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that the velocity c plays a fundamental role crystallization of this newly real property for the distant in this theory.”6 particle. In Bell’s recapitulation of the argument, though, for c The speed of light plays a fundamental role vis-`a-vis Einstein, Podolsky, and Rosen (EPR) this causality in SR because of the relativity of simultaneity. For two events A and B with space-like separation (i.e., “simply showed that [Bohr, Heisenberg, and such that a signal connecting A and B would have to Jordan] had been hasty in dismissing the re- propagate faster than c), the time ordering is ambiguous: ality of the microscopic world. In particu- different inertial observers will disagree about whether A lar, Jordan had been wrong in supposing that precedes B in time, or vice versa. There is thus, accord- nothing was real or fixed in that world be- ing to SR, no objective matter of fact about which event fore observation. For after observing only one occurs first – and hence no possibility of a causal relation particle the result of subsequently observing between them, since the relation between a cause and its the other (possibly at a very remote place) effect is necessarily time-asymmetric. As J.S. Bell put is immediately predictable. Could it be that this point, “To avoid causal chains going backward in the first observation somehow fixes what was time in some frames of reference, we require them to go unfixed, or makes real what was unreal, not slower than light in any frame of reference.”7 only for the near particle but also for the re- After the advent of SR, it didn’t take long for the mote one? For EPR that would be an un- relativistic sense of local causality to be deployed in a thinkable ‘spooky action at a distance’. To criticism of other developing theories, much as the pre- avoid such action at a distance [one has] to relativistic concept had been used against Newton’s the- attribute, to the space-time regions in ques- ory. Indeed, it was Einstein himself – in both the fa- tion, real properties in advance of observa- mous, but widely misunderstood, EPR paper9 and sev- tion, correlated properties, which predeter- eral related but less widely known arguments10 – who mine the outcomes of these particular obser- first pointed out that the developing Copenhagen quan- vations. Since these real properties, fixed in tum theory violated SR’s locality constraint. In particu- advance of observation, are not contained in lar, according to Einstein, that theory’s account of mea- quantum formalism, that formalism for EPR surement combined with Niels Bohr’s completeness doc- is incomplete. It may be correct, as far as it trine committed the theory to precisely the sort of non- goes, but the usual quantum formalism can- local causation that was, at least according to Einstein, not be the whole story.”14 prohibited by SR. Einstein thus rejected Bohr’s complete- Bell thus agreed with Einstein that the local hidden vari- ness doctrine and supported (something like) what is now ables program constituted the only hope for a locally (unfortunately11) called the local “hidden variables” pro- causal re-formulation of quantum theory. gram. Bell’s historic contribution, however, was a 1964 theo- Note the interesting parallel to Newtonian gravity rem establishing that no such local hidden variable theory here, with the non-locality in some candidate theory be- – and hence no local theory of any kind – could gener- ing rendered either real or merely apparent, depending ate the correct empirical predictions for a certain class on whether or not one interprets the theory as providing of experiment.15 According to Bell, we must therefore a complete description of the physical processes in ques- accept the real existence, in nature, of faster-than-light tion. Einstein’s assessment of Copenhagen quantum the- causation – in apparent conflict with the requirements of ory with respect to local causality is thus logically parallel SR: to Newton’s analysis of his own theory of gravitation: the theory, if regarded as complete, violates locality – hence “For me then this is the real problem with upholding locality requires denying completeness. quantum theory: the apparently essential This brings us to the main subject of the present paper: conflict between any sharp formulation and the work of J.S. Bell. Bell (unlike many commentators) fundamental relativity. That is to say, we accepted Einstein’s proof of the non-locality of Copen- have an apparent incompatibility, at the hagen quantum theory. In particular, Bell accepted as deepest level, between the two fundamental valid “the EPR argument from locality to deterministic pillars of contemporary theory...”16 hidden variables.”14 The setup for this argument involves Indeed, Bell even went so far as to suggest, in response to a pair of specially-prepared particles which are allowed his theorem and the relevant experimental data,17,18 the to separate to remote locations. An observation of some rejection of “fundamental relativity” and the return to property of one particle then permits the observer to a Lorentzian view in which there is a dynamically priv- learn something about a corresponding property of the ileged (though probably empirically undetectable) refer- distant particle. According to the Copenhagen view, the ence frame: distant particle fails to possess a definite value for the property in question prior to the observation, and so it “It may well be that a relativistic version is precisely the observation of the nearby particle which of [quantum] theory, while Lorentz invari- – in apparent violation of local causality – triggers the ant and local at the observational level, may 3

be necessarily non-local and with a preferred review article cited in Ref. 24, for example, doesn’t ac- frame (or aether) at the fundamental level.”12 knowledge Bell’s formulation of local causality at all but instead proposes an alternative formulation very differ- And elsewhere: ent from Bell’s.) It is thus not terribly surprising that “...I would say that the cheapest resolution so many commentators and textbook authors have sum- is something like going back to relativity marized the upshot of Bell’s theorem in ways so different as it was before Einstein, when people like from Bell’s own. Typically, for example, one encounters Lorentz and Poincar´ethought that there was the claim that Bell’s inequality follows not from local an aether – a preferred frame of reference – causality alone, but from the conjunction of local causal- but that our measuring instruments were dis- ity with some additional premises; some of the usual torted by motion in such a way that we could suspects here include “hidden variables,” “determinism,” not detect motion through the aether. Now, “realism,” “counter-factual definiteness”, or an improper in that way you can imagine that there is insistence on a vaguely-defined “classical” way of think- a preferred frame of reference, and in this ing. One or more of these (rather than relativistic local preferred frame of reference things do go causality) is then invariably blamed for the inconsistency faster than light. .... Behind the apparent with experiment.29–36 Lorentz invariance of the phenomena, there is A full presentation of Bell’s alternative to these a deeper level which is not Lorentz invariant... widespread views would include a systematic treatment [This] pre-Einstein position of Lorentz and of point (iii). We will sketch this derivation in Section VI, Poincar´e, Larmor and Fitzgerald, was per- but the bulk of our discussion will focus instead on Bell’s fectly coherent, and is not inconsistent with formulation of local causality, i.e., his views on point (ii). relativity theory. The idea that there is an This discussion will be based primarily on (but will also aether, and these Fitzgerald contractions and in several ways interpret and extend beyond) Bell’s 1990 Larmor dilations occur, and that as a result paper “La nouvelle cuisine” (published in the same year the instruments do not detect motion through as his untimely death). Clearly explaining Bell’s for- the aether – that is a perfectly coherent point mulation of locality, however, will require also sketching of view.”19,20 Bell’s interesting and refreshingly unorthodox views on Here our intention is not to lobby for this radical view, a number of related issues in the foundations of quan- but simply to explain Bell’s rationale for contemplating tum theory. The discussion will thus be elaborated and it. supported with excerpts from Bell’s many other papers. This rationale involves a complex chain of reasoning The main audience for the paper is students and physi- including at least these four steps: (i) arguing that SR cists with little or no prior knowledge of Bell’s theorem prohibits causal influences between space-like separated beyond what they’ve read in textbooks. It should be un- events, (ii) constructing a mathematically precise formu- derstood, though, that virtually all of the issues raised lation of this prohibition, i.e., of relativistic local causal- here are included because some kind of misunderstand- ity, (iii) deriving an empirically-testable inequality from ing (or simple ignorance) of them has been present and this formulation of local causality, and then (iv) estab- influential in the Bell literature. We will provide occa- lishing that the inequality is inconsistent with actual em- sional citations to works which we think exemplify the pirical data. There is an extensive “Bell literature” in various important misunderstandings. But length con- which each of these steps is subjected to probing critical siderations (and the desire to keep this a self-contained, analysis. The time-asymmetric character of causal rela- positive presentation of Bell’s views) forbid any exten- tions – used in the argument for (i) that was sketched sive polemical discussions. Still, those familiar with the above – has for example been challenged by Huw Price21 Bell literature will have no trouble appreciating where and, in a rather different way, by recent work of Roderich Bell’s own views challenge the conventional wisdom. It Tumulka22 (which was, incidentally, based on earlier is hoped that simply bringing Bell’s views more out into work by Bell23). And there remain certain “loopholes” the open will stimulate fruitful debates among those with in the experiments demonstrating violations of Bell’s in- interests in these areas. equality, such that one might conceivably doubt (iv).24 The paper is organized as follows. In the following sec- For the most part, though, physicists do not seriously tion, we jump quickly from some of Bell’s preliminary, question (i) and regard (iv) as established with reason- qualitative statements to his final, quantitative formula- able conclusiveness. The controversies about the mean- tion of relativistic local causality. Then, in Sections III- ing and implications of Bell’s theorem have thus centered V, we will highlight and explore various aspects by clari- on points (ii) and (iii). Clearly, though, what one says fying some perhaps-unfamiliar or suspicious terms which about point (iii) – the question of whether and how a appear in Bell’s formulation and by contrasting them to Bell-type inequality is entailed by local causality – will various other ideas with which they have sometimes been depend strongly on whether and how one has addressed confused. Section VI will show how local causality (as (ii). And sadly, Bell’s own views in regard to (ii) have formulated by Bell) can be used to derive an empirically been almost entirely invisible in the Bell literature. (The testable Bell-type inequality, and also how it can be used 4

1 2

effects time

3 1 2

space causes FIG. 2: “Full specification of what happens in 3 makes events in 2 irrelevant for predictions about 1 in a locally causal the- ory.” (Figure and caption are from Ref. 7).

FIG. 1: “Space-time location of causes and effects of events in region 1.” (Figure and caption are from Ref. 7.) remember that this is, at this point, merely a ‘teaser’ which those to whom it is not already familiar should only expect to understand after further reading.) to recapitulate the EPR argument. Finally, in Section “A theory will be said to be locally causal VII we will summarize the arguments presented and ac- if the probabilities attached to values of local knowledge some limitations of and open questions about beables in a space-time region 1 are unaltered Bell’s formulation. by specification of values of local beables in a space-like separated region 2, when what happens in the backward light cone of 1 is II. LOCAL CAUSALITY: OVERVIEW already sufficiently specified, for example by a full specification of local beables in a space- Let us begin with a qualitative formulation of Bell’s time region 3...”7 concept of local causality. In a 1988 interview, in an- swer to the question “What does locality mean?” Bell The space-time regions referred to are illustrated in Fig- responded:37 ure 2. We may translate Bell’s formulation into mathe- matical form as follows: “It’s the idea that what you do has con- P (b1 B3,b2)= P (b1 B3), (1) sequences only nearby, and that any conse- | | quences at a distant place will be weaker where bi refers to the value of some particular beable and will arrive there only after the time per- in space-time region i and Bi refers to a sufficient (for mitted by the velocity of light. Locality is example, a complete) specification of all beables in the the idea that consequences propagate contin- relevant region. The P s here are the probabilities as- 38 uously, that they don’t leap over distances.” signed to event b1 by the candidate theory in question. Eq. (1) thus asserts mathematically just what Bell states Bell gave a slightly more careful (but still qualitative) for- in the caption of his accompanying figure (reproduced mulation of what he called the “Principle of local causal- here as Figure 2): “full specification of [beables] in 3 ity” in his 1990 paper: makes events in 2 irrelevant for predictions about 1 in a 7 “The direct causes (and effects) of events are locally causal theory.” near by, and even the indirect causes (and Let us then jump right in to a closer examination of effects) are no further away than permitted the several perhaps-puzzling features of this formulation. by the velocity of light.”7 Then, citing a figure which we have reproduced here as III. BEABLES Figure 1, Bell continues: “Thus for events in a space-time region 1 ... For those to whom the term is new, the first question we would look for causes in the backward about the word “beable” is: how to pronounce it? The light cone, and for effects in the future light word has three syllables. It does not rhyme with “fee- cone. In a region like 2, space-like separated ble,” but with “agreeable.” Bell invented the word as a from 1, we would seek neither causes nor ef- contrast to the “observables” which play a fundamental fects of events in 1.”7 role in the formulation of orthodox quantum theory, so let us begin there. This should be relatively uncontroversial. Bell immedi- ately notes, however, that “[t]he above principle of lo- cal causality is not yet sufficiently sharp and clean for A. Beables vs. Observables mathematics.”7 Here, then, is Bell’s sharpened and cleaned formulation Beables (as contrasted to observables) are those ele- of special relativistic local causality. (The reader should ments of a theory which are supposed to correspond to 5 something that is physically real, independent of any ob- notions already implicit in, and basic to, or- servation. Bell elaborates: dinary quantum theory. For, in the words of Bohr, ‘it is decisive to recognize that, how- “The beables of the theory are those elements ever far the phenomena transcend the scope which might correspond to elements of real- of classical physical explanation, the account ity, to things which exist. Their existence of all evidence must be expressed in classical does not depend on ‘observation’. Indeed ob- terms.’ It is the ambition of the theory of servation and observers must be made out of 25 local beables to bring these ‘classical terms’ beables.” into the equations, and not relegate them en- 26 Or as he explains elsewhere, tirely to the surrounding talk.” The unprofessional vagueness and ambiguity of orthodox “The concept of ‘observable’ .... is a rather quantum theory, then, is related to the fact that its for- woolly concept. It is not easy to identify pre- mulation presupposes these (classical, macroscopic) be- cisely which physical processes are to be given ables, but fails to provide clear mathematical laws to the status of ‘observations’ and which are to describe them. As Bell explains, be relegated to the limbo between one obser- vation and another. So it could be hoped that “The kinematics of the world, in [the] ortho- some increase in precision might be possible dox picture, is given by a wavefunction ... for by concentration on the beables ... because the quantum part, and classical variables – they are there.”26 variables which have values – for the classi- cal part... [with the classical variables being] Bell’s reservations here (about the concept “observable” somehow macroscopic. This is not spelled appearing in the fundamental formulation of allegedly out very explicitly. The dynamics is not fundamental theories) are closely related to the so-called very precisely formulated either. It includes “measurement problem” of orthodox quantum mechan- a Schr¨odinger equation for the quantum part, ics, which Bell encapsulated by remarking that the ortho- 25 and some sort of classical mechanics for the dox theory is “unprofessionally vague and ambiguous” classical part, and ‘collapse’ recipes for their in so far as its fundamental dynamics is expressed in interaction.”27 terms of “words which, however legitimate and neces- sary in application, have no place in a formulation with There are thus two related problems. First, the posited any pretension to physical precision” – such words as ontology is rather different on the two sides of (what Bell “system, apparatus, environment, microscopic, macro- calls) “the shifty split”27 – that is, the division between scopic, reversible, irreversible, observable, information, “the quantum part” and “the classical part.” But then, measurement.”27 As Bell elaborates, as a whole, the posited ontology remains unavoidably vague so long as the split remains shifty – i.e., so long “The concepts ‘system’, ‘apparatus’, ‘envi- as the dividing line between the macroscopic and micro- ronment’, immediately imply an artificial di- scopic remains undefined. And second, the interaction vision of the world, and an intention to ne- across the split is problematic. Not only is the account of glect, or take only schematic account of, the this dynamics (the “collapse” process) inherently bound interaction across the split. The notions of up in concepts from Bell’s list of dubious terms, but the ‘microscopic’ and ‘macroscopic’ defy precise very existence of a special dynamics for the interaction definition. So also do the notions of ‘re- seems to imply inconsistencies with the dynamics already versible’ and ‘irreversible’. Einstein said that posited for the two realms separately. it is theory which decides what is ‘observable’. As Bell summarizes, I think he was right – ‘observable’ is a com- plicated and theory-laden business. Then the “I think there are professional problems [with notion should not appear in the formulation quantum mechanics]. That is to say, I’m a of fundamental theory.”27 professional theoretical physicist and I would like to make a clean theory. And when I As Bell points out, even Bohr (a convenient personi- look at quantum mechanics I see that it’s a fication of skepticism regarding the physical reality of dirty theory. The formulations of quantum unobservable microscopic phenomena) recognizes certain mechanics that you find in the books involve things (for example, the directly perceivable states of a dividing the world into an observer and an classical measuring apparatus) as unambiguously real, observed, and you are not told where that di- i.e., as beables: vision comes... So you have a theory which is fundamentally ambiguous...”19 “The terminology, be-able as against observ- able, is not designed to frighten with meta- The point of all this is to clarify the sort of theory Bell physic those dedicated to realphysic. It is had in mind as satisfying the relevant standards of pro- chosen rather to help in making explicit some fessionalism in physics. It is often thought, by those who 6 do not understand or do not accept Bell’s criticisms of ...corresponding to propagation with velocity orthodox quantum theory, that the very concept of “be- c. But the scalar potential, if one chooses to able” (in terms of which his concept of local causality is work in ‘Coulomb gauge’, satisfies Laplace’s formulated) commits one already to hidden variables or equation determinism or some sort of naive realism or some other 2φ =0 physically or philosophically dubious principle. − ∇ But this is not correct. The requirement here, ulti- ...corresponding to propagation with infinite mately, is only that candidate fundamental theories – at velocity. Because the potentials are only least, those “with any pretension to physical precision”27 mathematical conveniences, and arbitrary to – be formulated clearly and precisely. And this requires, a high degree, made definite only by the im- according to Bell, that the theories provide a uniform position of one convention or another, this and consistent candidate description of physical reality. infinitely fast propagation of the Coulomb- In particular, there should be no ambiguity or inconsis- gauge scalar potential disturbs no one. Con- tency regarding what a given candidate theory is fun- ventions can propagate as fast as may be con- damentally about (the beables), nor regarding precisely venient. But then we must distinguish in our how those posited physically real elements are posited to theory between what is convention and what act and interact (the laws). is not.”7 Thus, in order to cleanly decide whether a given theory B. Beables vs. Conventions is or is not consistent with local causality, “you must identify in your theory ‘local So far we have explained the term “beable” by con- beables’. The beables of the theory are those trasting it to the “observables” of orthodox quantum the- entities in it which are, at least tentatively, to ory. We must now also contrast the concept of “beables” be taken seriously, as corresponding to some- with those elements of a theory which are, to some de- thing real. The concept of ‘reality’ is now an gree, conventional: embarrassing one for many physicists.... But “The word ‘beable’ will also be used here to if you are unable to give some special sta- carry another distinction, that familiar al- tus to things like electric and magnetic fields ready in classical theory between ‘physical’ (in classical electromagnetism), as compared and ‘non-physical’ quantities. In Maxwell’s with the vector and scalar potentials, and electromagnetic theory, for example, the British sovereignty, then we cannot begin a 7 fields E and H are ‘physical’ (beables, we will serious discussion.” A say) but the potentials and φ are ‘non- This explains why, according to Bell: “It is in terms of physical’. Because of gauge invariance the local beables that we can hope to formulate some notion same physical situation can be described by of local causality.”26 very different potentials. It does not matter [i.e., it is not a violation of local causality] that in Coulomb gauge the scalar potential C. Beables and Candidate Theories propagates with infinite velocity. It is not re- ally supposed to be there. It is just a mathe- It is important to appreciate that a beable is only a be- matical convenience.”26 able relative to some particular candidate theory which Or, as Bell puts the same point in another paper, posits those elements as physically real (and, presumably, gives precise mathematical laws for their dynamics). For “...there are things which do go faster than example, the fields E and B (and not the potentials) are light. British sovereignty is the classical ex- beables according to classical Maxwellian electrodynam- ample. When the Queen dies in London (long ics as it is normally understood. But one could imagine may it be delayed) the Prince of Wales, lec- some alternative theory which (perhaps motivated by the turing on modern architecture in Australia, Aharanov-Bohm effect) posits the Coulomb gauge poten- becomes instantaneously King.... And there tials as beables instead. Note that, although it would are things like that in physics. In Maxwell’s be empirically and in some sense mathematically equiv- theory, the electric and magnetic fields in free alent to the usual theory, the alternative theory would space satisfy the wave equation evidently violate local causality (because, say, wiggling a charge would instantaneously affect the physically-real 1 ∂2E scalar potential at distant locations) while the usual, 2E = 0 c2 ∂t2 − ∇ Maxwellian theory would respect it. 1 ∂2B Thinking in terms of such candidate theories helps us 2B = 0 c2 ∂t2 − ∇ separate any questions about what the “real beables” are 7

– what really exists out there in physical reality – into anyone who is uncomfortable with the apparently “meta- two distinct parts: first, what elements does a given can- physical” positing of ultimate “elements of reality” (even didate theory posit as beables; and second, which candi- in a tentative way, through the tentative positing of a date theory do we think is true? The point is, you don’t candidate physical theory) should be relieved to find that have to be able to answer the second question in order the concept “beable” is merely a placeholder for what- to answer (for a given theory) the first. This should pro- ever entities we (tentatively) include in the class which vide some comfort to those who (perhaps influenced by already, by necessity, exists and includes certain basic, positivist or instrumentalist philosophy) think we can’t directly-perceivable features of the world around us. And (and/or shouldn’t try to) establish some theoretical pic- second, these particular beables – e.g., the settings of ture of external reality as true. Such people may still knobs and the positions of pointers – have a particularly accept Bell’s characterization of when “a theory will be central role to play in the derivation (from Bell’s con- said to be locally causal.” cept of local causality) of the empirically testable Bell But even those who are not skeptical on principle rec- inequalities. This will be developed in Section VI. ognize that, because of the complexity in practice of set- tling questions about the truth status of scientific theo- ries, some tentativeness is often in order. Bell recognizes IV. COMPLETENESS this too: Having clarified the concept of “beables” which ap- “I use the term ‘beable’ rather than some pears in Bell’s formulation of local causality, let us now more committed term like ‘being’ or ‘beer’ turn to the last phrase in that formulation, italicized to recall the essentially tentative nature of here: any physical theory. Such a theory is at best a candidate for the description of na- “A theory will be said to be locally causal ture. Terms like ‘being’, ‘beer’, ‘existent’, if the probabilities attached to values of lo- etc., would seem to me lacking in humility. cal beables in a space-time region 1 are un- In fact ‘beable’ is short for ‘maybe-able’.”25 altered by specification of values of local be- ables in a space-like separated region 2, when The crucial point is that the “maybe” here pertains to what happens in the backward light cone of 1 the epistemological status of a given candidate theory. is already sufficiently specified, for example by 26 By contrast, the “beable status” of certain elements a full specification of local beables in a space- of a theory relative to that theory should be completely time region 3...”7 straightforward and uncontroversial. If there is any ques- tion about what elements a theory posits as beables, it In a word, the key assumption here is “that events in 3 7 can only be because the proponents of the theory have be specified completely” (emphasis added). not (yet) sufficiently clarified what the theory is about, Let us first see why this requirement is necessary. Con- ¯ what the theory is. Whether the theory is true or false sider again Figure 2, and suppose that B3 denotes an in- is an orthogonal question. complete specification of beables in region 3. It can then All of that said, Bell does take certain elements largely be seen that a violation of for granted as beables – that is, as beables that any se- P (b1 B¯3,b2)= P (b1 B¯3) (2) rious candidate theory would have to recognize as such: | | “The beables must include the settings of switches and does not entail the existence of any super-luminal causal knobs on experimental equipment, the currents in coils, 26 influences. For suppose some event “X” in the overlap- and the readings of instruments.” As noted before, ping backwards light cones of regions 1 and 2 causally even Bohr acknowledges the real existence (the beable influences both b1 and b2. It might then be possible to status) of these sorts of things. And, as suggested by infer, from b2, something about X, from which one could Bohr, since our primary cognitive access to the world in turn infer something about b1. Suppose, though, that is through “switches and knobs on experimental equip- the incomplete description of events in region 3 – B¯3 – ment” and other such directly perceivable facts – i.e., omits precisely the “traces” of this past common cause since such facts must always constitute the primary ev- X. Then b2 could usefully supplement B¯3 – i.e., Eq. (2) idence on which we will have to rest any argument for could be violated – even in the presence of purely local the truth of a particular candidate physical theory – it causation. is hard to imagine a serious theory which doesn’t grant Thus, as Bell explains, in order for Eq. (1) to function such facts beable status: a theory which didn’t would ev- as a valid locality criterion, idently have to regard our perception as systematically delusional, and hence would have to regard any alleged “it is important that events in 3 be specified empirical evidence – for anything, including itself – as completely. Otherwise the traces in region 2 invalid. In short, such a theory would evidently be self- of causes of events in 1 could well supplement refuting.39 whatever else was being used for calculating We stress this point for two related reasons. First, probabilities about 1. The hypothesis is that 8

any such information about 2 becomes redun- of beables in some spacetime region, there is an impor- dant when 3 is specified completely.”7 tant reason why Bell explicitly leaves open the possibil- ity that a specification of “what happens in the back- And here is the same point from an earlier paper: ward light cone of 1” might be “sufficiently specified” “Now my intuitive notion of local causality by something less than a complete specification of the is that events in 2 should not be ‘causes’ of beables there. This has to do with the fact, to be dis- events in 1, and vice versa. But this does cussed more in Section VI, that in order to derive an not mean that the two sets of events should empirically-testable Bell-type inequality from the local be uncorrelated, for they could have common causality condition, one needs a subsidiary assumption, causes in the overlap of their backward light sometimes called “experimental freedom” or “no con- cones [in a local theory]. It is perfectly intel- spiracies”. This is in essence the assumption that, in ligible then that if [B3] in [region 3] does not the usual EPR-Bell kind of scenario in which a central contain a complete record of events in that source emits pairs of specially-prepared particles in oppo- [region], it can be usefully supplemented by site directions toward two spatially-separated measuring information from region 2. So in general it devices, it is possible for certain settings on the devices is expected that [P (b1 b2, B¯3) = P (b1 B¯3).] (determining which of several possible measurements are | | However, in the particular case that [B3] con- made on a given incoming particle) to be made “freely” tains already a complete specification of be- or “randomly” – that is, independently of the state of the ables in [region 3], supplementary informa- incident particle pair. tion from region 2 could reasonably be ex- pected to be redundant.”26 In the more recent versions of these experiments, the relevant settings are made using independent (quan- It is important to stress that, like the concept of beables tum) random number generators.18 According to ortho- itself, the idea of a sufficient (full or complete) specifi- dox QM, there is therefore nothing in the past light cone cation of beables is relative to a given candidate theory. of an individual measurement event foretelling which of What Bell’s local causality condition requires is that – the possible measurements will be performed. But there in order to assess the consistency between a given candi- of course exist alternative candidate theories (such as the date theory and the relativistic causal structure sketched de Broglie - Bohm pilot-wave theory, which is determin- in Figure 1 – we must include, in B3, everything that istic) according to which those same settings are influ- candidate theory says is present (or relevant) in region 3. enced by events in their pasts. But then, the relevant It is not, by contrast, necessary that we achieve omni- pasts of the device settings necessarily overlap with the science regarding what actually exists in some spacetime pre-measurement states of the particles being measured. region. A complete specification of beables in the relevant region The appearance of the word “completeness” tends to containing those pre-measurement states will therefore remind commentators of the EPR argument, and hence inevitably include facts relevant to (if not determining) apparently also tends to suggest that Bell smuggled into the device settings. And so, in deriving the Bell inequal- his definition of local causality the unwarranted assump- ity from local causality, there is a kind of subtle tension tion that orthodox quantum theory is incomplete. (See, between the requirement “that events in 3 be specified for example, Ref. 44.) As mentioned earlier, Bell did ac- completely”7 and the requirement that device settings cept the validity of the EPR argument. But this means can be made independent of the states of the particles- only that, according to Bell, local causality – plus some to-be-measured. of QM’s empirical predictions – entail the incompleteness of orthodox QM. His view on that point, however, is no To resolve the tension, one need merely allow that the part of his formulation of local causality. (See Section beables in the relevant region can be divided up into dis- VI for further discussion of the relation between local joint classes: those which are influenced by the prepara- causality and the EPR argument.) tion procedure at the source (and which thus encode the What Bell’s formulation does say is only this: whatever “state of the particle pair”) and those which are to be your theory posits as physically real (in region 3), make used in the setting of measurement apparatus parame- sure you include all of that when calculating the rele- ters. And note that these two classes are likely to be far vant probabilities to test whether your theory respects from jointly exhaustive: in any plausible candidate the- or violates Eq. (1), i.e., whether your theory is or isn’t ory, there will have to exist many additional beables (cor- locally causal in the sense of Figure 1. No assumptions responding, for example, to stray electromagnetic fields, are made about the type of theory to which the locality low energy relic neutrinos, etc.) which are in neither criterion can be applied. In particular, the incomplete- of the mentioned classes. One thus expects a consider- ness of orthodox quantum theory (i.e., the existence of able “causal distance” between the two classes of beables “hidden variables”) is not assumed. The virtue of Bell’s (at least in a well-designed experiment). This makes the formulation lies precisely in this generality. “freedom” or “no conspiracies” assumption – namely, the Although it is simplest to understand Bell’s local absense of correlations between the two classes of beables causality condition as requiring a complete specification – quite reasonable to accept. 9

This issue will be addressed in some more detail in Sec- the implication that there would exist also theories that tion VI. For now, we simply acknowledge its existence as are “non-causal.” A theory, by the very nature of what a way of explaining why Bell’s formulation of local causal- we mean by that term in this context, is automatically ity mentions “complete” descriptions of events in region causal. “Causal theory” is a redundancy. And so, as 3 as merely an example of the kind of description which noted earlier, one must understand Bell’s “definition of is “sufficient”. One might summarize the discussion here locally causal theories” as a criterion that theories – i.e., as follows: what is required for the validity of the local candidate descriptions of causal processes in nature – causality condition is a complete specification of beables must satisfy in order to be in accord with special rela- in region 3 – but only those beables which are relevant tivistic locality. In short, the causality in the “definition in some appropriate sense to the event b1 in question in of locally causal theories” is simply whatever a given can- region 1. didate theory says, about whatever it says it about. As Bell explains, the practical reason for defining lo- V. CAUSALITY cal causality in terms of the physical processes posited by some candidate theory (as opposed to the physical processes which actually exist in nature) has to do with Recall the transition from Bell’s preliminary, qualita- our relatively direct access to the one as opposed to the tive formulation of local causality to the final, “sharp and other: clean” version. And recall in particular Bell’s statement that the preliminary version was insufficiently sharp and clean for mathematics. What is it, exactly, that Bell con- sidered inadequate about the qualitative statement? It “I would insist here on the distinction be- seems likely that it was the presence there of the terms tween analyzing various physical theories, on “cause” and “effect” which are notoriously difficult to de- the one hand, and philosophising about the fine mathematically. Indeed, about his final formulation unique real world on the other hand. In this Bell says: “Note, by the way, that our definition of locally matter of causality it is a great inconvenience causal theories, although motivated by talk of ‘cause’ and that the real world is given to us once only. ‘effect’, does not in the end explicitly involve these rather 7 We cannot know what would have happened vague notions.” if something had been different. We cannot How exactly does Bell’s “definition of locally causal repeat an experiment changing just one vari- theories” fail to “explicitly involve” the “rather vague no- able; the hands of the clock will have moved, tions” of cause and effect? On its face this sounds para- and the moons of Jupiter. Physical theories doxical. But the resolution is simple: what Bell’s defini- are more amenable in this respect. We can tion actually avoids is any specific commitment about calculate the consequences of changing free what physically exists and how it acts. (Indeed, any elements in a theory, be they only initial con- such commitments would seriously restrict the general- ditions, and so can explore the causal struc- ity of the locality criterion, and hence undermine the ture of the theory. I insist that [the theory scope of Bell’s theorem.) Instead, Bell’s definition shifts of local beables, i.e., the local causality con- the burden of providing some definite account of causal cept] is primarily an analysis of certain kinds processes to theories and itself merely defines a space- of physical theory.”28 time constraint that must be met if the causal processes posited by a candidate theory are to be deemed locally causal in the sense of SR. The important mediating role of candidate theories vis- Bell’s view, contra several commentators45,46, is thus `a-vis causality will be further stressed and clarified in that no special philosophical account of causation is the following subsection. Subsequent subsections further needed to warrant the conclusion that violation of the lo- clarify the concept of “causality” in Bell’s “local causal- cality condition implies genuine non-local causation. For ity” by contrasting it with several other ideas with which Bell, it is a rather trivial matter to decide, for some (un- it has often been confused or conflated. ambiguously formulated) candidate physical theory, what is and is not a genuinely causal influence. We can simply “explore the causal structure of” the candidate theory. A. Causality and candidate theories This of course raises the question of how we might go from recognizing the non-locality of some particular can- As already discussed, according to Bell it is the job didate theory, to the claim that nature is non-local. But of physical theories to posit certain physically real struc- this is just Bell’s theorem: all candidate theories which tures (beables) and laws governing their interactions and respect the locality condition are inconsistent with ex- evolution. Thus Bell’s definition of locally causal the- periment. (See Section VI.) So the “one true theory” ories is not a specification of locality for a particular (whatever that turns out to be!) – and hence nature type of theory, namely, those that are “causal” – with itself – must violate relativistic local causality. 10

B. Causality vs. determinism i.e., b2 is irrelevant – not for determining what happens in region 1 because that, in a stochastic theory, is simply The previous subsection stressed that the “causal” in not determined – but rather for determining the prob- “locally causal theories” simply refers to the physically ability for possible happenings in region 1. Such prob- real existents and processes (beables and associated laws) abilities are the “output” of stochastic theories in the posited by some candidate theory, whatever exactly those same sense that the actual realized values of beables are might be. We in no way restrict the class of theories the “output” of deterministic theories. Thus, Bell’s lo- (whose locality can be assessed by Bell’s criterion) by in- cal causality condition for stochastic theories – Eq. (4) troducing “causality.” In particular, the word “causal” in – and the analogous condition – Eq. (3) – for determin- “locally causal theories” is not meant to imply or require istic theories, are imposing precisely the same locality that theories be deterministic as opposed to irreducibly requirement on the two kinds of theories: information stochastic. about region 2 should be irrelevant in regard to what the theory says about region 1, once the beables in region 3 “We would like to form some [notion] of local are sufficiently specified. causality in theories which are not determin- Of course, if one insists that any stochastic theory is istic, in which the correlations prescribed by ipso facto a stand-in for some (perhaps unknown) under- the theory, for the beables, are weaker.”26 lying deterministic theory (with the probabilities in the stochastic theory thus resulting not from indeterminism Bell thus uses the word “causal” quite deliberately as a in nature, but from our ignorance), Bell’s locality con- wider abstraction which subsumes but does not necessar- cept would cease to work. The requirement of a complete ily entail determinism. specification of beables in region 3 would then contradict This is manifested most clearly in the fact that Bell’s the allowance that such a specification does not neces- mathematical formulation of “local causality” – Eq. (1) sarily determine the events in region 1. But this is no – is stated in terms of probabilities. Indeed, in his 1976 objection to Bell’s concept of local causality. Bell is not paper “The Theory of Local Beables”26 Bell discusses asking us to accept that any particular theory (stochas- “Local Determinism” first, arguing that, in a “local de- tic or otherwise) is true; he’s just asking us to accept his terministic” theory the actual values of beables in region definition of what it would mean for a stochastic theory 1 (of Figure 2) will be determined by a complete speci- to respect relativity’s prohibition on superluminal cau- fication of beables in region 3 (with additional specifica- sation. And this requires us to accept, at least in prin- tion of beables from region 2 being redundant). In our ciple, that there could be such a thing as a genuinely, mathematical notation, local determinism means irreducibly stochastic theory, and that the way “causal- b (B ,b )= b (B ) (3) ity” appears in such a theory is that certain beables do, 1 3 2 1 3 and others do not, influence the probabilities for specific where, as before, b1 and b2 are the values of specific be- events. ables in regions 1 and 2, while B3 denotes a sufficient We have been stressing here that “causality” is wider (e.g., complete) specification of beables in region 3. than, and does not necessarily entail, determinism. Bell In a (local) stochastic theory, however, even a complete has deliberately and carefully formulated a local causal- specification of relevant beables in the past (e.g., those ity criterion that does not tacitly assume determinism, in region 3 of Figure 2) may not determine the realized and which is thus stated explicitly in terms of probabil- value of the beable in question (in region 1). Rather, the ities – the fundamental, dynamical probabilities assigned theory specifies only probabilities for the various possible by stochastic theories to particular happenings in space- values that might be realized for that beable. Of course, time. Note in particular that the probabilities in Eq. (1) determinism is not really an alternative to, but is rather are not subjective (in the sense of denoting the degree merely a special case of, stochasticity: of someone’s belief in a proposition about b1), they can- not be understood as reflecting partial ignorance about “Consider for example Maxwell’s equations, relevant beables in region 3, and they do not (primar- in the source-free case for simplicity. The ily) represent empirical frequencies for the appearance of E B fields and in region 1 are completely de- certain values of b1. They are, rather, the fundamental termined by the fields in region 3, regardless “output” of some candidate (stochastic) physical theory. of those in 2. Thus this is a locally causal theory in the present sense. The determinis- tic case is a limit of the probabilistic case, the C. Causality vs. correlation probabilities becoming delta functions.”7 The natural generalization of the above mathematical Everyone knows that correlation doesn’t imply causal- formulation of “local determinism” is precisely Bell’s lo- ity. Two events (say, the values taken on by beables b1 cal causality condition: and b2 in Bell’s spacetime regions 1 and 2, respectively) may be correlated without there necessarily being any P (b1 B3,b2)= P (b1 B3), (4) implication that b1 is the cause of b2 or vice versa: | | 11

1 2 But a more careful analysis shows that a violation of

P (b1 B3∗ ,b2)= P (b1 B3∗ ) (5) | | X 3∗ (the same as Eq. (1) but with region 3 of Figure 2 re- placed by region 3∗ of Figure 3) does not entail any non- local causation. Here, there is a perfectly local causal ∗ FIG. 3: Similar to Figure 2, except that region 3 (unlike re- mechanism by which b1 and b2 can be correlated, in a gion 3 of Figure 2) fails to shield off region 1 from the overlap- way that isn’t “screened off” by conditionalization on ping backward light cones of regions 1 and 2. Thus, (following B3∗ , thus violating Eq. (5) in a situation which involves the language of Figure 2’s caption) even full specification of ∗ no violation of relativistic local causation. The mecha- what happens in 3 does not make events in 2 irrelevant for predictions about 1 in a locally causal theory. nism is this: in a stochastic theory, an event may occur at the space-time point labelled “X” in Figure 3 which was not determined by the complete specification of beables ∗ (B3∗ ) in region 3 . But despite not having been deter- “Of course, mere correlation between distant mined by beables in its past, that event really comes into events does not by itself imply action at a dis- existence and may in principle have effects throughout its tance, but only correlation between the sig- future light cone – which includes both region 1 and re- nals reaching the two places.”14 gion 2. Event X may, so to speak, broadcast sub-luminal influences which bring about correlations between b1 and And Bell describes the issue motivating his 1990 paper b2, such that information about b2 is not redundant in “La nouvelle cuisine” as “the problem of formulating ... regard to defining what happens in region 1 (even after sharply in contemporary physical theory” “these notions, conditionalizing on B3∗ ). Thus we may have a violation of cause and effect on the one hand, and of correlation of Eq. (5) – i.e., the candidate theory in question could 7 on the other”. attribute different values to P (b1 B3∗ ,b2) and P (b1 B3∗ ) | | It is sometimes reported that Bell’s local causality con- – despite there being, according to the theory in ques- dition is really only a “no correlation” requirement, such tion, no non-local causation at work. While Eq. (5) may that the empirical violation of the resulting inequalities perhaps be described as some kind of “no correlations” establishes only “non-local correlations” (as opposed to condition for regions 1 and 2, it definitely fails as a “no non-local causation). (See, e.g., Ref. 33.) But this is a causality” condition. misconception. Bell uses the term “causality” (e.g., in If we return, however, to the original region 3 (of Fig- talking about his “definition of locally causal theories”) ure 2) which does “completely [shield] off from 1 the to highlight that a violation of this condition (by some overlap of the backward light cones of 1 and 2”7 it be- theory) means that the theory posits non-local causal in- comes clear that no such correlation-without-non-local- fluences, as opposed to mere “non-local correlations.” causality can occur. Here, if some X-like event (not de- It will be clarifying to illustrate this by relaxing a point termined by even a complete specification of beables in that Bell has carefully built into his formulation of lo- region 3) occurs somewhere in the future light cone of re- cal causality, and showing that violation of the resulting, gion 3, it will necessarily fail to lie in the overlapping past weakened condition may still entail correlations between light cones of regions 1 and 2 (which would be necessary space-like separated events, but no longer implies that for it to in turn locally influence both of those events). there are faster-than-light causal influences. Actually, Bell has carefully set things up so that a violation of we have done this once already, in the previous section, Eq. (1) entails that there is some non-local causation. It when we explained why a violation of Eq. (2) would not isn’t necessarily that something in region 2 is causally (unlike a violation of Eq. (1)) entail any violation of the influencing something in region 1, or vice versa. It is causal structure of Figure 1. We now consider a second always possible that there is some other event, neither in modified version of Bell’s criterion. region 1 nor region 2, which was not determined by B3, Consider again the spacetime diagram sketched in Fig- and which itself causally influences both b1 and b2. The ure 2. Bell notes that point is, though, that this causal influence would have to be non-local (i.e., would have to violate the special “It is important that region 3 completely relativistic causal structure sketched in Figure 1).47 shields off from 1 the overlap of the backward To summarize the point that a violation of Eq. (1) light cones of 1 and 2.”7 entails non-local causation (rather than mere correlations between space-like separated events) it is helpful to recall Why is this so important? For example, why couldn’t Bell’s example of the correlation between the ringing of a we replace region 3 of Figure 2 with a region like that kitchen alarm and the readiness of a boiling egg. That the labelled 3∗ in Figure 3? This region, just like 3 in Fig- alarm rings just as the egg is finished cooking obviously ure 2, closes off the back light cone of 1 and hence – it does not entail or even suggest that the ringing caused might seem – would be perfectly sufficient for defining the the egg to harden. Correlation does not imply causality. probabilites associated with b1 in a locally causal theory. As Bell completes the point, 12

“The ringing of the alarm establishes the argued by comparison to be perfectly “local” (where now readiness of the egg. But if it is already given only “local signaling” is meant or proved). Such reason- that the egg was nearly boiled a second be- ing, though, is obviously equivocal once one appreciates fore, then the ringing of the alarm makes the that “local causality” and “local signaling” simply mean readiness no more certain.”7 different things. Clearly differentiating these two notions does raise the Reading b2 for “the ringing of the alarm,” b1 for “the question of which, after all, SR should be understood to readiness of the egg,” and B3 for “the egg was nearly prohibit. But the idea that the relativistic causal struc- boiled a second before,” we have a simple intuitive exam- ture, sketched in Figure 1, should somehow apply ex- ple of Eq. (1): although b1 and b2 may be correlated such clusively to this narrowly human activity, seems highly that information about b2 can tell us something about b1, dubious: that information will be redundant (in a locally causal theory) once B3 is specified. “...the ‘no signaling...’ notion rests on con- cepts which are desperately vague, or vaguely applicable. The assertion that ‘we cannot sig- D. Causality vs. signaling nal faster than light’ immediately provokes the question: One final idea that is often confused with local causal- Who do we think we are? ity is local (i.e., exclusively slower-than-light) signaling.40 Signaling is, of course, a certain human activity in which We who can make ‘measurements’, we who one person transmits information, across some distance, can manipulate ‘external fields’, we who can to another person. Such transmission clearly requires a ‘signal’ at all, even if not faster than light? Do causal connection between the sending event and the re- we include chemists, or only physicsts, plants, ceiving event, but it requires more as well: namely, the or only animals, pocket calculators, or only ability of the people involved to send and receive the mainframe computers?”7 information. That is, signaling requires some measure of control (over appropriate beables) on the part of the That is, the idea that SR is compatible with non-local sender, and some measure of access (to appropriate be- causal influences (but only prohibits non-local signaling) ables) on the part of the recipient. seems afflicted by the same problem (reviewed in Sec- The requirement that theories prohibit the possibility tion III) that necessarily afflicts theories whose formu- of faster-than-light signaling – which incidentally is all lations involve words like “observable”, “microscopic”, that is imposed in relativistic quantum field theory by the “environment”, etc. In particular, the notion of “signal- requirement that field operators at spacelike separation ing” seems somehow too superficial, too anthropocentric, commute26 – is thus a much weaker condition than the to adequately capture the causal structure of Figure 1. prohibition of faster-than-light causal influences. Theo- ries can exhibit violations of relativistic local causality and yet (because certain beables are inadequately con- VI. IMPLICATIONS OF LOCAL CAUSALITY trollable by and/or inadequately accessible to humans) preclude faster-than-light signals. Orthodox quantum Having reviewed Bell’s careful formulation of relativis- mechanics (including ordinary relativistic quantum field tic local causality, let us now more briefly indicate some theory) is one example of such a theory. Another exam- of its important applications. ple is the pilot-wave theory of de Broglie and Bohm, in which A. Factorization “...the consequences of events at one place propagate to other places faster than light. In the typical EPR-Bell setup, we have separated This happens in a way that we cannot use for observers (traditionally Alice and Bob) making spin- signaling. Nevertheless it is a gross violation component measurements (using, say, Stern-Gerlach de- of relativistic causality.”16 vices oriented spatially along thea ˆ and ˆb directions, re- One of the most prevalent mistakes made by commenta- spectively) on each of a pair of spin-entangled particles. tors on Bell’s theorem is to conflate local causality with The outcomes of their individual measurements (mani- local signaling.41 Often this takes the form of a kind of fested in the final location of the particle, or the position double-standard in which alternatives to ordinary QM of some pointer, or some fact about some other beable) are dismissed as “non-local” (and therefore unacceptable) may be denoted by A and B respectively. on the grounds that they include (either manifestly, as The beables pertaining to a given run of the experi- in the case of the pilot-wave theory, or just in princi- ment may then be cataloged as in Figure 4. Roughly, we ple, as established by Bell’s theorem) “gross violations may think ofa ˆ and ˆb (which live in regions 1 and 2, re- of relativistic causality” – but then ordinary QM itself is spectively) as referring to the spatial orientations of the 13

A B “Very often such factorizability is taken as 1 2 the starting point of the analysis. Here we ˆb aˆ have preferred to see it not as the formula- 3 tion of ‘local causality’, but as a consequence 7 λ thereof.”

B. The EPR Argument FIG. 4: Space-time diagram illustrating the various beables of relevance for the EPR-Bell setup. (Cf. Bell’s diagram In their famous 1935 paper, Einstein, Podolsky and in Ref. 7.) Separated observers Alice (in region 1) and Bob Rosen argued that a local explanation for the perfect cor- (in region 2) make spin-component measurements (using ap- relations predicted by quantum theory (in a certain kind paratus settingsa ˆ and ˆb respectively) on a pair of spin- or polarization-entangled particles (represented by the dashed of situation of which the above EPR-Bell setup is an ex- A B ample) required the existence of locally pre-determined lines). The measurements have outcomes and respec- 9 tively. The state of the particle pair in region 3 is denoted values for the measurement outcomes. Since, as men- λ. Note that what we are here calling region 3 extends across tioned in the introduction, ordinary QM contains no such the past light cones of both regions 1 and 2. It thus not only elements of reality, EPR concluded that ordinary QM “completely shields off from 1 the overlap of the backward (and in particular the wave function) did not provide a 7 light cones of 1 and 2” , but also vice versa. Bell’s local complete description of physical reality. They suggested causality condition therefore requires both that ˆb and B are that an alternative, locally causal theory which did pro- irrelevant for predictions about the outcome A, and thata ˆ vide a complete description of physical reality might be and A are irrelevant for predictions about the outcome B, found. once λ is specified. Taking for granted that the relevant empirical predic- tions of quantum theory are correct, one can summarize the logic of EPR’s argument this way: two pieces of measuring apparatus (this being the basis for the notation), and λ (in region 3) as referring to the locality incompleteness (10) → state of the particle pair emitted by the source. (Note that the phrase “state of the particle pair” should not be where “incompleteness” means specifically the incom- taken too seriously; no actual assumption is made about pleteness of the orthodox quantum mechanical descrip- the existence, for example, of literal particles.) tion of the particles in question (in terms of their quan- Unlike region 3 of Figure 2, region 3 of Figure 4 ex- tum state alone). This is of course logically equivalent to tends across the past light cone not only of region 1, but the statement that of region 2 as well. It particular, this extended region completeness non-locality (11) 3 closes off the past light cones of both regions 1 and 2 → and shields both regions from their overlapping past light which explains why the EPR argument is sometimes cones. A complete specification of beables in this region 3 characterized as an argument for the incompleteness of will therefore, according to Bell’s concept of local causal- orthodox QM, and sometimes instead as pointing out the ity, “make events in 2 irrelevant for predictions about 1”7 non-locality of that candidate theory. and will also make events in 1 irrelevant for predictions In their 1935 paper, EPR appealed to an intuitive no- about 2: tion of local causality which was not precisely formulated. P (A a,ˆ ˆb, B, λ)= P (A a,ˆ λ) (6) It is therefore of interest that their argument can be reca- | | pitulated and made rigorous by using Bell’s formulation and of local causality. It is clarifying to begin with the EPR argument in the form of Statement 11. The proof then P (B a,ˆ ˆb, λ)= P (B ˆb, λ). (7) | | consists in simply using the formulated notion of local From these and the identity causality in its directly-intended way – namely, to assess whether a particular candidate theory is or is not local. P (A, B a,ˆ ˆb, λ)= P (A a,ˆ ˆb,B,λ) P (B a,ˆ ˆb, λ) (8) Take again the situation indicated in Figure 4. Here it | | | is important to appreciate that because of the structure the so-called “factorization” of the joint probability for of region 3 – and note that it could be extended into a A B outcomes and immediately follows: space-like hypersurface crossing through the region 3 de- P (A, B a,ˆ ˆb, λ)= P (A a,ˆ λ) P (B ˆb, λ). (9) picted there, and still satisfy the requirements discussed | | | earlier – the relevant complete specification of beables This factorization condition is widely recognized in the does not presuppose that the “state of the particle pair” Bell literature to be sufficient for the derivation of itself must “factorize” into two distinct and independent empirically-testable Bell-type inequalities. As Bell notes, states for the two particles. The state can instead be however, characterized in a way that is essentially inseparable, as 14 in ordinary QM, and the argument still goes through: “It the state λ produced by the preparation procedure at is notable that in this argument nothing is said about the the source. Instead, we allow that λ may perhaps take locality, or even localizability, of the variable λ. These several distinct values from one run of the experiment to variables could well include, for example, quantum me- another. Let us also continue to assume that Alice and chanical state vectors, which have no particular localiza- Bob both freely choose to make measurements along the tion in ordinary space-time. It is assumed only that the zˆ direction. outputs A and B, and the particular inputs a and b, are The argument is then simple: we have already shown well localized.”14 Let us suppose in particular that the that local causality entails the factorization of the joint “preparation procedure” at the particle source (the star probability for outcomes A, B, once λ is specified. Con- in Figure 4) gives rise to a pair in (what ordinary QM sidering for example the case A = +1,B = +1, whose describes as) the spin singlet state joint probability vanishes, we therefore have that, for each specific value of λ that might (with nonzero prob- 1 ψ = ( 1 2 1 2) (12) ability) be produced by the preparation procedure, one | √2 | ↑ | ↓ −| ↓ | ↑ of P (A = +1 a,ˆ λ) and P (B = +1 ˆb, λ) must vanish. But since there| are only two possible| outcomes for each where 1 means that particle 1 is spin-up along (say) the z-direction,| ↑ etc. measurement, each of these possibilities entails that the opposite outcome is pre-determined. For example, Suppose also thata ˆ = ˆb =z ˆ, i.e., both Alice and Bob (freely) choose to measure the spins of the incoming par- P (A = +1 a,ˆ λ)=0 P (A = 1 a,ˆ λ)=1 (15) ticles along the z-direction. Then QM predicts (letting | → − | A = +1 denote the result that Alice finds her particle to which means that those particular values of λ to which be spin up, etc.) that either A = +1 and B = 1 (with this applies must “contain” or “encode” the outcome probability 50%) or that A = 1 and B = +1− (with A = 1 which will then be revealed with certainty if a probability 50%). − measurement− alonga ˆ is performed. It is easy to see that Thus, noting that for orthodox QM λ in Figure 4 is the possible values of λ must therefore fall into two mu- simply the quantum mechanical wave function, we have tually exclusive and jointly exhaustive categories – those for example that which encode the pre-determined outcomes A = +1 and 1 B = 1, and those which encode the pre-determined P (A = +1 a,ˆ λ)= , (13) − 42 | 2 outcomes A = 1 and B = +1. Indeed, since− the measurement axis is assumed to be but also that “free” the same argument will establish that λ must en- code pre-determined outcomes for all possible measure- P (A = +1 a,ˆ ˆb, λ, B = 1)=1, (14) | − ment directions. One thus sees how theories of deter- in violation of Eq. (1). Orthodox QM is not a locally ministic hidden variables (or what N. David Mermin has 48 causal theory: dubbed “instruction sets” ) are in fact required by local causality. “The theory requires a perfect correlation of [results] on the two sides. So specification of the result on one side permits a 100% confi- C. CHSH Inequality dent prediction of the previously totally un- certain result on the other side. Now in ordi- It is well-known that a Bell-type inequality follows nary quantum mechanics there just is noth- from the assumption of local deterministic hidden vari- ing but the wavefunction for calculating prob- ables or “instruction sets”. That theories of this type are abilities. There is then no question of mak- actually required by locality (as explained in the previous ing the result on one side redundant on the sub-section) should therefore already clarify the serious- other by more fully specifying events in some ness with which Bell took the idea of a fundamental con- space-time region 3. We have a violation of flict between SR and the predictions of QM. This conflict local causality.”7 can, however, be brought out in an even more streamlined As pointed out, Statements 10 and 11 are logically way, by deriving a Bell-type inequality directly from the equivalent, so it is clear that a locally causal explanation factorization of the joint probability as in Eq. (9) – and for the perfect correlations predicted by QM will require hence from Bell’s local causality (without any additional a theory with more (or perhaps just different) beables discussion of determinism or pre-determined values). than just the quantum mechanical wave function. But it Assume that the measurement scenario indicated in is possible to show directly from Bell’s concept of local Figure 4 is repeated many times, with each setting being causality that one must in particular posit beables which selected “freely” or “randomly” on each run, from two pre-determine the outcomes of both measurements. possibilities:a ˆ aˆ1, aˆ2 , ˆb ˆb1, ˆb2 . The procedure To begin with, let us drop the assumption (which ap- which prepares or∈ {creates} the∈ “particles { } to be measured” plied to ordinary QM) that it is possible to fully control is held fixed for all runs of the experiment. As before, this 15 will not necessarily imply that λ is constant for all runs, D. The “Free Variables” Assumption since the relevant beables may be less than fully control- lable; we will assume, though, that the distribution of Let us finally return to the assumption that the set- different values of λ across the runs can be characterized tingsa ˆ and ˆb are “free” or “random”. Mathematically by a probability distribution ρ(λ). speaking, this was the assumption that the probability Next we define the correlation of outcomes A and B distribution ρ(λ) for the distribution of possible “states as the expected value of their product: of the particle pair” created by the source is independent of the apparatus settingsa ˆ and ˆb. For example, in de- C(ˆa, ˆb) = A B P (A a,ˆ λ) P (B ˆb, λ) ρ(λ) dλ Z | | riving Eq. (18) one assumes that the same probability A,BX distribution ρ(λ) characterizes runs in whicha ˆ1 and ˆb1 ˆ = A¯(ˆa, λ) B¯(ˆa, λ) ρ(λ) dλ (16) are measured, as characterizes runs in whicha ˆ1 and b2 Z are measured. As Bell writes in support of this assump- where tion, A¯(ˆa, λ)= P (A = +1 a,ˆ λ) P (A = 1 a,ˆ λ) (17) “we may imagine the experiment done on | − − | such a scale, with the two sides of the exper- satisfies A¯ 1, and similarly for B¯. iment separated by a distance of order light | |≤ Now we consider several combinations of correlations minutes, that we can imagine these settings involving different pairs of settings. To begin with, being freely chosen at the last second by two different experimental physicists.... If these C(ˆa1, ˆb1) C(ˆa1, ˆb2) ± last second choices are truly free ..., they are not influenced by the variables λ. Then the = A¯(ˆa1, λ) B¯(ˆb1, λ) B¯(ˆb2, λ) ρ(λ) dλ (18) Z  ±  resultant values for [ˆa] and [ˆb] do not give any information about λ. So the probability dis- so that tribution over λ does not depend on [ˆa] or ˆ 7 C(ˆa1, ˆb1) C(ˆa1, ˆb2) [b]...” ±

The real (as opposed to imagined) experiments, however, B¯(ˆb1, λ) B¯(ˆb2, λ) ρ(λ) dλ. (19) ≤ Z ± do not involve “settings being freely chosen at the last

second by two different experimental physicists” but in- Similarly, we have that stead involve physical random (or pseudo-random) num- ber generators. As mentioned earlier, though, this means C(ˆa2, ˆb1) C(ˆa2, ˆb2) ∓ that – at least in principle, for some possible candidate

theories – a complete description of beables in region 3 B¯(ˆb1, λ) B¯(ˆb2, λ) ρ(λ) dλ. (20) ≤ Z ∓ of Figure 4 will include not only a complete description of the “state of the particle pair” but also a complete Adding Equations (19) and (20), and noting that x | ± description of whatever physical degrees of freedom are y + x y is one of 2x, 2x, 2y, or 2y, we have that ˆ | | ∓ | − − determining or influencing the eventual settingsa ˆ and b – making it not only possible but rather likely that the can- C(ˆa1, ˆb1) C(ˆa1, ˆb2) + C(ˆa2, ˆb1) C(ˆa2, ˆb2) ± ∓ didate theory in question should exhibit (contrary to the assumption that was made) correlations between what 2 (21) ≤ we have called λ and those settings. which is the so-called Clauser-Horne-Shimony-Holt As suggested earlier, though, one can here appeal to (CHSH) inequality.44 This is in essence the relation tested the expectation that serious candidate theories will posit in the experiments, e.g., Refs. 17 and 18. Quantum the- an enormously large number of physical degrees of free- ory predicts that (for appropriate preparations of the dom in a spacetime region like 3, only some tiny fraction two-particle state and for appropriate choices ofa ˆ1,a ˆ2, of which are actually needed to completely specify the ˆb1, and ˆb2) the left hand side should be 2√2 – more than “state of the particle pair” – i.e., the beables which are 40% larger than the constraint implied by local causality. physically influenced by the preparation procedure at the And the experimental results are in excellent agreement source. There are then innumerable other beables in re- with the quantum predictions. gion 3 which might be used to determine/influence the Since the inequality is derived from the local causal- apparatus settings. The expectation – more precisely, ity condition, what follows from the experimental results in the context of the derivation of the locality inequal- is that any theory which makes empirically correct pre- ity, the assumption – is that this is done in some way dictions will have to violate the local causality condition. such that there are no correlations, no conspiratorial pre- As Bell writes, “The obvious definition of ‘local causality’ established harmonies, between the beables chosen to de- does not work in quantum mechanics, and this cannot be termine the apparatus settings and those which encode attributed to the ‘incompleteness’ of that theory.”7 the “state of the particle pair”. 16

As Bell acknowledges, therefore, one logical possibil- “we feel that it is wrong on methodological ity (in the face of the empirical violations of the CHSH grounds to worry seriously about [the pos- inequality) is that sibility of the kind of conspiracy that would render the assumption inapplicable] if no spe- “it is not permissible to regard the experi- cific causal linkage [between the beables λ mental settings [ˆa] and [ˆb] in the analyzers as and those which determine the apparatus set- independent of the supplementary variables tings] is proposed. In any scientific experi- λ, in that [ˆa] and [ˆb] could be changed without ment in which two or more variables are sup- changing the probability distribution ρ(λ). posed to be randomly selected, one can al- Now even if we have arranged that [ˆa] and ways conjecture that some factor in the over- [ˆb] are generated by apparently random ra- lap of the backward light cones has controlled dioactive devices, housed in separated boxes the presumably random choices. But, we and thickly shielded, or by Swiss national lot- maintain, skepticism of this sort will essen- tery machines, or by elaborate computer pro- tially dismiss all results of scientific experi- grammes, or by apparently free willed experi- mentation. Unless we proceed under the as- mental physicists, or by some combination of sumption that hidden conspiracies of this sort all of these, we cannot be sure that [ˆa] and [ˆb] do not occur, we have abandoned in advance are not significantly influenced by the same the whole enterprise of discovering the laws factors λ that influence A and B. But this of nature by experimentation.”43 way of arranging quantum mechanical cor- relations would be even more mind boggling Imagine, for example, an experimental drug trial in which than one in which causal chains go faster than patients are randomly selected to receive either the drug light. Apparently separate parts of the world or a placebo. It is always logically possible that the sup- would be deeply and conspiratorially entan- posedly random selections (made, say, by flipping a coin) gled, and our apparent free will would be en- are in fact correlated with some pre-existing facts about tangled with them.”14 the health of patients. And such a correlation could skew the results of the trial – say, resulting in a statistically sig- And here is another relevant statement from Bell: nificant improvement in the health of the patients given the genuine drug even though in fact the drug is impotent “An essential element in the reasoning here or worse. The suggestion is that, in general, unless there ˆ is that [ˆa] and [b] are free variables. One can is some plausible causal mechanism that might conceiv- envisage then theories in which there just are ably produce the correlations in question (for example, no free variables for the polarizer angles to be instead of flipping coins, the drug/placebo assignments coupled to. In such ‘superdeterministic’ the- are made on the basis of patients’ blood pressures) it is ories the apparent free will of experimenters, reasonable to assume that the conspiratorial correlations and any other apparent randomness, would are absent. be illusory. Perhaps such a theory could be That is, the additional assumption (beyond local both locally causal and in agreement with causality) which is needed to derive the CHSH inequal- quantum mechanical predictions. However I ity “is no stronger than one needs for experimental rea- do not expect to see a serious theory of this soning generically”.43 The “no conspiracies” assumption kind. I would expect a serious theory to per- thus falls into the same category as some other things mit ‘deterministic chaos’ or ‘pseudorandom- (for example, the validity of logic, and certain mathe- ness’, for complicated subsystems (e.g. com- matical operations) which, while used in the derivation, puters) which would provide variables suffi- are not on the table as seriously challengeable. This ex- ciently free for the purpose at hand. But I do 7 plains why we sometimes do not even bother to mention not have a theorem about that.” this assumption – as, for example, when writing that the CHSH inequality follows from Bell’s concept of local It is sometimes erroneously thought that the “freedom” causality alone. or “no conspiracies” assumption being discussed here does – or should – follow from local causality. For exam- ple, in their illuminating “Exchange [with Bell] on Local Beables” A. Shimony, M.A. Horne, and J.F. Clauser crit- VII. SUMMARY AND OPEN QUESTIONS icized Bell’s derivation for using (in our notation) the as- sumption ρ(λ a,ˆ ˆb) = ρ(λ) which, they correctly pointed We have reviewed J.S. Bell’s formulation of relativis- out, does not| follow from local causality. Bell subse- tic local causality, including a careful survey of its con- quently clarified that it was a separate assumption, not ceptual background and a sketch of its most important supposed to follow from local causality. As Shimony, implications. We have stressed in particular that Bell’s Horne, and Clauser noted, however, the additional as- formulation does not presuppose determinism or the ex- sumption seems eminently reasonable: istence of hidden variables (or any of the other sorts of 17 things that are sometimes blamed for the empirical vio- Bohm pilot-wave theory is probably the clearest example lation of Bell-type inequalities) but instead seems plau- here: its posited ontology includes both particles (which sibly to just capture the intuitive idea, widely taken as follow definite trajectories in 3-space and are pre-eminent an implication of special relativity, indicated in Figure examples of local beables) and a guiding wave (which is 1 – namely, the idea that causal influences cannot prop- simply the usual quantum mechanical wave function but agate faster than light. And as we have seen, taking interpreted unambiguously as a beable). But, for an N- now the “no conspiracies” assumption for granted, the particle system (the universe, say) the wave function is a empirically-violated CHSH inequality can be cleanly de- function on the 3N-dimensional configuration space. It rived from Bell’s concept of local causality without the is thus, for this theory, a non-local beable.49 need for further assumptions involving determinism, hid- den variables, “realism” or “classicality” (whatever ex- As mentioned previously, Bell’s region 3 can be ex- actly those ideas mean), etc. tended into a space-like hypersurface without spoiling This hopefully clarifies why Bell disagreed with the any of the arguments that have been given in this pa- widespread opinion that his theorem – and the associ- per. We may then generously include, where Bell’s for- ated experiments – somehow vindicate ordinary quan- mulation instructs us to use a complete specification of tum theory as against hidden variable alternative theo- the local beables in region 3, also values for any non- ries or somehow vindicate Bohr’s philosophy as against local beables which, like wave functions, can be associ- Einstein’s. Instead, the reader can now appreciate why, ated with hypersurfaces. And it is important that, even for Bell, “the real problem with quantum theory” is the when including information about non-local beables in “apparently essential conflict between any sharp formu- this way, such theories still violate the condition, i.e., are lation and relativity [, i.e., the] apparent incompatibility, diagnosable as non-local. Still, as formulated, Bell’s no- at the deepest level, between the two fundamental pillars tion of local causality seems to presuppose that we are of contemporary theory...”16 dealing with theories positing exclusively local beables.50 It should be noted, however, that the presentation here It can be stretched to accomodate certain extant theo- has been of J.S. Bell’s formulation of relativistic local ries which posit also non-local beables, but how to do causality. Although we have argued strongly for its rea- this with complete generality, and what other issues (like sonableness, this particular formulation should not nec- those encountered above in the case of non-Markovian essarily be regarded as definitive. theories) may arise in the attempt, remains unclear. On Here we briefly indicate several points on which its ap- the other hand, it is also unclear how seriously one can plicability to various sorts of “exotic” theories could be or should take theories with non-local beables in the first questioned and where a more general or distinct formu- place. In particular: should such theories even be consid- lation of local causality might be sought. ered candidates for “locally causal” status? And perhaps For example, one might worry that a theory with more importantly, could a theory positing non-local be- a non-Markovian character (that is, a theory in which ables possibly count as genuinely consistent with special causal influences can jump discontinuously from one time relativity? to a later time) could violate Bell’s local causality con- dition despite positing no strictly faster-than-light influ- Such questions will certainly not be answered here. We ences. The idea would be that, in such a theory, influ- raise them simply to give the reader some sense of the ences could “hop over” region 3 of Figure 2, leading to concerns that one might have about Bell’s formulation of correlations in regions 1 and 2 but leaving no trace in local causality. Their admittedly exotic character should 3. This could seemingly be addressed by requiring that however help explain why Bell felt driven to contemplate Bell’s region 3 extend farther back in time, i.e., cover a re- “unspeakable” deviations from conventional wisdom. In gion of spacetime so “thick” that hopping non-Markovian particular, one can now appreciate how simple everything influences cannot make it across. In the limit of arbitrar- would become if we simply dropped the insistence on rec- ily large violations of Markovianity, this would evidently onciling the Bell experiments with “fundamental relativ- require region 3 to encompass the entire past light cone ity” and instead returned to the (empirically equivalent) of the region 3 pictured in Figure 2. But this maneuvre pre-Einstein view according to which there exists a (hid- raises further issues: the more of spacetime that gets in- den) preferred frame of reference. As explained by Bell cluded in region 3, the more one might start to doubt in the quotes given in the Introduction, such a view can the reasonableness of the “no conspiracies” assumption, easily accomodate faster-than-light causal influences in a and the more one might worry that the condition would way that Einsteinian relativity, seemingly, can’t. fail to detect certain kinds of non-localities (and so would function merely as a necessary condition for, rather than Again, though, our goal here is not to lobby for this a formulation of, locality). view, but merely to explain Bell’s rationale for taking Similar problems arise when one contemplates the pos- it seriously, as a possibility warranting careful attention sibility of theories which posit not only local beables – not just by philosophers and commentators, but by (i.e., those “being associated with definite positions in physicists interested in addressing the puzzles of yester- space”25) but also non-local beables. The de Broglie - day, today, and tomorrow. 18

Acknowledgments formulation of local causality, and to several anonymous referees for a number of helpful suggestions on earlier Thanks to Shelly Goldstein, Daniel Tausk, Nino drafts of the paper. Zanghi, and Roderich Tumulka for discussions on Bell’s

∗ Electronic address: [email protected] American Journal of Physics, 72(9), 1220-1226 (2004) 1 The Born-Einstein Letters, Irene Born, trans., (Walker and 12 J. S. Bell, “Quantum mechanics for cosmologists,” in Company, New York, 1971) Quantum Gravity 2, C. Isham, R. Penrose, and D. Sciama, 2 Mary B. Hesse, Forces and Fields: The Concept of Action eds., (Oxford, Clarendon Press, 1981); also reprinted in at a Distance in the History of Physics, (Dover Publica- Ref. 8, pp. 117-138. tions, Mineola, New York, 2005) 13 J.S. Bell, “On the impossible pilot wave,” Foundations of 3 Ernan McMullin, “The Explanation of Distant Action: Physics, 12 (1982), pp. 989-99; also reprinted in Ref. 8, Historical Notes,” in James T. Cushing and Ernan Mc- pp. 159-168. Mullin, Eds., Philosophical Consequences of Quantum 14 J.S. Bell, “Bertlmann’s socks and the nature of reality,” Theory, (University of Notre Dame Press, Indiana, 1989) Journal de Physique, Colloque C2, suppl. au numero 3, 4 Isaac Newton’s February 25, 1693 letter to Richard Tome 42 (1981) pp. C2 41-61; reprinted in Ref. 8, pp. 139- Bentley, quoted (e.g.) in Andrew Janiak, “Newton’s 158. Philosophy”, The Stanford Encyclopedia of Philoso- 15 J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,” phy (Winter 2009 Edition), Edward N. Zalta, Ed., Physics 1 (1964) 195-200; reprinted in Ref. 8, pp. 14-21. ¡http://plato.stanford.edu/archives/win2009/entries/newton-philosophy/16 J. S. Bell,¿ “Speakable and unspeakable in quantum me- 5 I. Bernard Cohen, “A Guide to Newton’s Principia” in chanics,” introduction remarks at Naples-Amalfi meeting, Isaac Newton, The Principia, I.B. Cohen and Anne Whit- May 7, 1984; reprinted in Ref. 8, pp. 169-172. man, trans., (University of California Press, 1999). See es- 17 A. Aspect et al., “Experimental Test of Bell’s Inequali- pecially pp. 60-4 and 277-80. ties Using Time-Varying Analyzers,” Phys. Rev. Lett., 49, 6 Albert Einstein, Relativity: The Special and the General 1804-1807 (1982). Theory, (Penguin Classics, 2006), p. 47 18 Gregor Weihs et al., “Violation of Bell’s inequality un- 7 John S. Bell, “La nouvelle cuisine,” in Between Science der strict Einstein locality conditions,” Phys. Rev. Lett. and Technology, A. Sarlemijn and P. Kroes, eds., (Elsevier 81 5039-5043 (1998). Science Publishers, 1990); also reprinted in Ref. 8, pp. 232- 19 P.C.W. Davies and J.R. Brown, eds., The Ghost in the 48. Atom, Chapter 3: interview with J.S. Bell, Cambridge Uni- 8 John S. Bell, Speakable and Unspeakable in Quantum Me- versity Press, 1986 chanics, 2nd ed., (Cambridge University Press, 2004). 20 See also Bell’s paper “How to teach special relativity,” 9 Albert Einstein, Boris Podolsky, and Nathan Rosen, “Can Progress in Scientific Culture, Vol. 1, No. 2, summer 1976; quantum-mechanical description of reality be considered reprinted in Ref. 8, pp.67-80 complete?” Phys. Rev. 47, 777-780, 1935 21 H. Price, Time’s Arrow and Archimedes’ Point, Oxford 10 Travis Norsen, “Einstein’s Boxes”, American Journal of University Press, 1997. Physics, 73(2), 164-176 (2005) 22 Roderich Tumulka, “A Relativistic Version of the Ghirardi- 11 The terminology of “hidden variables” is unfortunate be- Rimini-Weber Model” J. Stat. Phys. 125 (2006) 821-840; cause, at least in the one clear extant example of a “hid- see also Tim Maudlin, “Non-local correlations in quantum den variables theory,” the “hidden variables” are precisely theory: how the trick might be done” in W. L. Craig and the variables which are not, in fact, hidden. About this Q. Smith, eds., Einstein, Relativity and Absolute Simul- theory (the de Broglie - Bohm “pilot wave” theory, a.k.a. taneity, Routledge, 2008 “Bohmian Mechanics,” which adds to the standard quan- 23 J.S. Bell, “Are there quantum jumps?” in Schr¨odinger: tum mechanical wave function definite particle positions Centenary of a polymath, Cambridge, 1987; reprinted in obeying a deterministic evolution law) Bell remarked: “it Ref. 8, pp. 201-212. would be appropriate to refer to the xs as ‘exposed vari- 24 For a recent review and many relevant references see A. ables’ and to ψ as a ‘hidden variable’. It is ironic that the Shimony, “Bell’s Theorem,” The Stanford Encyclopedia of traditional terminology is the reverse of this.”12 And simi- Philosophy (Summer 2009 Edition), Edward N. Zalta, Ed., larly: “Although [in Bohmian Mechanics] Ψ is a real field it ¡http://plato.stanford.edu/archives/sum2009/entries/bell-theorem/¿ does not show up immediately in the result of a single ‘mea- 25 J.S. Bell, “Beables for quantum field theory,” CERN-TH surement,’ but only in the statistics of many such results. 4035/84, Aug. 2, 1984; reprinted in Ref. 8, pp. 173-180. It is the de Broglie - Bohm variable X that shows up im- 26 J.S. Bell, “The theory of local beables,” Epistemological mediately each time. That X rather than Ψ is historically Letters, March 1976; reprinted in Ref. 8, pp. 52-62 called a ‘hidden’ variable is a piece of historical silliness.”13 27 J.S. Bell, “Against ‘measurement’,” in 62 Years of Un- It is also relevant that the wave function ψ is “hidden” certainty, Erice, 5-14 August 1989, Plenum Publishers; – in the sense of being not accessible via experiment – reprinted in Ref. 8, pp. 213-231 even in orthodox quantum theory (which is suppsed to be 28 J. S. Bell, “Free variables and local causality,” Epistemo- the primary example of a non-hidden-variable theory). For logical Letters, Feb. 1977; reprinted in Ref. 8, pp. 100-104 some illuminating further discussion, see: Roderich Tu- 29 Eugene P. Wigner, “Interpretion of Quantum Mechan- mulka, “Understanding Bohmian Mechanics: A Dialogue” ics,” (1976), in Quantum Theory and Measurement, J.A. 19

Wheeler and W.H. Zurek, eds., (Princeton University 40 What we here call “local signaling” is also sometimes called Press, 1983). “no signaling” or “signal locality”. 30 N. David Mermin, “Hidden Variables and the Two Theo- 41 See, for example, the discussion in References 33 and34 and rems of John Bell,” Rev. Mod. Phys., 65(3), July 1993, pp. also L.E. Ballentine and Jon P. Jarrett, “Bell’s Theorem: 803-815. Does quantum mechanics contradict relativity?” American 31 Jon Jarrett, “Bell’s Theorem: A Guide to the Implica- Journal of Physics 55(8), 696-701 (1987) tions” in J. Cushing and E. McMullin, eds., Philosophical 42 For a more detailed version of this argument, see T. Consequences of Quantum Theory, (University of Notre Norsen, “Bell Locality and the Nonlocal Character of Na- Dame Press, Indiana, 1989). ture”, Foundations of Physics Letters, 19, pp. 633-655 32 Anton Zeilinger, “The message of the quantum,” Nature, (2006) 438, 743 (8 December, 2005) 43 A. Shimony, M.A. Horne, J.F. Clauser, “Comment on ‘The 33 Greenstein and Zajonc, The Quantum Challenge, 2nd edi- Theory of Local Beables’,” from Epistemological Letters tion, (Jones and Bartlett, 2005) Hidden Variables and quantum Uncertainty, F. Bonsack, 34 D.J. Griffiths, Introduction to Quantum Mechanics, (Pren- ed., No. 9 (1976); reprinted in “An Exchange on Local tice Hall, NJ, 1995) Beables”, Dialectica, Vol. 39, pp. 97-101 (1985) 35 J.S. Townsend, A Modern Approach to Quantum Mechan- 44 J.F. Clauser, M.A. Horne, A. Shimony, and R.A. Holt, ics, (University Science Books, McGraw-Hill, 1992) “Proposed Experiment to Test Local Hidden-Variable The- 36 J.J. Sakurai, Modern Quantum Mechanics, (Addison- ories,” Phys. Rev. Lett. 23, 880-884 (1969) Wesley, 1994) 45 Jeremy Butterfield, “Bell’s Theorem: What it Takes,” 37 For Bell, there was no important distinction between the British Journal for the Philosophy of Science 42, pp. 41-83 terms “locality” and “local causality”. For example, Bell 46 H. R. Brown, Physical Relativity, (Oxford University Press, first uses the phrase “local causality” in print in his 1976 2005) paper “The theory of local beables”. The paper’s section 2 47 Tim Maudlin, Quantum Non-Locality and Relativity, Sec- (entitled “Local causality”) presents an early version of his ond Edition, (Blackwell, 2002) formulation of this idea. But then later in the same paper 48 N. D. Mermin, “Bringing home the atomic world: Quan- he refers to the inequality (which he has shown how to tum mysteries for anybody” Am. J. Phys., 49, 940-3 (1981) derive from “local causality”) as “the locality inequality” 49 About the de Broglie - Bohm theory, Bell writes: “No one and even remarks that the detailed discussion of “local can understand this theory until he is willing to think of causality” in section 2 had “been an attempt to be rather ψ as a real objective field rather than just a ‘probability explicit and general about the notion of locality, along lines amplitude’. Even though it propagates not in 3-space but only hinted at in previous publications.”26 . This usage is in 3N-space.”12 Some proponents of the de Broglie - Bohm consistent with his later publications: see, for example, pilot wave theory, however, prefer to interpret the wave his 1987 preface to the first edition of his collected papers function in that theory not as a beable at all, but rather (see pp. xi-xii of Ref. 8) and his 1990 paper “La nouvelle as a law. See, for example, S. Goldstein and N. Zanghi, cuisine.”7 “Reality and the Role of the Wavefunction in Quantum 38 Charles Mann and Robert Crease, “John Bell, Particle Theory,” arXiv:1101.4575 Physicist” (Interview), Omni, Vol. 10 No. 8, 84-92 and 50 T. Norsen, “The theory of (exclusively) local beables”, 121 (May 1988) Foundations of Physics 40 1858 (2010) 39 T. Norsen, “Against ‘Realism’,” Foundations of Physics, 37(3), 311-340 (2007)