Local Causality and Completeness-Bell vs. Jarrett - Travis Norsen Copy

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Local Causality and Completeness: Bell vs. Jarrett

Travis NorsenMarlboro College

Marlboro, VT [email protected]

(Dated: August 14, 2008)

J.S. Bell believed that his famous theorem entailed a deep and troubling conflict between the

empirically verified predictions of quantum theory and the notion of local causality that is motivatedby relativity theory. Yet many physicists continue to accept, usually on the reports of textbookwriters and other commentators, that Bells own view was wrong, and that, in fact, the theoremonly brings out a conflict with determinism or the hidden-variables program or realism or someother such principle that (unlike local causality), allegedly, nobody should have believed anyway.(Moreover, typically such beliefs arise without the person in question even being aware that theview they are accepting differs so radically from Bells own.) Here we try to shed some light on thesituation by focusing on the concept of local causality that is the heart of Bells theorem, and, inparticular, by contrasting Bells own understanding with the analysis of Jon Jarrett which has beenthe most influential source, in recent decades, for the kinds of claims mentioned previously. Wepoint out a crucial difference between Jarretts and Bells own understanding of Bells formulationof local causality, which turns out to be the basis for the erroneous claim, made by Jarrett and manyothers, that Bell misunderstood the implications of his own theorem.

I. INTRODUCTION

In 1964, J.S. Bell proved the result now known as BellsTheorem: any physical theory of a certain type mustmake predictions (for a certain class of experiment) whichrespect a so-called Bell Inequality. [1] Quantum Mechan-ics (QM) predicts violations of the inequality, and sub-sequent experiments establish convincingly (though notwithout loopholes) that the Quantum Mechanical pre-dictions are correct i.e., the experiments establish thatthe type of theory Bell showed must respect the Inequal-ity, cannot be empirically viable, i.e., cannot be true. [2]But the question (which has given rise to an enormous

literature) remains: what type of theory is it, exactly,that Bells Theorem (combined with the associated ex-periments) refutes?

Bells own view, expressed already in the opening linesof his 1964 paper and subsequently clarified and defendedin virtually all of his later writings, was that It is therequirement of locality ... that creates the essential diffi-culty. (Bell, 1964, p. 14) By locality Bell here meansthe prohibition, usually taken to be an implication of spe-cial relativity (SR), of super-luminal (faster than light)causation. Bell thus took his own theorem to establish atroubling conflict between (the empirically verified pre-dictions of ) QM (i.e., betweenexperiment) and SR:

For me then this is the real problem withquantum theory: the apparently essentialconflict between any sharp formulation andfundamental relativity. That is to say, wehave an apparent incompatibility, at thedeepest level, between the two fundamentalpillars of contemporary theory... (Bell, 1984,p. 172)

Most practicing physicists, however, (and most philoso-

phers of physics) have disagreed with Bell and continuedto believe in the unproblematic consistency of QM andSR. Where do they think Bell went wrong?

One can divide reasons for disagreement (with Bellsown interpretation of the significance of his theorem)into two classes. First, there are those who assert thatthe derivation of a Bell Inequality relies not just onthe premise of locality, but on some additional premisesas well. The usual suspects here include Realism,Hidden Variables, Determinism, and Counter-Factual-Definiteness. (Note that the items on this list are highlyoverlapping, and often commentators use them inter-changeably.) The idea is then that, since it is only the

conjunction of locality with some other premise whichis in conflict with experiment, and since locality is sostrongly motivated by SR, we should reject the otherpremise. Hence the widespread reports that Bells the-orem finally refutes the hidden variables program, theprinciple of determinism, the philosophical notion of re-alism, etc. [3]

Here is how Bell responded to this first class of dis-agreement:

My own first paper on this subject ...starts with a summary of the EPR argu-ment from locality to deterministic hiddenvariables. But the commentators have almost

universally reported that it begins with de-terministic hidden variables. (Bell, 1981, p.157)

Here (a footnote, but also in the main text of the ar-ticle in question) Bell goes out of his way to stress theoverall logical structure of his two-part argument: first,an argumentfromlocality and certain predictions of QM(namely, perfect anti-correlation for parallel spin mea-surements on a pair of spin 1/2 particles in the spin sin-glet state) to the existence of deterministic local hidden


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variables; and then second, from such variables to theinequality, i.e., to a disagreement with certain other pre-dictions of QM. This whole first class of disagreementwith Bell, then, rests on a simple confusion about Bellsargument. [4]

The more interesting and more subtle second classof disagreement includes those who accept that theempirically-violated Bell inequality can be derived from

Bells locality condition alone, but who argue that thislocality condition is too strong, i.e., that it smuggles insome extra requirements beyond those minimally neces-sary to respect SRs prohibition on superluminal causa-tion. At the head of this class is Jon Jarrett, whose 1983PhD thesis and subsequent 1984 paper [5] argued thatBells own local causality condition (which Jarrett callsstrong locality) is logically equivalent to the conjunc-tion of two subsidiary conditions, which Jarrett describedrespectively as locality and completeness.

Roughly speaking, Jarretts locality is the require-ment that the outcome of a measurement on one particlebe independent of the type of measurement performed (at

spacelike separation) on a second particle (which, in theinteresting sorts of cases, is described by QM as being en-tangled with the first particle). Jarretts completenesson the other hand requires the outcome of the first mea-surement to be independent of the outcome of the second,spacelike separated measurement. He then argues that aviolation of locality would entail the ability to sendsuperluminal signals (Alice could learn something aboutthe setting of Bobs measurement device by examiningthe outcome of her own experiment) which would allowBob to transmit a signal across a spacelike interval, some-thing clearly forbidden by SR. By contrast, a violationof completeness indicates no conflict with SR. Thus,Jarrett argues, in the face of the empirical data conflict-ing with strong locality we may reject completenessand thereby achieve contra Bell a kind of PeacefulCoexistence between QM and SR. [6]

Jarretts project has been widely hailed and widely dis-cussed. It was the immediate stimulus for almost every-thing in the Bell literature for about a decade afterits appearance, and continues to set a broadly influentialcontext for much ongoing work in this area. [7]

The purpose of the present paper is to critically assessJarretts analysis and the conclusions it led him to, bycomparing his project side-by-side with Bells own discus-sions of the relevant issues. In particular, I will claim thatJarrett simply missed a crucial aspect (having somethingto do with completeness) of Bells formulation of localcausality; this turns out to be the heart of the thinkingbehind Jarretts (prima facie rather puzzling) terminol-ogy for his two sub-conditions, as well as his central claimthat violations of his completeness criterion indicate noconflict with special relativistic local causality. The mainconclusion is thus that, contrary to Jarrett and his fol-lowers, Bells own local causality criterion is in no sensetoo strong. And this of course undermines the attemptto establish the Peaceful Coexistence of QM and SR, i.e.,

it supports Bells own interpretation of the meaning ofhis theorem.

The following two sections present, respectively, Bellsown (final and most careful) formulation of the localitypremise, and then Jarretts analysis. Section IV includessome comparative discussion, highlighting especially therelation of Jarretts thinking to the EPR argument. Abrief final section then summarizes and concludes.

Before launching into this, however, it is appropriateto briefly survey earlier criticisms of Jarretts analysis ofBells locality concept. To my knowledge, the fullest crit-ical discussion of Jarretts project is in Maudlins book.[8] Maudlins main points vis-a-visJarrett are as follows:(a) Jarretts identification of his locality sub-conditionwith the prohibition on superluminal signals is wrong; (b)Jarretts identification of superluminal signaling with su-perluminal causation is wrong; (c) Jarretts claim that aviolation of his completeness condition does not entailany nonlocal causation is wrong. [9]

The arguments for (a) and (b) are made clearly andcompellingly by Maudlin, who thus really demolishes Jar-

retts erroneous identification of his locality with therelevant requirements of SR. But the case for (c) is madeonly indirectly, essentially by dismissing Jarrett and re-asserting Bells claim to the contrary. So, to be a bitmore precise about the goal of the present paper, theaim is to fill this gap by exploring in detail how Jarrettscompleteness condition relates to Bells local causal-ity criterion and how Jarretts misunderstanding of thelatter led him to the various erroneous conclusions.

I should note at the outset, however, that the viewto be presented here as Jarretts is almost certainly abit misleading as to his (or those I consider his follow-ers) fully-considered views. Jarrett does actually say allthe things I attribute to him, but my gloss will perhaps

minimize the extent to which Jarrett (I would argue, in-consistently) also acknowledges points in conflict withthe views I will attribute to him. It is probably best,therefore, to understand the Jarrett discussed here asa rhetorically clarifying construct, which may or may notcorrespond to the views of the actual Jon Jarrett.

II. BELLS CONCEPT OF LOCAL CAUSALITY

Bells fullest and evidently most-considered discussionof local causality occurs in his last published paper, Lanouvelle cuisine(1990, 232-248). We will here essentiallyfollow that discussion, supplementing it occasionally withthings from his earlier papers.

Bell first introduces what he calls the Principle of lo-cal causality as follows: The direct causes (and effects)of events are near by, and even the indirect causes (andeffects) are no further away than permitted by the veloc-ity of light. Then, referencing what has been reproducedhere as Figure 1, Bell elaborates: Thus, for events in aspace-time region 1 ... we would look for causes in thebackward light cone, and for effects in the future light


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1 2

effects

causes

time

s ace

FIG. 1: Space-time location of causes and effects of events inregion 1. (Figure and caption are from Bell, 1990, p. 239.)

cone. In a region like 2, space-like separated from 1, wewould seek neither causes nor effects of events in 1. Ofcourse this does not mean that events in 1 and 2 mightnot be correlated... (1990, p. 239)

After remarking that this formulation is not yet suffi-

ciently sharp and clean for mathematics, Bell then pro-poses the following version, referencing what has beenreproduced here as Figure 2:

A theory will be said to be locally causalif the probabilities attached to values of localbeables in a space-time region 1 are unalteredby specification of values of local beables ina space-like separated region 2, when whathappens in the backward light cone of 1 isalready sufficiently specified, for example bya full specification of local beables in a space-time region 3... (1990, 239-40)

Although Bell doesnt immediately formulate this math-ematically (which is curious, since he has just advertisedit as a formulation which issufficiently sharp and cleanfor mathematics), we may do so in a way that is clearly(as evidenced by what comes later in the paper) what hehad in mind:

P(b1|B3, b2) = P(b1|B3). (1)

Herebirefers to some beable (or more precisely, its value)in region i, and Bi refers to a full specification of be-ables in region i. This simply asserts mathematicallywhat Bell states in the caption of his accompanying fig-ure: full specification of [beables] in 3 makes events

in 2 irrelevant for predictions about 1. Note that Bellhere uses the term (which he had earlier coined) beable(rhymes with agreeable) to denote whatever is posited,by the candidate theory in question, to be physically real:

The beables of the theory are those elementswhich might correspond to elements of real-ity, to things which exist. Their existencedoes not depend on observation. Indeed ob-servation and observers must be made out ofbeables. (1984, p. 174)

1 2

3

FIG. 2: Full specification of what happens in 3 makes eventsin 2 irrelevant for predictions about 1 in a locally causal the-ory. (Figure and caption are from Bell, 1990, p. 240.)

For further discussion of beables see Bells The Theoryof Local Beables(1975, pages 52-3),Beables for quantum

field theory (1984, pages 174-6), and La nouvelle cuisine(1990, pages 234-5).

Bell then adds the following clarificatory remarks:

It is important that region 3 completelyshields off from 1 the overlap of the back-ward light cones of 1 and 2. And it is im-portant that events in 3 be specified com-pletely. Otherwise the traces in region 2 ofcauses of events in 1 could well supplementwhatever else was being used for calculatingprobabilities about 1. The hypothesis is thatany such information about 2 becomes redun-dant when 3 is specified completely. (1990,p. 240)

It will be crucial to understand these remarks, so we shallbriefly elaborate.

First, suppose that the region labeled 3 in Figure 2sliced across the backwards light cone of 1 at an earliertime, such that it failed to completely shield offfrom 1

the overlap of the backward light cones of 1 and 2. See,for example, the region labeled 3 in Figure 3. Why thenwould a violation of Equation 1 (but with 3 3) fail tonecessarily indicate the presence of nonlocal causation?Suppose we are dealing with a non-deterministic (i.e., ir-reducibly stochastic, genuinely chancy) theory. And sup-pose that some event (a beable we shall call X) comesinto existence in the future of this region 3, such that itlies in the overlapping backwards light cones of 1 and 2.By assumption, the theory in question did not allow theprediction of this beable on the basis of a full specifica-tion of beables in 3. Yet once X comes into existence,it can (in a way that is perfectly consistent with localcausality) influence events in both 1 and 2. And there istherefore the possibility that specification of events from2 could allow one to infer something about X from whichone could in turn infer moreabout goings-on in 1 thanone could have inferred originally from just the full spec-ification of beables in 3. In other words (Bells): thetraces in region 2 of causes [such as our X] of events in 1could well supplement whatever else was being used forcalculating probabilities about 1. This mechanism forproducing violations of Equation 1 without any superlu-minal causation, however, will clearly not be available so


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X

1 2

3*

FIG. 3: Similar to Figure 2, except that region 3

(unlike re-gion 3 of Figure 2) fails to shield offregion 1 from the overlap-ping backward light cones of regions 1 and 2. Thus, (followingthe language of Figure 2s caption) even full specification ofwhat happens in 3 does not necessarily make events in 2irrelevant for predictions about 1 in a locally causal theory.

long as region 3 completely shields offfrom 1 the overlapof the backward light cones of 1 and 2.

Bells other clarification is also crucial. Suppose thatevents in region 3 of Figure 2 are not specified completely.We may denote such an incomplete description by B3.Then does a violation of Equation 1 (but withB3 B3)

necessarily imply the existence of any nonlocal causa-tion? No, for it would then be possible that some eventX (again in the overlapping past light cones of 1 and 2)influences both 1 and 2 such that information about 2could tell us something about X which in turn could tellus something about 1 which we couldnt infer from B3.We only need stipulate that the beables in region 3 whichcarry the causal influence from X to 1 are (among)those omitted by B3. But since there is, by definition,no such omission in the complete specification B3, thiseventuality cannot arise, and a violation of Equation 1mustindicate the existence of some nonlocal causation,i.e., causal influences not respecting Bells original Prin-

ciple of local causality (as displayed in Figure 1). [10]It is worth noting that we cannot necessarily infer,from a violation of Equation 1, that b2 exerts any di-rect or indirect causal influence on b1. It might be, forexample, that a violation of Equation 1 is produced bysome X-like event lying in the future of region 3, whichcausally influences both 1 and 2. But in order to exert alocalcausal influence on 1, such an X would have to lieoutsidethe past light cone of 2 and vice versa. Whatis ensured by a violation of Equation 1 is thus only thatsomeviolation of local causality (as sketched in Figure1) is being posited somewhere by the theory in question.Whether something in 2 is exerting a causal influence on1 (or vice versa, or neither) the mere violation of Equa-tion 1 doesnt permit us to say. [11]

Note that everything in the above discussion refers tosome particular candidate physical theory. For example,there is a tendency for misplaced skepticism to arise fromBells use of the concept of beables in the formulationof local causality. This term strikes the ears of those in-fluenced by orthodox quantum philosophy as having ametaphysical character and/or possibly committing one(already, in the very definition of what it means for a the-ory to respect relativistic local causality) to something

unorthodox like realism or hidden variables. Suchconcerns, however, are based on the failure to appreciatethat the concept beable is theory-relative. Beablerefers not to what isphysically real, but to what somecandidate theory posits as being physically real. Bellwrites: I use the term beable rather than some morecommitted term like being or beer to recall the essen-tially tentative nature of any physical theory. Such a

theory is at best a candidate for the description of na-ture. Terms like being, beer, existent, etc., wouldseem to me lacking in humility. In fact beable is shortfor maybe-able. (1984, p. 174)

Similar considerations apply to the notion of com-pleteness that is, as stressed above, essential to Bellsformulation. A complete specification of beables in somespacetime region simply means a specification of every-thing (relevant) that is posited by the candidate theoryin question. There is no presumption that such a fullspecification actually correspond to what really exists inthe relevant spacetime region, i.e., no presumption thatthe candidate theory in question is true. And the samegoes for the probabilities in Equation 1 that Bells lo-cality criterion is formulated in terms of. These shouldbe read not as empirical frequencies or subjective mea-sures of expectation, but as the fundamental dynamicalprobabilities described by the candidate theory in ques-tion (which we assume, without loss of generality, to beirreducibly stochastic). [12]

Since all the crucial aspects of Bells formulation oflocality are thus meaningful only relative to some candi-date theory, it is perhaps puzzling how Bell thought wecould say anything about the locally causal character ofNature. Wouldnt the locality condition only allow usto assess the local character of candidate theories? It isimportant to understand that the answer is essentially

(at least initially): Yes! Indeed, note that Bell beginsthe formulation with A theorywill be said to be locallycausal if... (emphasis added). Let us state it openly andexplicitly: Bells locality criterion is a way of distinguish-ing local theories from nonlocal ones:

I would insist here on the distinction be-tween analyzing various physical theories, onthe one hand, and philosophising about theunique real world on the other hand. In thismatter of causality it is a great inconveniencethat the real world is given to us once only.We cannot know [by looking] what wouldhave happened if something had been differ-ent. We cannot repeat an experiment chang-ing just one variable; the hands of the clockwill have moved, and the moons of Jupiter.Physical theories are more amenable in thisrespect. We can calculate the consequencesof changing free elements in a theory, be theyonly initial conditions, and so can explore thecausal structure of the theory. I insist that[my concept of local causality] is primarily ananalysis of certain kinds of physical theory.


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(1977, p. 101)

How then did Bell think we could end up saying some-thing interesting about Nature? That is precisely thebeauty of Bells theorem, which shows that no theoryrespecting the locality condition (no matter what otherproperties it may or may not have e.g., hidden variablesor only the non-hidden sort, deterministic or stochastic,particles or fields or both or neither, etc.) can agree

with the empirically-verified QM predictions for certaintypes of experiment. That is (and leaving aside the var-ious experimental loopholes), no locally causal theory inBells sense can agree with experiment, can be empiri-cally viable, can be true. Which means the true theory(whatever it might be) necessarily violatesBells localitycondition. Nature is not locally causal. [13]

For ease of future reference and to fix some terminol-ogy, it will be helpful to lay out here a bit more explicitlythe type of setup involved in the Bell experiments, andto indicate precisely how one gets from locality as formu-lated by Bell to the somewhat different-looking math-ematical condition (sometimes called factorizability)

from which standard derivations of Bells inequality pro-ceed.

The setup relevant to Bells theorem involves a parti-cle source which emits pairs of spin-correlated particles,and two spatially separated devices each of which allowsmeasurement of one of several spin components on therespective incident particle. (In actual experiments, theparticles are typically photons with polarization playingthe role of spin.) Two experimenters, traditionally Al-ice and Bob, man the two devices. We use the symbolsa and b to refer, respectively, to the settings of Al-ices and Bobs apparatus (one usually thinks here of anaxis in space along which the polarizer or Stern-Gerlach

magnetic field is oriented), and A and B to refer to theoutcomes of their respective spin-component measure-ments. Finally, we will use the symbol to refer to thestate of the particle pair. The scare-quotes around thevarious terms here are an advertisement for the followingdiscussion.

First, note that all of the symbols just introduced re-fer to beables. There is a tendency in the literature forall of these things (the apparatus settings, the outcomes,and the physical state of the particle pair) to remainneedlessly abstract. But all of these things are perfectlyconcrete, at least relative to some particular candidatetheory. The setting of Alices apparatus, for example,refers to something like the spatial orientation of a Stern-Gerlach device, or some sort of knob or lever on somemore black-box-ish device. Thus, this setting ulti-mately comes down to the spatial configuration of somephysically real matter, i.e., it must be reflected somehowin the beables posited by any serious candidate theory.[14]

Likewise, the outcome of Alices experiment is notsome ethereal event taking place in Alices conscious-ness or some other place about which serious candidatephysical theories fail to speak directly. Rather, the out-

1 2

3a

ba !

B3b

FIG. 4: Space-time diagram illustrating the various beablesof relevance for a discussion of factorization. Separated ob-servers Alice and Bob make spin-component measurements(using apparatus characterized by variables a and b respec-tively) of a pair of entangled spin-1/2 particles. The stateof the particles (and/or any other appropriately associatedbeables) is denoted , and the outcomes of the two measure-ments are represented by the beables A (in region 1) and B(in region 2). Note that and ajointly constitute a completespecification of beables in space-time region 3a, which shieldsoffregion 1 from the overlap of the past light cones of 1 and 2.Likewise, and b jointly constitute a complete specificationof beables in space-time region 3b, which shields offregion 2from the overlap of the past light cones of 2 and 1. Thus thejoint specification of and a will in a locally causal theory

make b and B redundant for a theorys predictions aboutA (and likewise, specification of and b will render a and Aredundant for predictions about B ).

come should be thought of as being displayed in the post-measurement position of a pointer (or the arrangementof some ink on a piece of paper, etc.) in short, theoutcome too is just a convenient way of referring to somephysically real and directly observable configuration ofmatter, and so will necessarily be reflected in the beablesposited by any serious candidate theory. The case with is a little different, because this is not something to

which we have any sort of direct observational access.But this only means there is significantly more freedomabout what sort of beables might refer to in variousdifferent candidate theories.

The basic space-time structure of the setup in questionis sketched in Figure 4, and the overall logic is explainedin the caption. The idea is simply to apply Bells localitycondition to the two measurement outcomes A and B inorder to assert that the probability assigned to a givenoutcome (by a locally causal candidate theory) shouldbe independent of both the setting and outcome of thedistant experiment. That is, in a locally causal theory,we must have

P(A|a, b,B, ) = P(A|a, ) (2)

and

P(B|a, b,A, ) = P(B|b, ). (3)

It is then trivial to apply the definition of conditionalprobability to arrive at the conclusion that the joint prob-ability (for outcomes A and B ) factorizes:

P(A, B|a, b, ) = P(A|a, ) P(B |b, ). (4)


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Bell writes: Very often such factorizability is taken asthe starting point of the analysis. Here we have preferredto see it not as the formulationof local causality, butas a consequence thereof. (1990, p. 243)

It is also important that the apparatus settings a andb are in some sense free. This is often discussed interms of Alice and Bob making literal last-minute free-will choices about how to orient their devices. What is

actually required for the proof is merely the assumptionthat aand bare (stochastically) independent of the par-ticle pair state . This comes up in the course of thederivation of Bells inequality when we write an expres-sion for predicted empirical frequency of a certain jointoutcome as a weighted average over the candidate the-orys predictions for a given that is,

E(A, B|a, b) =

d P(A, B|a, b, ) P() (5)

where P() is the probability that a given particle pairstate is produced by the preparation procedure used atthe source.

If the probability distribution P(

) actually dependedon aor b as one would expect if the particle pair exertedsome causal influence on the settings (or vice versa!) or ifthe particle pair and the settings were mutually causallyinfluenced in some non-trivial way by events farther backin the past then the above expression for the empiricalfrequency would be invalid and the derivation of the Bellinequality wouldnt go through. [15]

This might seem like cause for concern, especially con-sidering that (as displayed in Figure 4) the past light

cones of a and b overlap with the region containing and by definition is supposed to contain a completespecification of beables in this region. Given all that,

one wonders how a andb could possibly not be causallyinfluenced by (in a locally causal theory).

Here we see the reason for another aspect of Bellscarefully-phrased formulation of locality, which requiresthat beables in (the relevant) region 3 (which will ofcourse be two different regions for the two measurements)be sufficiently specified, for example by a full specifica-tion... Here there is the implication that a specificationcould be sufficient withoutbeing complete. For example,some candidate theory (and this is actually true of everyserious extant candidate theory) might provide a specifi-cation of the state of the particle pair which is sufficientin the relevant sense, even though it leaves out some fact(say, the millionth digit of the energy of some relic mi-crowave background photon that happens to fly into thedetection region just prior to the measurement) whichactually exists in the relevant spacetime region. Sucha fact could then be allowed to determine the setting awithout introducing even the slightest evidence for theproblematic sort of correlation between aand . [16] In-deed, this is just an exaggerated version of what happensin the actual experiments, where carefully-isolated andindependent pseudo-random-number generators are usedto produce the settings at the two stations.

Finally, as Shimony, Horne, and Clauser have pointedout,

In any scientific experiment in which two ormore variables are supposed to be randomlyselected, one can always conjecture that somefactor in the overlap of the backward lightcones has controlled the presumably randomchoices. But, we maintain, skepticism of thissort will essentially dismiss all results of sci-entific experimentation. Unless we proceedunder the assumption that hidden conspira-cies of this sort do not occur, we have aban-doned in advance the whole enterprise of dis-covering the laws of nature by experimenta-tion. [Hence, the extra] supposition needed[to derive Bells inequality from Bells localitycondition] is no stronger than one needs forexperimental reasoning generically, and nev-ertheless just strong enough to yield the de-sired inequality. [17]

So in the end there is really nothing worth worryingabout here, i.e., nothing which, in the face of the experi-mental data conflicting with Bells inequalities, one mightreasonably reject as an alternative to rejecting Bells localcausality. What is important is that Equation 4 (alongwith the freedom or no conspiracies assumption justdiscussed) entails the Bell inequality. The derivation isstandard and will not be repeated here. [18] We willinstead simply note that (starting in 1975) Bell almostalways referred to the empirically testable inequality asthe locality inequality. One can hopefully now appre-ciate why.

One final caveat. There is a sense in which our ver-bal description of the meanings of the relevant beables(a, b, A, B, and ) is potentially confusing. For exam-ple, there is no sense in which Bell makes some dubiousextra assumption that, e.g., particlesfly from the par-ticle source to the measurement devices. [19] A theorywhich posits no particle beables, but only (say) wavesor a wave function or whatever is perfectly fine and theformal locality criterion will apply to it just the sameway. Only the words would need to change. Relatedly,many papers in the Bell literature raise issues about hid-den variables associated not with the particle pair, butwith the measurement devices. Have we excluded suchvariables, since our arefers only to some knob setting onAlices device and refers only to the state of the particle

pair? No. The only important distinction here is that aand brefer to beables which are free (or random) inthe above sense, while the variables are somehow setby past events.

This is no doubt a fuzzy distinction, but nothing im-portant hinges on it. The point is that any microscopicfeatures of Alices device (hidden or not) or anythingelse relevant to the candidate theorys predictions for theprobabilities in question can be included under the set-ting variables (a,b)orthe particle state variables (),


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whichever seems more natural. In short, Bells formula-tion of locality is significantly more general than mightotherwise be suggested by some of the words used to de-scribe it.

III. JARRETTS ANALYSIS

Jon Jarretts influential analysis of Bells locality cri-terion begins with Equations 2 and 3, which he dubsstrong locality. (Of course, we are innocuously chang-ing and occasionally simplifying Jarretts notation tomake it consistent with that introduced above.) For sim-plicity, let us focus the discussion on Equation 2. Jarrettdefines two sub-conditions which, he subsequently proves,are jointly equivalent to this strong locality.

The first sub-condition Jarrett dubs locality. (It isalso sometimes referred to as simple locality or param-eter independence or remote context independence inthe literature.)

P(A|a, b, ) = P(A|a, ). (6)

As Jarrett explains, Locality requires that the probabil-ity for the outcome [A] ... be determined locally; i.e.,that it depend only on the state ... of the two-particlesystem and on the state [a] of the measuring device. Inparticular, that probability must be independent of which(if any) component of spin the distant measuring deviceis set to measure. [20]

Jarretts second sub-condition, which he dubs com-pleteness (and is also known as predictive complete-ness, outcome independence, remote outcome inde-pendence and conditional outcome independence) isthe following:

P(A|a, b,B, ) =P(A|a, b, ) (7)

which asserts the stochastic independence of the twooutcomes in each pair of spin measurements. [21]

It is easy to see that, indeed, locality and com-pleteness are jointly equivalent to Bells locality condi-tion as expressed in Equation 2. First, P(A|a, b,B, )is, under the assumption of completeness, equal toP(A|a, b, ). And this, in turn, is equal to P(A|a, )under the assumption of locality. So locality andcompleteness jointly entail strong locality. And like-wise strong locality clearly entails both locality andcompleteness. So the two sub-conditions are, indeed,equivalent to Equation 2.

The alleged significance of this decomposition, how-ever, emerges only from Jarretts discussion of the phys-ical interpretation of his two sub-conditions.

First, Jarrett argues that locality is equivalent tothe prohibition of superluminal signaling, and hence ex-presses just what relativity requires of other theories. Aswas mentioned in the introduction, both steps of this ar-gument have been found wanting. First, it is only in

combination with some assumptions about the control-labilityof various beables (notably ) that Jarretts lo-cality is equivalent to the prohibition on superluminalsignaling. There is at least one extant, empirically viabletheory (Bohmian Mechanics) which violates Jarretts lo-cality condition and yet doesnt permit the possibilityof superluminal signaling, precisely because the relevantstates cannot (as a matter of principle, as predicted by

the theory) be sufficiently controlled. And second, it isdubious to claim that the prohibition of superluminal sig-nals adequately captures relativitys fundamental speedlimit. This would, for example, render Bohmian Me-chanics consistent with SR despite its need to postulatea dynamically privileged reference frame a gross vio-lation of relativistic causality according to Bell. (1984,p. 171) But since these problems have been discussedelsewhere in the literature, we leave them aside here andfocus instead on Jarretts physical interpretation of hissecond sub-condition, completeness.

Jarrett elaborates the meaning of his completenesscondition with the following example:

A simple (and incorrect) model for the Bell-type correlated spin phenomena may serve asa useful illustration. Suppose, purely for thesake of illustration, that spin is correctly rep-resented as an ordinary classical angular mo-mentum. Suppose further that when a pair ofparticles is prepared in the singlet state, thespin vectors for the two particles are alignedexactly anti-parallel to each other. Moreover,given an ensemble of such two-particle sys-tems, suppose that each direction in space isequally likely to be the direction of alignmentfor an arbitrarily selected member of the en-

semble. Finally, if the unit vector [a] gives thedirection along which the axis of the Stern-Gerlach apparatus is aligned ... and if [s] isthe spin vector of the particle ... then the out-come of that measurement is +1 if [as >0]and 1 if [as


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outcome of Alices is fixed. Suppose, for example, thatB = +1. Then A = +1 is forbidden, i.e.,

P(A = +1|a, b,B, ) = 0. (9)

Comparing the previous two equations, we see that com-pleteness is clearly violated. [22]

Jarrett elaborates:

The probabilities specified for this model aregrounded in a blatantly incomplete descrip-tion of the two-particle state. In the contextof this model, if the theory assigns probabil-ities only on the basis of the occupancy bythe two-particle system of the singlet state,then conditioning on the outcome [B] of a [b]-component spin measurement on [Bobs par-ticle] may well yield a different probabilityfor the outcome of a spin measurement on[Alices particle] than would have been givenby the corresponding unconditioned proba-bility (i.e., 1/2). This is so because, if the

outcome of the measurement on [Bobs par-ticle] is +1 [and if a = b], then it may beinferred that [a sA < 0] (with probability1), where [sA] is the spin of Alices particle....The outcome of the measurement on [Bobsparticle thus] provides information about [Al-ices particle] which was not included in theincomplete state description []. [23]

The important conclusion is this: the fact that the prob-ability assigned to a certain outcome for Alices exper-iment depends, in violation of Equation 7, on the out-come of Bobs experiment, does not mean that there is

any relativistically-forbidden superluminal causal influ-ence (e.g., from Bobs outcome to Alices). That is thewhole point of the illustrative example, in which (by as-sumption) the outcome of Alices experiment is deter-mined exclusively by factors (namely a and sA) whichare present at her location. No nonlocal causal influenceexists. Instead, the violation of Equation 7 indicates onlythat we were dealing with incomplete state descriptions,such that Bobs outcome provides someinformation(use-fully supplementing what was already contained in )which warrants an updating of probabilities.

On the basis of this example, Jarrett thus urges the fol-lowing physical interpretation of his two sub-conditions:a violation of locality would allow the possibility ofsending superluminal signals, and hence indicates clearlythe existence of some relativity-violating superluminalcausal influences; on the other hand, a violation ofcompleteness does not indicate the existence of anyrelativity-violating influences, but instead suggests onlythat the state descriptions of the theory in question arenot complete. It is clear that these physical interpreta-tions of the two conditions were the basis for Jarrettsdecision to name them locality and completeness re-spectively.

Here is Jarretts summary of the cash value of this de-composition vis-a-visBells theorem and the associatedexperiments: these together provide very strong evidencethat strong locality cannot be satisfied by any empiri-cally adequate theory. Since locality is contravened onlyon pain of a serious conflict with relativity theory (whichis extraordinarily well-confirmed independently), it is ap-propriate to assign the blame to the completeness con-

dition. ...[O]ne must conclude that certain phenomenasimply cannot be adequately represented by any theorywhich ascribes properties to the entities it posits in such away that no measurement on the system may yield infor-mation which is both non-redundant (not deducible fromthe state descriptions) and predictively relevant for dis-tant measurements. That information is not (neitherexplicitly nor implicitly) contained in the incompletestate description. [24]

IV. COMPARISON

The fundamental origin of the disagreement betweenBell and Jarrett should now be clear: the two authorsdo not understand (e.g.) Equation 2 in the same way.For Bell, the variables in this formula (together with aas per the previous discussion) constitute a complete (orperhaps merely sufficient) specification of beables in somespace-time region that has the same relation to Alicesexperiment that region 3 (of Figure 1) had to region 1.Jarrett, by contrast, is agnostic about the completenessof the description afforded by .

Strictly speaking, therefore, Jarretts decomposition ofEquation 2 is not a decomposition of Bells locality con-dition, but, rather, a decomposition of some sort of no-correlation condition

P(b1|B3, b2) = P(b1|B3) (10)

which is analogous to Equation 1, except that, followingthe notation of Section II, the variables B3 are not as-sumed to provide a complete specification of beables inthe relevant spacetime region. But as we have alreadydiscussed in Section II, and as Bell was perfectly aware,a violation of Equation 10 does not necessarily indicatethe presence of any nonlocal causation in the candidatetheory in question. Indeed, it is precisely to close offthe avenue eventually taken by Jarrett (that is, blam-ing a violation of this no-correlation condition on theincompleteness of the specified beables) that Bell specifi-cally stresses the importance that events in [the relevantregion] be specified completely. (1990, p. 240)

This confusion this departure from Bells actual lo-cality criterion is the ultimate basis for Jarretts choiceof terminology, and also for any initial plausibility of hisproject of establishing Peaceful Coexistence by show-ing that a violation of the locality criterion needed forBells inequality (in particular, a violation of his com-pleteness sub-condition) need not indicate any conflictwith relativistic local causality.


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Of course, one could simply apply Jarretts decomposi-tion strategy to Bells actual locality condition. That is,it is true that, as Jarrett claimed to have shown, Bells lo-cality condition can be decomposed into two Jarrett-likesub-conditions namely, our Equations 6 and 7 but nowwith the requirement (inherited from Bells actual local-ity condition) that provide a complete (or sufficient)specification of relevant beables (as posited by some can-

didate theory whose locality is being assessed by the lo-cality condition). But thisdecomposition fails to haveany of the physical implications urged by Jarrett. In par-ticular, a violation of the (strengthened) completenesscondition

P(A|a, b,B, ) =P(A|a, b, ) (11)

(formally equivalent to Jarretts completeness sub-condition, but now, with Bell but contra Jarrett, withthe insistence that and ajointly provide a complete de-scription of beables in some spacetime region through thepast ofAwhich dividesAoffcompletely from the overlapof the past light cones ofA and B) has absolutely noth-

ing to do with the completeness of state descriptions, butinstead indicates the presence of some nonlocal causation(in violation of the causal structure outlined in Figure 1)in the candidate theory in question.

The most that could be said to distinguish the two sub-conditions is that, since b is (by definition) controllableand B (most likely) isnt, a violation of 6 is (all otherthings being equal) more likely to yield the possibility ofsuperluminal signaling than a violation of 7. But thatonly matters if we drop what Bell calls fundamentalrelativity and instead read SR instrumentally, as pro-hibiting superluminal signalling but allowing in principlesuperluminal causation (so long as it cant be harnessed

by humans to transmit messages). That is, at best, adubious and controversial reading of SR, as already men-tioned. [25]

It is sometimes suggested [26] that the relation betweenBell and Jarrett on this point is one of basic agreement,since both held that the validity of Equation 7 has some-thing to do with the completeness of the physical theoryin question. This is doubly incorrect. First, the mathe-matical condition in question is, for Jarrett, a formulationof completeness; it is, for Bell, (part of) a formulation oflocal causality which functions appropriately as suchonlyif the relevant symbols in the formula stand for a completespecification of the relevant beables. According to Jarrett,Equation 7 is supposed to tell us whether completenessholds. For Bell, on the other hand, the equation (cor-rectly understood) already presupposes that complete-ness holds. (If it doesnt hold, the condition is uselessfor his purposes, and states only that the two outcomesare uncorrelated.) And second, where Jarrett (and mostsubsequent commentators) regard the completeness inquestion as a property oftheories, Bell regards his com-pleteness as a property of a certain specification of be-ables relative to some candidate theory. In this context,the separate question of whether the theory itself is com-

plete (i.e., whether its posited beables capture everythingthat really exists) simply doesnt come up.

Note also that, even on his own terms, i.e., even leavingaside his departure from Bells actual locality criterion,Jarretts formulation of completeness actually fails as acriterion for assessing the completeness of the relevantphysical state descriptions.

Consider again Equation 7 where for the moment we

follow Jarrett and interpret here as providing some kindof state description, but not necessarily a complete one.Jarrett has shown (with the example discussed in SectionIII) that a violation of this condition can sometimes beblamed on the use of incomplete state descriptions, withno implication of any superluminal causation. But it isequally easy to display an example in which violationof Jarretts condition cannot be blamed on incompletestate descriptions, but instead indicates the presence ofsuperluminal causation.

Consider a toy model discussed by Maudlin, in whicheach particle in the pair is indeterminate (in regard to itsspin along any particular direction) until one of the par-

ticles encounters a spin-measurement apparatus; at thispoint, this first particle flips a coin to decide whetherto emerge from the +1 or the 1 port of the apparatus,and sends an instantaneous tachyon signal to the otherparticle in the pair, instructing it as to how it shouldbehave in order to give rise to the correct quantum me-chanical joint outcomes. [27] Assuming, as before, that

a=b, and that Bobs particle arrives at its detector first[28] with the result B = +1, this model makes the samepredictions, Equations 8 and 9, as the local deterministicmodel discussed by Jarrett. And so this model, like Jar-retts, violates Jarretts completeness condition. Andyet clearly with this model there is no blaming that viola-

tion on the use of incomplete state descriptions insteadhere it is obvious (by construction) that the violation isdue to the presence of superluminal causal influences (inthe particular form of tachyon signals).

This should not be surprising. We have already ar-gued that, if one follows Bell in requiring to constitutea complete state description, then a violation of Jarrettscompleteness canonlybe understood as indicating thepresence of nonlocal causation. The point here is that,even if we follow Jarrett in remaining agnostic about thecompleteness of the description afforded by , we can-not necessarilysay that a violation of completeness iscompatible with relativitys prohibition on superluminalcausation. It might be (as shown by Jarretts model).

But it might not be (as shown by Maudlins).The correct conclusion is therefore as follows: a vio-

lation of Jarretts completeness condition (where weare openly agnostic about the completeness of the statedescription ) means either that we have relativity-violating nonlocal causation, or that we were dealingwith incomplete state descriptions. And thisdilemma isprecisely that posed already in 1935 by Einstein, Podol-sky, and Rosen: eitherwe concede Bohrs claim that theQM description of states is complete and accept the re-


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ality of the nonlocal causation present in that theory; orwe reject the completeness claim and adopt a different(hidden variables) theory which (the authors thought)could restore locality. [29]

Jarrett also sees a similarity between his conclusionand that of EPR (namely that QM is not complete). Butwhere EPR considered this a defect to be corrected insome alternative hidden variable theory (which they

hoped would also restore local causality), Jarrett arguesthat incompleteness is not a defect of orthodox QM, but,rather, a fact of nature:

By separating out the relativistic compo-nent of the strong locality condition ... thereemerges a clarification of that class of theo-ries excluded by the Bell arguments: the classof theories which satisfy completeness. Al-though the term incompleteness may con-note a defect (as if, as was the case for themodel discussed [above], all incomplete theo-ries may be completed), incomplete theories(e.g., quantum mechanics) are by no means

ipso facto defective. On the contrary, whenthe result of Bell-type experiments are takeninto account, the truly remarkable implica-tion of Bells Theorem is that incomplete-ness, in some sense, is a genuine feature ofthe world itself. [30]

This sort of claim is echoed also in Jarretts later writings.[31]

Ballentine and Jarrett also give an argument that thecompleteness of interest to EPR is a stronger form ofJarretts completeness in the sense that the former en-tails the latter. (This occurs in the context of their agree-

ing with EPR that QM is incomplete, which they estab-lish by arguing that, since the correct conclusion fromBell/experiment is that Jarretts completeness condi-tion fails, then the stronger EPR completeness conditionmust also fail, just as the EPR authors claimed.) Butthis association is mistaken, since the argument Ballen-tine and Jarrett display sneaks in the additional premiseof (EPRs version of) local causality. [32] And this isthe premise that does (literally) all the work in the EPRargument.

There is thus no apparent sense at all in which EPRscompleteness has anything to do with Jarretts, exceptthat it was precisely in the face of QMs violation of Jar-retts completeness that EPR argued (correctly) thatQM was a non-local theory which, perhaps, could be re-placed by a locally causal alternative theory by addinghidden variables (or jettisoning the description in termsof wave functions entirely). Indeed, at the end of the day,it is pretty clear that Jarretts condition has nothing todo with the completeness of physical state descriptions.In concluding (from Bells Theorem and the associatedexperiments) that reality itself is in some sense incom-plete, Jarrett makes clear that he is no longer using theterm with anything like its ordinary meaning namely, a

description which leaves nothing (relevant) out. By whatstandard, exactly, could the world itself be supposedto have left something out?

It is worth stepping back, therefore, and clarifyingwhat, if anything, one can say about completeness.There are two related senses on the table. First of all,a theory may be said to be complete or incompletein relation to external physical reality. In this sense, a

theory is complete if and only if it captures or describeseverything (relevant) that in fact really exists. This isof course just the sense of completeness of interest toEPR. Their argument, in essence, was that relativisticlocal causality (which they simply took for granted) com-bined with certain empirical predictions of QM entailedthe existence of some elements of reality which had nocounterpart in the QM description of reality. That is,the QM description left something out; it was hence anincomplete theory.

Since this usage of completeness involves a compar-ison between theories and external reality (to which ourbest access is precisely through theories!), there is a ten-

dency for it to be regarded as metaphysical. Perhapsthis apparently metaphysical flavor is responsible for Jar-retts suppression of Bells requirement that containa complete specification of the relevant beables. Sincethered be no way to verify whether a given specificationwas or wasnt complete in this sense, one might think,Bells requirement is meaningless and might as well justbe dropped.

But this attitude fails to appreciate one of Bells im-portant advances namely, that his formulation of localcausality is a criterion for assessing the locality ofcandi-date theories. As already discussed in Section II, Bellscomplete specification of beables simply does not meana specification that captures everything which in fact re-ally exists; rather, it means a specification which captureseverything which is posited to exist by some candidatetheory. There is thus nothing the least bit metaphysicalor obscure about Bells requirement. For any unambigu-ously formulated candidate theory, there should be noquestion about what is being posited to exist. And sothere will be no ambiguity about what a complete de-scription of relevant beables should consist of, and henceno ambiguity about the status vis-a-vis local causality of a given well-formulated theory.

There will of course still be difficult questions abouthow to decide whether a given candidate theory is true,and hence whether the particular sortof non-local cau-sation contained in it accurately describes some aspectof Nature. But the miracle of Bells argument is that weneed not know which theory is true, in order to know thatthe true theory (whatever it turns out to be) will have toexhibit non-local, super-luminal causation. There is thusno escaping Bells conclusion that some sort of non-localcausation (in violation of the structure displayed in Fig-ure 1) exists in Nature in apparent conflict with whatmost physicists take to be the requirements of SR.


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V. DISCUSSION

In the previous section, we interpreted Jarrett as claim-ing that a violation of his completeness condition inno way implied the presence of non-local causation, butinstead only implied that the state descriptions used inthe test had been incomplete. This is both fair and un-fair fair because Jarrett does hang his entire case for

the plausibility of his terminology and his physical in-terpretation of the two sub-conditions on precisely thisview, but also unfair because Jarrett also later seems toacknowledge that, ultimately, his completeness condi-tion has to be understood very differently. For exam-ple, he remarks in a footnote that completeness, too,has the character of a locality condition. [33] Andthe trend in the Bell literature since Jarretts paper hascertainly been to concede that a violation of Jarrettscompleteness cannot be quite so trivially written off(as involving a mere updating of information in the faceof having previously used incomplete state descriptions),but rather must be understood as indicating some sort

of non-locality or holism or non-separability or non-causal passion at a distance.We do not, therefore, wish to claim that Jarrett (and

those who follow him in thinking his decomposition is insome way or other helpful in understanding Bells localitycondition and/or in establishing the peaceful coexistenceof SR and QM [34] ) fully commits precisely the mistakepresented (through the stark contrast to Bells views) inthe previous section. Rather, we intend only the weakerclaim that Jarrett et al. have been led, by Jarretts ini-tial analysis, down a path which is obviously untenableonce one clearly understands Bells own formulation oflocal causality (including especially the parallel status namely, both arebeables of settings and outcomes,and the crucial distinction between superluminal causa-tion and superluminal signaling).

It turns out to be a rather subtle question whether ornot SR genuinely requires local causality in the sense ofFigure 1. [35] But if one grants this (and virtually allphysicists and commentators do ), then it really is possi-ble to establish an

essential conflict between any sharp formu-lation [of QM] and fundamental relativity.That is to say, we have an apparent incom-patibility, at the deepest level, between thetwo fundamental pillars of contemporary the-

ory... (Bell, 1984, p. 172)The widespread claims to the contrary i.e., the claimsthat instead Bells theorem refutes only some already-dubious, dogmatic, philosophically-motivated programto restore determinism or classicality or realism(and I mean here both classes of such claims mentioned

in the introduction) turn out inevitably to have theirroots in a failure to appreciate some aspect of Bells ownarguments.

There is, in particular, a tendency for a relatively su-perficial focus on the relatively formal aspects of Bells ar-guments, to lead commentators astray. For example, howmany commentators have too-quickly breezed throughthe prosaic first section of Bells 1964 paper (p. 14-

21) where his reliance on the EPR argument fromlocality to deterministic hidden variables is made clear and simply jumped ahead to section 2s Equation 1(p. 15), hence erroneously inferring (and subsequentlyreporting to other physicists and ultimately teaching tostudents) that the derivation begins with deterministichidden variables? (1981, p. 157) Likewise, we have hereexplored in detail a similar case of too-quickly acceptingsome formal version of a premise used in Bells derivation(such as factorizability) while failing to appreciate therich conceptual context that gives it the precise meaningBell intended.

Our final conclusion, therefore, is a plea directed at

physicists in general, but commentators on Bells theo-rem, textbook writers, and students in particular tosimply read (and not just read, but read) Bells writings.They are truly a model of clarity and physical insight,and almost always convey the essential ideas much morelucidly and tersely than anything in the secondary Bellliterature. (I have no doubt this applies even to the cur-rent essay!) Bell himself, in the preface to the first edi-tion of his compiled papers (Speakable and Unspeakable inQuantum Mechanics) suggests that even quantum ex-perts might begin with [chapter] 16, Bertlmanns socksand the nature of reality, not skipping the slightly moretechnical material at the end. It is hard to disagree withthat advice, although a strong case could be made alsofor Bells 1990 essay (written after the first edition of thebook, and hence included only in the more recent secondedition) La nouvelle cuisine, in which the central impor-tance and meaning of local causality is emphasized inlucid detail.

If more physicists would only study Bells papers in-stead of relying on dubious secondary reports, theywould, I think, come to appreciate that there really ishere a serious inconsistency to worry about. A muchhigher-level inconsistency between quantum theory and(general) relativity has been the impetus, in recentdecades, for enormous efforts spent pursuing (what Bellonce referred to as) presently fashionable string theo-ries of everything. (1990, p. 235) How might a reso-lution of the more basic inconsistency identified by Bellshed light on (or radically alter the motivation and con-text for) attempts to quantize gravity? We cant possiblyknow until (perhaps long after) we face up squarely toBells important insights.

[1] John S. Bell,Speakable and Unspeakable in Quantum Me-chanics, 2nd ed., Cambridge University Press, 2004. Sub-

sequent references to Bells writings in the text will be


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given in-line with the year of the referenced paper andpage numbers from the book.

[2] For a review of recent experiments and associatedloopholes, see, e.g., Abner Shimony, Bells The-orem, The Stanford Encyclopedia of Philosophy(Fall 2006 Edition), Edward N. Zalta (ed.), URL =http://plato.stanford.edu/archives/fall2006/entries/bell-theorem/

[3] See, for example: N. David Mermin, What is quantum

mechanics trying to tell us? AmJPhys, 66(9), Septem-ber 1998, pg 753-767; Marek Zukowski, On the para-doxical book of Bell, Stud. Hist. Phil. Mod. Phys., 36(2005) 566-575; A. Zeilinger, The message of the quan-tum,Nature 438, 743 (8 December, 2005); Daniel Styer,The Strange World of Quantum Mechanics (page 42),Cambridge, 2000; George Greenstein and Arthur Zajonc,The Quantum Challenge (Second Edition), Jones andBartlett Publishers, Sudbury, Massachusetts, 2006; JohnTownsend, A Modern Approach to Quantum Mechanics,McGraw-Hill, 1992; Herbert Kroemer Quantum Mechan-ics, Prentice Hall, New Jersey, 1994; Richard Liboff, In-troductory Quantum Mechanics (2nd edition), Addison-Wesley, Reading, Massachusetts, 1992

[4] For further discussion, see any of Bells papers and,e.g.: Tim Maudlin, Quantum Non-Locality and Relativ-ity(Second Edition), Blackwell, Malden, Massachusetts,2002; Travis Norsen, Bell Locality and the NonlocalCharacter of Nature, Found. Phys. Lett., 19(7), 633-655(Dec. 2006)

[5] Jon Jarrett, On the Physical Significance of the LocalityConditions in the Bell Arguments, Nous 18 (1984) 569-589

[6] Peaceful Coexistence is Abner Shimonys term: Meta-physical problems in the foundations of quantum me-chanics, International Philosophical Quarterly18, 3-17.

[7] See, for example, M.L.G. Redhead,Incompleteness, Non-locality, and Realism: A Prolegomenon to the Philosophyof Quantum MechanicsOxford, 1987; J. Cushing and E.

McMullin, eds., Philosophical Consequences of QuantumTheory, Notre Dame, 1989 (see especially the contribu-tions of Paul Teller and Don Howard); Brandon Fogel,Formalizing the separability condition in Bells theo-rem, Stud. Hist. Phil. Mod. Phys., 38 (2007), 920-937

[8] Op cit., pp. 93-98[9] Similar points are made in Jeremy Butterfield, Bells

Theorem: What it Takes Brit. J. Phil. Sci., 43 (1992)41-83. Butterfield, however, (unlike Maudlin) ends upaccepting both the validity/meaningfulness of Jarrettsanalysis and Jarretts ultimate conclusion about Peace-ful Coexistence, even though he disagrees with some ofJarretts arguments. Since I disagree with Butterfieldsconclusions, I find him a less convincing overall critic ofJarretts project than Maudlin. My objection to Butter-

fields proposed route to Peaceful Coexistence which,like the present discussion of Jarrett, involves the claimthat the commentator has missed or misunderstood acrucial element of Bells concept of local causality willbe presented elsewhere.

[10] Note that Bell stresses the need for a complete speci-fication of beables in the relevant space-time region al-ready in his 1975 paperThe theory of local beables: Nowmy intuitive notion of local causality is that events in 2should not be causes of events in 1, and vice versa. Butthis does not mean that the two sets of events should

be uncorrelated, for they could have common causes inthe overlap of their backward light cones. It is perfectlyintelligible then that if [B3] in [our Equation 1] does notcontain a complete record of events ... it can be usefullysupplemented by information from region 2. So in gen-eral it is expected that [P(b1|B3, b2) = P(b1|B3).] How-ever, in the particular case that [B3] contains already acompletespecification of beables ... supplementary infor-mation from region 2 could reasonably be expected to be

redundant. (1975, p. 54) Emphasis in original. This is es-pecially relevant since we will eventually criticize Jarrettfor failing to appreciate (in his 1984 paper) this particu-lar aspect. So it shouldnt be thought that we are criticiz-ing him for something Bell only understood and clarifiedlater. It is worth noting, however, that there are some in-teresting differences between Bells 1975 and 1990 formu-lations of local causality; these will be explored elsewhere,though, since they do not bear on Jarretts analysis.

[11] This is why, despite being an improvement over Jarrettsterminology for the two sub-conditions to be discussedin Section III, Abner Shimonys terminology (parame-ter independence for what Jarrett calls locality andoutcome independence for what Jarrett calls com-pleteness) is also dubious. For the terminology impliesthat a violation of one of the conditions entails that theevent in question causally depends on the distant pa-rameter or outcome respectively. But this need notbe the case.

[12] Determinism is simply a special case in which all proba-bilities are either zero or unity.

[13] This sometimes comes as a shock to adherents of ortho-dox quantum theory, who are used to thinking of theirown theory especially in its allegedly relativistic vari-ants as perfectly consistent with SR. But the nonlocal-ity of orthodox QM is quite obvious, if one knows whereto look. The key here is that the theory is not definedexclusively by the Schrodinger (or equivalent) dynamicalequation, but also by some version of a collapse postu-

late. And this latter is explicitly nonlocal. Indeed, ortho-dox (collapse) QM is even more nonlocal than certainalternative theories, like Bohmian Mechanics, which areoften maligned precisely for displaying an obvious kindof nonlocality. The simplest type of example which suf-fices to make this point is the Einsteins Boxes sce-nario. (See Travis Norsen, AmJPhys 73(2), Feb. 2005,pages 164-176.) Bell explains beautifully how this sce-nario manifests the nonlocal causation inherent in or-thodox QM: Suppose ... we have a radioactive nucleuswhich can emit a single -particle, surrounded at a con-siderable distance by -particle counters. So long as it isnot specified that some other counter registers, there isa chance for a particular counter that itregisters. But ifit is specified that some other counter does register, even

in a region of space-time outside the relevant backwardlight cone, the chance that the given counter registers iszero. We simply do not have [Equation 1]. (1975, p. 55)Of course, one might contemplate an alternative theoryin which the relevant complete specification of beablesincluded something else in addition to (or instead of) theQM wave function; such an alternative may or may notexhibit any nonlocal causation. This illustrates the pointmade earlier: what we diagnose as nonlocal by apply-ing Bells criterion is some particular candidate theory(as opposed to immediately inferring, from either the em-


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pirical predictions of a theory or a given set of empiricalobservations, that some non-local causation is occuring inNature.) The point here is that orthodox QMs accountof the Einsteins Boxes scenario involves non-local cau-sation; whether any nonlocality in fact occurs in Naturewhen one performs the indicated experiment involvingan -particle, however, is a very different question. If,for example, Bohmian Mechanics (rather than orthodoxQM) is true, the answer would be no.

[14] A candidate theory which posited no beables correspond-ing to such things as knobs and levers should not, andprobably could not, be taken seriously. Bell stresses in hisvery first discussion of beables that: The beables mustinclude the settings of switches and knobs on experimen-tal equipment ... and the readings of instruments. (1975,p. 52) For elaboration of the sense of the term seriousbeing used here, see Bells (1986, pp. 194-5).

[15] See Bell, 1990, pp. 243-4.[16] See Bells discussion in his 1977, pages 100-104.[17] A. Shimony, M.A. Horne, and J.F. Clauser, An Ex-

change on Local Beables, Dialectica, 39 (1985) 86-110[18] See, e.g., Bell 1975, pages 55-57.[19] Bell, 1981, p. 150[20] Jon Jarrett,op cit., p. 573[21] Ibid., p. 578[22] Ibid., p. 580[23] Ibid., p. 580[24] Ibid., p. 585[25] See Bells discussions of controllable beables and cau-

sation vs. signaling in his 1975, pp. 60-61; 1984, p. 171;and 1990, pp. 244-246.

[26] See, e.g., p. 153 of Harvey Browns half of Nonlocalityin Quantum Mechnics, Michael Redhead and HarveyBrown, Proceedings of the Aristotelian Society, Supple-mentary Volumes, Vol. 65 (1991), pp. 119-159.

[27] See Maudlin, op cit., pg 82. After presenting this modelas a simple example of how the observed spin correla-tions might arise, Maudlin uses it to counterexample the

claim, made by Don Howard and others, that Jarrettscompleteness and locality can be mapped, respec-tively, onto the separability and locality conditionswhich emerge from some of Einsteins comments aboutlocal causality. As Maudlin points out, the toy model isperfectly separable in the sense of Einstein, and yet vio-lates what Howard et al.would have us take as a mathe-matical formulation of Einsteins separability (namely,

Jarretts completeness.) Curiously, however, Maudlindoes not mention this model in his (earlier) discussion ofJarrett.

[28] Of course, since Alices and Bobs measurements are,by hypothesis, space-like separated, there is no relativis-tically unambigous meaning to first. But that is re-ally the whole point. This model explicitly involves anti-relativistic superluminal causation, so part of the modelis that relativity is wrong and there exists some dynam-

ically privileged reference frame which gives an unam-biguous meaning to this first (and also to the instan-taneous in the description of the tachyon signal).

[29] A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be consid-ered complete? Phys. Rev. 47 (1935), 777-780. Forsome more recent discussion, see T. Norsen, EinsteinsBoxes, op cit. and references therein.

[30] Op cit., p. 585[31] L.E. Ballentine and Jon P. Jarrett, Bells theorem: Does

quantum mechanics contradict relativity? Am.J.Phys.55(8) (August 1987), 696-701; Jon Jarrett, Bells Theo-rem: A Guide to the Implications in J. Cushing and E.McMullin, eds., op cit.

[32] But this prediction was made without in any way dis-turbing particleL, since the Rdevice is at spacelike sepa-ration from it... This mistake was also noted by AndrewElby, Harvey R. Brown, and Sara Foster in What Makesa Theory Physically Complete? Found. Phys. 23(7),971-985 (1993). Note also that this error leads Ballentineand Jarrett to remark, in passing, that EPRs concept ofcompleteness entails determinism all by itself, which issurely a misrepresentation of the worries of Einstein andhis followers. See Section 1 of Bells 1981, pp. 139-145,and the important footnote 10 already partially quotedin the introduction on p. 157.

[33] Jon Jarrett, op cit., p. 589[34] For a systematic review of the recent literature, see

Berkovitz, Joseph, Action at a Distance in Quantum

Mechanics, The Stanford Encyclopedia of Philosophy(Spring 2007 Edition), Edward N. Zalta (ed.), URL =http://plato.stanford.edu/archives/spr2007/entries/qm-action-distance/

[35] Much of Maudlins excellent book, op cit., is dedicatedto exploring just this question.

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