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«Contents 1 Introduction 1 2 Jones and Clifton’s Argument Part 1: The Formal Situation 5 3 Jarrett’s Argument 10 4 Jones and Clifton’s Argument ...»

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The Delicacy of Causal Ascription and Bell’s Theorem

Erik Curiel

August 24, 2009

Contents

1 Introduction 1

2 Jones and Clifton’s Argument Part 1: The Formal Situation 5

3 Jarrett’s Argument 10

4 Jones and Clifton’s Argument Part 2: The Delicacy of Causal Ascription 16 5 Conclusion 19 The utility of a notion testifies not to its clarity but rather to the philosophical importance of clarifying it.

Nelson Goodman Fact, Fiction and Forecast 1 Introduction Quantum mechanics predicts startling behavior for pairs of certain types of particles (those described as having ‘spin- 1 ’) in a certain joint state (the ‘singlet’ state) under certain experimental conditions.

Such an experimental set-up is as follows. In the middle of a (very large) room, there is a device for producing pairs of spin- 1 particles in the singlet state. The particles of each pair shoot off in opposite directions, one to the left, the other to the right, toward measuring apparatuses waiting at each end of the room to measure the spin of the particles in a particular direction. According to quantum mechanics, the measuring apparatuses will always record either a value of ‘spin-up’ or a value of ‘spin-down’ for each particle, each with a probability of 50%: quantum mechanics does not allow one to predict what value will be measured for any given particle, though it does demand that for any large enough ensemble of particles the two values will be measured with roughly the same frequency. If the apparatuses on opposite ends of the room are set up to measure spin in the same direction, moreover, then quantum mechanics also predicts that measurements on particles of the same pair will be strictly anti-correlated: if up is measured for the particle passing through the left measuring apparatus, for instance, then down will be measured for the other particle with probability 1. If the apparatuses on opposite ends of the room are poised to measure spin in directions different from each other, then quantum mechanics still predicts that the results of the measurements on particles of the same pair will be produced with particular correlations, though not with perfect anti-coorelation: when the directions are close, then the measurement results will be nearly perfectly anti-coorelated; as the directions get farther apart, the measurement results for particles in the same pair will become less strongly anti-correlated, until the directions become perpendicular, at which point there will be no correlation become the measurement results on opposite sides at all—the results will be statistically independent of each other; finally, as the difference in the directions of the spin measurements on the opposite apparatuses continue to increase beyond 90 degrees, the measurement results will begin to be correlated with each other, achieving perfect correlation when the spin measurements are made in exactly opposite directions.

This results by themselves are not terribly surprising. That the results of certain measurements on the particles should exhibit strong correlations, even when the measurements are made when the particles are far apart, can surely be explained by the fact that the particles have a common origin—they were produced in the same spot, in pairs, and so some aspect of their interaction with each other when they were created, or of their respective interactions with a common element of the production device, ought to be able to explain the information each particle appears to have about how its mate will behave under certain circumstances. In more evocative terms, the correlated behavior of the particles should be explicable by positing a common cause in their joint histories.

Such a postulated common cause is usually referred to as a ‘hidden-variable’, since there is no explicit feature of quantum mechanics that plays the role this postulated common cause is supposed to play.

The truly surprising fact is that such a common cause cannot be found. That is the gist of Bell’s Theorem. John Bell showed in 1964, roughly speaking, that the correlations in these sorts of experimental results could be explained by reference to such a common cause only if the measurement statistics satisfied a certain set of inequalities, the so-called Bell inequalities. Not only do the predictions of quantum mechanics not satisfy the Bell inequalities, but, more what is more striking, experiments designed to test them indicate that the real world does not either.1 Because the measurements on the separated particles can be made as far apart as one likes, the upshot is that the real world appears to delight in the existence of striking correlations between arbitrarily separated phenomena, while refusing to countenance any explanation for these correlations in terms of the common causal past of the phenomena. One wonders mopre physicists are not conspiracy theorists in their politics.

In 1984 Jon Jarrett published an analysis of the “strong locality” condition used to derive Belltype inequalities for stochastic, contextual local (models of) hidden-variable theories that reproduce the quantum mechanical spin statistics. It was known before this that Kochen-Specker takes care of all non-contextual theories (local or not), and the derivation of the Kochen-Specker theorem was 1 Cf. Aspect, Dalibard, and Roger (1982).

Erik Curiel 2 August 24, 2009 uncontroversial. Likewise, there was no serious debate about the physical significance of the weaker locality condition needed to derive Bell-type inequalities for local deterministic theories. There was, however, no general agreement in the literature at the time on the physical significance (if any) of strong locality over and above its statistical requirements.2 Since the results of experiments strongly suggest that the physical world violates Bell-type inequalities, by modus tollens one (at least) of the assumptions behind the inequalities had to go, and the faulty premise was almost universally adjudged to be strong locality—but, since no one was able to offer a convincing physical explication of strong locality, no one was able to provide a deeper physical analysis of what is involved in abandoning it.3 Jarrett offered a precise characterization of what may be involved in dispensing with strong locality by demonstrating that it is logically equivalent to the conjunction of two other (logically





independent) conditions, “locality” and “completeness”, each of which, he argued, has more perspicuous physical content than strong locality. Jarrett’s analysis thus offered a refined set of alternatives:

we can reject strong locality, and so (in accord with reality) avoid a physical theory satisfying the Bell inequalities, by dispensing either with locality or with completeness (or both). By arguing that locality is practically equivalent to a ban on the possibility of superluminal signaling in the context of Bell-type experimental arrangements, Jarrett concluded that any adequate physical theory should violate completeness, and not locality, since superluminal signaling4 would prima facie constitute a serious conflict with relativity theory,5 an infelicitous state of affairs.

In this paper, I shall neither attack nor defend Jarrett’s conclusions concerning adequacy constraints appropriate to (models of) physical theories mooted in discussions of Bell’s Theorem. Rather I shall attempt to clarify what sorts of arguments can and cannot coherently be made when Jarretttype premises are accepted (or are accepted at least for the sake of argument). In particular I want to show that certain recent construals of Jarrett’s 1984 argument that focus on the notion of causality not only are beside the point of Jarrett’s argument but, more important, obfuscate what salutary can be gleaned from his argument. Martin Jones and Rob Clifton (1993) among recent commentators on Jarrett not only are notable for their clarity, precision and thoroughness of argument, but they also are typical in what, I suspect, is the main culprit behind confused arguments that are concerned with issues of causality, viz. a careless deployment of the notion of ‘causality’ itself. For this reason 2 Cf. Hellman (1983), especially pp. 603–6, for a helpful survey of the issues as understood at the time.

3 This paper will ignore the possibility of another of the assumptions behind Bell-type inequalities failing, as consideration of it would take us too far afield; for what it is worth, the prevailing philosophical winds are certainly against such an option.

4 To avoid tedious repetition, it should be understood throughout this paper unless explicitly stated otherwise that by ‘superluminal signaling’ I always and only mean the possibility of superluminal signaling specifically in the context of Bell-type experimental arrangements.

5 There is some controversy concerning the precise nature of the relationship between the possibility of superluminal signaling and the demands of relativity theory. For a thorough discussion of the issues involved see Friedman (1983, ch.

4, §§5–7). The arguments of this paper do not depend on the correctness of of any of the positions in this controversy.

I thus should be understood to use ‘superluminal’ only as a convenient word to refer to any process that would connect events in spacelike relation to one another, with no express commitment as to whether there is ‘really something there moving faster than light’.

Erik Curiel 3 August 24, 2009 I have chosen them as a foil, precisely because theirs seems to me the clearest, best case for what I take to be at bottom a confused way both of thinking of Jarrett’s argument in particular, and of deploying causal arguments in general.

Jones and Clifton argue against Jarrett in the following way: by producing a model of a theory that can accommodate the possibility of superluminal signaling and in which the signaling seems to be possible ‘because of’ completeness violations, Jones and Clifton argue that locality does not in fact have the privileged relation to superluminal signaling that Jarrett ascribes to it. I shall argue that Jones and Clifton’s proposed model, while flawless technically, cannot serve as a counter-example to Jarrett’s analysis. Nothing in Jarrett’s paper rules out the model that Jones and Clifton propose as a counter-example—so far from it, Jones and Clifton’s model is wholly consistent with, and can even be taken to support Jarrett’s conclusions, in so far as locality violations necessarily accompany all superluminal signaling in Jones and Clifton’s proposed model, even though it is a novel kind of model, one not explicitly considered by Jarrett in his arguments.

Their arguments fail because they misapprehend what it is that Jarrett argues the significant relation between locality and superluminal signaling to be. Jones and Clifton believe that determining which of the conditions, locality or completeness, can be causally implicated in superluminal signaling is the crux of the debate as Jarrett framed it, but I shall argue that in this they are mistaken.6 Jarrett argues only for a logical connection between locality and superluminal signaling, completely sidestepping the nasty issue of what is ‘responsible for’ or ‘causes’ the signaling, or ‘in virtue of’ what is signaling possible, and indeed this logical connection is all he requires for the argument to work. Whatever else is the case about the real world, he argues, and no matter what mechanism, process or connection, causal or otherwise, one may wish to dream up to explain superluminal signaling, the possibility of superluminal signaling and of locality violations (in principle) imply each other, whereas the possibility of superluminal signaling and of completeness violations have no such logical connection. In any particular model that violates completeness superluminal signaling may be possible, or it may not—from the simple fact of completeness violations in a model, we can say nothing about the possibility of superluminal signaling in that model without further information, such information essentially, as I shall argue, consisting of the status of locality in the proposed model. From this it is easy to conclude that we should look for physical theories that violate completeness but respect locality, so long as we wish both to avoid possible relativistic headaches and to have a theory that violates Bell’s inequality.

One need not (and indeed ought not) refer to causality in any way to make this argument. Jones and Clifton’s introduction of causal considerations into Jarrett-type arguments not only obfuscates the fundamental issues but ultimately detracts from the cogency of their own analysis. Among other things, it is not clear at all what they mean by claiming that, in their model, there is superluminal signaling ‘because of’ completeness violations. Such carelessness derives, I think, in part from an overly formal approach, which obscures the important physical differences between completeness and locality as conditions on physical theories, and thus also obscures the respective power of each 6 Kronz (1990), Butterfield (1992) and Maudlin (1994, pp. 94–6) misread Jarrett in a very similar way, and come to conclusions similar to those of Jones and Clifton.

Erik Curiel 4 August 24, 2009 to justify causal conclusions drawn from a set of simple correlations between distant physical events.

The problem of giving a general criterion for the presence of causality is famously murky, but there is no doubt that one always does stand in need of something else besides correlation to justify causal assertions. In specific cases, it may reasonably be hoped that one can specify what else is required besides the presence of correlations to make warranted causal judgments, even in the absence of a larger, more general theory of causality. Such a thought is at the heart of much work on Bell’s Theorem: no matter what else, if anything, one thinks may mark the presence of a causal connection, surely the ability to send a signal bespeaks causality, at least in any everyday sense in which one uses the word.7 Once so much has been said, though, there is still the problem of delineating precisely what one means by ‘causality’ in this context, and examining one’s causal conclusions to ensure that they are consistent with the constraints on one’s notion of causality.



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