of a wavefunction. However, they formulated this method to deal with measurements in the usual quantum mechanics, and did not apply it directly to the GRW theory (Ghirardi & Bassi, 1999, pp. 56–60). Thus, we see that many elements of the preceding analysis were already present in Ghirardi and Bassi’s paper. Unfortunately, these elements were not clearly emphasised or stated in their work. It is small wonder, then, that Clifton and Monton re-examined Lewis’s original conclusions and contested Ghirardi and Bassi’s analysis. In essence, Clifton and Monton (1999) argued that Lewis’s critique of the GRW theory is correct, but that in principle, it could never become manifest. In short, they argued that in order to actually count the n marbles, one would have to operationalise the procedure by entangling the individual marbles with some sort of pointer, which would always yield a definite result concerning the number of marbles in the box. Their analysis of Lewis’s arguments excluded MDI, and focused solely on the standard GRW interpretation. Given Ghirardi and Bassi’s claims that their arguments directed toward the standard GRW interpretation were equally applicable to the MDI, one can hardly fault Clifton and Monton for concentrating on the standard GRW interpretation. Yet Bassi and Ghirardi were highly critical in their response. In their retort, Bassi and Ghirardi (1999) now claimed that the Counting Anomaly is only analysable through MDI. They call Clifton and Monton’s analysis ‘‘not appropriate and superfluous’’ (p. 719). If the MDI is the only viable method of treating Lewis’s arguments, then this is the case. But this was not previously assumed. In fact, one can show that Clifton and Monton’s elaborate line of reasoning is superfluous (but not inappropriate) since they consider the standard
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GRW interpretation in their first paper (1999). Consider their argument, abridged and adapted to MDI: if one can only talk of the total mass in the box, then the mass of the individual marbles is entangled with a pointer variable, viz., the mass of the entire system. The Counting Anomaly cannot arise since weighing the box will always yield a definite result. In order to operationalise the procedure, one measures the total weight of the box, which is naturally correlated under the MDI with the weight of the individual marbles. Under MDI, Clifton and Monton’s argument takes on a particularly simple form. But the essential disagreement remains: Clifton and Monton claim that the result of such a measurement would not record n marbles in the box, while Bassi and Ghirardi claim that it would. Bassi and Ghirardi’s (1999) new constructive argument against the Counting Anomaly added nothing to the debate. They simply reproduced the R measure for determining the accessibility of a mass distribution, and claimed that, given the ontology of the interpretation, ‘‘the marbles can therefore be claimed to be all within the box, and the total mass within the box is actually the one corresponding to all of them being in the boxythe very structure of the theory guarantees that this is the case’’ (p. 725). As I have argued above, this is essentially right, but doesn’t address where Lewis went wrong. Lewis plainly acknowledged the fact that the R measure claims that all the marbles are in the box, but argued that this cannot be so because the chances of finding them all there on observation is very small, and the D measure states otherwise. Obviously, Bassi and Ghirardi have done little to further the argument, aside from shifting the focus to the MDI. This leads to the next chapter in the debate. Clifton and Monton (2000) looked at MDI in terms of their previous arguments and concluded, ‘‘either the Counting Anomaly arises, or some other (equally surprising) anomaly will arise’’ (p. 158). Now it may be the case that some other problems arise from MDI, but the Counting Anomaly will not be one of them. Let us try to make this clear. First, Clifton and Monton posited that ‘‘one can treat the semantics for dynamical reduction theories as not adding anything of ontological import to them’’ (2000, p. 155) and question the meaning of the terms ‘‘accessibility’’ and ‘‘objectivity’’ of a mass density when the condition R51 obtains. Bassi and Ghirardi were clear on what is meant by these terms, as the quotation cited earlier from p. 725 in their 1999 paper shows. This seems to be the essential disagreement between the two sides of the debate: Bassi and Ghirardi maintained that a marble is ‘‘in the box’’ just in case it is accessible or objectively localised within the box, for this is precisely what it means for a marble to be ‘‘in the box.’’ Conversely, Clifton and Monton argued that the marble is not truly ‘‘in the box’’ when the R measure obtains, but only works as a semantic rule for property assignment. This is surely an issue that requires resolution, but it has absolutely nothing to do with the Counting Anomaly (since the problem can be stated equally well for a one marble system as it can for an n marble system). The argument put forward in this paper assumes the validity of MDI in order to show that the Counting Anomaly fails under it. Again, the Counting Anomaly poses no problems for MDI, and the virtues and faults of the interpretation should be discussed without reference to Lewis’s arguments.
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Second, Clifton and Monton (2000) assumed that if the total mass inside the box is given by jaj2 nm under MDI, then n can be chosen such that the mass in the box is not greater than mðn xÞ; for some arbitrary natural number x: In this situation, Clifton and Monton tried to place the hypothetical mass mðn xÞ (one not allowed by MDI) inside the box. This manner of comparing hypothetical masses and potentially realisable masses merely reformulates Lewis’s discussion of the D measure, where an unrealisable mass distribution is compared to a realisable one, and this objection is thus met by my earlier argument that it is invalid to compare possible mass distributions within a box to impossible ones under the D function. The hypothetical mass does not exist in our world, and cannot denote a real state of affairs.
6. Conclusion I have argued in this paper that Lewis’s Counting Anomaly does not obtain under the MDI. The structure of MDI clearly prevents the anomaly from arising. It is unfortunate that the chronology of the debate between Bassi and Ghirardi, and Clifton and Monton masked some crucial points related to the differences between structures of the standard GRW interpretation and MDI. Furthermore, no one had before bothered to critically evaluate Lewis’s arguments against MDI, choosing instead to focus on the notions of accessibility and objective localizability. I conclude that MDI is sufficient, on its own terms, to block the Counting Anomaly.
Acknowledgements I would like to thank Joseph Berkovitz, Jeffrey Bub, Roman Frigg and several anonymous referees for their insightful comments on earlier drafts of this paper.
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