1. Norsen. John S. Bell’s concept of local causality

8/2/2019 1. Norsen. John S. Bells concept of local causality

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John S. Bells concept of local causality

Travis Norsena)

Department of Physics, Smith College, McConnell Hall, Northampton, Massachusetts 01063

(Received 15 August 2008; accepted 6 August 2011)

John Stewart Bells famous theorem is widely regarded as one of the most important developments

in the foundations of physics. Yet even as we approach the 50th anniversary of Bells discovery, its

meaning and implications remain controversial. Many workers assert that Bells theorem refutes the

possibility suggested by Einstein, Podolsky, and Rosen (EPR) of supplementing ordinary quantumtheory with hidden variables that might restore determinism and/or some notion of an observer-

independent reality. But Bell himself interpreted the theorem very differentlyas establishing an

essential conflict between the well-tested empirical predictions of quantum theory and relativistic

local causality. Our goal is to make Bells own views more widely known and to explain Bells little-

known formulation of the concept of relativistic local causality on which his theorem rests. We also

show precisely how Bells formulation of local causality can be used to derive an empirically testable

Bell-type inequality and to recapitulate the EPR argument.VC 2011 American Association of Physics Teachers.

[DOI: 10.1119/1.3630940]

I. INTRODUCTION

In its most general sense, local causality is the idea thatphysical influences propagate continuously through spacethat what Einstein famously called spooky actions at a dis-tance are impossible.1 In addition to originating this catchyphrase, Einstein was chiefly responsible for the relativisticsense of local causality, according to which causal influencesshould not only propagate continuously (never hoppingacross a gap in which no trace is left) but also do so alwaysat the speed of light or slower. The elaboration and formula-tion of this idea will be our central concern.

The pre-relativistic no action at a distance sense of localcausality has played an important role in the constructionand assessment of theories throughout the history ofphysics.2,3 For example, some important objections to New-

tons theory of gravitation centered on the theorys allegedpositing of non-local action at a distance. Newtons ownview seems to have been that although his theory claimed(for example) that the Sun exerted causal influences on thedistant planets, this influence was consistent with local cau-sality, which he strongly endorsed:

It is inconceivable that inanimate brute mattershould, without the mediation of something elsewhich is not material, operate upon and affectother matter without mutual contact Thatgravity should be innate, inherent, and essential tomatter, so that one body may act upon another at adistance through a vacuum, without the mediation

of anything else, by and through which their actionand force may be conveyed from one to another, isto me so great an absurdity that I believe no manwho has in philosophical matters a competentfaculty of thinking can ever fall into it.

4

Newtons idea was evidently that his gravitational theorydidnt provide a complete description of the underlying (andpresumably local) mechanism by which [massive bodies]action and force may be conveyed from one to another.5

Such debates had a philosophical character because at thetime nothing was unambiguously excluded by the require-ment of locality. Any apparent action at a distance in a

theory could be rendered compatible with local causality byfollowing Newton and by denying that the theory in question

provided a complete description of the relevant phenomena.This changed in 1905 with Einsteins discovery of special

relativity, which for the first time identified a class of causalinfluencesthose that propagate faster than lightas incon-sistent with local causality. As Einstein explained,

The success of the Faraday-Maxwell interpreta-tion of electromagnetic action at a distance resultedin physicists becoming convinced that there are nosuch things as instantaneous action at a distance(not involving an intermediary medium) of thetype of Newtons law of gravitation. According tothe theory of relativity, action at a distance withthe velocity of light always takes the place of in-

stantaneous action at a distance or of action at adistance with an infinite velocity of transmission.This is connected with the fact that the velocity cplays a fundamental role in the theory.6

The speed of light c plays a fundamental role regarding cau-sality because of the relativity of simultaneity. For two events

A and B with space-like separation (that is, such that a signalconnecting A and B would have to propagate faster than c),the time ordering is ambiguous: different inertial observerswill disagree about whether A precedes B in time or viceversa. According to special relativity, there is no objectivefact about which event occurs first, and hence no possibility ofa causal relation between them, because the relation between

a cause and its effect is necessarily time-asymmetric. As Bellexplained, To avoid causal chains going backward in time insome frames of reference, we require them to go slower thanlight in any frame of reference.7,8

After the advent of special relativity, the relativistic senseof local causality was soon used to critique other developingtheories, much as the pre-relativistic concept had been usedagainst Newtons theory. Indeed, it was Einstein himselfinboth the Einstein, Podolsky, and Rosen (EPR) paper9 andseveral related but less widely known arguments10whofirst pointed out that Copenhagen quantum theory violatedspecial relativitys locality constraint. According to Einstein,that theorys account of measurement combined with Bohrs

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completeness doctrine committed the theory to the sort ofnon-local causation which was, according to Einstein, pro-hibited by special relativity. Einstein thus rejected Bohrscompleteness doctrine and supported something like what isnow (unfortunately11) called the local hidden variablesprogram.

Note the parallel to Newtonian gravity, with the non-locality in a candidate theory being rendered as either real ormerely apparent, depending on whether or not we interpret

the theory as providing a complete description of the physi-cal processes in question. Einsteins assessment of Copenha-gen quantum theory with regard to local causality thusparallels Newtons analysis of his own theory of gravitation:the theory, if regarded as complete, violates locality, andhence upholding locality requires denying completeness.

This brings us to the main subject of the paper: the workof J. S. Bell. Bell accepted Einsteins proof of the non-locality of Copenhagen quantum theory. In particular, Bellaccepted as valid the EPR argument from locality to deter-ministic hidden variables.13 This argument involves a pairof specially prepared particles that are allowed to separate toremote locations. An observation of some property of oneparticle permits the observer to learn something about a cor-

responding property of the distant particle. According to theCopenhagen view, the distant particle fails to possess a defi-nite value for the property in question prior to the observa-tion; it is precisely the observation of the nearby particlewhichin apparent violation of local causalitytriggers thecrystallization of this newly real property for the distantparticle.

In Bells recapitulation of the argument, for EPR this

showed that [Bohr, Heisenberg, and Jordan] hadbeen hasty in dismissing the reality of themicroscopic world. In particular, Jordan had beenwrong in supposing that nothing was real or fixedin that world before observation. For after

observing only one particle the result ofsubsequently observing the other (possibly at avery remote place) is immediately predictable.Could it be that the first observation somehowfixes what was unfixed, or makes real what wasunreal, not only for the near particle but also forthe remote one? For EPR that would be anunthinkable spooky action at a distance. To avoidsuch action at a distance [one has] to attribute, tothe space-time regions in question, real propertiesin advance of observation, correlated properties,which predetermine the outcomes of these particu-lar observations. Because these real properties,fixed in advance of observation, are not contained

in [the] quantum formalism, that formalism forEPR is incomplete. It may be correct, as far as itgoes, but the usual quantum formalism cannot bethe whole story.13

Bell thus agreed with Einstein that the local hidden varia-bles program constituted the only hope for a locally causalre-formulation of quantum theory. Bells historic contribu-tion was a theorem establishing that no such local hiddenvariable theoryand hence no local theory of any kindcould generate the correct empirical predictions for a certainclass of experiments.14 According to Bell, we must thereforeaccept the real existence of faster-than-light causation and

hence an apparent conflict with the requirements of specialrelativity: For me then this is the real problem with quan-tum theory: the apparently essential conflict between anysharp formulation and fundamental relativity. That is to say,we have an apparent incompatibility, at the deepest level,between the two fundamental pillars of contemporarytheory.15

Bell even suggested, in response to his theorem and rele-vant experiments,16,17 the rejection of fundamental rela-

tivity and the return to a Lorentzian view in which there is adynamically privileged, though probably empirically unde-tectable, reference frame: It may well be that a relativisticversion of [quantum] theory, while Lorentz invariant andlocal at the observational level, may be necessarily non-localand with a preferred frame (or aether) at the fundamentallevel.18 And elsewhere he remarked:

I would say that the cheapest resolution issomething like going back to relativity as it wasbefore Einstein, when people like Lorentz andPoincare thought that there was an aetherapreferred frame of referencebut that ourmeasuring instruments were distorted by motion in

such a way that we could not detect motionthrough the aether. Now, in that way you canimagine that there is a preferred frame ofreference, and in this preferred frame of referencethings do go faster than light. Behind theapparent Lorentz invariance of the phenomena,there is a deeper level which is not Lorentzinvariant [This] pre-Einstein position of Lorentzand Poincare, Larmor and Fitzgerald, was perfectlycoherent, and is not inconsistent with relativitytheory. The idea that there is an aether, and theseFitzgerald contractions and Larmor dilations occur,and that as a result the instruments do not detectmotion through the aetherthat is a perfectly

coherent point of view.19,20

Our intention is not to argue for this radical view, but toexplain Bells rationale for contemplating it. This rationaleinvolves a complex chain of reasoning involving at leastthese four steps: (1) arguing that special relativity prohibitscausal influences between space-like separated events, (2)constructing a precise formulation of this prohibition, that is,of relativistic local causality, (3) deriving of an empiricallytestable inequality from this formulation of local causality,and (4) establishing that the inequality is inconsistent withempirical data.

There is an extensive literature in which each of thesesteps is subjected to a critical analysis. The time-asymmetric

character of causal relations, which was used in the argumentfor (1) that we have sketched, has, for example, been chal-lenged by Price21 and (in a very different way, based on ear-lier work by Bell22) Tumulka.23 And there remain loopholesin the experiments which test Bells inequality, such that onemight conceivably doubt claim (4).24 But for the most part,physicists do not seriously question (1) and regard (4) ashaving been established with reasonable conclusiveness. Thecontroversies about the meaning and implications of Bellstheorem have thus centered on (2) and (3).

But what is said about (3)the question of whether and howa Bell-type inequality is entailed by local causalitydepends onwhether and how (2) has been addressed. And sadly, Bells own

1262 Am. J. Phys., Vol. 79, No. 12, December 2011 Travis Norsen 1262

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views on (2) have been almost invisible in the literature. (Ref.24, for example, does not acknowledge Bells formulation oflocal causality, and instead proposes an alternative formulationvery different from Bells.) It is thus not surprising that manyhave summarized the implications of Bells theorem in waysvery different from Bells own. Usually, it is claimed that Bellsinequality follows not from local causality alone, but from theconjunction of local causality with some additional assumptionsuch as realism or determinism. One or more of these

assumptions, rather than relativistic local causality, is then typi-cally blamed for the inconsistency with experiment.2532

The bulk of our discussion will focus on Bells formulationof local causality, that is, his views on (2). This discussion willbe based primarily on Ref. 7, published in 1990, the same yearas his untimely death. Explaining Bells formulation of localitywill require also sketching Bells interesting and refreshinglyunorthodox views on several related issues in the foundationsof quantum theory. The discussion will be elaborated and sup-ported with excerpts from Bells many other papers.

The main audience for the paper is readers with little or noprior knowledge of Bells theorem beyond what they have readin textbooks. Almost all of the issues raised are discussedbecause some kind of misunderstanding or ignorance of them is

present in the literature. We will provide occasional citations toworks that exemplify the various important misunderstandings.But length considerations and the desire to keep the paper self-contained do not allow any extensive polemical discussions.

The paper is organized as follows. In Sec. II, we jumpquickly from some of Bells preliminary, qualitative state-ments to his quantitative formulation of relativistic local cau-sality. In Secs. IIIV, we clarify some controversial orunfamiliar terms that appear in Bells formulation and contrastthem to other ideas with which they have sometimes beenconfused. Section VI shows how local causality as formulatedby Bell can be used to derive an empirically testable Bell-typeinequality, and how it can be used to recapitulate the EPRargument in a rigorous way. In Sec. VII, we will summarize

the arguments presented and acknowledge some of the limita-tions and open questions regarding Bells formulation.

II. LOCAL CAUSALITY: OVERVIEW

We begin with a qualitative formulation of Bells conceptof local causality. In answer to an interview question aboutthe meaning of locality, Bell responded:33

Its the idea that what you do has consequencesonly nearby, and that any consequences at a distantplace will be weaker and will arrive there only afterthe time permitted by the velocity of light. Localityis the idea that consequences propagate

continuously, that they dont leap over distances.34

Bell gave a more careful but still qualitative formulationof what he called the principle of local causality in 1990:The direct causes (and effects) of events are near by, andeven the indirect causes (and effects) are no further awaythan permitted by the velocity of light.7 Then, citing a fig-ure which is reproduced here in Fig. 1, Bell continues:

Thus for events in a space-time region 1 wewould look for causes in the backward light cone,and for effects in the future light cone. In a regionlike 2, space-like separated from 1, we would seekneither causes nor effects of events in 1.7

This formulation should be uncontroversial. Bell noted,however, that [t]he above principle of local causality is notyet sufficiently sharp and clean for mathematics.7

Here is Bells sharpened formulation. (The reader shouldunderstand that this formulation is, at this point, a teaserwhich those to whom it is not familiar should expect tounderstand only after further reading.)

A theory will be said to be locally causal if theprobabilities attached to values of local beables ina space-time region 1 are unaltered by specifica-tion of values of local beables in a space-like sepa-rated region 2, when what happens in the backwardlight cone of 1 is already sufficiently specified, forexample by a full specification of local beables ina space-time region 37

The space-time regions referred to are illustrated in Fig. 2.We can express Bells formulation mathematically as

Pb1jB3; b2 Pb1jB3; (1)where bi refers to the value of a particular beable in space-time region i and Bi refers to a sufficient (for example, acomplete) specification of all beables in the relevant region.(See Sec. III for the meaning of beable.) P is the probabil-ity assigned to event b1 by the theory in question, condi-tioned on the information specified after the vertical bar.Equation (1) captures just what Bell states in the caption ofhis accompanying figure (see Fig. 2): full specification of[beables] in 3 makes events in 2 irrelevant for predictionsabout 1 in a locally causal theory.7

III. BEABLES

The first question about the word beable is: how to pro-nounce it? The word does not rhyme with feeble, but withagreeable. Bell invented the word as a contrast to theobservables which play a fundamental role in the formula-tion of orthodox quantum theory.

FIG. 1. Space-time location of causes and effects of events in region 1.

FIG. 2. Full specification of what happens in 3 makes events in 2 irrelevant

for predictions about 1 in a locally causal theory.


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A. Beables versus observables

Beables are those elements of a theory that are supposedto correspond to something that is physically real, independ-ent of any observation: The beables of the theory are thoseelements which might correspond to elements of reality, tothings which exist. Their existence does not depend on ob-servation. Indeed observation and observers must be madeout of beables.

35As Bell explained,

The concept of observable is a rather woollyconcept. It is not easy to identify precisely whichphysical processes are to be given the status ofobservations and which are to be relegated to thelimbo between one observation and another. So itcould be hoped that some increase in precisionmight be possible by concentration on thebeables because they are there.36

Bells reservations about the concept of observableappearing in the formulation of a fundamental theory areclosely related to the measurement problem of orthodoxquantum mechanics, which Bell encapsulated by remarkingthat the orthodox theory is unprofessionally vague andambiguous35 in so far as its fundamental dynamics isexpressed in terms of words which, however legitimate andnecessary in application, have no place in a formulation withany pretension to physical precision.37 As Bell elaborated,

The concepts system, apparatus, environment,immediately imply an artificial division of theworld, and an intention to neglect, or take onlyschematic account of, the interaction acrossthe split. The notions of microscopic andmacroscopic defy precise definition. So also dothe notions of reversible and irreversible.Einstein said that it is theory which decides what isobservable. I think he was rightobservation isa complicated and theory-laden business. Then the

notion should not appear in the formulation of fun-damental theory.37

As Bell pointed out, even Bohr (a convenient personifica-tion of skepticism regarding the physical reality of unobserv-able microscopic phenomena) recognized certain objects (forexample, the directly perceivable states of a classical meas-uring apparatus) as unambiguously real, that is, as beables:

The terminology, be-able as against observ-able,is not designed to frighten with metaphysic thosededicated to realphysic. It is chosen rather to helpin making explicit some notions already implicitin, and basic to, ordinary quantum theory. For, inthe words of Bohr, it is decisive to recognize that,

however far the phenomena transcend the scope ofclassical physical explanation, the account of allevidence must be expressed in classical terms. It isthe ambition of the theory of local beables to bringthese classical terms into the equations, and notrelegate them entirely to the surrounding talk.36

The vagueness and ambiguity of orthodox quantum theoryis related to the fact that its formulation presupposes theseclassical, macroscopic beables, but fails to provide clearlaws to describe them. As Bell explained,

The kinematics of the world, in [the] orthodoxpicture, is given by a wavefunction for the

quantum part, and classical variablesvariableswhich have valuesfor the classical part [with theclassical variables being] somehow macroscopic.This is not spelled out very explicitly. The dynamicsis not very precisely formulated either. It includes aSchrodinger equation for the quantum part, andsome sort of classical mechanics for the classicalpart, and collapse recipes for their interaction.37

There are thus two related problems. First, the world as

described by the theory is different on the two sides of whatBell called the shifty split37that is, the division betweenthe quantum part and the classical part. The nature ofthe world posited by the theory thus remains vague as longas the dividing line between the macroscopic and micro-scopic remains undefined. Also the interaction across thesplit is problematic. Not only is the account of this dynamics(the collapse process) inherently bound up in conceptsfrom Bells list of dubious terms, but also the existence of aspecial dynamics for the interaction seems to imply inconsis-tencies with the dynamics already posited for the two realmsseparately. As Bell summarized,

I think there are professional problems [with

quantum mechanics]. That is to say, Im aprofessional theoretical physicist and I would like tomake a clean theory. And when I look at quantummechanics I see that its a dirty theory. Theformulations of quantum mechanics that you find inthe books involve dividing the world into anobserver and an observed, and you are not toldwhere that division comes So you have a theorywhich is fundamentally ambiguous

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This discussion should clarify the sort of theory Bell hadin mind as satisfying the relevant standards of professional-ism. It is often thought by those who do not understand or donot accept Bells criticisms of orthodox quantum theory, thatthe concept of beable, in terms of which his concept oflocal causality is formulated, commits one to hidden varia-bles or determinism or some sort of naive realism or someother physically or philosophically dubious principle. Butthis is not correct. The requirement is only that fundamentaltheories, those with any pretension to physical precision,37

be formulated clearly and precisely. According to Bell, suchclarity and precision requires that the theories provide a uni-form and consistent candidate description of physical reality.In particular, there should be no ambiguity or inconsistencyregarding what a theory is fundamentally about (the beables),nor regarding how those posited physically real elements areassumed to act and interact (the laws).

B. Beables versus conventionsSo far, we have explained the term beable by contrast-

ing it to the observables of orthodox quantum theory. Wemust now also contrast the concept of beables with thoseelements of a theory which are conventional:

The word beable will also be used here to carryanother distinction, that familiar already inclassical theory between physical and non-physical quantities. In Maxwells electromagnetictheory, for example, the fields E and H are physi-cal (beables, we will say) but the potentials A andu are non-physical. Because of gauge invariance


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the same physical situation can be described byvery different potentials. It does not matter [that is,it is not a violation of local causality] that in Cou-lomb gauge the scalar potential propagates with in-finite velocity. It is not really supposed to be there.It is just a mathematical convenience.36

Or, as Bell explained it in another paper,

there are things which do go faster than light.

British sovereignty is the classical example. Whenthe Queen dies in London (long may it be delayed)the Prince of Wales, lecturing on modernarchitecture in Australia, becomes instantaneouslyKing And there are things like that in physics. InMaxwells theory, the electric and magnetic fieldsin free space satisfy the wave equation

1

c2@2E

@t2 r2E 0

1

c2@2B

@t2 r2B 0

corresponding to propagation with velocity c.But the scalar potential, if one chooses to work inthe Coulomb gauge, satisfies Laplaces equation

r2/ 0

corresponding to propagation with infinitevelocity. Because the potentials are onlymathematical conveniences, and arbitrary to a highdegree, made definite only by the imposition ofone convention or another, this infinitely fastpropagation of the Coulomb-gauge scalar potentialdisturbs no one. Conventions can propagate as fastas may be convenient. But then we must distin-

guish in our theory between what is conventionand what is not.7

Consequently, to decide whether a given theory is or is notconsistent with local causality,

you must identify in your theory local beables.The beables of the theory are those entities in itwhich are, at least tentatively, to be takenseriously, as corresponding to something real. Theconcept of reality is now an embarrassing one formany physicists. But if you are unable to givesome special status to things like electric andmagnetic fields (in classical electromagnetism), ascompared with the vector and scalar potentials,

and British sovereignty, then we cannot begin aserious discussion.7

This explains why, for Bell, It is in terms of local beablesthat we can hope to formulate some notion of localcausality.36

C. Beables and candidate theories

It is important to appreciate that a beable is only a beablerelative to some particular candidate theory which positsthose elements as physically real (and gives precise lawsfor their dynamics). For example, the fields E and B (and

not the potentials) are beables according to classical Max-wellian electrodynamics as it is usually understood. But, wecould imagine an alternative theory which (perhaps moti-vated by the Aharanov-Bohm effect) posits the Coulombgauge potentials as beables instead. Although this alterna-tive theory would be empirically and mathematically equiv-alent to the usual theory, they would not have the samestatus regarding local causality. Where the usual Maxwel-lian theory respects local causality, the alternative theory

would violate it: wiggling a charge would instantaneouslyaffect the physically-real scalar potential at distantlocations.

Thinking in terms of such candidate theories helps us toseparate questions about what the real beables arewhatreally exists out there in physical realityinto two parts:what elements does a candidate theory posit as beables, andwhich candidate theory do we think is true? The point is thatyou do not have to be able to answer the second part to an-swer the first. This should provide some comfort to thosewho think we cannot establish a theoretical picture of exter-nal reality as true. Such people may still accept Bells char-acterization of when a theory will be said to be locallycausal.7

But even those who are not skeptical on principle recog-nize that, because of the complexity in practice of settlingquestions about the truth status of scientific theories, sometentativeness is often in order. Bell recognizes this too:

I use the term beable rather than some morecommitted term like being or beer to recall theessentially tentative nature of any physical theory.Such a theory is at best a candidate for thedescription of nature. Terms like being, beer,existent, etc., would seem to me lacking inhumility. In fact beable is short for maybe-able.35

The crucial point is that maybe pertains to the epistemo-

logical status of a given candidate theory. In contrast, thebeable status36 of certain elements of a theory should bestraightforward and uncontroversial. If there is any questionabout what elements a theory posits as beables, it can onlybe because the theory has not (yet) been presented in a suffi-ciently clear way. Whether the theory is true or false is a dif-ferent question.

Bell did, however, take certain elements largely forgranted as beables that any serious candidate theory wouldhave to recognize as such: The beables must include the set-tings of switches and knobs on experimental equipment, thecurrents in coils, and the readings of instruments.36 Asnoted, even Bohr acknowledged the real existence (thebeable status) of these sorts of things. And as suggested by

Bohr, because our primary cognitive access to the world isthrough switches and knobs on experimental equipmentand other such directly observable facts, it is difficult toimagine how one might take seriously a theory which didntgrant such facts beable status.

38

We stress this point for two related reasons. First, anyonewho is uncomfortable with the metaphysical positing ofultimate elements of reality should be relieved to find thatthe concept beable is merely a placeholder for whateverentities we tentatively include in the class which already, bynecessity, exists and includes certain basic, directly perceiva-ble features of the world around us. And second, these partic-ular beables, for example, the settings of knobs and the



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positions of pointers, have a particularly central role to playin the derivation (see Sec. VI) of the empirically testableBell inequalities.

IV. COMPLETENESS

We now turn to the last phrase in Bells formulation oflocal causality:

A theory will be said to be locally causal if the

probabilities attached to values of local beables ina space-time region 1 are unaltered by specifica-tion of values of local beables in a space-like sepa-rated region 2, when what happens in the backwardlight cone of 1 is already sufficiently specified, forexample by a full specification of local beables ina space-time region 37

The key assumption here is that events in 3 be specifiedcompletely7 (emphasis added).

Let us first see why this requirement is necessary. Supposethat B3 denotes an incomplete specification of beables inregion 3 (see Fig. 2). It can be seen that a violation of

Pb1j

B3; b2 Pb1j

B3 (2)does not entail the existence of any super-luminal causalinfluences. Suppose an event X in the overlapping backwardlight cones of regions 1 and 2 causally influences both b1 andb2. It might then be possible to infer from b2, somethingabout X, from which we could in turn infer something aboutb1. Suppose, though, that the incomplete description ofevents in region 3, B3, omits precisely the traces of thispast common cause X. Then, b2 could usefully supplementB3 for predictions about 1; that is, Eq. (2) could be violatedeven in the presence of purely local causation.

Thus, as Bell explained, for Eq. (1) to function as a validlocality criterion,

it is important that events in 3 be specifiedcompletely. 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 that anysuch information about 2 becomes redundant when3 is specified completely.7

Or as Bell explained in an earlier paper:

Now my intuitive notion of local causality is thatevents in 2 should not be causes of events in 1,and vice versa. But this does not mean that the twosets of events should be uncorrelated, for theycould have common causes in the overlap of theirbackward light cones [in a local theory]. It isperfectly intelligible then that if [B3] in [region 3]does not contain a complete record of events inthat [region], it can be usefully supplemented byinformation from region 2. So in general it isexpected that [Pb1jb2; B3 6 Pb1j B3.] However,in the particular case that [B3] contains already acomplete specification of beables in [region 3],supplementary information from region 2 couldreasonably be expected to be redundant.36

Like the concept of beables itself, the idea of a sufficient(full or complete) specification of beables is relative to a

given candidate theory. Bells local causality conditionrequires that, to assess the consistency between a giventheory and the relativistic causal structure sketched in Fig. 1,we must include in B3 everything that the theory says is pres-ent (or relevant) in region 3. It is not necessary that weachieve omniscience regarding what actually exists in somespacetime region.

The appearance of the word completeness often remindspeople of the EPR argument and suggests to some that Bell

smuggled into his definition of local causality the unwar-ranted assumption that orthodox quantum theory is incom-plete (see, for example, Ref. 39). As mentioned, Bell didaccept the validity of the EPR argument. But this acceptancemeans only that, according to Bell, local causality (togetherwith some of quantum mechanics empirical predictions)entails the incompleteness of orthodox quantum mechanics.His view on this point is, however, no part of his formulationof local causality.

Although it is simplest to understand Bells local causalitycondition as requiring a complete specification of beables insome spacetime region, there is an important reason whyBell explicitly left open the possibility that what happens inthe backward light cone of 1 might be sufficiently spec-

ified by something less than a complete specification of thebeables there. This has to do with the fact (see Sec. VI) thatto derive an empirically testable Bell-type inequality fromthe local causality condition, a subsidiary assumption isneeded, sometimes called experimental freedom or noconspiracies. This is in essence the assumption that, in theusual EPR-Bell kind of scenario in which a central sourceemits pairs of specially prepared particles in opposite direc-tions toward two spatially separated measuring devices, it ispossible for certain settings on the devices (determiningwhich of several possible measurements are made on a givenincoming particle) to be made freely or randomlythatis, independently of the state of the incident particle pair.

In more recent versions of these experiments, the relevant

settings are made using independent (quantum) random num-ber generators.17 According to orthodox quantum mechanics,the outputs of such devices are genuinely, irreducibly ran-dom. Thus, for orthodox quantum mechanics, there is noth-ing in the past light cone of an individual measurement eventforetelling which of the possible measurements will be per-formed. But there exist alternative candidate theories (suchas the de Broglie-Bohm pilot-wave theory, which is deter-ministic) according to which those same settings are influ-enced by events in their past. But then, the relevant pasts ofthe device settings necessarily overlap with the pre-measurement states of the particles being measured. A com-plete specification of beables in the relevant region contain-ing those pre-measurement states will therefore inevitably

include facts relevant to (if not determining) the device set-tings. And so, in deriving the Bell inequality from local cau-sality, there is a subtle tension between the requirement thatevents in 3 be specified completely7 and the requirementthat device settings can be made independent of the states ofthe particles-to-be-measured.

To resolve the tension, we need merely allow that thebeables in the relevant region can be divided up into disjointclasses: those that are influenced by the preparation proce-dure at the source (and which thus encode the state of theparticle pair) and those that are to be used in the setting ofthe measurement apparatus parameters. Note that these twoclasses are likely to be far from jointly exhaustive: in any



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plausible candidate theory, there will have to exist manyadditional beables (corresponding, for example, to stray elec-tromagnetic fields and low energy relic neutrinos) which arein neither of these classes. We thus expect a considerablecausal distance between the two classes of beables (at leastin a well-designed experiment). Such distance makes thefreedom or no conspiracies assumptionnamely, theabsence of correlations between the two classes of beablesreasonable to accept.

This issue will be addressed in some more detail in Sec.VI. For now, we acknowledge its existence as a way ofexplaining why Bells formulation of local causality men-tions complete descriptions of events in region 3 as merelyan example of the kind of description which is sufficient.We summarize the discussion here as follows: what isrequired for the validity of the local causality condition is acomplete specification of beables in region 3but only thosebeables that are relevant in some appropriate sense to theevent b1 in question in region 1.

V. CAUSALITY

Recall the transition from Bells preliminary, qualitative

formulation of local causality to the final version. And recall,in particular, Bells statement that the preliminary versionwas insufficiently sharp and clean for mathematics. What didBell consider inadequate about the qualitative statement? Itseems likely that it was the presence of the terms causeand effect, which are notoriously difficult to define mathe-matically. About his final formulation Bell wrote: Note, bythe way, that our definition of locally causal theories,although motivated by talk of cause and effect, does notin the end explicitly involve these rather vague notions.7

How does Bells definition of locally causal theories failto explicitly involve the rather vague notions of causeand effect? On its face, this sounds paradoxical. But the reso-lution is simple: what Bells definition actually avoids is any

specific commitment about what physically exists and how itacts. (Any such commitments would seriously restrict thegenerality of the locality criterion, and hence undermine thescope of Bells theorem.) Instead, Bells definition shifts theburden of providing some definite account of causal proc-esses to candidate theories and merely defines a space-timeconstraint that must be met if the causal processes posited bya candidate theory are to be deemed locally causal in thesense of special relativity.

The important mediating role of candidate theories regard-ing causality will be further stressed and clarified in Sec. VA. We then further clarify the concept of causality inBells local causality by contrasting it with several otherideas with which it has often been confused or conflated.

A. Causality and candidate theories

As discussed, according to Bell it is the job of physicaltheories to posit certain physically real structures (beables)and laws governing their evolution and interactions. Thus,Bells definition of locally causal theories is not a specifica-tion of locality for a particular type of theory, namely, thosethat are causalwith the implication that there would existalso theories that are non-causal. A theory, by the very na-ture of what we mean by that term in this context, is auto-matically causal. Causal theory is a redundancy. And so,as noted, we must understand Bells definition of locally

causal theories as a criterion that theories, that is, candidatedescriptions of causal processes in nature, must satisfy to bein accord with special relativistic locality. As Bell explained,the practical reason for defining local causality in terms ofthe physical processes posited by some candidate theory (incontrast to the physical processes that actually exist in na-ture) has to do with our relatively direct access to the one asopposed to the other:

I would insist here on the distinction between

analyzing various physical theories, on the onehand, and philosophising about the unique realworld on the other hand. In this matter of causalityit is a great inconvenience that the real world isgiven to us once only. We cannot know whatwould have happened if something had beendifferent. We cannot repeat an experimentchanging 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 consequences ofchanging free elements in a theory, be they onlyinitial conditions, and so can explore the causalstructure of the theory. I insist that [my

formulation of the local causality concept] isprimarily an analysis of certain kinds of physicaltheory.40

Bells view, contrary to several commentators,41,42 is thatno special philosophical account of causation is needed towarrant the conclusion that violation of the locality conditionimplies genuine non-local causation. For Bell, it is a trivialmatter to decide, for some (unambiguously formulated) can-didate physical theory, what is and is not a causal influence.We can simply explore the causal structure of the candi-date theory. This raises the question of how we might gofrom recognizing the non-locality of some particular candi-date theory to the claim that nature is non-local. But that is

precisely Bells theorem: all candidate theories whichrespect the locality condition are inconsistent with experi-ment (see Sec. VI). Therefore, the one true theory (what-ever that turns out to be) and hence nature itself must violaterelativistic local causality.

B. Causality versus determinism

Section V A stressed that the causal in locally causaltheories simply refers to the physically real existents andprocesses (beables and associated laws) posited by some can-didate theory, whatever those might be. We in no wayrestrict the class of theories (whose locality can be assessed

by Bells criterion) by introducing causality. In particular,the word causal in locally causal theories is not meant toimply or require that theories be deterministic in contrast toirreducibly stochastic:

We would like to form some [notion] of localcausality in theories which are not deterministic, inwhich the correlations prescribed by the theory, forthe beables, are weaker.36

Bell thus deliberately used the word causal as a widerabstraction that subsumes but does not necessarily entaildeterminism. This use is manifested most clearly in the factthat Bells mathematical formulation of local causality,



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Eq. (1), is stated in terms of probabilities. In Ref. 36, Belldiscussed local determinism first, arguing that, in a localdeterministic theory, the actual values of beables in region1 (of Fig. 2) will be determined by a complete specificationof beables in region 3 (with additional specification ofbeables from region 2 being redundant). In our notation,local determinism means

b1B3; b2 b1B3; (3)

where b1 and b2 are the values of specific beables in regions1 and 2, and B3 denotes a sufficient (for example, complete)specification of beables in region 3.

In a (local) stochastic theory, even a complete specificationof relevant beables in the past (for example, those in region 3of Fig. 2) might not determine the realized value of the beablein question in region 1. Rather, the theory specifies only prob-abilities for the various possible values that might be realizedfor that beable. Note that determinism is not an alternative tobut is rather a special case of stochasticity:

Consider for example Maxwells equations, in thesource-free case for simplicity. The fields E and Bin region 1 are completely determined by the fields

in region 3, regardless of those in 2. Thus this is alocally causal theory in the present sense. Thedeterministic case is a limit of the probabilisticcase, the probabilities becoming delta functions.7

The natural generalization of our mathematical formula-tion of local determinism is Bells local causalitycondition:

Pb1jB3; b2 Pb1jB3: (4)

That is, b2 is irrelevantnot for determining what happensin region 1 because that, in a stochastic theory, is not deter-minedbut rather for determining the probability for pos-

sible occurrences in region 1. Such probabilities are theoutput of stochastic theories in the same sense that theactual realized values of beables are the output of deter-ministic theories. Thus, Bells local causality condition forstochastic theories, Eq. (4), and the analogous condition,Eq. (3), for deterministic theories, impose the same localityrequirement on the two kinds of theories: informationabout region 2 is irrelevant in regard to what the theorysays about region 1, once the beables in region 3 are suffi-ciently specified.

If we insist that any stochastic theory is a stand-in forsome (perhaps unknown) underlying deterministic theory(with the probabilities in the stochastic theory resulting notfrom indeterminism in nature, but from our ignorance),

Bells locality concept would cease to work. The require-ment of a complete specification of beables in region 3would contradict the allowance that such a specification doesnot necessarily determine the events in region 1. But this isno objection to Bells concept of local causality. Bell did notask us to accept that any particular theory (stochastic or oth-erwise) is true. Instead he just asked us to accept his defini-tion of what it would mean for a stochastic theory to respectrelativitys prohibition on superluminal causation. And thisrequires us to accept, at least in principle, that there could bean irreducibly stochastic theory and that the way causalityappears in such a theory is that certain beables do, and othersdo not, influence the probabilities for specific events.

We stress here that the meaning of causality is broaderthan, and does not necessarily entail, determinism. Bell care-fully formulated a local causality criterion that does not tac-itly assume determinism, and which is stated explicitly interms of probabilitiesthe fundamental, dynamical proba-bilities assigned by stochastic theories to particular events inspace-time. The probabilities in Eq. (1) are not subjective inthe sense of denoting the degree of someones belief in aproposition about b1; they cannot be understood as reflecting

partial ignorance about the relevant beables in region 3; andthey do not represent empirical frequencies for the appear-ance of certain values of b1. They are, rather, the fundamen-tal output of some candidate (stochastic) physical theory.

C. Causality versus correlation

Correlation doesnt imply causality. Two events (say, thevalues taken by beables b1 and b2 in Bells spacetime regions1 and 2, respectively) may be correlated without there neces-sarily being any implication that b1 is the cause ofb2 or viceversa: Of course, mere correlation between distant eventsdoes not by itself imply action at a distance, but only correla-tion between the signals reaching the two places.13 Bell

described the issue motivating his 1990 paper as the prob-lem of formulating sharply in contemporary physical the-ory these notions, of cause and effect on the one hand, andof correlation on the other.7

It is sometimes reported that Bells local causality condi-tion is really only a no correlation requirement, such thatthe empirical violation of the resulting inequalities estab-lishes only non-local correlations (rather than non-localcausation) (see, for example, Ref. 29). But this is a miscon-ception. Bell used the term causality (for example, in hisdefinition of locally causal theories) to highlight that a vio-lation of this condition by some theory means that the theoryposits non-local causal influences, rather than mere non-local correlations.

It is helpful to illustrate this point by relaxing a require-ment that Bell carefully incorporated into his formulation oflocal causality and showing that violation of the resultingweakened condition may still entail correlations betweenspace-like separated events, but no longer implies that thereare faster-than-light causal influences. We have done thisonce already in Sec. IV, where we explained why a violationof Eq. (2) would notunlike a violation of Eq. (1)entail aviolation of the causal structure of Fig. 1. We now consider asecond modified version of Bells criterion.

Consider again the spacetime diagram in Fig. 2. Bell notedthat It is important that region 3 completely shields offfrom 1 the overlap of the backward light cones of 1 and 2. 7

FIG. 3. Similar to Fig. 2, except that region 3* (unlike region 3 of Fig. 2)

fails to shield off region 1 from the overlapping backward light cones of

regions 1 and 2. Thus, following the caption of Fig. 2, even full specification

of what happens in 3*

does not necessarily make events in 2 irrelevant for

predictions about 1 in a locally causal theory.



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Why is this complete shielding so important? For example,why can we not replace region 3 of Fig. 2 with a region likethat labeled 3* in Fig. 3? This region, just like 3 in Fig. 2,closes off the back light cone of 1. So, it might seem like itwould be sufficient for defining the probabilities associatedwith b1 in a locally causal theory.

But a more careful analysis shows that a violation of

Pb1jB3 ; b2 Pb1jB3 (5)(the same as Eq. (1) but with region 3 of Fig. 2 replaced byregion 3* of Fig. 3) does not entail any non-local causation.Here, there is a perfectly local causal mechanism by whichb1 and b2 can be correlated, in a way that isnt screened offby conditionalization on B3 , thus violating Eq. (5) in a situa-tion that involves no violation of relativistic local causation.The mechanism is the following. In a stochastic theory, anevent may occur at the space-time point labeled X in Fig. 3which was not determined by the complete specification ofbeables B3 in region 3

*. But despite not having been deter-mined by beables in its past, that event really comes into ex-istence and may have effects throughout its future light cone,which includes both regions 1 and 2. Event X may broadcast

sub-luminal influences which bring about correlationsbetween b1 and b2, such that information about b2 is notredundant in regard to defining what happens in region 1(even after conditionalizing on B3 ). Thus, we may have aviolation of Eq. (5)that is, a candidate theory could attrib-ute different values to P b1jB3 ; b2 and P b1jB3 despitethere being, according to the theory, no non-local causationat work. Although Eq. (5) may be described as a nocorrelations condition for regions 1 and 2, it definitely failsas a no causality condition.

If we return to the original region 3 of Fig. 2 which doescompletely [shield] off from 1 the overlap of the backwardlight cones of 1 and 2,7 it is clear that no such correlationwithout non-local-causality can occur. Here, if some X-like

event, not determined by even a complete specification ofbeables in region 3, occurs somewhere in the future lightcone of region 3, it will necessarily fail to lie in the overlap-ping past light cones of regions 1 and 2, which would be nec-essary for it to in turn locally influence both of those events.

Bell carefully set things up so that a violation of Eq. (1)entails that there is some non-local causation. It is not neces-sarily that an event in region 2 causally influences events inregion 1 or vice versa. It is possible, for example, that thereis some other event, neither in region 1 nor region 2, whichwas not determined by B3 and which causally influencesboth b1 and b2. The point is that this causal influence wouldhave to be non-local; that is, it would have to violate the spe-cial relativistic causal structure in Fig. 1.43

To summarize the point that a violation of Eq. (1) entailsnon-local causation rather than mere correlations betweenspace-like separated events, it is helpful to recall Bells exam-ple of the correlation between the ringing of a kitchen alarmand the readiness of a boiling egg. That the alarm rings just asthe egg is finished cooking does not entail or suggest that theringing caused the egg to harden. Correlation does not implycausality. As Bell completes the point, The ringing of thealarm establishes the readiness of the egg. But if it is alreadygiven that the egg was nearly boiled a second before, then theringing of the alarm makes the readiness no more certain.7

If we replace b2 for the ringing of the alarm, b1 for thereadiness of the egg, and B3 for the egg was nearly boiled

a second before, we have a simple example of Eq. (1):although b1 and b2 may be correlated such that informationabout b2 can tell us something about b1, that information isredundant in a locally causal theory once B3 is specified.

D. Causality versus signaling

An idea that is often confused with local causality is local(that is, exclusively slower-than-light) signaling.44 Signalingis a human activity in which one person transmits informa-tion, across some distance, to another person. Such transmis-sion requires a causal connection between the sending eventand the receiving event and requires the ability of the twopeople to send and receive the information. That is, signalingrequires some measure of control over appropriate beableson the part of the sender and some measure of access toappropriate beables on the part of the recipient.

The requirement that theories prohibit the possibility offaster-than-light signaling, which is all that is imposed in rel-ativistic quantum field theory by the requirement that fieldoperators at spacelike separation commute,36 is a muchweaker condition than the prohibition of faster-than-lightcausal influences. Theories can exhibit violations of relativis-tic local causality and yet, because certain beables are inad-equately controllable by and/or inadequately accessible tohumans, preclude faster-than-light signals. Orthodox quan-tum mechanics including ordinary relativistic quantum fieldtheory is an example of such a theory. Another example isthe pilot-wave theory of de Broglie and Bohm, in which the consequences of events at one place propagate to otherplaces faster than light. This happens in a way that we cannotuse for signaling. Nevertheless it is a gross violation of rela-tivistic causality.15 One of the most common mistakesmade by commentators on Bells theorem is to conflate localcausality with local signaling.45 Often this conflation takesthe form of a double-standard in which alternatives to ordi-

nary quantum mechanics are dismissed as non-local andtherefore unacceptable on the grounds that they include (ei-ther manifestly, as in pilot-wave theory, or in principle, asestablished by Bells theorem) gross violations of relativis-tic causality. But ordinary quantum mechanics is argued bycomparison to be perfectly local, where now only local sig-naling is meant. Such reasoning is clearly equivocal once

FIG. 4. Space-time diagram illustrating the various beables of relevance for

the EPR-Bell setup (see Bells diagram in Ref. 7). Separated observers Alice

(in region 1) and Bob (in region 2) make spin-component measurements

using apparatus settings a and b, respectively, on a pair of spin- or

polarization-entangled particles as indicated by the dashed lines. The meas-

urements have outcomes A and B, respectively. The state of the particle pair

in region 3 is denoted by k. Note that region 3 extends across the past light

cones of both regions 1 and 2. It thus not only completely shields off from

1 the overlap of the backward light cones of 1 and 27 but does so also for

region 2. Bells local causality condition therefore requires both that b and B

are irrelevant for predictions about the outcome A, and that a and A are irrel-

evant for predictions about the outcome B, once k is specified.


mo:mai qui)

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we appreciate that local causality and local signaling havedifferent meanings.

Differentiating these two notions raises the question ofwhat special relativity should be understood to prohibit. Butthe idea that the relativistic causal structure, sketched in Fig.1, should somehow apply exclusively to the narrowly humanactivity of signaling, seems highly dubious:

the no signaling notion rests on conceptswhich are desperately vague, or vaguely applicable.

The assertion that we cannot signal faster thanlight immediately provokes the question:

Who do we think we are?

We who can make measurements, we who canmanipulate external fields, we who can signalat all, even if not faster than light? Do we includechemists, or only physicsts, plants, or onlyanimals, pocket calculators, or only mainframecomputers?7

That is, the idea that special relativity is compatible withnon-local causal influences (but only prohibits non-localsignaling) seems afflicted by the same problem that afflicts the-

ories whose formulations involve words such as observable,microscopic, and environment. In particular, the notion ofsignaling seems too superficial and too anthropocentric toadequately capture the causal structure of Fig. 1.

VI. IMPLICATIONS OF LOCAL CAUSALITY

Having reviewed Bells careful formulation of relativisticlocal causality, let us now indicate some of its importantconsequences.

A. Factorization

A typical EPR-Bell setup involves separated observers(Alice and Bob) making spin-component measurementsusing Stern-Gerlach devices oriented spatially along the aand b directions, respectively, on each of a pair of spin-entangled particles. The outcomes of their individual meas-urementsmanifested in the final location of the particle orthe position of some pointer or some fact about some otherbeableare denoted by A and B, respectively.

The beables relevant to a given run of the experiment maybe cataloged as in Fig. 4. We may roughly think of a and b,which are in regions 1 and 2, respectively, as referring to thespatial orientations of the two pieces of the measuring appa-ratus and k in region 3 as referring to the state of the particlepair emitted by the source. (The phrase state of the particlepair should not be taken too seriously, because no actual

assumption is made about the existence, for example, of lit-eral particles.)Unlike region 3 of Fig. 2, region 3 of Fig. 4 extends across

the past light cone not only of region 1, but of region 2 aswell. It particular, this extended region closes off the pastlight cones of regions 1 and 2 and shields both regions fromtheir overlapping past light cones. A complete specificationof beables in region 3 will thus, according to Bells conceptof local causality, make events in 2 irrelevant for predic-tions about 1,7 and will also make events in 1 irrelevant forpredictions about 2:

PAja; b;B; k PAja; k; (6)

and

PBja; b; k PBjb; k: (7)From Eqs. (6), (7), and the identity

PA;Bja; b; k PAja; b;B; k PBja; b; k; (8)the factorization of the joint probability for outcomes A and

B immediately follows:

PA;Bja; b; k PAja; k PBjb; k (9)This factorization condition is widely recognized to be suffi-cient for the derivation of empirically testable Bell-typeinequalities. As Bell notes, however, Very often such factor-izability is taken as the starting point of the analysis. Here, wehave preferred to see it not as the formulation of local causal-ity, but as a consequence thereof.7

B. The EPR argument

In their famous paper, Einstein, Podolsky, and Rosen

argued that a local explanation for the perfect correlationspredicted by quantum theory required the existence oflocally pre-determined values for the measurement out-comes.9 Because ordinary quantum mechanics contains nosuch elements of reality, EPR concluded that ordinary quan-tum mechanics (and the wave function in particular) did notprovide a complete description of physical reality. They sug-gested that an alternative, locally causal theory which pro-vides a complete description of physical reality might befound.

If we assume that the relevant empirical predictions ofquantum theory are correct, we can summarize the logic ofEPRs argument as

locality ! incompleteness; (10)where incompleteness means the incompleteness of theorthodox quantum mechanical description of the particles inquestion (in terms of their quantum state alone). This state-ment is logically equivalent to the statement that

completeness ! non-locality (11)which explains why the EPR argument is sometimes charac-terized as an argument for the incompleteness of orthodoxquantum mechanics and sometimes as pointing out the non-locality of this theory.

In their argument, EPR appealed to an intuitive notion of

local causality which was not precisely formulated; but theargument can be made rigorous by using Bells formulationof local causality. It is clarifying to begin with the EPR argu-ment in the form of statement (11). The proof consists inusing the notion of local causality in its directly intendedway, namely, to assess whether a particular candidate theoryis or is not local.

Take again the situation indicated in Fig. 4. Because ofthe structure of region 3note that it could be extended intoa space-like hypersurface crossing through the region 3depicted in Fig. 4 and still satisfy the requirements discussedearlierthe relevant complete specification of beables doesnot presuppose that the state k of the particle pair must



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factorize into two distinct and independent states for the twoparticles. The state can instead be characterized in a way thatis inseparable, as in ordinary quantum mechanics, and theargument still holds: It is notable that in this argument noth-ing is said about the locality, or even localizability, of thevariable k. These variables could well include, for example,quantum mechanical state vectors, which have no particularlocalization in ordinary space-time. It is assumed only thatthe outputs A and B, and the particular inputs a and b, are

well localized.

13

Let us suppose that the preparation procedure at the parti-cle source (the star in Fig. 4) gives rise to a particle pair inthe spin singlet state as described by ordinary quantummechanics

jwi 1ffiffiffi2

p j "i1j #i2 j #i1j "i2 ; (12)

where j "i1 means that particle 1 is spin-up along the z-direc-tion, etc. (The full quantum state of the particle pair willinclude also spatial degrees of freedom. These will not enterinto the argument, though, and so are suppressed forsimplicity.)

Suppose for example that^

a ^

b ^z, that is, both Aliceand Bob choose to measure the spins of the incoming par-ticles along the z-direction. Then quantum mechanics pre-dicts (letting A 1 denote that Alice finds her particle tobe spin up, etc.) that either A 1 and B 1 (with proba-bility 50%) orA 1 and B 1 (with probability 50%).

For orthodox quantum mechanics k in Fig. 4 is just thequantum state of Eq. (12), and we have, for example, that

PA 1ja; k 12

(13)

while

PA 1j^a;^

b; k;B 1 1; (14)in violation of Eq. (1). As Bell explains

The theory requires a perfect correlation of[results] on the two sides. So specification of theresult on one side permits a 100% confidentprediction of the previously totally uncertain resulton the other side. Now in ordinary quantummechanics there just is nothing but thewavefunction for calculating probabilities. There isthen no question of making the result on one sideredundant on the other by more fully specifyingevents in some space-time region 3. We have aviolation of local causality.7

As mentioned, Statements (10) and (11) are logicallyequivalent, so a locally causal explanation for the perfectcorrelations predicted by quantum mechanics requires atheory with more (or different) beables than just the wavefunction. It is possible to show directly from Bells conceptof local causality that we must, in particular, posit beableswhich pre-determine the outcomes of both measurements.

To begin, we drop the assumption, which holds for ordi-nary quantum mechanics, that it is possible to fully controlthe state k produced by the preparation procedure at thesource. Instead, we allow that k may take several distinctvalues from one run of the experiment to another. We also

assume, for simplicity, that Alice and Bob freely choose tomake measurements along the z direction.

The argument is then simple: we have already shown thatlocal causality entails the factorization of the joint probabilityfor outcomes A and B once k is specified. Consider a jointevent, such as A 1, B 1, whose joint probability van-ishes. Factorization then implies that, for each value of k thatmight with nonzero probability be produced by the prepara-tion procedure, either PA 1ja; k or PB 1jb; kmust vanish.But because there are only two possible outcomes for eachmeasurement, each of these possibilities entails that the op-posite outcome is pre-determined. For example,

PA 1ja; k 0 ! PA 1ja; k 1; (15)which means that those values of k to which this statementapplies must encode the outcome A 1, which will then berevealed with certainty if a measurement along a is per-formed. The possible values ofk must therefore fall into twomutually exclusive and jointly exhaustive categoriesthosethat encode the pre-determined outcomes A 1 and

B 1 and those that encode the pre-determined outcomesA

1 and B

1.46

Furthermore, because the measurement axes are assumedto be free, the same argument can establish that k mustencode pre-determined outcomes for all possible measure-ment directions. We thus see how theories of deterministichidden variables (or what Mermin has dubbed instructionsets47) are required, by local causality, to explain the perfectcorrelations predicted by ordinary quantum mechanics.

C. The Clauser-Horne-Shimony-Holt inequality

It is well known that a Bell-type inequality follows fromthe assumption of local deterministic hidden variables orinstruction sets. That theories of this type are actuallyrequired by locality (as explained in Sec. VI B) should there-

fore already explain the seriousness with which Bell took theidea of a fundamental conflict between relativistic localityand the predictions of quantum mechanics. This conflict canbe brought out in an even more direct way by deriving aBell-type inequality directly from the factorization of thejoint probability as in Eq. (9)and hence from Bells localcausality (without any additional discussion of determinismor pre-determined values).

Assume that the measurement scenario indicated in Fig. 4is repeated many times, with each setting being selected ran-domly on each run from two possibilities: a 2 fa1; a2g,b 2 fb1; b2g. The procedure that prepares the particles to bemeasured is held fixed for all runs of the experiment. Asbefore, this does not necessarily imply that k is constant forall runs, because the relevant beables may be less than fullycontrollable. We will assume that the distribution of differentvalues ofk for the runs can be characterized by a probabilitydistribution q(k).

We define the correlation of (61-valued) outcomes A andB as the expected value of their product:

Ca; b X

A;B

A B P Aja; kPBjb; kqkdk (16a)

Aa; k Ba; kqkdk; (16b)

http://-/?-http://-/?-


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where Aa; k PA 1ja; k PA 1ja; k satisfiesj Aj 1 and similarly for B.

Now we consider several combinations of correlationsinvolving different pairs of settings. To begin with,

Ca1; b16Ca1; b2

Aa1; k Bb1; k6 Bb2; k

qkdk (17)

so that

Ca1; b16Ca1; b2

Bb1; k6 Bb2; k

qkdk: (18)Similarly, we have that

Ca2; b1 Ca2; b2

Bb1;k Bb2;k

qkdk: (19)By adding Eqs. (18) and (19) and noting that x6y j jx yj jequals 2x, 2x, 2y, or 2y, it follows immediately that

Ca1; b16Ca1; b2

Ca2; b1 Ca2; b2

2; (20)

which is the Clauser-Horne-Shimony-Holt inequality.39 Thisinequality is in essence the relation tested in experimentssuch as those discussed in Refs. 17 and 18. Quantum theorypredicts that (for appropriate preparations of the two-particlestate and for appropriate choices of a1, a2, b1, and b2) theleft-hand side of Eq. (20) should be 2

ffiffiffi2

p, which is more than

40% larger than the constraint implied by local causality.The experimental results are in excellent agreement with thequantum predictions.

Because the inequality is derived from the local causalitycondition, it follows from the experimental results that anytheory which makes empirically correct predictions will

have to violate the local causality condition. As Bell wrote,The obvious definition of local causality does not work inquantum mechanics, and this cannot be attributed to theincompleteness of that theory.7

D. The free variables assumption

We return to the assumption that the settings a and b arerandom or free. In terms of the derivation we have just pre-sented, the assumption is that the probability distributionq(k) for the distribution of possible states of the particle paircreated by the source is independent of the apparatus settingsa and b. For example, in deriving Eq. (17), we assumed thatthe same probability distribution q(k) characterizes the runsin which a1 and b1 are measured, as characterizes the runs inwhich a1 and b2 are measured. As Bell explained,

we may imagine the experiment done on such ascale, with the two sides of the experimentseparated by a distance of order light minutes, thatwe can imagine these settings being freely chosenat the last second by two different experimentalphysicists. If these last second choices are trulyfree, they are not influenced by the variables k.Then the resultant values for a and b do notgive any information about k. So the probabilitydistribution over k does not depend on a or

b

7

Of course, the real experiments do not involve settingsbeing freely chosen at the last second by two different experi-mental physicists, but instead involve physical random num-ber generators. As mentioned, this means that at least inprinciple, for some possible candidate theories, a completedescription of beables in region 3 of Fig. 4 includes not only acomplete description of the state of the particle pair but also acomplete description of whatever physical degrees of freedomwill determine the eventual settings a and bmaking it not

only possible but likely that the candidate theory should ex-hibit (contrary to the assumption that was made) correlationsbetween what we have called k and those settings.

As suggested earlier, though, we can appeal to the expecta-tion that serious candidate theories will posit an enormouslylarge number of physical degrees of freedom in a spacetimeregion such as 3, only some tiny fraction of which are actuallyneeded to completely specify the state of the particle pair, thatis, the beables that are physically influenced by the preparationprocedure at the source. There are then still many other beablesin region 3 which might be used to determine/influence the ap-paratus settings. The free variables assumption is that thesesettings are somehow made such that there are no correlationsbetween the beables used to determine the apparatus settings

and those that encode the state of the particle pair.As Bell acknowledged, one logical possibility in the faceof the empirical violations of the Clauser-Horne-Shimony-Holt inequality is that

it is not permissible to regard the experimentalsettings a and b in the analyzers as independentof the supplementary variables k, in that a and b could be changed without changing the probabilitydistribution q(k). Now even if we have arrangedthat a and b are generated by apparentlyrandom radioactive devices, housed in separatedboxes and thickly shielded, or by Swiss nationallottery machines, or by elaborate computerprogrammes, or by apparently free willedexperimental physicists, or by some combinationof all of these, we cannot be sure that a and b are not significantly influenced by the same factorsk that influence A and B. But this way of arrangingquantum mechanical correlations would be evenmore mind boggling than one in which causalchains go faster than light. Apparently separateparts of the world would be deeply andconspiratorially entangled, and our apparent freewill would be entangled with them.13

Bell introduced the term superdeterministic to describetheories which explain the empirically observed correlationsby denying that the apparatus settings can be treated as free:

An essential element in the reasoning here is that a and b

are free variables. One can envisage then

theories in which there just are no free variables forthe polarizer angles to be coupled to. In suchsuperdeterministic theories the apparent free will ofexperimenters, and any other apparent randomness,would be illusory. Perhaps such a theory could beboth locally causal and in agreement with quantummechanical predictions. However I do not expect tosee a serious theory of this kind. I would expect aserious theory to permit deterministic chaos orpseudorandomness, for complicated subsystems



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(for example computers) which would providevariables sufficiently free for the purpose at hand.But I do not have a theorem about that.7

It is sometimes erroneously thought that the freedom orno conspiracies assumption follows (or should follow)from local causality. For example, Shimony, Horne, andClauser48 criticized Bells derivation for using (in our nota-tion) the assumption qkja; b qk which, they correctlypointed out, does not follow from local causality. Bell subse-

quently clarified that it was a separate assumption, not sup-posed to follow from local causality. And, as articulated bythe discussants in Ref. 48, the additional assumption seemseminently reasonable:

we feel that it is wrong on methodologicalgrounds to worry seriously about [the possibility ofthe kind of conspiracy that would render theassumption inapplicable] if no specific causal linkage[between the beables k and those which determinethe apparatus settings] is proposed. In any scientificexperiment in which two or more variables aresupposed to be randomly selected, one can alwaysconjecture that some factor in the overlap of the

backward light cones has controlled the presumablyrandom choices. But, we maintain, skepticism of thissort will essentially dismiss all results of scientificexperimentation. Unless we proceed under theassumption that hidden conspiracies of this sort donot occur, we have abandoned in advance the wholeenterprise of discovering the laws of nature byexperimentation.48

Imagine, for example, an experimental drug trial in whichpatients are randomly selected to receive either the drug or aplacebo. It is logically possible that the supposedly randomselections (made, say, by flipping a coin) are correlated withsome pre-existing facts about the health of the patients. Such acorrelation could skew the results of the trial, resulting say ina statistically significant improvement in the health of thepatients given the genuine drug even though the drug is impo-tent or worse. The suggestion is that unless there is some plau-sible causal mechanism that might conceivably produce thecorrelations in question, it is reasonable to assume that theconspiratorial correlations are absent. That is, the additionalassumption beyond local causality which is needed to derivethe Clauser-Horne-Shimony-Holt inequality is no strongerthan one needs for experimental reasoning generically.48 Theno conspiracies assumption thus falls into the same categoryas, for example, the validity of logic and certain mathematicaloperations, which, although used in the derivation, are notseriously challengeable. This explains why we sometimes do

not even bother to mention this assumption as, for example,when writing that the Clauser-Horne-Shimony-Holt inequalityfollows from Bells concept of local causality alone.

VII. SUMMARY AND OPEN QUESTIONS

We have reviewed Bells formulation of relativistic localcausality, including a survey of its conceptual backgroundand a sketch of its most important implications. We havestressed that Bells formulation does not presuppose deter-minism or the existence of hidden variables, but insteadseems perfectly to capture the intuitive idea, widely taken asan implication of special relativity, that causal influences

cannot propagate faster than light. And as we have seen, nowtaking the no conspiracies assumption for granted, theempirically violated Clauser-Horne-Shimony-Holt inequalitycan be derived from Bells concept of local causality alone,without the need for further assumptions involving determin-ism, hidden variables, realism, or anything of that sort.

This hopefully clarifies why Bell disagreed with the wide-spread opinion that his theorem and the associated experi-ments vindicate ordinary quantum theory as against hidden

variable theories or vindicate Bohrs philosophy as againstEinsteins. Instead, for Bell, the real problem with quantumtheory is the apparently essential conflict between anysharp formulation and relativity [that is, the] apparent incom-patibility, at the deepest level, between the two fundamentalpillars of contemporary theory.15

Although we have argued strongly for the reasonablenessof Bells formulation of relativistic local causality, this par-ticular formulation should not necessarily be regarded as de-finitive. We briefly indicate several points on which itsapplicability to various sorts of exotic theories could bequestioned, and where a more general or distinct formulationof local causality might be sought. For example, we mightworry that a theory with a non-Markovian character (that is,

a theory in which causal influences can jump discontinuouslyfrom one time to a later time) could violate Bells local cau-sality condition despite positing no strictly faster-than-lightinfluences. The idea would be that influences could hopover region 3 of Fig. 2, leading to correlations in regions 1and 2 but leaving no trace in 3. This shortcoming in the for-mulation could seemingly be addressed by requiring thatBells region 3 cover a region of spacetime so thick (in thetemporal direction) that hopping non-Markovian influencescould not make it across. In the limit of arbitrarily large vio-lations of the Markov property, this change would requireregion 3 to encompass the entire past light cone of the region3 in Fig. 2. But this fix would come at a price: the more ofspacetime that is included in region 3, the more difficult it

will be to argue for the reasonableness of the no conspir-acies assumption, and the more we might worry that thecondition could fail to detect certain kinds of non-localitiessuch that it would function, no longer as a formulation of lo-cality, but merely as a necessary condition for locality.

Similar problems arise when we contemplate the possibil-ity of theories that posit not only local beables, that is, thoseassociated with definite positions in space35 but also non-local beables. The de Broglie-Bohm pilot-wave theory isprobably the clearest example: its posited ontology includesboth particles (which follow definite trajectories in 3-spaceand are pre-eminent examples of local beables) and a guidingwave (which is just the usual quantum mechanical wavefunction, interpreted as a beable). For an N-particle system,

the wave function is a function on the 3N-dimensional con-figuration space, so if it is a beable, it is a non-local beable.49

As mentioned, Bells region 3 can be extended into aspace-like hypersurface without spoiling any of the argu-ments that have been given in this paper. We may theninclude, as well, where Bells formulation instructs us to usea complete specification of the local beables in region 3, val-ues for any non-local beables which, like wave functions, canbe associated with hypersurfaces. And it is important that,even when including information about non-local beables inthis way, the local causality condition is violated by the pilot-wave theory. (Note that the argument in Sec. VI B for thenon-local character of orthodox quantum mechanics was of


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just this type.) So again, for theories involving non-localbeables, Bells formulation can be easily tweaked to yield anecessary condition for local causality, which condition isunambiguously violated by various extant and obviously-nonlocal theories.

Still, as formulated, Bells concept of local causalityseems to presuppose that we are dealing with theories posit-ing exclusively local beables.50 It can be stretched to accom-modate certain extant theories which also posit non-local

beables, but how to formulate the concept with completegenerality and what other issues (like those encountered fornon-Markovian theories) may arise in the attempt, remainsunclear. Of course, it is also unclear how seriously we shouldtake theories with non-local beables in the first place. In par-ticular, should such theories even be considered candidatesfor locally causal status? And could a theory positing non-local beables be genuinely consistent with special relativity?

Such questions will not be answered here. We raise themonly to give readers some sense of the concerns that theymight have about Bells formulation of local causality. Theiradmittedly exotic character should help explain why Bell feltdriven to contemplate unspeakable deviations from con-ventional wisdom. In particular, we can now appreciate how

simple everything would become if we dropped the insist-ence on reconciling the Bell experiments with fundamentalrelativity and instead returned to the pre-Einstein viewaccording to which there exists a preferred frame of refer-ence. As explained by Bell, such a view can accommodatefaster-than-light causal influences much more easily than theusual Einsteinian understanding of relativity.

Our goal here, though, is not to lobby for this view, butmerely to explain Bells rationale for taking it seriously as apossibility warranting attention, not just by philosophers, butby physicists interested in addressing the puzzles of yester-day, today, and tomorrow.

ACKNOWLEDGMENTS

Thanks to Shelly Goldstein, Daniel Tausk, Nino Zanghi,and Roderich Tumulka for discussions on Bells formulationof local causality and to several anonymous referees for anumber of helpful suggestions on earlier drafts of the paper.

a)Electronic address: [email protected]

Max Born, The Born-Einstein Letters, translated by Irene Born (Walker

and Company, New York, 1971).2

Mary B. Hesse, Forces and Fields: The Concept of Action at a Distance in

the History of Physics (Dover, Mineola, NY, 2005).3

Ernan McMullin, The Explanation of Distant Action: Historical Notes,

in Philosophical Consequences of Quantum Theory, edited by James T.

Cushing and E. McMullin (University of Notre Dame Press, Notre Dame,

1989).4Isaac Newton, February 25, 1693 letter to Richard Bentley. See, for exam-

ple, Andrew Janiak, Newtons philosophy, in The Stanford Encyclope-

dia of Philosophy, edited by Edward N. Zalta, .5I. Bernard Cohen, A guide to Newtons Principia, in Isaac Newton, The

Principia, translated by I. B. Cohen and Anne Whitman (University of

California Press, Berkeley, 1999). See especially pp. 6064 and 277280.6

Albert Einstein, Relativity: The Special and the General Theory (Penguin

Classics, New York, 2006), p. 47.7John S. Bell, La nouvelle cuisine, in Between Science and Technology,

edited by A. Sarlemijn and P. Kroes (Elsevier Science, Amsterdam, 1990);

reprinted in Ref. 8, pp. 232248.8

John S. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd ed.

(Cambridge U.P., Cambridge, 2004).

9Albert Einstein, Boris Podolsky, and Nathan Rosen, Can quantum-

mechanical description of reality be considered complete?, Phys. Rev.

47, 777780 (1935).10Travis Norsen, Einsteins boxes, Am. J. Phys. 73(2), 164176 (2005).11

The terminology of hidden variables is unfortunate because, at least in

the one existing example of a serious hidden variables theory (the de

Broglie-Bohm pilot wave theory, which adds to the standard quantum

mechanical wave function definite particle positions obeying a determinis-

tic evolution law), the hidden variables are not hidden. In Ref. 18, Bell

remarked that it would be appropriate to refer to the s as exposed vari-ables and to w as a hidden variable. It is ironic that the traditional termi-

nology is the reverse of this. Similarly in Ref. 12, he writes Although [inBohmian mechanics] W is a real field it does not show up immediately in

the result of a single measurement, but only in the statistics of many such

results. It is the de Broglie-Bohm variable X that shows up immediately

each time. That X rather than W is historically called a hidden variable is

a piece of historical silliness.12 It is also relevant that the wave function

w is hidden in the sense of being not accessible via experiment, even in

orthodox quantum theory, which is the primary example of a non-hidden-

variable theory. See Roderich Tumulka, Understanding Bohmian

mechanics: A dialogue, Am. J. Phys. 72(9), 12201226 (2004).12John S. Bell, On the impossible pilot wave, Found. Phys. 12, 98999

(1982); reprinted in Ref. 8, pp. 159168.13John S. Bell, Bertlmanns socks and the nature of reality, J. Physique,

Colloque C2, suppl. au numero 3, Tome 42, C2 4161 (1981); reprinted in

Ref. 8, pp. 139158.14John S. Bell, On the Einstein-Podolsky-Rosen paradox, Physics 1,

195200 (1964); reprinted in Ref. 8, pp. 1421.15John S. Bell, Speakable and unspeakable in quantum mechanics, intro-

ductory remarks at Naples-Amalfi meeting, May 7, 1984; reprinted in R

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