Action at a Distance in Quantum Mechanics

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Action at a Distance in Quantum

Mechanics

First published Fri 26 Jan, 2007

In the quantum realm, there are curious correlations between the properties of distantsystems. An example of such correlations is provided by the famous Einstein-Podolsky-Rosen/Bohm experiment. The correlations in the EPR/B experiment strongly suggest thatthere are non-local influences between distant systems, i.e., systems between which nolight signal can travel, and indeed orthodox quantum mechanics and its variousinterpretations postulate the existence of such non-locality. Yet, the question of whetherthe EPR/B correlations imply non-locality and the exact nature of this non-locality is amatter of ongoing controversy. Focusing on EPR/B-type experiments, in this entry weconsider the nature of the various kinds of non-locality postulated by different

interpretations of quantum mechanics. Based on this consideration, we briefly discuss thecompatibility of these interpretations with the special theory of relativity.

1. Introduction 2. Bell's theorem and non-locality 3. The analysis of factorizability 4. Action at a distance, holism and non-separability

o 4.1 Action at a distanceo 4.2 Holismo 4.3 Non-separability

5. Holism, non-separability and action at a distance in quantum mechanicso

5.1 Collapse theorieso 5.2 Can action-at-a-distance co-exist with non-separability and holism?o 5.3 No-collapse theories

6. Superluminal causation 7. Superluminal signaling

o 7.1 Necessary and sufficient conditions for superluminal signalingo 7.2 No-collapse theorieso 7.3 Collapse theorieso 7.4 The prospects ofcontrollable probabilistic dependenceo 7.5 Superluminal signaling and action-at-a-distance

8. The analysis of factorizability: implications for quantum non-localityo 8.1 Non-separability, holism and action at a distanceo 8.2 Superluminal signalingo 8.3 Relativityo 8.4 Superluminal causationo 8.5 On the origin and nature of parameter dependence

9. Can there be local quantum theories? 10. Can quantum non-locality be reconciled with relativity?

o 10.1 Collapse theories
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o 10.2 No-collapse theorieso 10.3 Quantum causal loops and relativity

Bibliography Other Internet Resources Related Entries

1. Introduction

The quantum realm involves curious correlations between distant events. A well-knownexample is David Bohm's (1951) version of the famous thought experiment that Einstein,Podolsky and Rosen proposed in 1935 (henceforth, the EPR/B experiment). Pairs ofparticles are emitted from a source in the so-called spin singlet state and rush in oppositedirections (see Fig. 1 below). When the particles are widely separated from each other,they each encounter a measuring apparatus that can be set to measure their spincomponents along various directions. Although the measurement events are distant fromeach other, so that no slower-than-light or light signal can travel between them, themeasurement outcomes are curiously correlated.[1] That is, while the outcome of each ofthe distant spin measurements seems to be a matter of pure chance, they are correlatedwith each other: The joint probability of the distant outcomes is different from theproduct of their single probabilities. For example, the probability that each of theparticles will spin clockwise about thez-axis in az-spin measurement (i.e., ameasurement of the spin component along thezdirection) appears to be . Yet, theoutcomes of such measurements are perfectly anti-correlated: If the left-hand-side (L-)particle happens to spin clockwise (anti-clockwise) about thez-axis, the right-hand-side(R-) particle will spin anti-clockwise (clockwise) about that axis. And this is true even ifthe measurements are made simultaneously.

Figure 1: A schematic illustration of the EPR/B experiment. Particle pairs in the spin

singlet state are emitted in opposite directions and when they are distant from each other(i.e., space-like separated), they encounter measurement apparatuses that can be set tomeasure spin components along various directions.

The curious EPR/B correlations strongly suggest the existence of non-local influencesbetween the two measurement events, and indeed orthodox collapse quantummechanics supports this suggestion. According to this theory, before the measurementsthe particles do not have any definite spin. The particles come to possess a definite spin
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only with the first spin measurement, and the outcome of this measurement is a matter ofchance. If, for example, the first measurement is az-spin measurement on the L-particle,the L-particle will spin either clockwise or anti-clockwise about thez-axis with equalchance. And the outcome of the L-measurement causes an instantaneous change in thespin properties of the distant R-particle. If the L-particle spins clockwise (anti-clockwise)

about thez-axis, the R-particle will instantly spin anti-clockwise (clockwise) about thesame axis. (It is common to call spins in opposite directions spin up and spin down,where by convention a clockwise spinning may be called spin up and anti-clockwisespinning may be called spin down.)

It may be argued that orthodox quantum mechanics is false, and that the non-localitypostulated by it does not reflect any non-locality in the quantum realm. Alternatively, itmay be argued that orthodox quantum mechanics is a good instrument for predictionsrather than a fundamental theory of the physical nature of the universe. On thisinstrumental interpretation, the predictions of quantum mechanics are not an adequatebasis for any conclusion about non-locality: This theory is just an incredible oracle (or a

crystal ball), which provides a very successful algorithm for predicting measurementoutcomes and their probabilities, but it offers little information about ontological matters,such as the nature of objects, properties and causation in the quantum realm.

Einstein, Podolsky and Rosen (1935) thought that quantum mechanics is incomplete andthat the curious correlations between distant systems do not amount to action at a distancebetween them. The apparent instantaneous change in the R-particle's properties during theL-measurement is not really a change of properties, but rather a change of knowledge.(For more about the EPR argument, see the entry on the EPR argument, Redhead 1987,chapter 3, and Albert 1992, chapter 3. For discussions of the EPR argument in therelativistic context, see Ghirardi and Grassi 1994 and Redhead and La Riviere 1997.) On

this view, quantum states of systems do not always reflect their complete state. Quantumstates of systems generally provide information aboutsome of the properties that systemspossess and information about the probabilities of outcomes of measurements on them,and this information does not generally reflect the complete state of the systems. Inparticular, the information encoded in the spin singlet state is about the probabilities ofmeasurement outcomes of spin properties in various directions, about the conditionalprobabilities that the L- (R-) particle has a certain spin property given that the R- (L-)particle has another spin property, and about the anti-correlation between the spins thatthe particles may have in any given direction (for more details, see section 5.1). Thus, theoutcome of az-spin measurement on the L-particle and the spin singlet state (interpretedas a state of knowledge) jointly provide information about thez-spin property of the R-particle. For example, if the outcome of the L-measurement isz-spin up, we know thatthe R-particle hasz-spin down; and if we assume, as EPR did, that there is no curiousaction at a distance between the distant wings (and that the change of the quantum-mechanical state of the particle pair in the L-measurement is only a change in state ofknowledge), we could also conclude that the R-particle hadz-spin down even before theL-measurement occurs.


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How could the L-outcome change our knowledge/ignorance about the R-outcome if it hasno influence on it? The simplest and most straightforward reply is that the L- and the R-outcome have a common cause that causes them to be correlated, so that knowledge ofone outcome provides knowledge about the other.[2] Yet, the question is whether thepredictions of orthodox quantum mechanics, which have been highly confirmed by

various experiments, are compatible with the quantum realm being local in the sense ofinvolving no influences between systems between which light and slower-than-lightsignals cannot travel (i.e., space-like separated systems). More particularly, the questionis whether it is possible to construct a local, common-cause model of the EPR/Bexperiment, i.e., a model that postulates no influence between systems/events in thedistant wings of the experiment, and that the correlation between them are due to the stateof the particle pair at the source. In 1935, Einstein, Podolsky and Rosen believed that thisis possible. But, as John Bell demonstrated in 1964, this belief is difficult to uphold.

2. Bell's theorem and non-locality

In a famous theorem, John Bell (1964) demonstrated that granted some plausibleassumptions, any local model of the EPR/B experiment is committed to certaininequalities about the probabilities of measurement outcomes, the Bell inequalities,which are incompatible with the quantum-mechanical predictions. When Bell proved histheorem, the EPR/B experiment was only a thought experiment. But due to technologicaladvances, various versions of this experiment have been conducted since the 1970s, andtheir results have overwhelmingly supported the quantum-mechanical predictions (forbrief reviews of these experiments and further references, see the entry on Bell's theoremand Redhead 1987, chapter 4, section 4.3 and Notes and References). Thus, a wideconsensus has it that the quantum realm involves some type of non-locality.

The basic idea of Bell's theorem is as follows. A model of the EPR/B experimentpostulates that the state of the particle pair together with the apparatus settings to measure(or not to measure) certain spin properties determine the probabilities for single and jointspin-measurement outcomes. A local Bell model of this experiment also postulates thatprobabilities of joint outcomes factorize into the single probabilities of the L- and the R-outcomes: The probability of joint outcomes is equal to the product of the probabilities ofthe single outcomes. More formally, let denote the pair's state before any measurementoccurs. Let ldenote the setting of the L-measurement apparatus to measure spin along thel-axis (i.e., the l-spin of the L-particle), and let rdenote the setting of the R-measurementapparatus to measure spin along the r-axis (i.e., the r-spin of the R-particle). Letxlbe theoutcome of a l-spin measurement in the L-wing, and letyr be the outcome of a r-spin

measurement in the R-wing; wherexl is either the L-outcome l-spin up or the L-outcome l-spin down, andyr is either the R-outcome r-spin up or the R-outcome r-spin down. LetPl r(xl&yr) be the joint probability of the L- and the R-outcome, andPl(xl) andPr(yr) be the single probabilities of the L- and the R-outcome, respectively;where the subscripts , land rdenote the factors that are relevant for the probabilities ofthe outcomesxl andyr. Then, for any , l, r,xl andyr:[3]
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Factorizability

Plr(xl &yr) =Pl(xl) Pr(yr).

(Here and henceforth, for simplicity's sake we shall denote events and states, such as themeasurement outcomes, and the propositions that they occur by the same symbols.)

The state is typically thought of as the pair's state at the emission time, and it isassumed that this state does not change in any relevant sense between the emission andthe first measurement. It is (generally) a different state from the quantum-mechanicalpair's state . is assumed to be an incomplete state of the pair, whereas is supposed tobe a (more) complete state of the pair. Accordingly, pairs with the same state may havedifferent states which give rise to different probabilities of outcomes for the same typeof measurements. Also, the states may be unknown, hidden, inaccessible oruncontrollable.

Factorizability is commonly motivated as a locality condition. In non-local models of the

EPR/B experiment, the correlations between the distant outcomes are accounted for bynon-local influences between the distant measurement events. For example, in orthodoxquantum mechanics the first spin measurement on, say, the L-particle causes animmediate change in the spin properties of the R-particle and in the probabilities of futureoutcomes of spin measurements on this particle. By contrast, in local models of thisexperiment the correlations are supposed to be accounted for by a common causethepair's state (see Fig. 2 below): The pair's state and the L-setting determine theprobability of the L-outcome; the pair's state and the R-setting determine the probabilityof the R-outcome; and the pair's state and the L- and the R-setting determine theprobability of joint outcomes, which (as mentioned above) is simply the product of thesesingle probabilities. The idea is that the probability of each of the outcomes is determined

by local events, i.e., events that are confined to its backward light-cone, and which canonly exert subluminal or luminal influences on it (see Figure 3 below); and the distantoutcomes are fundamentally independent of each other, and thus their joint probabilityfactorizes. (For more about this reasoning, see sections 6 and 8-9.)

Figure 2: A schematic common-cause model of the EPR/B experiment. Arrows denotecausal connections.


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Figure 3: A space-time diagram of a local model of the EPR/B experiment. The circlesrepresent the measurement events, and the cones represent their backward light cones,i.e., the boundaries of all the subluminal and luminal influences on them. The dotted linesdenote the propagation of the influences of the pair's state at the emission and of thesettings of the measurement apparatuses on the measurement outcomes.

A Bell model of the EPR/B experiment also postulates that for each quantum-mechanicalstate there is a distribution over all the possible pair states , which is independent of

the settings of the apparatuses. That is, the distribution of the (complete) states depends on the (incomplete) state , and this distribution is independent of theparticular choice of measurements in the L- and R-wing (including the choice not tomeasure any quantity). Or formally, for any quantum-mechanical state , L-settings landl, and R-settings rand r:

-independencelr() = lr() = lr() = lr() = ()

where the subscripts denote the factors that are potentially relevant for the distribution ofthe states .

Although the model probabilities (i.e., the probabilities of outcomes prescribed by thestates ) are different from the corresponding quantum-mechanical probabilities ofoutcomes (i.e., the probabilities prescribed by the quantum-mechanical states ), thequantum mechanical probabilities (which have been systematically confirmed) arerecovered by averaging over the model probabilities. That is, it is supposed that thequantum-mechanical probabilitiesPl r(xl&yr),Pl(xl) andPr(yr) are obtained byaveraging over the model probabilitiesPl r(xl &yr),Pl(xl) andPr(yr), respectively: Forany , l, r,xl andyr,

Empirical AdequacyPlr(xl &yr) = Plr(xl &yr) lr()Pl(xl) = Pl(xl) l()Pr(yr) = Pr(yr) r().[4]The assumption of -independence is very plausible. It postulates that (complete) pairstates at the source are uncorrelated with the settings of the measurement apparatuses.And independently of one's philosophical view about free will, this assumption is
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strongly suggested by our experience, according to which it seems possible to prepare thestate of particle pairs at the source independently of the set up of the measurementapparatuses.

There are two ways to try to explain a failure of -independence. One possible

explanation is that pairs' states and apparatus settings share a common cause, whichalways correlates certain types of pairs' states with certain types of L- and R-setting.Such a causal hypothesis will be difficult to reconcile with the common belief thatapparatus settings are controllable at experimenters' will, and thus could be setindependently of the pair's state at the source. Furthermore, thinking of all the differentways one can measure spin properties and the variety of ways in which apparatus settingscan be chosen, the postulation of such common cause explanation for settings and pairs'states would seem highly ad hoc and its existence conspiratorial.

Another possible explanation for the failure of -independence is that the apparatussettings influence the pair's state at the source, and accordingly the distribution of the

possible pairs' states is dependent upon the settings. Since the settings can be made afterthe emission of the particle pair from the source, this kind of violation of -independencewould require backward causation. (For advocates of this way out of non-locality, seeCosta de Beauregard 1977, 1979, 1985, Sutherland 1983, 1998, 2006 and Price 1984,1994, 1996, chapters 3, 8 and 9.) On some readings of John Cramer's (1980, 1986)transactional interpretation of quantum mechanics (see Maudlin 1994, pp. 197-199), suchviolation of -independence is postulated. According to this interpretation, the sourcesends offer waves forward to the measurement apparatuses, and the apparatuses sendconfirmation waves (from the space-time regions of the measurement events) backwardto the source, thus affecting the states of emitted pairs according to the settings of theapparatuses. The question of whether such a theory can reproduce the predictions of

quantum mechanics is a controversial matter (see Maudlin 1994, pp. 197-199, Berkovitz2002, section 5, and Kastner 2006). It is noteworthy, however, that while the violation of-independence is sufficient for circumventing Bell's theorem, the failure of thisconditionper se does not substantiate locality. The challenge of providing a local modelof the EPR/B experiment also applies to models that violate -independence. (For moreabout these issues, see sections 9 and 10.3.)

In any case, as Bell's theorem demonstrates, factorizability, -independence andempirical adequacy jointly imply the Bell inequalities, which are violated by thepredictions of orthodox quantum mechanics (Bell 1964, 1966, 1971, 1975a,b). Grantedthe systematic confirmation of the predictions of orthodox quantum mechanics and theplausibility of -independence, Bell inferred that factorizability fails in the EPR/Bexperiment. Thus, interpreting factorizability as a locality condition, he concluded thatthe quantum realm is non-local. (For further discussions of Bell's theorem, the Bellinequalities and non-locality, see Bell 1966, 1971, 1975a,b, 1981, Clauseret al1969,Clauser and Horne 1974, Shimony 1993, chapter 8, Fine 1982a,b, Redhead 1987, chapter4, Butterfield 1989, 1992a, Pitowsky 1989, Greenberger, Horne and Zeilinger 1989,Greenberger, Horne, Shimony and Zeilinger 1990, Mermin 1990, and the entry on Bell'stheorem.)


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3. The analysis of factorizability

Following Bell's work, a broad consensus has it that the quantum realm involves sometype of non-locality (for examples, see Clauser and Horne 1974, Jarrett 1984,1989,Shimony 1984, Redhead 1987, Butterfield 1989, 1992a,b, 1994, Howard 1989, Healey

1991, 1992, 1994, Teller 1989, Clifton, Butterfield and Redhead 1990, Clifton 1991,Maudlin 1994, Berkovitz 1995a,b, 1998a,b, and references therein).[5] But there is anongoing controversy as to its exact nature and its compatibility with relativity theory. Oneaspect of this controversy is over whether the analysis of factorizability and the differentways it could be violated may shed light on these issues. Factorizability is equivalent tothe conjunction of two conditions (Jarrett 1984, 1989, Shimony 1984):[6]

Parameter independence. The probability of a distant measurement outcome in theEPR/B experiment is independent of the setting of the nearby measurement apparatus. Orformally, for any pair's state , L-setting l, R-setting r, L-outcomexland R-outcomeyr:PI

Pl r(xl) =Pl(xl) and Pl r(yr) =Pr(yr).Outcome independence. The probability of a distant measurement outcome in theEPR/B experiment is independent of the nearby measurement outcome. Or formally, forany pair's state , L-setting l, R-setting r, L-outcomexl and R-outcomeyr:

Pl r(xl /yr) =Pl r(xl) and Pl r(yr /xl) =Pl r(yr)

Pl r(yr) > 0 Pl r(xl) > 0,or more generally,OI

Pl r(xl&yr) =Pl r(xl) Pl r(yr).

Assuming -independence (see section 2), any empirically adequate theory will have toviolate OI or PI. A common view has it that violations of PI involve a different type ofnon-locality than violations of OI: Violations of PI involve some type of action-at-a-distance that is impossible to reconcile with relativity (Shimony 1984, Redhead 1987, p.108), whereas violations of OI involve some type of holism, non-separability and/orpassion-at-a-distance that may be possible to reconcile with relativity (Shimony 1984,Readhead 1987, pp. 107, 168-169, Howard 1989, Teller 1989).

On the other hand, there is the view that the analysis above (as well as other similaranalyses of factorizability[7]) is immaterial for studying quantum non-locality (Butterfield1992a, pp. 63-64, Jones and Clifton 1993, Maudlin 1994, pp. 96 and 149) and evenmisleading (Maudlin 1994, pp. 94-95 and 97-98). On this alternative view, the way toexamine the nature of quantum non-locality is to study the ontology postulated by thevarious interpretations of quantum mechanics and alternative quantum theories.[8] Insections 4-7, we shall follow this methodology and discuss the nature of non-localitypostulated by several quantum theories. The discussion in these sections will furnish theground for evaluating the above controversy in section 8.
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4. Action at a distance, holism and non-separability

4.1 Action at a distance

In orthodox quantum mechanics as well as in any other current quantum theory thatpostulates non-locality (i.e., influences between distant, space-like separated systems),the influences between the distant measurement events in the EPR/B experiment do notpropagate continuously in space-time. They seem to involve action at a distance. Yet, acommon view has it that these influences are due to some type of holism and/or non-separability of states of composite systems, which are characteristic of systems inentangled states (like the spin singlet state), and which exclude the very possibility ofaction at a distance. The paradigm case of action at a distance is the Newtoniangravitational force. This force acts between distinct objects that are separated by some(non-vanishing) spatial distance, its influence is symmetric (in that any two massiveobjects influence each other), instantaneous and does not propagate continuously inspace. And it is frequently claimed or presupposed that such action at a distance could

only exist between systems with separate states in non-holistic universes (i.e., universesin which the states of composite systems are determined by, or supervene upon the statesof their subsystems and the spacetime relations between them), which are commonlytaken to characterize the classical realm.[9]

In sections 4.2 and 4.3, we shall briefly review the relevant notions of holism and non-separability (for a more comprehensive review, see the entry on holism andnonseparability in physics and Healey 1991). In section 5, we shall discuss the nature ofholism and non-separability in the quantum realm as depicted by various quantumtheories. Based on this discussion, we shall consider whether the non-local influences inthe EPR/B experiment constitute action at a distance.

4.2 Holism

In the literature, there are various characterizations of holism. Discussions of quantumnon-locality frequently focus on property holism, where certain physical properties ofobjects are not determined by the physical properties of their parts. The intuitive idea isthat some intrinsic properties of wholes (e.g. physical systems) are not determined by theintrinsic properties of their parts and the spatiotemporal relations that obtain betweenthese parts. This idea can be expressed in terms of supervenience relations.

Property Holism. Some objects have intrinsic qualitative properties and/or relations thatdo not supervene upon the intrinsic qualitative properties and relations of their parts andthe spatiotemporal relations between these parts.

It is difficult to give a general precise specification of the terms intrinsic qualitativeproperty and supervenience. Intuitively, a property of an object is intrinsic just in casethat object has this property in and for itself and independently of the existence or thestate of any other object. A property is qualitative (as opposed to individual) if it does notdepend on the existence of any particular object. And the intrinsic qualitative properties
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of an object O supervene upon the intrinsic qualitative properties and relations of its partsand the spatiotemporal relations between them just in case there is no change in theproperties and relations ofO without a change in the properties and relations of its partsand/or the spatiotemporal relations between them. (For attempts to analyze the termintrinsic property, see for example Langton and Lewis 1998 and the entry on intrinsic

vs. extrinsic properties. For a review of different types of supervenience, see for exampleKim 1978, McLaughlin 1994 and the entry on supervenience.)

Paul Teller (1989, p. 213) proposes a related notion of holism, relational holism, whichis characterized as the violation of the following condition:

Particularism. The world is composed of individuals. All individuals have non-relationalproperties and all relations supervene on the non-relational properties of the relata.

Here, by a non-relational property Teller means an intrinsic property (1986a, p. 72); andby the supervenience of a relational property on the non-relational properties of the

relata, he means that if two objects, 1 and 2, bear a relationR to each other, then,necessarily, if two further objects, 1 and 2 have the same non-relational properties, then1 and 2 will also bear the same relationR to each other (1989, p. 213). Teller (1986b,pp. 425-7) believes that spatiotemporal relations between objects supervene upon theobjects intrinsic physical properties. Thus, he does not include the spatiotemporalrelations in the supervenience basis. This view is controversial, however, as many believethat spatiotemporal relations between objects are neither intrinsic nor supervene upon theintrinsic qualitative properties of these objects. But, if such supervenience does notobtain, particularism will also be violated in classical physics, and accordingly relationalholism will fail to mark the essential distinction between the classical and the quantumrealms. Yet, one may slightly revise Teller's definition of particularism as follows:

Particularism*. The world is composed of individuals. All individuals have non-relational properties and all relations supervene upon the non-relational properties of therelata and the spatiotemporal relations between them.

In what follows in this entry, by relational holism we shall mean a violation ofparticularism*.

4.3 Non-separability

Like holism, there are various notions of non-separability on offer. The most common

notion in the literature is state non-separability, i.e., the violation of the followingcondition:

State separability. Each system possesses a separate state that determines its qualitativeintrinsic properties, and the state of any composite system is wholly determined by theseparate states of its subsystems.


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The term wholly determined is vague. But, as before, one may spell it out in terms ofsupervenience relations: State separability obtains just in case each system possesses aseparate state that determines its qualitative intrinsic properties and relations, and thestate of any composite system is supervenient upon the separate states of its subsystems.

Another notion of non-separability is spatiotemporal non-separability. Inspired byEinstein (1948), Howard (1989, pp. 225-6) characterizes spatiotemporal non-separabilityas the violation of the following separability condition:

Spatiotemporal separability. The contents of any two regions of space-time separatedby a non-vanishing spatiotemporal interval constitute two separate physical systems.Each separated space-time region possesses its own, distinct state and the joint state ofany two separated space-time regions is wholly determined by the separated states ofthese regions.

A different notion of spatiotemporal non-separability, proposed by Healey (see the entry

on holism and nonseparability in physics), is process non-separability. It is the violationof the following condition:

Process separability. Any physical process occupying a spacetime regionR supervenesupon an assignment of qualitative intrinsic physical properties at spacetime points inR.

5. Holism, non-separability and action at a distance in

quantum mechanics

The quantum realm as depicted by all the quantum theories that postulate non-locality,

i.e., influences between distant (space-like separated) systems, involves some type ofnon-separability or holism. In what follows in this section, we shall consider the nature ofthe non-separability and holism manifested by various interpretations of quantummechanics. On the basis of this consideration, we shall address the question of whetherthese interpretations predicate the existence of action at a distance. We start with the so-called collapse theories.

5.1 Collapse theories

5.1.1 Orthodox quantum mechanics

In orthodox quantum mechanics, normalized vectors in Hilbert spaces represent states ofphysical systems. When the Hilbert space is of infinite dimension, state vectors can berepresented by functions, the so-called wave functions. In any given basis, there is aunique wave function that corresponds to the state vector in that basis. (For an entry levelreview of the highlights of the mathematical formalism and the basic principles ofquantum mechanics, see the entry on quantum mechanics, Albert 1992, Hughes 1989,Part I, and references therein; for more advanced reviews, see Bohm 1951 and Redhead1987, chapters 1-2 and the mathematical appendix.)


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For example, the state of the L-particle havingz-spin up (i.e., spinning up about thez-axis) can be represented by the vector |z-up> in the Hilbert space associated with the L-particle, and the state of the L-particle havingz-spin down (i.e., spinning down aboutthez-axis) can be represented by the orthogonal vector, |z-down>. Particle pairs may bein a state in which the L-particle and the R-particle have opposite spins, for instance

either a state |1> in which the L-particle hasz-spin up and the R-particle hasz-spindown, or a state |2> in which the L-particle hasz-spin down and the R-particle hasz-spin up. Each of these states is represented by a tensor product of vectors in the Hilbertspace of the particle pair: |1> = |z-up>L |z-down>R and |2> = |z-down>L |z-up>R; wherethe subscripts L and R refer to the Hilbert spaces associated with the L- and the R-particle, respectively. But particle pairs may also be in a superposition of these states, i.e.,a state that is a linear sum of the states |1> and |2>, e.g. the state represented by

|3> = 1/2 (|1> |2>) = 1/2 (|z-up>L |z-down>R |z-down>L |z-up>R).

In fact, this is exactly the case in the spin singlet state. In this state, the particles are

entangled in a non-separable state (i.e., a state that cannot be decomposed into a productof separate states of the L- and the R-particle), in which (according to the property-assignment rules of orthodox quantum mechanics) the particles do not possess anydefinitez-spin (or definite spin in any other direction). Thus, the condition of stateseparability fails: The state of the particle pair (which determines its intrinsic qualitativeproperties) is not wholly determined by the separate states of the particles (whichdetermine their intrinsic qualitative properties). Or more precisely, the pair's state is notsupervenient upon the separable states of the particles. In particular, the superpositionstate of the particle pair assigns a correlational property that dictates that the outcomesof (ideal)z-spin measurements on both the L- and the R-particle will be anti-correlated,and this correlational property is not supervenient upon properties assigned by any

separable states of the particles (for more details, see Healey 1992, 1994). For similarreasons, the spin singlet state also involves property and relational holism; for the abovecorrelational property of the particle pair also fails to supervene upon the intrinsicqualitative properties of the particles and the spatiotemporal relations between them.Furthermore, the process that leads to each of the measurement outcomes is also non-separable, i.e. process separability fails (see Healey 1994 and the entry on holism andnonseparability in physics).

This correlational property is also responsible for the action at a distance that theorthodox theory seems to postulate between the distant wings in the EPR/B experiment.Recall (section 1) that Einstein, Podolsky and Rosen thought that this curious action at adistance reflects the incompleteness of this theory rather than a state of nature. The EPRargument for the incompleteness of the orthodox theory is controversial. But the orthodoxtheory seems to be incomplete for a different reason. This theory postulates that in non-measurement interactions, the evolution of states obeys a linear and unitary equation ofmotion, the so-called Schrdinger equation (see the entry on quantum mechanics),according to which the particle pair in the EPR/B experiment remains in an entangledstate. This equation of motion also dictates that in a spin measurement, the pointers of themeasurement apparatuses get entangled with the particle pair in a non-separable state in


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which (according to the theory's property assignment, see below) the indefiniteness ofparticles spins is transmitted to the pointer's position: In this entangled state of theparticle pair and the pointer, the pointer lacks any definite position, in contradiction toour experience of perceiving it pointing to either up or down.

The above problem, commonly called the measurement problem, arises in orthodox no-collapse quantum mechanics from two features that account very successfully for thebehavior of microscopic systems: The linear dynamics of quantum states as described bythe Schrdinger equation and the property assignment rule called eigenstate-eigenvaluelink. According to the eigenstate-eigenvalue link, a physical observable, i.e., a physicalquantity, of a system has definite value (one of its eigenvalues) just in case the system isin the corresponding eigenstate of that observable (see the entry on quantum mechanics,section 4). Microscopic systems may be in a superposition state of spin components,energies, positions, momenta as well as other physical observables. Accordingly,microscopic systems may be in a state of indefinitez-spin, energy, position, momentumand various other quantities. The problem is that given the linear and unitary Schrdinger

dynamics, these indefinite quantities are also endemic in the macroscopic realm. Forexample, in az-spin measurement on a particle in a superposition state ofz-spin up andz-spin down, the position of the apparatuss pointer gets entangled with the indefinitez-spin of the particle, thus transforming the pointer into a state of indefinite position, i.e., asuperposition of pointing up and pointing down (see Albert 1992, chapter 4, and theentry on collapse theories, section 3). In particular, in the EPR/B experiment the L-measurement causes the L-apparatus pointer to get entangled with the particle pair,transforming it into a state of indefinite position:

|4> = 1/2 (|z-up>L |z-down>R |up>LA |z-down>L |z-up>R |down>LA)

where |up>LA and |down>LA are the states of the L-apparatus pointer displaying theoutcomesz-spin up andz-spin down, respectively. Since the above type ofindefiniteness is generic in orthodox no-collapse quantum mechanics, in this theorymeasurements typically have no definite outcomes, in contradiction to our experience.

In order to solve this problem, the orthodox theory postulates that in measurementinteractions, entangled states of measured systems and the corresponding measurementapparatuses do not evolve according to the Schrdinger equation. Rather, they undergo acollapse into product (non-entangled) states, where the systems involved have therelevant definite properties. For example, the entangled state of the particle pair and theL-apparatus in the EPR/B experiment may collapse into a product state in which the L-particle comes to possessz-spin up, the R-particle comes to possessz-spin down andthe L-apparatus pointer displaying the outcomez-spin up:

|5> = |z-up>L |z-down>R |up>LA.

The problem is that in the orthodox theory, the notions of measurement and the time,duration and nature of state collapses remain completely unspecified. As John Bell(1987b, p. 205) remarks, the collapse postulate in this theory, i.e., the postulate that


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dictates that in measurement interactions the entangled states of the relevant systems donot follow the Schrdinger equation but rather undergo a collapse, is no more thansupplementary, imprecise, verbal, prescriptions.

This problem of accounting for our experience of perceiving definite measurement

outcomes in orthodox quantum mechanics, is an aspect of the more general problem ofaccounting for the classical-like behavior of macroscopic systems in this theory.

5.1.2 Dynamical models for state vector reduction

The dynamical models for state-vector reduction were developed to account for statecollapses as real physical processes (for a review of the collapse models and a detailedreference list, see the entry on collapse theories). The origin of the collapse models maybe dated to Bohm and Bub's (1966) hidden variable theory and Pearle's (1976)spontaneous localization approach, but the program has received its crucial impetus withthe more sophisticated models developed by Ghirardi, Rimini and Weber in 1986 (see

also Bell 1987a and Albert 1992) and their consequent development by Pearle (1989)(see also Ghirardi, Pearle and Rimini 1990, and Butterfield et al. 1993). Similarly toorthodox collapse quantum mechanics, in the GRW models the quantum-mechanicalstate of systems (whether it is expressed by a vector or a wave function) provides acomplete specification of their intrinsic properties and relations. The state of systemsfollows the Schrdinger equation, except that it has a probability for spontaneouscollapse, independently of whether or not the systems are measured. The chance ofcollapse depends on the size of the entangled systemsin the earlier models the sizeof systems is predicated on the number of the elementary particles, whereas in latermodels it is measured in terms of mass densities. In any case, in microscopic systems,such as the particle pairs in the EPR/B experiment, the chance of collapse is very small

and negligiblethe chance of spontaneous state collapse in such systems is cooked up sothat it will occur, on average, every hundred million years or so. This means that thechance that the entangled state of the particle pair in the EPR/B experiment will collapseto a product state between the emission from the source and the first measurement isvirtually zero. In an earlier L-measurement, the state of the particle pair gets entangledwith the state of the L-measurement apparatus. Thus, the state of the pointer of the L-apparatus evolves from being ready to measure a certain spin property to an indefiniteoutcome. For instance, in az-spin measurement the L-apparatus gets entangled with theparticle pair in a superposition state of pointing to up and pointing to down(corresponding to the states of the L-particle havingz-spin up and havingz-spindown), and the R-apparatus remains un-entangled with these systems in the state ofbeing ready to measurez-spin. Or formally:

|6> = 1/2 (|z-up>L |up>AL |z-down>R |z-down>L |down>AL |z-up>R) |ready>AR

where, as before, |up>AL and |down>AL denote the states of the L-apparatus displaying theoutcomesz-spin up and down respectively, and |ready>AR denotes the state of the R-apparatus being ready to measurez-spin. In this state, a gigantic number of particles ofthe L-apparatus pointer are entangled together in the superposition state of being in the


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position (corresponding to pointing to) up and the position (corresponding to pointingto) down. For assuming, for simplicity of presentation, that the position of all particlesof the L-apparatus pointer in the state of pointing to up (down) is the same, the state |6> can be rewritten as:

|7> = 1/2 (|z-up>L |up>p1 |up>p2 |up>p3 |z-down>R |z-down>L |down>p1 |down>p2 |down>p3 |z-up>R) |ready >AR

wherepi denotes the i-particle of the L-apparatus pointer, and |up>pi (|down>pi) is the stateof the i-particle being in the position corresponding to the outcomez-spin up (down).[10] The chance that at least one of the vast number of the pointer's particles will endure aspontaneous localization toward being in the position corresponding to either theoutcomez-spin up or the outcomez-spin down within a very short time (a split of amicro second) is very high. And since all the particles of the pointer and the particle pairare entangled with each other, such a collapse will carry with it a collapse of theentangled state of the pointer of the L-apparatus and the particle pair toward either

|z-up>L|up>p1|up>p2|up>p3 |z-down>R

or

|z-down>L|down>p1|down>p2|down>p3 |z-up>R.

Thus, the pointer will very quickly move in the direction of pointing to either theoutcomez-spin up or the outcomez-spin down.

If (as portrayed above) the spontaneous localization of particles were to a precise

position, i.e., to the position corresponding to the outcome up or the outcome down,the GRW collapse models would successfully resolve the measurement problem.Technically speaking, a precise localization is achieved by multiplying |7> by a deltafunction centered on the position corresponding to either the outcome up or theoutcome down (see the entry on collapse theories, section 5 and Albert 1992, chapter5); where the probability of each of these mutually exhaustive possibilities is . Theproblem is that it follows from the uncertainty principle (see the entry on the uncertaintyprinciple) that in such localizations the momenta and the energies of the localizedparticles would be totally uncertain, so that gases may spontaneously heat up andelectrons may be knocked out of their orbits, in contradiction to our experience. To avoidthis kind of problems, GRW postulated that spontaneous localizations are characterized

by multiplications by Gaussians that are centered around certain positions, e.g. theposition corresponding to either the outcome up or the outcome down in the state |7>.This may be problematic, because in either case the state of the L-apparatus pointer at(what we characteristically conceive as) the end of the L-measurement would be asuperposition of the positions up and down. For although this superpositionconcentrates on either the outcome up or the outcome down (i.e. the peak of thewave function that corresponds to this state concentrates on one of these positions), italso has tails that go everywhere: The state of the L-apparatus is a superposition of an


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infinite number of different positions. Thus, it follows from the eigenstate-eigenvaluelink that the position observable of the L-apparatus has no definite value at the end of themeasurement. But if the position observable having a definite value is indeed required inorder for the L-apparatus to have a definite location, then the pointer will point to neitherup nor down, and the GRW collapse models will fail to reproduce the classical-like

behavior of such systems.[11]

In later models, GRW proposed to interpret the quantum state as a density of mass andthey postulated that if almost all the density of mass of a system is concentrated in acertain region, then the system is located in that region. Accordingly, pointers ofmeasurement apparatuses do have definite positions at the end of measurementinteractions. Yet, this solution has also given rise to a debate (see Albert and Loewer1995, Lewis 1997, 2003a, 2004, Ghirardi and Bassi 1999, Bassi and Ghirardi 1999, 2001,Clifton and Monton 1999, 2000, Frigg 2003, and Parker 2003).

The exact details of the collapse mechanism and its characteristics in the GRW/Pearle

models have no significant implications for the type of non-separability and holism theypostulateall these models basically postulate the same kinds of non-separability andholism as orthodox quantum mechanics (see section 5.1.1). And action at a distancebetween the L- and the R-wing will occur if the L-measurement interaction, a supposedlylocal event in the L-wing, causes some local events in the R-wing, such as the event ofthe pointer of the measurement apparatus coming to possess a definite measurementoutcome during the R-measurement. That is, action at a distance will occur if the L-measurement causes the R-particle to come to possess a definitez-spin and this in turncauses the pointer of the R-apparatus to come to possess the corresponding measurementoutcome in the R-measurement. Furthermore, if the L-measurement causes the R-particleto come to possess (momentarily) a definite position in the R-wing, then the action at a

distance between the L- and the R-wing will occur independently of whether the R-particle undergoes a spin measurement.

The above discussion is based on an intuitive notion of action at a distance and itpresupposes that action at a distance is compatible with non-separability and holism. Inthe next section we shall provide more precise characterizations of action at a distanceand in light of these characterizations reconsider the question of the nature of action at adistance in the GRW/Pearle collapse models.

5.2 Can action-at-a-distance co-exist with non-separability and holism?

The action at a distance in the GRW/Pearle models is different from the Newtonianaction at a distance in various respects. First, in contrast to Newtonian action at adistance, this action is independent of the distance between the measurement events.Second, while Newtonian action is symmetric, the action in the GRW/Pearle models is(generally) asymmetric: Either the L-measurement influences the properties of the R-particle or the R-measurement influences the properties of the L-particle, depending onwhich measurement occurs first (the action will be symmetric when both measurementsoccur simultaneously). Third (and more important to our consideration), in contrast to


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Newtonian action at a distance, before the end of the L-measurement the state of the L-apparatus and the R-particle is not separable and accordingly it is not clear that theinfluence is between separate existences, as the case is supposed to be in Newtoniangravity.

This non-separability of the states of the particle pair and the L-measurement apparatus,and more generally the fact that the non-locality in collapse theories is due to state non-separability, has led a number of philosophers and physicists to think that wave collapsesdo not involve action at a distance. Yet, the question of whether there is an action at adistance in the GRW/Pearle models (and various other quantum theories) depends onhow we interpret the term action at a distance. And, as I will suggest below, on a naturalreading of Isaac Newton's and Samuel Clarke's comments concerning action at a distance,there may be a peaceful coexistence between action at a distance and non-separabilityand holism.

Newton famously struggled to find out the cause of gravity.[12] In a letter to Bentley, dated

January 17 1692/3, he said:You sometimes speak of Gravity as essential and inherent to Matter. Pray do not ascribethat Notion to me, for the Cause of Gravity is what I do not pretend to know, andtherefore would take more Time to consider it. (Cohen 1978, p. 298)

In a subsequent letter to Bentley, dated February 25, 1692/3, he added:

It is inconceivable that inanimate Matter should, without the Mediation of somethingelse, which is not material, operate upon, and affect other matter without mutualContactThat Gravity should be innate, inherent and essential to Matter, so that one

body may act upon another at a distance thro a Vacuum, without the Mediation of anything else, by and through which their Action and Force may be conveyed from one toanother, is to me so great an Absurdity that I believe no Man who has in philosophicalMatters a competent Faculty of thinking can ever fall into it. Gravity must be caused byan Agent acting constantly according to certain laws; but whether this Agent be materialor immaterial, I have left to the Consideration of my readers. (Cohen 1978, pp. 302-3)

Samuel Clarke, Newton's follower, similarly struggled with the question of the cause ofgravitational phenomenon. In his famous controversy with Leibniz, he said:[13]

That one body attracts another without any intermediate means, is indeed not a miracle

but a contradiction; for 'tis supposing something to act where it is not. But the means bywhich two bodies attract each other, may be invisible and intangible and of a differentnature from mechanism

And he added:

That this phenomenon is not producedsans moyen, that is without a cause capable ofproducing such an effect, is undoubtedly true. Philosophers therefore can search after and


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discover that cause, if they can; be it mechanical or not. But if they cannot discover thecause, is therefore the effect itself, the phenomenon, or the matter of fact discovered byexperience ever the less true?

Newton's and Clarke's comments suggest that for them gravity was a law-governed

phenomenon, i.e., a phenomenon in which objects influence each other at a distanceaccording to the Newtonian law of gravity, and that this influence is due to some meanswhich may be invisible and intangible and of a different nature from mechanism. On thisconception of action at a distance, there seems to be no reason to exclude the possibilityof action at a distance in the quantum realm even if that realm is holistic or the state ofthe relevant systems is non-separable. That is, action at a distance may be characterizedas follows:

Action at a distance is a phenomenon in which a change in intrinsic properties of onesystem induces a change in the intrinsic properties of a distant system, independently ofthe influence of any other systems on the distant system, and without there being a

process that carries this influence contiguously in space and time.We may alternatively characterize action at a distance in a more liberal way:

Action* at a distance is a phenomenon in which a change in intrinsic properties of onesystem induces a change in the intrinsic properties of a distant system without there beinga process that carries this influence contiguously in space and time.

And while Newton and Clarke did not have an explanation for the action at a distanceinvolved in Newtonian gravity, on the above characterizations action at a distance in thequantum realm would be explained by the holistic nature of the quantum realm and/or

non-separability of the states of the systems involved. In particular, if in the EPR/Bexperiment the L-apparatus pointer has a definite position before the L-measurement andthe R-particle temporarily comes to possess definite position during the L-measurement,then the GRW/Pearle models involve action at a distance and thus also action* at adistance. On the other hand, if the R-particle never comes to possess a definite positionduring the L-measurement, then the GRW/Pearle models only involve action* at adistance.

5.3 No-collapse theories

5.3.1 Bohm's theory

In 1952, David Bohm proposed a deterministic, hidden variables quantum theory thatreproduces all the observable predictions of orthodox quantum mechanics (see Bohm1952, Bohm, Schiller and Tiomno 1955, Bell 1982, Dewdney, Holland and Kyprianidis1987, Drr, Goldstein and Zangh 1992a, 1997, Albert 1992, Valentini 1992, Bohm andHiley 1993, Holland 1993, Cushing 1994, and Cushing, Fine and Goldstein 1996; for anentry level review, see the entry on Bohmian mechanics and Albert 1992, chapter 5).


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In contrast to orthodox quantum mechanics and the GRW/Pearle collapse models, inBohm's theory wave functions always evolve according to the Schrdinger equation, andthus they never collapse. Wave functions do not represent the states of systems. Rather,they are states of a quantum field (on configuration space) that influences the states ofsystems.[14] Also, particles always have definite positions, and the positions of the

particles and their wave function at a certain time jointly determine the trajectories of theparticles at all future times. Thus, particles positions and their wave function determinethe outcomes of any measurements (so long as these outcomes are recorded in thepositions of some physical systems, as in any practical measurements).

There are various versions of Bohm's theory. In the minimal Bohm theory, formulatedby Bell (1982),[15] the wave function is interpreted as a guiding field (which has nosource or any dependence on the particles) that deterministically governs the trajectoriesof the particles according to the so-called guiding equation (which expresses thevelocities of the particles in terms of the wave function).[16] The states of systems areseparable (the state of any composite system is completely determined by the state of its

subsystems), and they are completely specified by the particles positions. Spins, and anyother properties which are not directly derived from positions, are not intrinsic propertiesof systems. Rather, they are relational properties that are determined by the systemspositions and the guiding field. In particular, each of the particles in the EPR/Bexperiment has dispositions to spin in various directions, and these dispositions arerelational properties of the particles they are (generally) determined by the guidingfield and the positions of the particles relative to the measurement apparatuses and toeach other.

Figure 4. The EPR/B experiment with Stern-Gerlach measurement devices. Stern-Gerlach 1 is on, set up to measure thez-spin of the L-particle, and Stern-Gerlach 2 is off.The horizontal lines in the left-hand-side denote the trajectories of six L-particles in thespin singlet state after an (impulsive)z-spin measurement on the L-particle, and the


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horizontal lines in the right-hand-side denote the trajectories of the corresponding R-particles. The center plane is aligned orthogonally to thez-axis, so that particles thatemerge above this plane correspond toz-spin up outcome and particles that emergebelow this plane correspond toz-spin down outcome. The little arrows denote thez-spincomponents of the particles in the non-minimal Bohm theory (where spins are intrinsic

properties of particles), and are irrelevant for the minimal Bohm theory (where spins arenot intrinsic properties of particles).

To see the nature of non-locality postulated by the minimal Bohm theory, consider againthe EPR/B experiment and suppose that the measurement apparatuses are Stern-Gerlach(S-G) magnets which are prepared to measurez-spin. In any run of the experiment, themeasurement outcomes will depend on the initial positions of the particles and the orderof the measurements. Here is why. In the minimal Bohm theory, the spin singlet statedenotes the relevant state of the guiding field rather than the intrinsic properties of theparticle pair. If the L-measurement occurs before the R-measurement, the guiding fieldand the position of the L-particle at the emission time jointly determine the disposition of

the L-particle to emerge from the S-G device either above or below a plane aligned in thez-direction; where emerging above (below) the plane means that the L-particlez-spinsup (down) about thez-axis and the L-apparatus pointer points to up (down) (seeFig. 4 above). All the L-particles that are emitted above the center plane alignedorthogonally to thez-direction, like the L-particles 1-3, will be disposed to spin up; andall the particles that are emitted below this plane, like the L-particles 4-6, will bedisposed to spin down. Similarly, if the R-measurement occurs before the L-measurement, the guiding field and the position of the R-particle at the emission timejointly determine the disposition of the R-particle to emerge either above thez-axis (i.e.,toz-spin up) or below thez-axis (i.e., toz-spin down) according to whether it is aboveor below the center plane, independently of the position of the L-particle along thez-axis.

But thez-spin disposition of the R-particle changes immediately after an (earlier)z-spinmeasurement on the L-particle: The R-particles 1-3 (see Fig. 4), which were previouslydisposed toz-spin up, will now be disposed toz-spin down, i.e., to emerge below thecenter plane aligned orthogonally to thez-axis; and the R-particles 4-6, which werepreviously disposed toz-spin down, will now be disposed toz-spin up, i.e., to emergeabove this center plane. Yet, the L-measurementper se does not have any immediateinfluence on the state of the R-particle: The L-measurement does not influence theposition of the R-particle or any other property that is directly derived from this position.It only changes the guiding field, and thus grounds new spin dispositions for the R-particle. But these dispositions are not intrinsic properties of the R-particle. Rather, theyare relational properties of the R-particle, which are grounded in the positions of bothparticles and the state of the guiding field.[17] (Note that in the particular case in which theL-particle is emitted above the center plane aligned orthogonally to thez-axis and the R-particle is emitted below that plane, an earlierz-spin on the L-particle will have noinfluence on the outcome of az-spin on the R-particle.)

While there is no contiguous process to carry the influence of the L-measurementoutcome on events in the R-wing, the question of whether this influence amounts to


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action at a distance depends on the exact characterization of this term. In contrast to theGRW/Pearle collapse models, the influence of the L-measurement outcome on theintrinsic properties of the R-particle is dependent on the R-measurement: Before thismeasurement occurs, there are no changes in the R-particle's intrinsic properties. Yet, theinfluence of the L-measurement on the R-particle is at a distance. Thus, the EPR/B

experiment as depicted by the minimal Bohm theory involves action* at a distance butnot action at a distance.

Bohm's theory portrays the quantum realm as deterministic. Thus, the single-caseobjective probabilities, i.e., the chances, it assigns to individual spin-measurementoutcomes in the EPR/B experiment are different from the corresponding quantum-mechanical probabilities. In particular, while in quantum mechanics the chances of theoutcomes up and down in an earlier L- (R-) spin measurement are both , in Bohm'stheory these chances are either one or zero. Yet, Bohm's theory postulates a certaindistribution, the so-called quantum-equilibrium distribution, over all the possiblepositions of pairs with the same guiding field. This distribution is computed from the

quantum-mechanical wave function, and it is typically interpreted as ignorance over theactual position of the pair; an ignorance that may be motivated by dynamicalconsiderations and statistical patterns exhibited by ensembles of pairs with the same wavefunction (for more details, see the entry on bohmian mechanics, section 9). And the sum-average (or more generally the integration) over this distribution reproduces all thequantum-mechanical observable predictions.

What is the status of this probability postulate? Is it a law of nature or a contingent fact (ifit is a fact at all)? The answers to these questions vary (see Section 7.2.1, Bohm 1953,Valentini 1991a,b, 1992, 1996, 2002, Valentini and Westman 2004, Drr, Goldstein andZangh 1992a,b, 1996, fn. 15, and Callender 2006).

Turning to the question of non-separability, the minimal Bohm theory does not involvestate non-separability. For recall that in this theory the state of a system does not consistin its wave function, but rather in the system's position, and the position of a compositesystem always factorizes into the positions of its subsystems. Here, the non-separabilityof the wave function reflects the state of the guiding field. This state propagates not inordinary three-space but in configuration space, where each point specifies theconfiguration of both particles. The guiding field of the particle pair cannot be factorizedinto the guiding field that governs the trajectory of the L-particle and the guiding fieldthat governs the trajectory of the R-particle. The evolution of the particles trajectories,properties and dispositions is non-separable, and accordingly the particles trajectories,properties and dispositions are correlated even when the particles are far away from eachother and do not interact with each other. Thus, process separability fails.

In the non-minimal Bohm theory[18], the behavior of anN-particle system is determinedby its wave function and the intrinsic properties of the particles. But, in contrast to theminimal theory, in the non-minimal theory spins are intrinsic properties of particles. Thewave function always evolves according to the Schrdinger equation, and it is interpretedas a quantum field (which has no sources or any dependence on the particles). The
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quantum field guides the particles via the quantum potential, an entity which isdetermined from the quantum field, and the evolution of properties is fully deterministic.[19]

Like in the minimal Bohm theory, the non-separability of the wave function in the EPR/B

experiment dictates that the evolution of the particles trajectories, properties anddispositions is non-separable, but the behavior of the particles is somewhat different. Inthe earlierz-spin measurement on the L-particle, the quantum potential continuouslychanges, and this change induces an immediate change in the z-spin of the R-particle. Ifthe L-particle starts to spin up (down) in thez-direction, the R-particle will start tospin down (up) in the same direction (see the little arrows in Fig. 4).[20] Accordingly,the L-measurement induces instantaneous action at a distance between the L- and the R-wing. Yet, similarly to the minimal Bohm theory, while the disposition of the R-particleto emerge above or below the center plane aligned orthogonally to thez-direction in az-spin measurement may change instantaneously, the actual trajectory of the R-particlealong thez-direction does not change before the measurement of the R-particle'sz-spin

occurs. Only during the R-measurement, the spin and the position of the R-particle getcorrelated and the R-particle's trajectory along thez-direction is dictated by the value ofits (intrinsic)z-spin.

Various objections have been raised against Bohm's theory (for a detailed list and replies,see the entry on Bohmian mechanics, section 15). One main objection is that in Bohmianmechanics, the guiding field influences the particles, but the particles do not influence theguiding field. Another common objection is that the theory is involved with a radical typeof non-locality, and that this type of non-locality is incompatible with relativity. While itmay be very difficult, or even impossible, to reconcile Bohm's theory with relativity, as isnot difficult to see from the above discussion, the type of non-locality that the minimal

Bohm theory postulates in the EPR/B experiment does not seem more radical than thenon-locality postulated by the orthodox interpretation and the GRW/Pearle collapsemodels.

5.3.2 Modal interpretations

Modal interpretations of quantum mechanics were designed to solve the measurementproblem and to reconcile quantum mechanics with relativity. They are no-collapse,(typically) indeterministic hidden-variables theories. Quantum-mechanical states ofsystems (which may be construed as denoting their states or information about thesestates) always evolve according to unitary and linear dynamical equations (theSchrdinger equation in the non-relativistic case). And the orthodox quantum-mechanicalstate description of systems is supplemented by a set of properties, which depends on thequantum-mechanical state and which is supposed to be rich enough to account for theoccurrence of definite macroscopic events and their classical-like behavior, butsufficiently restricted to escape all the known no-hidden-variables theorems. (For modalinterpretations, see van Fraassen 1973, 1981, 1991, chapter 9, Kochen 1985, Krips 1987,Dieks 1988, 1989, Healey 1989, Bub 1992, 1994, 1997, Vermaas and Dieks 1995, Clifton1995, Bacciagaluppi 1996, Bacciagaluppi and Hemmo 1996, Bub and Clifton 1996,
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Hemmo 1996b, Bacciagaluppi and Dickson 1999, Clifton 2000, Spekkens and Sipe2001a,b, Bene and Dieks 2002, and Berkovitz and Hemmo 2006a,b. For an entry-levelreview, see the entry on modal interpretations of quantum theory. For comprehensivereviews and analyses of modal interpretations, see Bacciagaluppi 1996, Hemmo 1996a,chapters 1-3, Dieks and Vermaas 1998, Vermaas 1999, and the entry on modal

interpretations of quantum theory. For the no-hidden-variables theorems, see Kochen andSpecker 1967, Greenberger, Horne and Zeilinger 1989, Mermin 1990 and the entry on theKochen-Specker theorem.)[21]

Modal interpretations vary in their property assignment. For simplicity, we shall focus onmodal interpretations in which the property assignment is based on the so-called Schmidtbiorthogonal-decomposition theorem (see Kochen 1985, Dieks 1989, and Healey 1989).Let S1 and S2 be systems associated with the Hilbert spacesHS1 andHS2, respectively.There exist bases {|i>} and {|i>} forHS1 andHS2 respectively such that the state ofS1+S2 can be expressed as a linear combination of the following form of vectors fromthese bases:

|8 >S1+S2 = i ci |i>S1 |i>S2.

When the absolute values of the coefficients ci are all unequal, the bases {|i>} and {|i>}and the above decomposition of |8 >S1+S2 are unique. In that case, it is postulated that S1has a determinate value for each observable associated withHS1 with the basis {|i>} andS2 has a determinate value for each observable associated withHS2 with the basis {|i>},and |ci|2 provide the (ignorance) probabilities of the possible values that these observablesmay have.[22] For example, suppose that the state of the L- and the R-particle in theEPR/B experiment before the measurements is:

|9>= (1/2+) |z-up>L| z-down>R (1/2-) |z-down>L| z-up>Rwhere 1/2 >> ,, (1/2+)2+(1/2-)2 = 1, and (as before) |z-up>L (|z-up>R) and | z-down>L (| z-down>R) denote the states of the L- (R-) particle havingz-spin 'up' andz-spin'down', respectively.[23] Then, either the L-particle spins up and the R-particle spinsdown in thez-direction, or the L-particle spins down and the R-particle spins up inthez-direction. Thus, in contrast to the orthodox interpretation and the GRW/Pearlecollapse models, in modal interpretations the particles in the EPR/B experiment may havedefinite spin properties even before any measurement occurs.

To see how the modal interpretation accounts for the curious correlations in EPR/B-type

experiments, let us suppose that the state of the particle pair and the measurementapparatuses at the emission time is:

|10> = ((1/2+) |z-up>L |z-down>R (1/2) |z-down>L |z-up>R) |ready>AL|ready>AR

where |ready>AL (|ready>AR) denotes the state of the L-apparatus (R-apparatus) beingready to measurez-spin. In this state, the L- and the R-apparatus are in the definite stateof being ready to measurez-spin, and (similarly to the state |9>) the L- and the R-


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particle have definitez-spin properties: Either the L-particle hasz-spin up and the R-particle hasz-spin down, or the L-particle hasz-spin down and the R-particle hasz-spin up,[24] where the probability of the realization of each of these possibilities isapproximately 1/2. In the (earlier)z-spin measurement on the L-particle, the state of theparticle pair and the apparatuses evolves to the state:

|11>= ((1/2+) |z-up>L|up>AL| z-down>R (1/2-) |z-down>L|down>AL| z-up>R) |ready>AR

where (as before) |up>AL and |down>AL denote the states of the L-apparatus pointing to theoutcomesz-spin up andz-spin down, respectively. In this state, either the L-particlehas az-spin up and the L-apparatus points to up, or the L-particle hasz-spin downand the L-apparatus points to down. And, again, the probability of each of thesepossibilities is approximately 1/2. The evolution of the properties from the state |10> tothe state |11> depends on the dynamical laws. In almost all modal interpretations, if theparticles have definitez-spin properties before the measurements, the outcomes ofz-spin

measurements will reflect these properties. That is, the evolution of the properties of theparticles and the measurement apparatuses will be deterministic, so that the spinproperties of the particles do not change in the L-measurement and the pointer of the L-apparatus comes to display the outcome that corresponds to thez-spin property that the L-particle had before the measurement. If, for example, before the measurements the L- andthe R-particle have respectively the propertiesz-spin up andz-spin down, the (earlier)z-spin measurement on the L-particle will yield the outcome up and the spin propertiesof the particles will remain unchanged. Accordingly, az-spin measurement on the R-particle will yield the outcome down. Thus, in this case the modal interpretationinvolves neither action at a distance nor action* at a distance.

However, if the measurement apparatuses are set up to measurex-spin rather thanz-spin,the evolution of the properties of the L-particle and the L-apparatus will beindeterministic. As before, the L-measurement will not cause any change in the actualspin properties of the R-particle. But the L-measurement outcome will cause an instantchange in the spin dispositions of the R-particle and the R-measurement apparatus. If, forexample, the L-measurement outcome isx-spin up and the L-particle comes to possesx-spin up, then the R-particle and the R-apparatus will have respectively the dispositionsto possessx-spin down and to display the outcomex-spin down on ax-spinmeasurement. Thus, like the minimal Bohm theory, the modal interpretation may involveaction* at a distance in the EPR/B experiment. But, unlike the minimal Bohm theory,here spins are intrinsic properties of particles.

In the above modal interpretation, property composition fails: The properties ofcomposite systems are not decomposable into the properties of their subsystems.Consider, again, the state |10>. As separated systems (i.e. in the decompositions of thecomposite system of the particle pair+apparatuses into the L-particle and the R-particle+apparatuses and into the R-particle and the L-particle+apparatuses) the L- andthe R-particle have definitez-spin properties. But, as subsystems of the composite systemof the particle pair (e.g. in the decomposition of the composite system of the particle


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pair+apparatuses into the particle pair and the apparatuses), they have no definitez-spinproperties.

A failure of property composition occurs also in the state |11>, where the L- and the R-particle have definitez-spin properties both as separated systems and as subsystems of

the particle pair (though in contrast with |10>, in |11> the range of the possibleproperties of the particles as separated systems and as subsystems of the pair is the same).For nothing in the above property assignment implies that in |11> the spin properties thatthe L-particle has as a separated system and the spin properties that it has as asubsystem of the particle pair be the same: The L-particle may havez-spin up as aseparated system andz-spin down as a subsystem of the particle pair.

Furthermore, the dynamics of the properties that the L-particle (R-particle) has as aseparated system and the dynamics of its properties as a subsystem of the particle pair aregenerally different.[25] Consider, again, the state |10>. In the (earlier)z-spin measurementon the L-particle, the spin properties that the L-particle has as a separated system follow a

deterministic evolution the L-particle has eitherz-spin up orz-spin down beforeand after the L-measurement; whereas as a subsystem of the particle pair, the spinproperties of the L-particle follow an indeterministic evolution the L-particle has nodefinite spin properties before the L-measurement and eitherz-spin up (withapproximately chance ) orz-spin down (with approximately chance ) after the L-measurement.

The failure of property composition implies that the quantum realm as depicted by theabove version of the modal interpretation involves state non-separability and propertyand relational holism. State separability fails because the state of the particle pair is notgenerally determined by the separate states of the particles. Indeed, as is easily shown,

the actual properties that the L- and the R-particle each has in the state |9> are alsocompatible with product states in which the L- and the R-particle are not entangled.Property and relational holism fail because in the state |9> the properties of the pair donot supervene upon the properties of its subsystems and the spatiotemporal relationsbetween them. Furthermore, process separability fails for similar reasons.

The failure of property composition in the modal interpretation calls for explanation. Itmay be tempting to postulate that the properties that a system (e.g. the L-particle) has, asa separated system, are the same as the properties that it has as a subsystem of compositesystems. But, as Bacciagaluppi (1995) and Clifton (1996a) have shown, such propertyassignment will be inconsistent: It will be subject to a Kochen and Specker-typecontradiction. Furthermore, as Vermaas (1997) demonstrates, the properties of compositesystems and the properties of their subsystems cannot be correlated (in ways compatiblewith the Born rule).

For what follows in the rest of this subsection, the views of different authors differwidely. Several variants of modal interpretations were developed in order to fix theproblem of the failure of property composition. The most natural explanation of thefailure of property composition is that quantum states assign relational rather than


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intrinsic properties to systems (see Kochen 1985, Bene and Dieks 2002, and Berkovitzand Hemmo 2006a,b). For example, in the relational modal interpretation proposed byBerkovitz and Hemmo (2006a,b), the main idea is that quantum states assign propertiesto systems only relative to other systems, and properties of a system that are related todifferent systems are generally different. In particular, in the state |10> the L-particle has

a definitez-spin property relative to the R-particle, the measurement apparatuses and therest of the universe, but (as a subsystem of the particle pair) it has no definitez-spinrelative to the measurement apparatuses and the rest of universe.[26] On this interpretation,the properties of systems are highly non-local by their very nature. Properties likepointing to up and pointing to down are not intrinsic to the measurement apparatuses.Rather, they are relations between the apparatuses and other systems. For example, theproperty of the L-apparatus pointing to up relative to the particle pair, the R-apparatusand the rest of the universe is not intrinsic to the L-apparatus; it is a relation between theL-apparatus and the particle pair, the R-apparatus and the rest of the universe. As such,this property is highly non-local: It is located in neither the L-wing nor any othersubregion of the universe. Yet, due to the dynamical laws, properties like the position of

pointers of measurement apparatuses, which appear to us to be local, behave like localproperties in any experimental circumstances, and accordingly this radical type of non-locality is unobservable (for

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Action at a Distance in Quantum Mechanics