Action and Distance in Quantum Mechanics

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Action at a Distance in Quantum Mechanics

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Joseph Berkovitz

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Action at a Distance in Quantum MechanicsFirst published Fri Jan 26, 2007

In the quantum realm, there are curious correlations between theproperties of distant systems. An example of such correlations is providedby the famous Einstein-Podolsky-Rosen/Bohm experiment. Thecorrelations in the EPR/B experiment strongly suggest that there are non-local influences between distant systems, i.e., systems between which nolight signal can travel, and indeed orthodox quantum mechanics and itsvarious interpretations postulate the existence of such non-locality. Yet,the question of whether the EPR/B correlations imply non-locality and theexact nature of this non-locality is a matter of ongoing controversy.Focusing on EPR/B-type experiments, in this entry we consider the natureof the various kinds of non-locality postulated by different interpretationsof quantum mechanics. Based on this consideration, we briefly discuss thecompatibility of these interpretations with the special theory of relativity.

1. Introduction2. Bell's theorem and non-locality3. The analysis of factorizability4. Action at a distance, holism and non-separability

4.1 Action at a distance4.2 Holism4.3 Non-separability

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

5.1 Collapse theories5.2 Can action-at-a-distance co-exist with non-separability andholism?5.3 No-collapse theories

6. Superluminal causation

1

7. Superluminal signaling7.1 Necessary and sufficient conditions for superluminalsignaling7.2 No-collapse theories7.3 Collapse theories7.4 The prospects of controllable probabilistic dependence7.5 Superluminal signaling and action-at-a-distance

8. The analysis of factorizability: implications for quantum non-locality

8.1 Non-separability, holism and action at a distance8.2 Superluminal signaling8.3 Relativity8.4 Superluminal causation8.5 On the origin and nature of parameter dependence

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

10.1 Collapse theories10.2 No-collapse theories10.3 Quantum causal loops and relativity

BibliographyOther Internet ResourcesRelated Entries

1. Introduction

The quantum realm involves curious correlations between distant events.A well-known example is David Bohm's (1951) version of the famousthought experiment that Einstein, Podolsky and Rosen proposed in 1935(henceforth, the EPR/B experiment). Pairs of particles are emitted from asource in the so-called spin singlet state and rush in opposite directions(see Fig. 1 below). When the particles are widely separated from eachother, they each encounter a measuring apparatus that can be set to


2 Stanford Encyclopedia of Philosophy

other, they each encounter a measuring apparatus that can be set tomeasure their spin components along various directions. Although themeasurement events are distant from each other, so that no slower-than-light or light signal can travel between them, the measurement outcomesare curiously correlated.[1] That is, while the outcome of each of thedistant spin measurements seems to be a matter of pure chance, they arecorrelated with each other: The joint probability of the distant outcomes isdifferent from the product of their single probabilities. For example, theprobability that each of the particles will spin clockwise about the z-axisin a z-spin measurement (i.e., a measurement of the spin component alongthe z direction) appears to be ½. Yet, the outcomes of such measurementsare perfectly anti-correlated: If the left-hand-side (L-) particle happens tospin clockwise (anti-clockwise) about the z-axis, the right-hand-side (R-)particle will spin anti-clockwise (clockwise) about that axis. And this istrue even if the measurements are made simultaneously.

The curious EPR/B correlations strongly suggest the existence of non-local influences between the two measurement events, and indeedorthodox ‘collapse’ quantum mechanics supports this suggestion.According to this theory, before the measurements the particles do nothave any definite spin. The particles come to possess a definite spin onlywith the first spin measurement, and the outcome of this measurement is a

Figure 1: A schematic illustration of the EPR/B experiment.Particle pairs in the spin singlet state are emitted in oppositedirections and when they are distant from each other (i.e., space-like separated), they encounter measurement apparatuses that canbe set to measure spin components along various directions.

Joseph Berkovitz

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with the first spin measurement, and the outcome of this measurement is amatter of chance. If, for example, the first measurement is a z-spinmeasurement on the L-particle, the L-particle will spin either clockwiseor anti-clockwise about the z-axis with equal chance. And the outcome ofthe L-measurement causes an instantaneous change in the spin propertiesof the distant R-particle. If the L-particle spins clockwise (anti-clockwise)about the z-axis, the R-particle will instantly spin anti-clockwise(clockwise) about the same axis. (It is common to call spins in oppositedirections ‘spin up’ and ‘spin down,’ where by convention a clockwisespinning may be called ‘spin up’ and anti-clockwise spinning may becalled ‘spin down.’)

It may be argued that orthodox quantum mechanics is false, and that thenon-locality postulated by it does not reflect any non-locality in thequantum realm. Alternatively, it may be argued that orthodox quantummechanics is a good instrument for predictions rather than a fundamentaltheory of the physical nature of the universe. On this instrumentalinterpretation, the predictions of quantum mechanics are not an adequatebasis for any conclusion about non-locality: This theory is just anincredible oracle (or a crystal ball), which provides a very successfulalgorithm for predicting measurement outcomes and their probabilities,but it offers little information about ontological matters, such as the natureof objects, properties and causation in the quantum realm.

Einstein, Podolsky and Rosen (1935) thought that quantum mechanics isincomplete and that the curious correlations between distant systems donot amount to action at a distance between them. The apparentinstantaneous change in the R-particle's properties during the L-measurement is not really a change of properties, but rather a change ofknowledge. (For more about the EPR argument, see the entry on the EPRargument, Redhead 1987, chapter 3, and Albert 1992, chapter 3. Fordiscussions of the EPR argument in the relativistic context, see Ghirardiand Grassi 1994 and Redhead and La Riviere 1997.) On this view,



and Grassi 1994 and Redhead and La Riviere 1997.) On this view,quantum states of systems do not always reflect their complete state.Quantum states of systems generally provide information about some ofthe properties that systems possess and information about the probabilitiesof outcomes of measurements on them, and this information does notgenerally reflect the complete state of the systems. In particular, theinformation encoded in the spin singlet state is about the probabilities ofmeasurement outcomes of spin properties in various directions, about theconditional probabilities that the L- (R-) particle has a certain spinproperty given that the R- (L-) particle has another spin property, andabout the anti-correlation between the spins that the particles may have inany given direction (for more details, see section 5.1). Thus, the outcomeof a z-spin measurement on the L-particle and the spin singlet state(interpreted as a state of knowledge) jointly provide information about thez-spin property of the R-particle. For example, if the outcome of the L-measurement is z-spin ‘up,’ we know that the R-particle has z-spin‘down’; and if we assume, as EPR did, that there is no curious action at adistance between the distant wings (and that the change of the quantum-mechanical state of the particle pair in the L-measurement is only achange in state of knowledge), we could also conclude that the R-particlehad z-spin ‘down’ even before the L-measurement occurs.

How could the L-outcome change our knowledge/ignorance about the R-outcome if it has no influence on it? The simplest and moststraightforward reply is that the L- and the R- outcome have a commoncause that causes them to be correlated, so that knowledge of one outcomeprovides knowledge about the other.[2] Yet, the question is whether thepredictions of orthodox quantum mechanics, which have been highlyconfirmed by various experiments, are compatible with the quantumrealm being local in the sense of involving no influences between systemsbetween which light and slower-than-light signals cannot travel (i.e.,space-like separated systems). More particularly, the question is whetherit is possible to construct a local, common-cause model of the EPR/B

Joseph Berkovitz


it is possible to construct a local, common-cause model of the EPR/Bexperiment, i.e., a model that postulates no influence betweensystems/events in the distant wings of the experiment, and that thecorrelation between them are due to the state of the particle pair at thesource. In 1935, Einstein, Podolsky and Rosen believed that this ispossible. But, as John Bell demonstrated in 1964, this belief is difficult touphold.

2. Bell's theorem and non-locality

In a famous theorem, John Bell (1964) demonstrated that granted someplausible assumptions, any local model of the EPR/B experiment iscommitted to certain inequalities about the probabilities of measurementoutcomes, ‘the Bell inequalities,’ which are incompatible with thequantum-mechanical predictions. When Bell proved his theorem, theEPR/B experiment was only a thought experiment. But due totechnological advances, various versions of this experiment have beenconducted since the 1970s, and their results have overwhelminglysupported the quantum-mechanical predictions (for brief reviews of theseexperiments and further references, see the entry on Bell's theorem andRedhead 1987, chapter 4, section 4.3 and ‘Notes and References’). Thus, awide consensus 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/Bexperiment postulates that the state of the particle pair together with theapparatus settings to measure (or not to measure) certain spin propertiesdetermine the probabilities for single and joint spin-measurementoutcomes. A local Bell model of this experiment also postulates thatprobabilities of joint outcomes factorize into the single probabilities of theL- and the R- outcomes: The probability of joint outcomes is equal to theproduct of the probabilities of the single outcomes. More formally, let λdenote the pair's state before any measurement occurs. Let l denote the



denote the pair's state before any measurement occurs. Let l denote thesetting of the L-measurement apparatus to measure spin along the l-axis(i.e., the l-spin of the L-particle), and let r denote the setting of the R-measurement apparatus to measure spin along the r-axis (i.e., the r-spinof the R-particle). Let xl be the outcome of a l-spin measurement in the L-wing, and let yr be the outcome of a r-spin measurement in the R-wing;where xl is either the L-outcome l-spin ‘up’ or the L-outcome l-spin‘down,’ and yr is either the R-outcome r-spin ‘up’ or the R-outcome r-spin ‘down.’ Let Pλ l r(xl & yr) be the joint probability of the L- and theR-outcome, and Pλ l(xl) and Pλ r(yr) be the single probabilities of the L-and the R-outcome, respectively; where the subscripts λ, l and r denotethe factors that are relevant for the probabilities of the outcomes xl and yr.Then, for any λ, l, r, xl and yr:[3]

(Here and henceforth, for simplicity's sake we shall denote events andstates, such as the measurement outcomes, and the propositions that theyoccur by the same symbols.)

The state λ is typically thought of as the pair's state at the emission time,and it is assumed that this state does not change in any relevant sensebetween the emission and the first measurement. It is (generally) adifferent state from the quantum-mechanical pair's state ψ. ψ is assumedto be an incomplete state of the pair, whereas λ is supposed to be a (more)complete state of the pair. Accordingly, pairs with the same state ψ mayhave different states λ which give rise to different probabilities ofoutcomes for the same type of measurements. Also, the states λ may beunknown, hidden, inaccessible or uncontrollable.

Factorizability is commonly motivated as a locality condition. In non-local models of the EPR/B experiment, the correlations between thedistant outcomes are accounted for by non-local influences between the

FactorizabilityPλ l r(xl & yr) = Pλ l(xl) · Pλ r(yr).

Joseph Berkovitz


distant outcomes are accounted for by non-local influences between thedistant measurement events. For example, in orthodox quantummechanics the first spin measurement on, say, the L-particle causes animmediate change in the spin properties of the R-particle and in theprobabilities of future outcomes of spin measurements on this particle. Bycontrast, in local models of this experiment the correlations are supposedto be accounted for by a common cause—the pair's state λ (see Fig. 2below): The pair's state and the L-setting determine the probability of theL-outcome; the pair's state and the R-setting determine the probability ofthe R-outcome; and the pair's state and the L- and the R-setting determinethe probability of joint outcomes, which (as mentioned above) is simplythe product of these single probabilities. The idea is that the probability ofeach of the outcomes is determined by ‘local events,’ i.e., events that areconfined to its backward light-cone, and which can only exert subluminalor luminal influences on it (see Figure 3 below); and the distant outcomesare fundamentally independent of each other, and thus their jointprobability factorizes. (For more about this reasoning, see sections 6 and8-9.)

Figure 2: A schematic common-cause model of the EPR/Bexperiment. Arrows denote causal connections.



A Bell model of the EPR/B experiment also postulates that for eachquantum-mechanical state ψ there is a distribution ρ over all the possiblepair states λ, which is independent of the settings of the apparatuses. Thatis, the distribution of the (‘complete’) states λ depends on the(‘incomplete’) state ψ, and this distribution is independent of theparticular choice of measurements in the L- and R-wing (including thechoice not to measure any quantity). Or formally, for any quantum-mechanical state ψ, L-settings l and l′, and R-settings r and r′:

where the subscripts denote the factors that are potentially relevant for thedistribution of the states λ.

Although the model probabilities (i.e., the probabilities of outcomesprescribed by the states λ) are different from the corresponding quantum-mechanical probabilities of outcomes (i.e., the probabilities prescribed bythe quantum-mechanical states ψ), the quantum mechanical probabilities(which have been systematically confirmed) are recovered by averaging

Figure 3: A space-time diagram of a local model of the EPR/Bexperiment. The circles represent the measurement events, and thecones represent their backward light cones, i.e., the boundaries ofall the subluminal and luminal influences on them. The dottedlines denote the propagation of the influences of the pair's state atthe emission and of the settings of the measurement apparatuseson the measurement outcomes.

λ-independence ρψ l r(λ) = ρψ l′ r(λ) = ρψ l r′(λ) = ρψ l′ r′(λ) = ρψ(λ)

Joseph Berkovitz


(which have been systematically confirmed) are recovered by averagingover the model probabilities. That is, it is supposed that the quantum-mechanical probabilities Pψ l r(xl & yr), Pψ l(xl) and Pψ r(yr) areobtained by averaging over the model probabilities Pλ l r(xl & yr), Pλ l(xl) and Pλ r(yr), respectively: For any ψ, l, r, xl and yr,

The assumption of λ-independence is very plausible. It postulates that(complete) pair states at the source are uncorrelated with the settings ofthe measurement apparatuses. And independently of one's philosophicalview about free will, this assumption is strongly suggested by ourexperience, according to which it seems possible to prepare the state ofparticle pairs at the source independently of the set up of the measurementapparatuses.

There are two ways to try to explain a failure of λ-independence. Onepossible explanation is that pairs' states and apparatus settings share acommon cause, which always correlates certain types of pairs' states λwith certain types of L- and R-setting. Such a causal hypothesis will bedifficult to reconcile with the common belief that apparatus settings arecontrollable at experimenters' will, and thus could be set independently ofthe pair's state at the source. Furthermore, thinking of all the differentways one can measure spin properties and the variety of ways in whichapparatus settings can be chosen, the postulation of such common causeexplanation for settings and pairs' states would seem highly ad hoc and itsexistence conspiratorial.

Another possible explanation for the failure of λ-independence is that theapparatus settings influence the pair's state at the source, and accordinglythe distribution of the possible pairs' states λ is dependent upon the

Empirical Adequacy Pψ l r(xl & yr) = ∫λ Pλ l r(xl & yr) · ρψ l r(λ)Pψ l(xl) = ∫λ Pλ l(xl ) · ρψ l(λ)Pψ r(yr) = ∫λ Pλ r(yr) · ρψ r(λ).[4]



the distribution of the possible pairs' states λ is dependent upon thesettings. Since the settings can be made after the emission of the particlepair from the source, this kind of violation of λ-independence wouldrequire backward causation. (For advocates of this way out of non-locality, see Costa de Beauregard 1977, 1979, 1985, Sutherland 1983,1998, 2006 and Price 1984, 1994, 1996, chapters 3, 8 and 9.) On somereadings of John Cramer's (1980, 1986) transactional interpretation ofquantum mechanics (see Maudlin 1994, pp. 197-199), such violation of λ-independence is postulated. According to this interpretation, the sourcesends ‘offer’ waves forward to the measurement apparatuses, and theapparatuses send ‘confirmation’ waves (from the space-time regions ofthe measurement events) backward to the source, thus affecting the statesof emitted pairs according to the settings of the apparatuses. The questionof whether such a theory can reproduce the predictions of quantummechanics is a controversial matter (see Maudlin 1994, pp. 197-199,Berkovitz 2002, section 5, and Kastner 2006). It is noteworthy, however,that while the violation of λ-independence is sufficient for circumventingBell's theorem, the failure of this condition per se does not substantiatelocality. The challenge of providing a local model of the EPR/Bexperiment also applies to models that violate λ-independence. (For moreabout these issues, see sections 9 and 10.3.)

In any case, as Bell's theorem demonstrates, factorizability, λ-independence and empirical adequacy jointly imply the Bell inequalities,which are violated by the predictions of orthodox quantum mechanics(Bell 1964, 1966, 1971, 1975a,b). Granted the systematic confirmation ofthe predictions of orthodox quantum mechanics and the plausibility of λ-independence, Bell inferred that factorizability fails in the EPR/Bexperiment. Thus, interpreting factorizability as a locality condition, heconcluded that the quantum realm is non-local. (For further discussions ofBell's theorem, the Bell inequalities and non-locality, see Bell 1966, 1971,1975a,b, 1981, Clauser et al 1969, Clauser and Horne 1974, Shimony1993, chapter 8, Fine 1982a,b, Redhead 1987, chapter 4, Butterfield 1989,

Joseph Berkovitz


1993, chapter 8, Fine 1982a,b, Redhead 1987, chapter 4, Butterfield 1989,1992a, Pitowsky 1989, Greenberger, Horne and Zeilinger 1989,Greenberger, Horne, Shimony and Zeilinger 1990, Mermin 1990, and theentry on Bell's theorem.)

3. The analysis of factorizability

Following Bell's work, a broad consensus has it that the quantum realminvolves some type of non-locality (for examples, see Clauser and Horne1974, 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 withrelativity theory. One aspect of this controversy is over whether theanalysis of factorizability and the different ways it could be violated mayshed light on these issues. Factorizability is equivalent to the conjunctionof two conditions (Jarrett 1984, 1989, Shimony 1984):[6]

Parameter independence. The probability of a distantmeasurement outcome in the EPR/B experiment is independent ofthe setting of the nearby measurement apparatus. Or formally, forany pair's state λ, L-setting l, R-setting r, L-outcome xl and R-outcome yr:

PIPλ l r(xl) = Pλ l(xl) and Pλ l r(yr) = Pλ r(yr).

Outcome independence. The probability of a distantmeasurement outcome in the EPR/B experiment is independent ofthe nearby measurement outcome. Or formally, for any pair's stateλ, L-setting l, R-setting r, L-outcome xl and R-outcome yr:



Assuming λ-independence (see section 2), any empirically adequatetheory will have to violate OI or PI. A common view has it that violationsof PI involve a different type of non-locality than violations of OI:Violations of PI involve some type of action-at-a-distance that isimpossible to reconcile with relativity (Shimony 1984, Redhead 1987, p.108), whereas violations of OI involve some type of holism, non-separability and/or passion-at-a-distance that may be possible to reconcilewith relativity (Shimony 1984, Readhead 1987, pp. 107, 168-169, Howard1989, Teller 1989).

On the other hand, there is the view that the analysis above (as well asother similar analyses of factorizability[7]) is immaterial for studyingquantum non-locality (Butterfield 1992a, pp. 63-64, Jones and Clifton1993, Maudlin 1994, pp. 96 and 149) and even misleading (Maudlin 1994,pp. 94-95 and 97-98). On this alternative view, the way to examine thenature of quantum non-locality is to study the ontology postulated by thevarious interpretations of quantum mechanics and alternative quantumtheories.[8] In sections 4-7, we shall follow this methodology and discussthe nature of non-locality postulated by several quantum theories. Thediscussion in these sections will furnish the ground for evaluating theabove controversy in section 8.

4. Action at a distance, holism and non-separability

Pλ l r(xl / yr) = Pλ l r(xl) and Pλ l r(yr / xl) = Pλ l r(yr)

Pλ l r(yr) > 0 Pλ l r(xl) > 0,

or more generally,

OIPλ l r(xl & yr) = Pλ l r(xl) · Pλ l r(yr).

Joseph Berkovitz


4.1 Action at a distance

In orthodox quantum mechanics as well as in any other current quantumtheory that postulates non-locality (i.e., influences between distant, space-like separated systems), the influences between the distant measurementevents in the EPR/B experiment do not propagate continuously in space-time. They seem to involve action at a distance. Yet, a common view hasit that these influences are due to some type of holism and/or non-separability of states of composite systems, which are characteristic ofsystems in entangled states (like the spin singlet state), and which excludethe very possibility of action at a distance. The paradigm case of action ata distance is the Newtonian gravitational force. This force acts betweendistinct objects that are separated by some (non-vanishing) spatialdistance, its influence is symmetric (in that any two massive objectsinfluence each other), instantaneous and does not propagate continuouslyin space. And it is frequently claimed or presupposed that such action at adistance could only exist between systems with separate states in non-holistic universes (i.e., universes in which the states of composite systemsare determined by, or supervene upon the states of their subsystems andthe spacetime relations between them), which are commonly taken tocharacterize the classical realm.[9]

In sections 4.2 and 4.3, we shall briefly review the relevant notions ofholism and non-separability (for a more comprehensive review, see theentry on holism and nonseparability in physics and Healey 1991). Insection 5, we shall discuss the nature of holism and non-separability in thequantum realm as depicted by various quantum theories. Based on thisdiscussion, we shall consider whether the non-local influences in theEPR/B experiment constitute action at a distance.

4.2 Holism



In the literature, there are various characterizations of holism. Discussionsof quantum non-locality frequently focus on property holism, wherecertain physical properties of objects are not determined by the physicalproperties of their parts. The intuitive idea is that some intrinsic propertiesof wholes (e.g. physical systems) are not determined by the intrinsicproperties of their parts and the spatiotemporal relations that obtainbetween these parts. This idea can be expressed in terms of superveniencerelations.

It is difficult to give a general precise specification of the terms ‘intrinsicqualitative property’ and ‘supervenience.’ Intuitively, a property of anobject is intrinsic just in case that object has this property in and for itselfand independently of the existence or the state of any other object. Aproperty is qualitative (as opposed to individual) if it does not depend onthe existence of any particular object. And the intrinsic qualitativeproperties of an object O supervene upon the intrinsic qualitativeproperties and relations of its parts and the spatiotemporal relationsbetween them just in case there is no change in the properties andrelations of O without a change in the properties and relations of its partsand/or the spatiotemporal relations between them. (For attempts toanalyze the term ‘intrinsic property,’ see for example Langton and Lewis1998 and the entry on intrinsic vs. extrinsic properties. For a review ofdifferent types of supervenience, see for example Kim 1978, McLaughlin1994 and the entry on supervenience.)

Paul Teller (1989, p. 213) proposes a related notion of holism, ‘relationalholism,’ which is characterized as the violation of the followingcondition:

Property Holism. Some objects have intrinsic qualitativeproperties and/or relations that do not supervene upon the intrinsicqualitative properties and relations of their parts and thespatiotemporal relations between these parts.

Joseph Berkovitz


condition:

Here, by a non-relational property Teller means an intrinsic property(1986a, p. 72); and by ‘the supervenience of a relational property on thenon-relational properties of the relata,’ he means that ‘if two objects, 1and 2, bear a relation R to each other, then, necessarily, if two furtherobjects, 1′ and 2′ have the same non-relational properties, then 1′ and 2′will also bear the same relation R to each other’ (1989, p. 213). Teller(1986b, pp. 425-7) believes that spatiotemporal relations between objectssupervene upon the objects’ intrinsic physical properties. Thus, he doesnot include the spatiotemporal relations in the supervenience basis. Thisview is controversial, however, as many believe that spatiotemporalrelations between objects are neither intrinsic nor supervene upon theintrinsic qualitative properties of these objects. But, if such superveniencedoes not obtain, particularism will also be violated in classical physics,and accordingly relational holism will fail to mark the essentialdistinction between the classical and the quantum realms. Yet, one mayslightly revise Teller's definition of particularism as follows:

In what follows in this entry, by relational holism we shall mean aviolation of particularism*.

4.3 Non-separability

Particularism. The world is composed of individuals. Allindividuals have non-relational properties and all relationssupervene on the non-relational properties of the relata.

Particularism*. The world is composed of individuals. Allindividuals have non-relational properties and all relationssupervene upon the non-relational properties of the relata and thespatiotemporal relations between them.



Like holism, there are various notions of non-separability on offer. Themost common notion in the literature is state non-separability, i.e., theviolation of the following condition:

The term ‘wholly determined’ is vague. But, as before, one may spell itout in terms of supervenience relations: State separability obtains just incase each system possesses a separate state that determines its qualitativeintrinsic properties and relations, and the state of any composite system issupervenient upon the separate states of its subsystems.

Another notion of non-separability is spatiotemporal non-separability.Inspired by Einstein (1948), Howard (1989, pp. 225-6) characterizesspatiotemporal non-separability as the violation of the followingseparability condition:

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 violation of the following condition:

State separability. Each system possesses a separate state thatdetermines its qualitative intrinsic properties, and the state of anycomposite system is wholly determined by the separate states ofits subsystems.

Spatiotemporal separability. The contents of any two regions ofspace-time separated by a non-vanishing spatiotemporal intervalconstitute 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 theseparated states of these regions.

Process separability. Any physical process occupying aspacetime region R supervenes upon an assignment of qualitativeintrinsic physical properties at spacetime points in R.

Joseph Berkovitz


5. Holism, non-separability and action at a distancein quantum mechanics

The quantum realm as depicted by all the quantum theories that postulatenon-locality, i.e., influences between distant (space-like separated)systems, involves some type of non-separability or holism. In whatfollows in this section, we shall consider the nature of the non-separabilityand holism manifested by various interpretations of quantum mechanics.On the basis of this consideration, we shall address the question ofwhether these interpretations predicate the existence of action at adistance. 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 spacesrepresent states of physical systems. When the Hilbert space is of infinitedimension, state vectors can be represented by functions, the so-called‘wave functions.’ In any given basis, there is a unique wave function thatcorresponds to the state vector in that basis. (For an entry level review ofthe 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 Redhead 1987, chapters 1-2 and the mathematicalappendix.)

For example, the state of the L-particle having z-spin ‘up’ (i.e., spinning‘up’ about the z-axis) can be represented by the vector |z-up> in theHilbert space associated with the L-particle, and the state of the L-particlehaving z-spin ‘down’ (i.e., spinning ‘down’ about the z-axis) can be

intrinsic physical properties at spacetime points in R.



having z-spin ‘down’ (i.e., spinning ‘down’ about the z-axis) can berepresented by the orthogonal vector, |z-down>. Particle pairs may be in astate in which the L-particle and the R-particle have opposite spins, forinstance either a state |ψ1> in which the L-particle has z-spin ‘up’ and theR-particle has z-spin ‘down,’ or a state |ψ2> in which the L-particle hasz-spin ‘down’ and the R-particle has z-spin ‘up.’ Each of these states isrepresented by a tensor product of vectors in the Hilbert space of theparticle pair: |ψ1> = |z-up>L |z-down>R and |ψ2> = |z-down>L |z-up>R;where the subscripts L and R refer to the Hilbert spaces associated withthe L- and the R-particle, respectively. But particle pairs may also be in asuperposition of these states, i.e., a state that is a linear sum of the states |ψ1> and |ψ2>, e.g. the state represented by

In fact, this is exactly the case in the spin singlet state. In this state, theparticles are entangled in a non-separable state (i.e., a state that cannot bedecomposed into a product of separate states of the L- and the R-particle),in which (according to the property-assignment rules of orthodoxquantum mechanics) the particles do not possess any definite z-spin (ordefinite spin in any other direction). Thus, the condition of stateseparability fails: The state of the particle pair (which determines itsintrinsic qualitative properties) is not wholly determined by the separatestates of the particles (which determine their intrinsic qualitativeproperties). Or more precisely, the pair's state is not supervenient uponthe separable states of the particles. In particular, the superposition stateof the particle pair assigns a ‘correlational’ property that dictates that theoutcomes of (ideal) z-spin measurements on both the L- and the R-particle will be anti-correlated, and this correlational property is notsupervenient upon properties assigned by any separable states of theparticles (for more details, see Healey 1992, 1994). For similar reasons,the spin singlet state also involves property and relational holism; for the

|ψ3> = 1/√2 (|ψ1> − |ψ2>)= 1/√2 (|z-up>L |z-down>R − |z-down>L |z-up>R).

Joseph Berkovitz


the spin singlet state also involves property and relational holism; for theabove correlational property of the particle pair also fails to superveneupon the intrinsic qualitative properties of the particles and thespatiotemporal relations between them. Furthermore, the process thatleads 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 adistance that the orthodox theory seems to postulate between the distantwings in the EPR/B experiment. Recall (section 1) that Einstein, Podolskyand Rosen thought that this curious action at a distance reflects theincompleteness of this theory rather than a state of nature. The EPRargument for the incompleteness of the orthodox theory is controversial.But the orthodox theory seems to be incomplete for a different reason.This theory postulates that in non-measurement interactions, the evolutionof states obeys a linear and unitary equation of motion, the so-calledSchrödinger equation (see the entry on quantum mechanics), according towhich 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 the measurement apparatuses get entangled with theparticle pair in a non-separable state in which (according to the theory'sproperty assignment, see below) the indefiniteness of particles’ spins is‘transmitted’ to the pointer's position: In this entangled state of theparticle pair and the pointer, the pointer lacks any definite position, incontradiction to our experience of perceiving it pointing to either ‘up’ or‘down.’

The above problem, commonly called ‘the measurement problem,’ arisesin orthodox no-collapse quantum mechanics from two features thataccount very successfully for the behavior of microscopic systems: Thelinear dynamics of quantum states as described by the Schrödingerequation and the property assignment rule called ‘eigenstate-eigenvalue



equation and the property assignment rule called ‘eigenstate-eigenvaluelink.’ According to the eigenstate-eigenvalue link, a physical observable,i.e., a physical quantity, of a system has definite value (one of itseigenvalues) just in case the system is in the corresponding eigenstate ofthat 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 indefinite z-spin,energy, position, momentum and various other quantities. The problem isthat given the linear and unitary Schrödinger dynamics, these indefinitequantities are also endemic in the macroscopic realm. For example, in a z-spin measurement on a particle in a superposition state of z-spin ‘up’ andz-spin ‘down,’ the position of the apparatus’s pointer gets entangled withthe indefinite z-spin of the particle, thus transforming the pointer into astate of indefinite position, i.e., a superposition of pointing ‘up’ andpointing ‘down’ (see Albert 1992, chapter 4, and the entry on collapsetheories, section 3). In particular, in the EPR/B experiment the L-measurement causes the L-apparatus pointer to get entangled with theparticle pair, transforming it into a state of indefinite position:

|ψ4> = 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 pointerdisplaying the outcomes z-spin ‘up’ and z-spin ‘down,’ respectively.Since the above type of indefiniteness is generic in orthodox no-collapsequantum mechanics, in this theory measurements typically have nodefinite outcomes, in contradiction to our experience.

In order to solve this problem, the orthodox theory postulates that inmeasurement interactions, entangled states of measured systems and thecorresponding measurement apparatuses do not evolve according to theSchrödinger equation. Rather, they undergo a ‘collapse’ into product(non-entangled) states, where the systems involved have the relevant

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(non-entangled) states, where the systems involved have the relevantdefinite properties. For example, the entangled state of the particle pairand the L-apparatus in the EPR/B experiment may collapse into a productstate in which the L-particle comes to possess z-spin ‘up,’ the R-particlecomes to possess z-spin ‘down’ and the L-apparatus pointer displayingthe outcome z-spin ‘up’:

The problem is that in the orthodox theory, the notions of measurementand the time, duration and nature of state collapses remain completelyunspecified. As John Bell (1987b, p. 205) remarks, the collapse postulatein this theory, i.e., the postulate that dictates that in measurementinteractions the entangled states of the relevant systems do not follow theSchrödinger equation but rather undergo a collapse, is no more than‘supplementary, imprecise, verbal, prescriptions.’

This problem of accounting for our experience of perceiving definitemeasurement outcomes in orthodox quantum mechanics, is an aspect ofthe more general problem of accounting for the classical-like behavior ofmacroscopic systems in this theory.

5.1.2 Dynamical models for state vector reduction

The dynamical models for state-vector reduction were developed toaccount for state collapses as real physical processes (for a review of thecollapse models and a detailed reference list, see the entry on collapsetheories). The origin of the collapse models may be dated to Bohm andBub's (1966) hidden variable theory and Pearle's (1976) spontaneouslocalization approach, but the program has received its crucial impetuswith the more sophisticated models developed by Ghirardi, Rimini andWeber in 1986 (see also Bell 1987a and Albert 1992) and theirconsequent development by Pearle (1989) (see also Ghirardi, Pearle andRimini 1990, and Butterfield et al. 1993). Similarly to orthodox collapse

|ψ5> = |z-up>L |z-down>R |up>LA.



Rimini 1990, and Butterfield et al. 1993). Similarly to orthodox collapsequantum mechanics, in the GRW models the quantum-mechanical state ofsystems (whether it is expressed by a vector or a wave function) providesa complete specification of their intrinsic properties and relations. Thestate of systems follows the Schrödinger equation, except that it has aprobability for spontaneous collapse, independently of whether or not thesystems are measured. The chance of collapse depends on the ‘size’ of theentangled systems—in the earlier models the ‘size’ of systems ispredicated on the number of the elementary particles, whereas in latermodels it is measured in terms of mass densities. In any case, inmicroscopic systems, such as the particle pairs in the EPR/B experiment,the chance of collapse is very small and negligible—the chance ofspontaneous state collapse in such systems is cooked up so that it willoccur, on average, every hundred million years or so. This means that thechance that the entangled state of the particle pair in the EPR/Bexperiment will collapse to a product state between the emission from thesource and the first measurement is virtually zero. In an earlier L-measurement, the state of the particle pair gets entangled with the state ofthe L-measurement apparatus. Thus, the state of the pointer of the L-apparatus evolves from being ‘ready’ to measure a certain spin propertyto an indefinite outcome. For instance, in a z-spin measurement the L-apparatus gets entangled with the particle pair in a superposition state ofpointing to ‘up’ and pointing to ‘down’ (corresponding to the states of theL-particle having z-spin ‘up’ and having z-spin ‘down’), and the R-apparatus remains un-entangled with these systems in the state of beingready to measure z-spin. Or formally:

|ψ6> = 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 the outcomes z-spin ‘up’ and ‘down’ respectively,and |ready>AR denotes the state of the R-apparatus being ready to

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and |ready>AR denotes the state of the R-apparatus being ready tomeasure z-spin. In this state, a gigantic number of particles of the L-apparatus pointer are entangled together in the superposition state ofbeing in the position (corresponding to pointing to) ‘up’ and the position(corresponding to pointing to) ‘down.’ For assuming, for simplicity ofpresentation, that the position of all particles of the L-apparatus pointer inthe state of pointing to ‘up’ (‘down’) is the same, the state |ψ6> can berewritten as:

where pi denotes the i-particle of the L-apparatus pointer, and |up>pi(|down>pi) is the state of the i-particle being in the position correspondingto the outcome z-spin ‘up’ (‘down’).[10] The chance that at least one ofthe vast number of the pointer's particles will endure a spontaneouslocalization toward being in the position corresponding to either theoutcome z-spin ‘up’ or the outcome z-spin ‘down’ within a very shorttime (a split of a micro second) is very high. And since all the particles ofthe pointer and the particle pair are entangled with each other, such acollapse will carry with it a collapse of the entangled state of the pointerof the L-apparatus and the particle pair toward either

or

Thus, the pointer will very quickly move in the direction of pointing toeither the outcome z-spin ‘up’ or the outcome z-spin ‘down.’

If (as portrayed above) the spontaneous localization of particles were to aprecise position, i.e., to the position corresponding to the outcome ‘up’ orthe outcome ‘down,’ the GRW collapse models would successfully

|ψ7> = 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

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

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



the outcome ‘down,’ the GRW collapse models would successfullyresolve the measurement problem. Technically speaking, a preciselocalization is achieved by multiplying |ψ7> by a delta function centeredon the position corresponding to either the outcome ‘up’ or the outcome‘down’ (see the entry on collapse theories, section 5 and Albert 1992,chapter 5); where the probability of each of these mutually exhaustivepossibilities is ½. The problem is that it follows from the uncertaintyprinciple (see the entry on the uncertainty principle) that in suchlocalizations the momenta and the energies of the localized particleswould be totally uncertain, so that gases may spontaneously heat up andelectrons may be knocked out of their orbits, in contradiction to ourexperience. To avoid this kind of problems, GRW postulated thatspontaneous localizations are characterized by multiplications byGaussians that are centered around certain positions, e.g. the positioncorresponding to either the outcome ‘up’ or the outcome ‘down’ in thestate |ψ7>. This may be problematic, because in either case the state ofthe L-apparatus pointer at (what we characteristically conceive as) theend of the L-measurement would be a superposition of the positions ‘up’and ‘down.’ For although this superposition ‘concentrates’ on either theoutcome ‘up’ or the outcome ‘down’ (i.e., the peak of the wave functionthat corresponds to this state concentrates on one of these positions), italso has ‘tails’ that go everywhere: The state of the L-apparatus is asuperposition of an infinite number of different positions. Thus, it followsfrom the eigenstate-eigenvalue link that the position observable of the L-apparatus has no definite value at the end of the measurement. But if theposition observable having a definite value is indeed required in order forthe L-apparatus to have a definite location, then the pointer will point toneither ‘up’ nor ‘down,’ and the GRW collapse models will fail toreproduce the classical-like behavior of such systems.[11]

In later models, GRW proposed to interpret the quantum state as a densityof mass and they postulated that if almost all the density of mass of asystem is concentrated in a certain region, then the system is located in

Joseph Berkovitz


system is concentrated in a certain region, then the system is located inthat region. Accordingly, pointers of measurement apparatuses do havedefinite positions at the end of measurement interactions. Yet, thissolution has also given rise to a debate (see Albert and Loewer 1995,Lewis 1997, 2003a, 2004, Ghirardi and Bassi 1999, Bassi and Ghirardi1999, 2001, Clifton and Monton 1999, 2000, Frigg 2003, and Parker2003).

The exact details of the collapse mechanism and its characteristics in theGRW/Pearle models have no significant implications for the type of non-separability and holism they postulate—all these models basicallypostulate the same kinds of non-separability and holism as orthodoxquantum mechanics (see section 5.1.1). And action at a distance betweenthe L- and the R-wing will occur if the L-measurement interaction, asupposedly local event in the L-wing, causes some local events in the R-wing, such as the event of the pointer of the measurement apparatuscoming to possess a definite measurement outcome 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 definite z-spinand this in turn causes the pointer of the R-apparatus to come to possessthe corresponding measurement outcome in the R-measurement.Furthermore, if the L-measurement causes the R-particle to come topossess (momentarily) a definite position in the R-wing, then the action ata distance between the L- and the R-wing will occur independently ofwhether the R-particle undergoes a spin measurement.

The above discussion is based on an intuitive notion of action at adistance and it presupposes that action at a distance is compatible withnon-separability and holism. In the next section we shall provide moreprecise characterizations of action at a distance and in light of thesecharacterizations 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 andholism?

The action at a distance in the GRW/Pearle models is different from theNewtonian action at a distance in various respects. First, in contrast toNewtonian action at a distance, this action is independent of the distancebetween the measurement events. Second, while Newtonian action issymmetric, 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 on which measurement occurs first (the action will besymmetric when both measurements occur simultaneously). Third (andmore important to our consideration), in contrast to Newtonian action at adistance, before the end of the L-measurement the state of the L-apparatus and the R-particle is not separable and accordingly it is notclear that the influence is between separate existences, as the case issupposed to be in Newtonian gravity.

This non-separability of the states of the particle pair and the L-measurement apparatus, and more generally the fact that the non-localityin collapse theories is due to state non-separability, has led a number ofphilosophers and physicists to think that wave collapses do not involveaction 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 on how we interpret the term ‘action at a distance.’ And, as I willsuggest below, on a natural reading of Isaac Newton's and SamuelClarke's comments concerning action at a distance, there may be apeaceful coexistence between action at a distance and non-separabilityand holism.

Newton famously struggled to find out the cause of gravity.[12] In a letterto Bentley, dated January 17 1692/3, he said:

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

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

And he added:

You sometimes speak of Gravity as essential and inherent toMatter. Pray do not ascribe that Notion to me, for the Cause ofGravity is what I do not pretend to know, and therefore would takemore Time to consider it. (Cohen 1978, p. 298)

It is inconceivable that inanimate Matter should, without theMediation of something else, which is not material, operate upon,and affect other matter without mutual Contact…That Gravityshould be innate, inherent and essential to Matter, so that one bodymay act upon another at a distance thro’ a Vacuum, without theMediation of any thing else, by and through which their Actionand Force may be conveyed from one to another, is to me so greatan Absurdity that I believe no Man who has in philosophicalMatters a competent Faculty of thinking can ever fall into it.Gravity must be caused by an Agent acting constantly according tocertain laws; but whether this Agent be material or immaterial, Ihave left to the Consideration of my readers. (Cohen 1978, pp.302-3)

That one body attracts another without any intermediate means, isindeed not a miracle but a contradiction; for 'tis supposingsomething to act where it is not. But the means by which twobodies attract each other, may be invisible and intangible and of adifferent nature from mechanism …



Newton's and Clarke's comments suggest that for them gravity was a law-governed phenomenon, i.e., a phenomenon in which objects influenceeach other at a distance according to the Newtonian law of gravity, andthat this influence is due to some means which may be invisible andintangible and of a different nature from mechanism. On this conceptionof action at a distance, there seems to be no reason to exclude thepossibility of action at a distance in the quantum realm even if that realmis holistic or the state of the relevant systems is non-separable. That is,action at a distance may be characterized as follows:

We may alternatively characterize action at a distance in a more liberalway:

And while Newton and Clarke did not have an explanation for the actionat a distance involved in Newtonian gravity, on the abovecharacterizations action at a distance in the quantum realm would be

That this phenomenon is not produced sans moyen, that is withouta cause capable of producing such an effect, is undoubtedly true.Philosophers therefore can search after and discover that cause, ifthey can; be it mechanical or not. But if they cannot discover thecause, is therefore the effect itself, the phenomenon, or the matterof fact discovered by experience … ever the less true?

Action at a distance is a phenomenon in which a change inintrinsic properties of one system induces a change in the intrinsicproperties of a distant system, independently of the influence ofany other systems on the distant system, and without there being aprocess that carries this influence contiguously in space and time.

Action* at a distance is a phenomenon in which a change inintrinsic properties of one system induces a change in the intrinsicproperties of a distant system without there being a process thatcarries this influence contiguously in space and time.

Joseph Berkovitz


characterizations action at a distance in the quantum realm would beexplained by the holistic nature of the quantum realm and/or non-separability of the states of the systems involved. In particular, if in theEPR/B experiment the L-apparatus pointer has a definite position beforethe L-measurement and the R-particle temporarily comes to possessdefinite position during the L-measurement, then the GRW/Pearle modelsinvolve action at a distance and thus also action* at a distance. On theother hand, if the R-particle never comes to possess a definite positionduring the L-measurement, then the GRW/Pearle models only involveaction* at a distance.

5.3 No-collapse theories

5.3.1 Bohm's theory

In 1952, David Bohm proposed a deterministic, ‘hidden variables’quantum theory that reproduces all the observable predictions of orthodoxquantum mechanics (see Bohm 1952, Bohm, Schiller and Tiomno 1955,Bell 1982, Dewdney, Holland and Kyprianidis 1987, Dürr, Goldstein andZanghì 1992a, 1997, Albert 1992, Valentini 1992, Bohm and Hiley 1993,Holland 1993, Cushing 1994, and Cushing, Fine and Goldstein 1996; foran entry level review, see the entry on Bohmian mechanics and Albert1992, chapter 5).

In contrast to orthodox quantum mechanics and the GRW/Pearle collapsemodels, in Bohm's theory wave functions always evolve according to theSchrödinger equation, and thus they never collapse. Wave functions donot represent the states of systems. Rather, they are states of a ‘quantumfield (on configuration space)’ that influences the states of systems.[14]

Also, particles always have definite positions, and the positions of theparticles and their wave function at a certain time jointly determine thetrajectories of the particles at all future times. Thus, particles’ positionsand their wave function determine the outcomes of any measurements (so



and their wave function determine the outcomes of any measurements (solong as these outcomes are recorded in the positions of some physicalsystems, as in any practical measurements).

There are various versions of Bohm's theory. In the ‘minimal’ Bohmtheory, formulated by Bell (1982),[15] the wave function is interpreted asa ‘guiding’ field (which has no source or any dependence on the particles)that deterministically governs the trajectories of the particles according tothe so-called ‘guiding equation’ (which expresses the velocities of theparticles in terms of the wave function).[16] The states of systems areseparable (the state of any composite system is completely determined bythe state of its subsystems), and they are completely specified by theparticles’ positions. Spins, and any other properties which are not directlyderived from positions, are not intrinsic properties of systems. Rather,they are relational properties that are determined by the systems’ positionsand the guiding field. In particular, each of the particles in the EPR/Bexperiment has dispositions to ‘spin’ in various directions, and thesedispositions are relational properties of the particles— they are (generally)determined by the guiding field and the positions of the particles relativeto the measurement apparatuses and to each other.

Figure 4. The EPR/B experiment with Stern-Gerlach

Joseph Berkovitz


To see the nature of non-locality postulated by the minimal Bohm theory,consider again the EPR/B experiment and suppose that the measurementapparatuses are Stern-Gerlach (S-G) magnets which are prepared tomeasure z-spin. In any run of the experiment, the measurement outcomeswill depend on the initial positions of the particles and the order of themeasurements. Here is why. In the minimal Bohm theory, the spin singletstate denotes the relevant state of the guiding field rather than the intrinsicproperties of the particle pair. If the L-measurement occurs before the R-measurement, the guiding field and the position of the L-particle at theemission time jointly determine the disposition of the L-particle toemerge 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-particle z-spins ‘up’ (‘down’) about the z-axis and the L-apparatus‘pointer’ points to ‘up’ (‘down’) (see Fig. 4 above). All the L-particlesthat are emitted above the center plane aligned orthogonally to the z-direction, like the L-particles 1-3, will be disposed to spin ‘up’; and allthe particles that are emitted below this plane, like the L-particles 4-6,

Figure 4. The EPR/B experiment with Stern-Gerlachmeasurement devices. Stern-Gerlach 1 is on, set up to measure thez-spin of the L-particle, and Stern-Gerlach 2 is off. The horizontallines in the left-hand-side denote the trajectories of six L-particlesin the spin singlet state after an (impulsive) z-spin measurementon the L-particle, and the horizontal lines in the right-hand-sidedenote the trajectories of the corresponding R-particles. The centerplane is aligned orthogonally to the z-axis, so that particles thatemerge above this plane correspond to z-spin ‘up’ outcome andparticles that emerge below this plane correspond to z-spin ‘down’outcome. The little arrows denote the z-spin components of theparticles in the ‘non-minimal’ Bohm theory (where spins areintrinsic properties of particles), and are irrelevant for the‘minimal’ Bohm theory (where spins are not intrinsic properties ofparticles).



the particles that are emitted below this plane, like the L-particles 4-6,will be disposed to spin ‘down.’ Similarly, if the R-measurement occursbefore the L-measurement, the guiding field and the position of the R-particle at the emission time jointly determine the disposition of the R-particle to emerge either above the z-axis (i.e., to z-spin ‘up’) or below thez-axis (i.e., to z-spin ‘down’) according to whether it is above or belowthe center plane, independently of the position of the L-particle along thez-axis.

But the z-spin disposition of the R-particle changes immediately after an(earlier) z-spin measurement on the L-particle: The R-particles 1-3 (seeFig. 4), which were previously disposed to z-spin ‘up,’ will now bedisposed to z-spin ‘down,’ i.e., to emerge below the center plane alignedorthogonally to the z-axis; and the R-particles 4-6, which were previouslydisposed to z-spin ‘down,’ will now be disposed to z-spin ‘up,’ i.e., toemerge above this center plane. Yet, the L-measurement per se does nothave any immediate influence on the state of the R-particle: The L-measurement does not influence the position of the R-particle or anyother property that is directly derived from this position. It only changesthe guiding field, and thus grounds new spin dispositions for the R-particle. But these dispositions are not intrinsic properties of the R-particle. Rather, they are relational properties of the R-particle, which aregrounded in the positions of both particles and the state of the guidingfield.[17] (Note that in the particular case in which the L-particle isemitted above the center plane aligned orthogonally to the z-axis and theR-particle is emitted below that plane, an earlier z-spin on the L-particlewill have no influence on the outcome of a z-spin on the R-particle.)

While there is no contiguous process to carry the influence of the L-measurement outcome on events in the R-wing, the question of whetherthis influence amounts to action at a distance depends on the exactcharacterization of this term. In contrast to the GRW/Pearle collapsemodels, the influence of the L-measurement outcome on the intrinsic

Joseph Berkovitz


models, the influence of the L-measurement outcome on the intrinsicproperties of the R-particle is dependent on the R-measurement: Beforethis measurement occurs, there are no changes in the R-particle's intrinsicproperties. Yet, the influence of the L-measurement on the R-particle is ata distance. Thus, the EPR/B experiment as depicted by the minimal Bohmtheory involves action* at a distance but not action at a distance.

Bohm's theory portrays the quantum realm as deterministic. Thus, thesingle-case objective probabilities, i.e., the chances, it assigns toindividual spin-measurement outcomes in the EPR/B experiment aredifferent from the corresponding quantum-mechanical probabilities. Inparticular, while in quantum mechanics the chances of the outcomes ‘up’and ‘down’ in an earlier L- (R-) spin measurement are both ½, in Bohm'stheory these chances are either one or zero. Yet, Bohm's theory postulatesa certain distribution, the so-called ‘quantum-equilibrium distribution,’over all the possible positions of pairs with the same guiding field. Thisdistribution is computed from the quantum-mechanical wave function,and it is typically interpreted as ignorance over the actual position of thepair; an ignorance that may be motivated by dynamical considerations andstatistical patterns exhibited by ensembles of pairs with the same wavefunction (for more details, see the entry on bohmian mechanics, section9). And the sum-average (or more generally the integration) over thisdistribution reproduces all the quantum-mechanical observablepredictions.

What is the status of this probability postulate? Is it a law of nature or acontingent fact (if it 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, Dürr, Goldstein and Zanghì 1992a,b, 1996,fn. 15, and Callender 2006).

Turning to the question of non-separability, the minimal Bohm theorydoes not involve state non-separability. For recall that in this theory the



does not involve state non-separability. For recall that in this theory thestate of a system does not consist in its wave function, but rather in thesystem's position, and the position of a composite system alwaysfactorizes into the positions of its subsystems. Here, the non-separabilityof the wave function reflects the state of the guiding field. This statepropagates not in ordinary three-space but in configuration space, whereeach point specifies the configuration of both particles. The guiding fieldof the particle pair cannot be factorized into the guiding field that governsthe trajectory of the L-particle and the guiding field that governs thetrajectory of the R-particle. The evolution of the particles’ trajectories,properties and dispositions is non-separable, and accordingly theparticles’ trajectories, properties and dispositions are correlated evenwhen the particles are far away from each other and do not interact witheach other. Thus, process separability fails.

In the non-minimal Bohm theory[18], the behavior of an N-particle systemis determined by its wave function and the intrinsic properties of theparticles. But, in contrast to the minimal theory, in the non-minimaltheory spins are intrinsic properties of particles. The wave functionalways evolves according to the Schrödinger equation, and it isinterpreted as a ‘quantum field’ (which has no sources or any dependenceon the particles). The quantum field guides the particles via the ‘quantumpotential,’ an entity which is determined from the quantum field, and theevolution of properties is fully deterministic.[19]

Like in the minimal Bohm theory, the non-separability of the wavefunction in the EPR/B experiment dictates that the evolution of theparticles’ trajectories, properties and dispositions is non-separable, but thebehavior of the particles is somewhat different. In the earlier z-spinmeasurement on the L-particle, the quantum potential continuouslychanges, and this change induces an immediate change in the z-spin of theR-particle. If the L-particle starts to spin ‘up’ (‘down’) in the z-direction,the R-particle will start to spin ‘down’ (‘up’) in the same direction (see

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the R-particle will start to spin ‘down’ (‘up’) in the same direction (seethe little arrows in Fig. 4).[20] Accordingly, the L-measurement inducesinstantaneous action at a distance between the L- and the R-wing. Yet,similarly to the minimal Bohm theory, while the disposition of the R-particle to emerge above or below the center plane aligned orthogonally tothe z-direction in a z-spin measurement may change instantaneously, theactual trajectory of the R-particle along the z-direction does not changebefore the measurement of the R-particle's z-spin occurs. Only during theR-measurement, the spin and the position of the R-particle get correlatedand the R-particle's trajectory along the z-direction is dictated by thevalue of its (intrinsic) z-spin.

Various objections have been raised against Bohm's theory (for a detailedlist and replies, see the entry on Bohmian mechanics, section 15). Onemain objection is that in Bohmian mechanics, the guiding field influencesthe particles, but the particles do not influence the guiding field. Anothercommon objection is that the theory is involved with a radical type ofnon-locality, and that this type of non-locality is incompatible withrelativity. While it may be very difficult, or even impossible, to reconcileBohm's theory with relativity, as is not difficult to see from the abovediscussion, the type of non-locality that the minimal Bohm theorypostulates in the EPR/B experiment does not seem more radical than thenon-locality postulated by the orthodox interpretation and theGRW/Pearle collapse models.

5.3.2 Modal interpretations

Modal interpretations of quantum mechanics were designed to solve themeasurement problem and to reconcile quantum mechanics withrelativity. They are no-collapse, (typically) indeterministic hidden-variables theories. Quantum-mechanical states of systems (which may beconstrued as denoting their states or information about these states)always evolve according to unitary and linear dynamical equations (the



always evolve according to unitary and linear dynamical equations (theSchrödinger equation in the non-relativistic case). And the orthodoxquantum-mechanical state description of systems is supplemented by a setof properties, which depends on the quantum-mechanical state and whichis supposed to be rich enough to account for the occurrence of definitemacroscopic events and their classical-like behavior, but sufficientlyrestricted to escape all the known no-hidden-variables theorems. (Formodal interpretations, 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, Clifton 1995, Bacciagaluppi 1996,Bacciagaluppi and Hemmo 1996, Bub and Clifton 1996, Hemmo 1996b,Bacciagaluppi and Dickson 1999, Clifton 2000, Spekkens and Sipe2001a,b, Bene and Dieks 2002, and Berkovitz and Hemmo 2006a,b. Foran entry-level review, see the entry on modal interpretations of quantumtheory. For comprehensive reviews and analyses of modal interpretations,see Bacciagaluppi 1996, Hemmo 1996a, chapters 1-3, Dieks and Vermaas1998, Vermaas 1999, and the entry on modal interpretations of quantumtheory. For the no-hidden-variables theorems, see Kochen and Specker1967, Greenberger, Horne and Zeilinger 1989, Mermin 1990 and the entryon the Kochen-Specker theorem.)[21]

Modal interpretations vary in their property assignment. For simplicity,we shall focus on modal interpretations in which the property assignmentis based on the so-called Schmidt biorthogonal-decomposition theorem(see Kochen 1985, Dieks 1989, and Healey 1989). Let S1 and S2 besystems associated with the Hilbert spaces HS1 and HS2, respectively.There exist bases {|αi>} and {|βi>} for HS1 and HS2 respectively suchthat the state of S1+S2 can be expressed as a linear combination of thefollowing form of vectors from these bases:

|ψ8 >S1+S2 = ∑i ci |α i>S1 |βi>S2.

Joseph Berkovitz


When the absolute values of the coefficients ci are all unequal, the bases{|αi>} and {|βi>} and the above decomposition of |ψ8 >S1+S2 are unique.In that case, it is postulated that S1 has a determinate value for eachobservable associated with HS1 with the basis {|αi>} and S2 has adeterminate value for each observable associated with HS2 with the basis{|βi>}, and |ci|2 provide the (ignorance) probabilities of the possiblevalues that these observables may have.[22] For example, suppose that thestate of the L- and the R-particle in the EPR/B experiment before themeasurements is:

|ψ9> = (1/√2+ε) |z-up>L| z-down>R − (1/√2-ε′) |z-down>L| z-up>R

where 1/√2 >> ε,ε′, (1/√2+ε)2+(1/√2-ε′)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 having z-spin ‘up’ and z-spin ‘down’, respectively.[23] Then,either the L-particle spins ‘up’ and the R-particle spins ‘down’ in the z-direction, or the L-particle spins ‘down’ and the R-particle spins ‘up’ inthe z-direction. Thus, in contrast to the orthodox interpretation and theGRW/Pearle collapse models, in modal interpretations the particles in theEPR/B experiment may have definite spin properties even before anymeasurement occurs.

To see how the modal interpretation accounts for the curious correlationsin EPR/B-type experiments, let us suppose that the state of the particlepair and the measurement apparatuses at the emission time is:

|ψ10> = ((1/√2+ε) |z-up>L |z-down>R − (1/√2−ε′) |z-down>L |z-up>R) |ready>AL|ready>AR

where |ready>AL (|ready>AR) denotes the state of the L-apparatus (R-apparatus) being ready to measure z-spin. In this state, the L- and the R-apparatus are in the definite state of being ready to measure z-spin, and(similarly to the state |ψ9>) the L- and the R-particle have definite z-spin



(similarly to the state |ψ9>) the L- and the R-particle have definite z-spinproperties: Either the L-particle has z-spin ‘up’ and the R-particle has z-spin ‘down,’ or the L-particle has z-spin ‘down’ and the R-particle has z-spin ‘up,’[24] where the probability of the realization of each of thesepossibilities is approximately 1/2. In the (earlier) z-spin measurement onthe L-particle, the state of the particle pair and the apparatuses evolves tothe state:

where (as before) |up>AL and |down>AL denote the states of the L-apparatus pointing to the outcomes z-spin ‘up’ and z-spin ‘down’,respectively. In this state, either the L-particle has a z-spin ‘up’ and the L-apparatus points to ‘up,’ or the L-particle has z-spin ‘down’ and the L-apparatus points to ‘down.’ And, again, the probability of each of thesepossibilities is approximately 1/2. The evolution of the properties fromthe state |ψ10> to the state |ψ11> depends on the dynamical laws. Inalmost all modal interpretations, if the particles have definite z-spinproperties before the measurements, the outcomes of z-spin measurementswill reflect these properties. That is, the evolution of the properties of theparticles and the measurement apparatuses will be deterministic, so thatthe spin properties of the particles do not change in the L-measurementand the pointer of the L-apparatus comes to display the outcome thatcorresponds to the z-spin property that the L-particle had before themeasurement. If, for example, before the measurements the L- and the R-particle have respectively the properties z-spin ‘up’ and z-spin ‘down’, the(earlier) z-spin measurement on the L-particle will yield the outcome ‘up’and the spin properties of the particles will remain unchanged.Accordingly, a z-spin measurement on the R-particle will yield theoutcome ‘down’. Thus, in this case the modal interpretation involvesneither action at a distance nor action* at a distance.

|ψ11> = ((1/√2+ε) |z-up>L|up>AL| z-down>R − (1/√2-ε′) |z-down>L|down>AL| z-up>R) |ready>AR

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However, if the measurement apparatuses are set up to measure x-spinrather than z-spin, the evolution of the properties of the L-particle and theL-apparatus will be indeterministic. As before, the L-measurement willnot cause any change in the actual spin properties of the R-particle. Butthe L-measurement outcome will cause an instant change in the spindispositions of the R-particle and the R-measurement apparatus. If, forexample, the L-measurement outcome is x-spin ‘up’ and the L-particlecomes to posses x-spin ‘up,’ then the R-particle and the R-apparatus willhave respectively the dispositions to possess x-spin ‘down’ and to displaythe outcome x-spin ‘down’ on a x-spin measurement. Thus, like theminimal Bohm theory, the modal interpretation may involve action* at adistance 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: Theproperties of composite systems are not decomposable into the propertiesof their subsystems. Consider, again, the state |ψ10>. As ‘separated’systems (i.e., in the decompositions of the composite system of theparticle pair+apparatuses into the L-particle and the R-particle+apparatuses and into the R-particle and the L-particle+apparatuses) the L- and the R-particle have definite z-spinproperties. But, as subsystems of the composite system of the particle pair(e.g. in the decomposition of the composite system of the particlepair+apparatuses into the particle pair and the apparatuses), they have nodefinite z-spin properties.

A failure of property composition occurs also in the state |ψ11>, wherethe L- and the R-particle have definite z-spin properties both as‘separated’ systems and as subsystems of the particle pair (though incontrast with |ψ10>, in |ψ11> the range of the possible properties of theparticles as separated systems and as subsystems of the pair is the same).For nothing in the above property assignment implies that in |ψ11> the



For nothing in the above property assignment implies that in |ψ11> thespin properties that the L-particle has as a ‘separated’ system and the spinproperties that it has as a subsystem of the particle pair be the same: TheL-particle may have z-spin ‘up’ as a separated system and z-spin ‘down’as a subsystem of the particle pair.

Furthermore, the dynamics of the properties that the L-particle (R-particle) has as a separated system and the dynamics of its properties as asubsystem of the particle pair are generally different.[25] Consider, again,the state |ψ10>. In the (earlier) z-spin measurement on the L-particle, thespin properties that the L-particle has as a separated system follow adeterministic evolution — the L-particle has either z-spin ‘up’ or z-spin‘down’ before and after the L-measurement; whereas as a subsystem ofthe particle pair, the spin properties of the L-particle follow anindeterministic evolution — the L-particle has no definite spin propertiesbefore the L-measurement and either z-spin ‘up’ (with approximatelychance ½) or z-spin ‘down’ (with approximately chance ½) after the L-measurement.

The failure of property composition implies that the quantum realm asdepicted by the above version of the modal interpretation involves statenon-separability and property and relational holism. State separabilityfails because the state of the particle pair is not generally determined bythe separate states of the particles. Indeed, as is easily shown, the actualproperties that the L- and the R-particle each has in the state |ψ9> arealso compatible with product states in which the L- and the R-particle arenot entangled. Property and relational holism fail because in the state |ψ9> the properties of the pair do not supervene upon the properties of itssubsystems and the spatiotemporal relations between them. Furthermore,process separability fails for similar reasons.

The failure of property composition in the modal interpretation calls forexplanation. It may be tempting to postulate that the properties that asystem (e.g. the L-particle) has, as a separated system, are the same as the

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system (e.g. the L-particle) has, as a separated system, are the same as theproperties that it has as a subsystem of composite systems. But, asBacciagaluppi (1995) and Clifton (1996a) have shown, such propertyassignment will be inconsistent: It will be subject to a Kochen andSpecker-type contradiction. Furthermore, as Vermaas (1997)demonstrates, the properties of composite systems and the properties oftheir subsystems cannot be correlated (in ways compatible with the Bornrule).

For what follows in the rest of this subsection, the views of differentauthors differ widely. Several variants of modal interpretations weredeveloped in order to fix the problem of the failure of propertycomposition. The most natural explanation of the failure of propertycomposition is that quantum states assign relational rather than intrinsicproperties to systems (see Kochen 1985, Bene and Dieks 2002, andBerkovitz and Hemmo 2006a,b). For example, in the relational modalinterpretation proposed by Berkovitz and Hemmo (2006a,b), the mainidea is that quantum states assign properties to systems only relative toother systems, and properties of a system that are related to differentsystems are generally different. In particular, in the state |ψ10> the L-particle has a definite z-spin property relative to the R-particle, themeasurement apparatuses and the rest of the universe, but (as a subsystemof the particle pair) it has no definite z-spin relative to the measurementapparatuses and the rest of universe.[26] On this interpretation, theproperties of systems are highly non-local by their very nature. Propertieslike pointing to ‘up’ and pointing to ‘down’ are not intrinsic to themeasurement apparatuses. Rather, they are relations between theapparatuses and other systems. For example, the property of the L-apparatus pointing to ‘up’ relative to the particle pair, the R-apparatus andthe rest of the universe is not intrinsic to the L-apparatus; it is a relationbetween the L-apparatus and the particle pair, the R-apparatus and therest of the universe. As such, this property is highly non-local: It islocated in neither the L-wing nor any other subregion of the universe.



located in neither the L-wing nor any other subregion of the universe.Yet, due to the dynamical laws, properties like the position of pointers ofmeasurement apparatuses, which appear to us to be local, behave likelocal properties in any experimental circumstances, and accordingly thisradical type of non-locality is unobservable (for more details, seeBerkovitz and Hemmo 2006b, sections 8.1 and 9).

Another way to try to explain the failure of property composition is tointerpret the properties of composite systems as holistic, non-decomposable properties. On this interpretation, the z-spin ‘up’ propertythat the L-particle has as a subsystem of the particle pair in the state |ψ9>is completely different from the z-spin ‘up’ property that the L-particlehas as a separated system, and the use of the term ‘z-spin up’ in bothcases is misleading (for more details, see Berkovitz and Hemmo2006a).[27]

The relational and holistic interpretations of properties mark a radicalshift from the standard interpretation of properties in orthodox quantummechanics. Other advocates of the modal interpretation have chosen notto follow this interpretation, and opted for a modal interpretation that doesnot violate property composition. While the property assignment abovedoes not assume any preferred partition of the universe (the partition ofthe universe into a particle pair and the rest of the universe is as good asthe partition of the universe into the L-particle and the rest of theuniverse), proponents of property composition postulated that there is apreferred partition of the universe into ‘atomic’ systems and accordingly apreferred factorization of the Hilbert space of the universe. This preferredfactorization is supposed to be the basis for the ‘core’ propertyassignment: Properties are prescribed to atomic systems according to aproperty assignment that is a generalization of the bi-orthogonaldecomposition property assignment.[28] And the properties of complexsystems are postulated to be compositions of the properties of their atomicsystems (see the entry on modal interpretations of quantum theory,

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systems (see the entry on modal interpretations of quantum theory,section 2, and Bacciagaluppi and Dickson 1999). The challenge for thisatomic modal interpretation is to justify the assumption that there is apreferred partition of the universe, and to provide some idea about howsuch factorization should look like.

Finally, while the modal interpretation was designed to solve themeasurement problem and reconcile quantum mechanics with specialrelativity, it faces challenges on both accounts. First, in certain imprefectmeasurements (where there are imprefections in the coupling between themeasured system and the pointer of the measurement apparatus and/or thepointer and the environment), modal interpretations that are based on theSchmidt biorthogonal-decomposition theorem (and more generally thespectral decomposition theorem) fail to account for definite measurementoutcomes, in contradiction to our experience (see Bacciagaluppi andHemmo 1996 and Bacciagaluppi 2000). For versions of the modalinterpretations that seem to escape this problem, see Van Fraassen (1973,1991), Bub (1992, 1997), Bene and Dieks (2002) and Berkovitz andHemmo (2006a,b). Second, as we shall see in section 10.2, a number ofno-go theorems challenge the view that modal interpretations could begenuinely relativistic.

5.3.3 Everett-like interpretations

In 1957, Everett proposed a new no-collapse interpretation of orthodoxquantum mechanics (see Everett 1957a,b, 1973, Barrett 1999, the entry onEverett's relative-state formulation of quantum mechanics, the entry onthe many-worlds interpretation of quantum mechanics, and referencestherein). The Everett interpretation is a no-collapse interpretation ofquantum mechanics, where the evolution of quantum states is alwaysaccording to unitary and linear dynamical equations (the Schrödingerequation in the non-relativistic case). In this interpretation, quantum statesare fundamentally relative. Systems have relative states, which are



are fundamentally relative. Systems have relative states, which arederivable from the various branches of the entangled states. For example,consider again |ψ11>.

In this quantum-mechanical state, the L-apparatus is in the state ofpointing to the outcome z-spin ‘up’ relative to the L-particle being in thestate z-spin ‘up,’ the R-particle being in the state z-spin ‘down’ and theR-apparatus being ready to measure z-spin; and in the state of pointing tothe outcome z-spin ‘down’ relative to the L-particle being in the state z-spin ‘down,’ the R-particle being in the state z-spin ‘up’ and the R-apparatus being ready to measure z-spin. Likewise, the L-particle is in thestate z-spin ‘up’ relative to the L-apparatus being in the state of pointingto the outcome z-spin ‘up,’ the R-particle being in the state z-spin ‘down’and the R-apparatus being ready to measure z-spin; and in the state z-spin‘down’ relative to the L-apparatus being in the state of pointing to theoutcome z-spin ‘down,’ the R-particle being in the state z-spin ‘up’ andthe R-apparatus being ready to measure z-spin. And similarly, mutatismutandis, for the relative state of the R-particle and the R-apparatus.

Everett's original formulation left the exact meaning of these relativestates and their relations to observers’ experience and beliefs open, andthere have been different Everett-like interpretations of these states.Probably the most popular reading of Everett is the splitting-worldsinterpretation (see DeWitt 1971, Everett's relative-state formulation ofquantum mechanics, Barrett 1999, and references therein). In thesplitting-worlds interpretation, each of the branches of the state |ψ11>refers to a different class of worlds (all of which are real) where the statesof the L-apparatus, R-apparatus and the particles are all separable: Class-1 worlds in which the L-particle is in the state z-spin ‘up,’ the R-particleis in the state z-spin ‘down,’ the L-apparatus is in the state of pointing to

|ψ11> = (1/√2+ε) |z-up>L|up>AL| z-down>R |ready>AR − (1/√2-ε′) |z-down>L|down>AL| z-up>R |ready>AR.

Joseph Berkovitz


is in the state z-spin ‘down,’ the L-apparatus is in the state of pointing tothe outcome z-spin ‘up’ and the R-apparatus in the state of being ready tomeasure z-spin; and class-2 worlds in which the L-particle is in the statez-spin ‘down,’ the R-particle is in the state z-spin ‘up,’ the L-apparatus isin the state of pointing to the outcome z-spin ‘down’ and the R-apparatusis in the state of being ready to measure z-spin. More generally, each termin state of the universe, as represented in a certain preferred basis, reflectsthe states of its systems in some class of worlds; where the range of thedifferent classes of worlds increases whenever the number of the terms inthe quantum state (in the preferred basis) increases (this process is called‘splitting’).

The splitting-worlds reading of Everett faces a number of challenges.First, supporters of the Everett interpretation frequently motivate theirinterpretation by arguing that it postulates the existence of neither acontroversial wave collapse nor hidden variables, and it leaves the simpleand elegant mathematical structure of quantum mechanics intact. But, thesplitting-worlds interpretation adds extra structure to no-collapse orthodoxquantum mechanics. Further, this interpretation marks a radical shift fromorthodox quantum mechanics. A scientific theory is not constituted onlyby its mathematical formalism, but also by the ontology it postulates, theway it depicts the physical realm and the way it accounts for ourexperience. The many parallel worlds ontology of the splitting-worldsinterpretation and its account of our experience are radically differentfrom the ontology of the intended interpretation of orthodox quantummechanics and its account for our experience. Second, relative states arewell defined in any basis, and the question arises as to which basis shouldbe preferred and the motivation for selecting one particular basis overothers. Third, in the splitting-worlds interpretation each of the worlds inthe universe may split into two or more worlds, and the problem is that(similarly to the collapse in orthodox collapse quantum mechanics) thereare no clear criteria for when a splitting occurs and how long it takes.Fourth, there is the question of how the splitting-worlds interpretation



Fourth, there is the question of how the splitting-worlds interpretationaccounts for the statistical predictions of the orthodox theory. In theEverett-like interpretations in general, and in the splitting-worldsinterpretation in particular, all the possible measurement outcomes in theEPR/B experiment are realized and may be observed. Thus, the questionarises as to the meaning of probabilities in this interpretation. Forexample, what is the meaning of the statement that in the state |ψ10> (seesection 5.3.2) the probability of the L-measurement apparatus pointing tothe outcome ‘up’ in an earlier z-spin measurement on the L-particle is(approximately) ½? In the splitting-worlds interpretation the probabilityof that outcome appears to be 1! Furthermore, setting aside the problem ofinterpretation, there is also the question of whether the splitting-worldsinterpretation, and more generally Everett-like interpretations, canaccount for the particular values of the quantum probabilities ofmeasurement outcomes. Everett claimed to derive the Born probabilitiesin the context of his interpretation. But this derivation has beencontroversial. (For discussions of the meaning of probabilities, or moreprecisely the meaning of the coefficients of the various terms in quantumstates, in Everett-like interpretations, see Butterfield 1996, Lockwood1996a,b, Saunders 1998, Vaidman 1998, Barnum et al. 2000,Bacciagaluppi 2002, Gill 2003, Hemmo and Pitowsky 2003, 2005,Wallace 2002, 2003, 2005a,b, Greaves 2004 and Saunders 2004, 2005.)

Other readings of Everett include the many-minds interpretation (Albertand Loewer 1988, Barrett 1999, chapter 7), the consistent-historiesapproach (Gell-Mann and Hartle 1990), the Everett-like relationalinterpretation (Saunders 1995, Mermin 1998) and (what may be called)the many-structures interpretation (Wallace 2005c). While these readingsaddress more or less successfully the problems of the preferred basis andsplitting, except for the many-minds interpretation of Albert and Loewerthe question of whether there could be a satisfactory interpretation ofprobabilities in the context of these theories and the adequacy of thederivation of the Born probabilities are still a controversial issue (see

Joseph Berkovitz


derivation of the Born probabilities are still a controversial issue (seeDeutsch 1999, Wallace 2002, 2003, Lewis 2003, Graves 2004, Saunders2004, Hemmo and Pitowsky 2005, and Price 2006).

What kind of non-locality do Everett-like interpretations involve?Unfortunately, the answer to this question is not straightforward, as itdepends on one's particular reading of the Everett interpretation. Indeed,all the above readings of Everett seem to treat the no-collapse wavefunction of the universe as a real physical entity that reflects the non-separable state of the universe, and accordingly they involve state non-separability. But, one may reasonably expect that different readings depictdifferent pictures of physical reality and accordingly might postulatedifferent kinds of non-locality. Thus, any further analysis of the type ofnon-locality postulated by each of these readings requires a detailed studyof their ontology (which we plan to conduct in future updates of thisentry).

For example, the question of action at a distance in the EPR/B experimentmay arise in the context of the splitting-worlds interpretation, but not inthe context of Albert and Loewer's many-minds interpretation. Albert andLoewer's interpretation takes the bare no-collapse orthodox quantummechanics to be the complete theory of the physical realm. Accordingly,the L-apparatus in the state |ψ11> does not display any definite outcome.Yet, in order to account for our experience of a classical-like world,where at the end of measurements observers are typically in mental statesof perceiving definite outcomes, the many-minds interpretation appeals toa dualism of mind-body. Each observer is associated with a continuousinfinity of non-physical minds. And while the physical state of the worldevolves in a completely deterministic manner according to theSchrödinger evolution, and the pointers of the measurement apparatusesin the EPR/B experiment display no definite outcomes, states of mindsevolve in a genuinely indeterministic fashion so as to yield an experienceof perceiving definite measurement outcomes. For example, consider



of perceiving definite measurement outcomes. For example, consideragain, the state |ψ10>. While in a first z-spin L-measurement, this stateevolves deterministically into the state |ψ11>, minds of observers evolveindeterministically into either the state of perceiving the outcome z-spin‘up’ or the state of perceiving the outcome z-spin ‘down’ with the usualBorn-rule probabilities (approximately 50% chance for each of theseoutcomes). Since in this state the L-particle has no definite spin propertiesand the L-apparatus points to no definite measurement outcome, and sincein the later z-spin measurement on the R-particle the R-particle does notcome to possess any definite spin properties and the R-apparatus points tono definite spin outcome, the question of whether there is action at adistance between the L-particle and the L-apparatus on the one hand andthe R-particle and the R-apparatus on the other does not arise.

6. Superluminal causation

In all the above interpretations of quantum mechanics, the failure offactorizability (i.e., the failure of the joint probability of the measurementoutcomes in the EPR/B experiment to factorize into their singleprobabilities) involves non-separability, holism and/or some type ofaction at a distance. As we shall see below, non-factorizability alsoimplies superluminal causal dependence according to certain accounts ofcausation.

First, as is not difficult to show, the failure of factorizability impliessuperluminal causation according to various probabilistic accounts ofcausation that satisfy Reichenbach's (1956, section 19) principle of thecommon cause (for a review of this principle, see the entry onReichenbach's principle of the common cause).

Here is why. Reichenbach's principle may be formulated as follows:

Joseph Berkovitz


The above formulation of PCC is mainly intended to cover cases in whichx and y have no partial, non-common causes. But PCC can be generalizedas follows:

PCC (Principle of the Common Cause). For any correlationbetween two (distinct) events which do not cause each other, thereis a common cause that screens them off from each other. Orformally: If distinct events x and y are correlated, i.e.,

and they do not cause each other, then their common cause,CC(x,y), screens them off from each other, i.e.,

Accordingly, CC(x,y) renders x and y probabilisticallyindependent, and the joint probability of x and y factorizes uponCC(x,y):

(Correlation) P(x & y) ≠ P(x) · P(y),

(Screening Off)

PCC(x,y)(x/y) = PCC(x,y)(x) PCC(x,y)(y) ≠ 0

PCC(x,y)(y/x) = PCC(x,y)(y) PCC(x,y)(x) ≠ 0.[29]

PCC(x,y)(x & y) = PCC(x,y)(x) · PCC(x,y)(y).

PCC*. The joint probability of any distinct, correlated events, xand y, which are not causally connected to each other, factorizesupon the union of their partial (separate) causes and their commoncause. That is, let CC(x,y) denote the common causes of x and y,and PC(x) and PC(y) denote respectively their partial causes.Then, the joint probability of x and y factorizes upon the Union oftheir Causal Pasts (henceforth, FactorUCP), i.e., on the union ofPC(x), PC(y) and CC(x,y):



Like PCC, the basic idea of FactorUCP is that the objective probabilitiesof events that do not cause each other are determined by their causalpasts, and given these causal pasts they are probabilistically independentof each other. As is not difficult to see, factorizability is a special case ofFactorUCP. That is, to obtain factorizability from FactorUCP, substitute λfor CC(x,y), l for PC(x) and r for PC(y). FactorUCP and the assumptionthat the probabilities of the measurement-outcomes in the EPR/Bexperiment are determined by the pair's state and the settings of themeasurement apparatuses jointly imply factorizability. Thus, given thislater assumption, the failure of factorizability implies superluminalcausation between the distant outcomes in the EPR/B experimentaccording to any account of causation that satisfies FactorUCP (for someexamples of such accounts, see Butterfield 1989 and Berkovitz 1995a,1995b, section 6.7, 1998b).[30]

Superluminal causation between the distant outcomes also existsaccording to various counterfactual accounts of causation, includingaccounts that do not satisfy FactorUCP. In particular, in Lewis's (1986)influential account, counterfactual dependence between distinct eventsimplies causal dependence between them. And as Butterfield (1992b) andBerkovitz (1998b) demonstrate, the violation of Factorizability involves acounterfactual dependence between the distant measurement outcomes inthe EPR/B experiment.

But the violation of factorizability does not imply superluminal causationaccording to some other accounts of causation. In particular, in processaccounts of causation there is no superluminal causation in the EPR/Bexperiment. In such accounts, causal dependence between events isexplicated in terms of continuous processes in space and time that

PC(x), PC(y) and CC(x,y):

FactorUCP PPC(x) PC(y) CC(x,y) (x & y) = PPC(x) CC(x,y) (x) · PPC(y) CC(x,y) (y).

Joseph Berkovitz


explicated in terms of continuous processes in space and time thattransmit ‘marks’ or conserved quantities from the cause to the effect (seeSalmon 1998, chapters 1, 12, 16 and 18, Dowe 2000, the entry on causalprocesses, and references therein). Thus, recalling (sections 1, 2, 4 and 5)that none of the interpretations of quantum mechanics and alternativequantum theories postulates any (direct) continuous process between thedistant measurement events in the EPR/B experiment, there is nosuperluminal causation between them according to process accounts ofcausation.

7. Superluminal signaling

Whether or not the non-locality predicted by quantum theories may beclassified as action at a distance or superluminal causation, the questionarises as to whether this non-locality could be exploited to allowsuperluminal (i.e., faster-than-light) signaling of information. Thisquestion is of particular importance for those who interpret relativity asprohibiting any such superluminal signaling. (We shall return to discussthis interpretation in section 10.)

Superluminal signaling would require that the state of nearby controllablephysical objects (say, a keyboard in my computer) superluminallyinfluence distant observable physical phenomena (e.g. a pattern on acomputer screen light years away). The influence may be deterministic orindeterministic, but in any case it should cause a detectable change in thestatistics of some distant physical quantities.

It is commonly agreed that in quantum phenomena, superluminalsignaling is impossible in practice. Moreover, many believe that suchsignaling is excluded in principle by the so-called ‘no-signaling theorem’(for proofs of this theorem, see Eberhard 1978, Ghirardi, Rimini andWeber 1980, Jordan 1983, Shimony 1984, Redhead 1987, pp. 113-116and 118). It is thus frequently claimed with respect to EPR/B experiments



and 118). It is thus frequently claimed with respect to EPR/B experimentsthat there is no such thing as a Bell telephone, namely a telephone thatcould exploit the violation of the Bell inequalities for superluminalsignaling of information.[31]

The no-signaling theorem demonstrates that orthodox quantum mechanicsexcludes any possibility of superluminal signaling in the EPR/Bexperiment. According to this theory, no controllable physical factor inthe L-wing, such as the setting of the L-measurement apparatus, can takeadvantage of the entanglement between the systems in the L- and the R-wing to influence the statistics of the measurement outcomes (or anyother observable) in the R-wing. As we have seen in section 5.1.1, theorthodox theory is at best incomplete. Thus, the fact that it excludessuperluminal signaling does not imply that other quantum theories orinterpretations of the orthodox theory also exclude such signaling. Yet, ifthe orthodox theory is empirically adequate, as the consensus has it, itsstatistical predictions obtain, and accordingly superluminal signaling willbe excluded as a matter of fact; for if this theory is empirically adequate,any quantum theory will have to reproduce its statistics, including theexclusion of any actual superluminal signaling.

But the no-signaling theorem does not demonstrate that superluminalsignaling would be impossible if orthodox quantum mechanics were notempirically adequate. Furthermore, this theorem does not show thatsuperluminal signaling is in principle impossible in the quantum realm asdepicted by other theories, which actually reproduce the statistics oforthodox quantum mechanics but do not prohibit in theory the violationof this statistics. In sections 7.2-7.3, we shall consider the in-principlepossibility of superluminal signaling in certain collapse and no-collapseinterpretations of quantum mechanics. But, first, we need to consider thenecessary and sufficient conditions for superluminal signaling.

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7.1 Necessary and sufficient conditions for superluminalsignaling

To simplify things, in our discussion we shall focus on non-factorizablemodels of the EPR/B experiment that satisfy λ-independence (i.e., theassumption that the distribution of the states λ is independent of thesettings of the measurement apparatuses). Superluminal signaling in theEPR/B experiment would be possible in theory just in case the value ofsome controllable physical quantity in the nearby wing could influencethe statistics of measurement outcomes in the distant wing. And in non-factorizable models that satisfy λ-independence this could happen just incase the following conditions obtained:

Four comments: (i) In controllable probabilistic dependence, the term'probabilities of measurement outcomes' refers to the model probabilities,i.e., the probabilities that the states λ prescribe for measurementoutcomes. (ii) Our discussion in this entry focuses on models of the EPR/Bexperiment in which probabilities of measurement outcomes depend onlyon the pair's state λ and the settings of the measurement apparatuses tomeasure certain properties. In such models, parameter dependence (i.e.,the dependence of the probability of the distant measurement outcome onthe setting of the nearby measurement apparatus) is a necessary andsufficient condition for controllable probabilistic dependence. But, recall

Controllable probabilistic dependence. The probabilities ofdistant measurement outcomes depend on some nearbycontrollable physical quantity.

λ-distribution. There can be in theory an ensemble of particlepairs the states of which deviate from the quantum-equilibriumdistribution; where the quantum-equilibrium distribution of pairs'states is the distribution that reproduces the predictions oforthodox quantum mechanics.



sufficient condition for controllable probabilistic dependence. But, recall(footnote 3) that in some models of the EPR/B experiment, in addition tothe pair's state and the setting of the L- (R-) measurement apparatus thereare other local physical quantities that may be relevant for the probabilityof the L- (R-) measurement outcome. In such models, parameterdependence is not a necessary condition for controllable probabilisticdependence. Some other physical quantities in the nearby wing may berelevant for the probability of the distant measurement outcome. (That is,let α and β denote all the relevant local physical quantities, other than thesettings of the measurement apparatuses, that may be relevant for theprobability of the L- and the R-outcome, respectively. Then, controllableprobabilistic dependence would obtain if for some pairs' states λ, L-setting l, R-setting r and local physical quantities α and β, Pλ l r α β(yr) ≠Pλ l r β(yr) obtained.) For the relevance of such models to the question ofthe in-principle possibility of superluminal signalling in some currentinterpretations of quantum mechanics, see sections 7.3 and 7.4. (iii) The quantum-equilibrium distribution will not be the same in allmodels of the EPR/B experiment; for in general the states λ will not bethe same in different models. (iv) In models that actually violate both controllable probabilisticdependence and λ-distribution, the occurrence of controllable probabilisticdependence would render the actual distribution of λ states as non-equilbrium distribution. Thus, if controllable probabilistic dependenceoccurred in such models, the actual distribution of λ states would satisfyλ-distribution.

The argument for the necessity of controllable probabilistic dependenceand λ-distribution is straightforward. Granted λ-independence, if theprobabilistic dependence of the distant outcome on a nearby physicalquantity is not controllable, there can be no way to manipulate thestatistics of the distant outcome so as to deviate from the statisticalpredictions of quantum mechanics. Accordingly, superluminaltransmission of information will be impossible even in theory. And if λ-

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transmission of information will be impossible even in theory. And if λ-distribution does not hold, i.e., if the quantum-equilibrium distributionholds, controllable probabilistic dependence will be of no use forsuperluminal transmission of information. For, averaging over the modelprobabilities according to the quantum-equilbrium distribution, the modelwill reproduce the statistics of orthodox quantum mechanics. That is, thedistribution of the λ-states will be such that the probabilistic dependenceof the distant outcome on the nearby controllable factor will be washedout: In some states the nearby controllable factor will raise the probabilityof the distant outcome and in others it will decrease this probability, sothat on average the overall statistics of the distant outcome will beindependent of the nearby controllable factor (i.e., the same as thestatistics of orthodox quantum mechanics). Accordingly, superluminalsignaling will be impossible.

The argument for the sufficiency of these conditions is alsostraightforward. If λ-distribution held, it would be possible in theory toarrange ensembles of particle pairs in which controllable probabilisticdependence would not be washed out, and accordingly the statistics ofdistant outcomes would depend on the nearby controllable factor. (For aproof that these conditions are sufficient for superluminal signaling incertain deterministic hidden variables theories, see Valentini 2002.)

Note that the necessary and sufficient conditions for superluminalsignaling are different in models that do not exclude in theory theviolation of λ-independence. In such models controllable probabilisticdependence is not a necessary condition for superluminal signaling. Thereasoning is as follows. Consider any empirically adequate model of theEPR/B experiment in which the pair's state and the settings of themeasurement apparatuses are the only relevant factors for theprobabilities of measurement outcomes, and the quantum-equilibriumdistribution is λ-independent. In such a model, parameter independenceimplies the failure of controllable probabilistic dependence, yet the



implies the failure of controllable probabilistic dependence, yet theviolation of λ-independence would imply the possibility of superluminalsignaling: If λ-independence failed, a change in the setting of the nearbymeasurement apparatus would cause a change in the distribution of thestates λ, and a change in this distribution would induce a change in thestatistics of the distant (space-like separated) measurement outcome.

Leaving aside models that violate λ-independence, we now turn toconsider the prospects of controllable probabilistic dependence and λ-distribution, starting with no-collapse interpretations.


7.2.1 Bohm's Theory

Bohm's theory involves parameter dependence and thus controllableprobabilistic dependence: The probabilities of distant outcomes depend onthe setting of the nearby apparatus. In some pairs' states λ, i.e., in someconfigurations of the positions of the particle pair, a change in theapparatus setting of the (earlier) say L-measurement will induce animmediate change in the probability of the R-outcome: e.g. the probabilityof R-outcome z-spin ‘up’ will be 1 if the L-apparatus is set to measure z-spin and 0 if the L-apparatus is switched off (see section 5.3.1). Thus, thequestion of superluminal signaling turns on whether λ-distributionobtains.

Now, recall (section 5.3.1) that Bohm's theory reproduces the quantumstatistics by postulating the quantum-equilibrium distribution over thepositions of particles. If this distribution is not an accidental fact aboutour universe, but rather obtains as a matter of law, superluminal signalingwill be impossible in principle. Dürr, Goldstein and Zanghì (1992a,b,1996, fn. 15) argue that, while the quantum-equilibrium distribution is nota matter a law, other distributions will be possible but atypical. Thus, theyconclude that although superluminal signaling is not impossible in theory,

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conclude that although superluminal signaling is not impossible in theory,it may occur only in atypical worlds. On the other hand, Valentini(1991a,b, 1992, 1996, 2002) and Valentini and Westman 2004) argue thatthere are good reasons to think that our universe may well have started offin a state of quantum non-equilibrium and is now approaching gradually astate of equilibrium, so that even today some residual non-equilibriummust be present.[32] Yet, even if such residual non-equilbrium existed, thequestion is whether it would be possible to access any ensemble ofsystems in a non-equilbrium distribution.

7.2.2 Modal interpretations

The presence or absence of parameter independence (and accordingly thepresence or absence of controllable probabilistic dependence) in themodal interprtation is a matter of controversy, perhaps due in part to themultiplicity of versions of this interpretation. Whether or not modalinterpretations involve parameter dependence would probably depend onthe dynamics of the possessed properties. At least some of the currentmodal interpretations seem to involve no parameter dependence. But, asthe subject editor pointed out to the author, some think that the no-gotheorem for relativistic modal interpretation due to Dickson and Clifton(1998) implies the existence of parameter dependence in all theinterpretations to which this theorem is applicable. Do modalinterpretations satisfy λ-distribution? The prospects of this conditiondepend on whether the possessed properties that the modal interpretationassigns in addition to the properties prescribed by the orthodoxinterpretation, are controllable. If these properties were controllable atleast in theory, λ-distribution would be possible. For example, if thepossessed spin properties that the particles have at the emission from thesource in the EPR/B experiment were controllable, then λ-distributionwould be possible. The common view seems to be that these propertiesare uncontrollable.




7.3.1 Dynamical models for state-vector reduction

In the GRW/Pearle collapse models, wave functions represent the mostexhaustive, complete specification of states of individual systems. Thus,pairs prepared with the same wave function have always the same λ state— a state that represents their quantum-equilbrium distribution for theEPR/B experiment. Accordingly, λ-distribution fails. Do these modelsinvolve controllable probabilistic dependence?

Recall (section 5.1.2) that there are several models of state reduction inthe literature. One of these models is the so-called non-linear ContinuousStochastic Localization (CSL) models (see Pearle 1989, Ghirardi, Pearleand Rimini 1990, Butterfield et al. 1993, and Ghirardi et al. 1993).Butterfield et al. (1993) argue that in these models there is a probabilisticdependence of the outcome of the R-measurement on the process thatleads to the (earlier) outcome of the L-measurement. In these models, theprocess leading to the L-outcome (either z-spin ‘up’ or z-spin ‘down’)depends on the interaction between the L-particle and the L-apparatus(which results in an entangled state), and the specific realization of thestochastic process that strives to collapse this macroscopic superpositioninto a product state in which the L-apparatus displays a definite outcome.And the probability of the R-outcome depends on this process. Forexample, if this process is one that gives rise to a z-spin ‘up’ (or rendersthat outcome more likely), the probability of R-outcome z-spin ‘up’ is 0(more likely to be 0); and if this process is one that gives rise to a z-spin‘down’ (or renders that outcome more likely), the probability of R-outcome z-spin ‘down’ is 0 (more likely to be 0). The question is whetherthere are controllable factors that influence the probability of realizationsof stochastic processes that lead to a specific L-outcome, so that it wouldbe possible to increase or decrease the probability of the R-outcome. If

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be possible to increase or decrease the probability of the R-outcome. Ifsuch factors existed, controllable probabilistic dependence would bepossible at least in theory. And if this kind of controllable probabilisticdependence existed, λ-distribution would also obtain; for if suchdependence existed, the actual distribution of pairs' states (in which thepair always have the same state, the quantum-mechanical state) wouldcease to be the quantum-equilbrium distribution.

7.4 The prospects of controllable probabilistic dependence

In section 7.3.1, we discussed the question of the in-principlecontrollability of local measurement processes and in particular theprobability of their outcome, and the implications of such controllabilityfor the in-principle possibility of superluminal signaling in the context ofthe CSL models. But this question is not specific to the CSL model and(more generally) the dynamical models for state-vector reduction. Itseems likely to arise also in other quantum theories that modelmeasurements realistically. Here is why. Real measurements take time.And during that time, some physical variable, other than the state of themeasured system and the setting of the measurement apparatus, mightinfluence the chance (i.e., the single-case objective probability) of themeasurement outcome. In particular, during the L-measurement in theEPR/B experiment, the chance of the L-outcome z-spin ‘up’ (‘down’)might depend on the value of some physical variable in the L-wing, otherthan the state of the particle pair and the setting of the L-measurementapparatus. If so, it will follow from the familiar perfect anti-correlation ofthe singlet state that the chance of R-outcome z-spin ‘up’ (‘down’) willdepend on the value of such variable (for details, see Kronz 1990a,b,Jones and Clifton 1993, pp. 304-305, and Berkovitz 1998a, section 4.3.4).Thus, if the value of such a variable were controllable, controllableprobabilistic dependence would obtain.



7.5 Superluminal signaling and action-at-a-distance

If superluminal signaling were possible in the EPR/B experiment in anyof the above theories, it would not require any continuous process inspacetime to mediate the influences between the two distant wings.Indeed, in all the current quantum theories in which the probability of theR-outcome depends on some controllable physical variable in the L-wing,this dependence is not due to a continuous process. Rather, it is due tosome type of ‘action’ or (to use Shimony's (1984) terminology) ‘passion’at a distance, which is the ‘result’ of the holistic nature of the quantumrealm, the non-separability of the state of entangled systems, or the non-separable nature of the evolution of the properties of systems.

8. The analysis of factorizability: implications forquantum non-locality

In sections 5-7, we considered the nature of quantum non-locality asdepicted by theories that violate factorizability, i.e., the assumption thatthe probability of joint measurement outcomes factorizes into the singleprobabilities of these outcomes. Recalling section 3, factorizability can beanalyzed into a conjunction of two conditions: OI (outcomeindependence)—the probability of a distant measurement outcome in theEPR/B experiment is independent of the nearby measurement outcome;and PI (parameter independence)—the probability of a distantmeasurement outcome in the EPR/B experiment is independent of thesetting of the nearby measurement apparatus. Bohm's theory violates PI,whereas other mainstream quantum theories satisfy this condition butviolate OI. The question arises as to whether violations of PI involve adifferent kind of non-locality than violations of OI. So far, ourmethodology was to study the nature of quantum non-locality byanalyzing the way various quantum theories account for the curiouscorrelations in the EPR/B experiment. In this section, we shall focus onthe question of whether quantum non-locality can be studied in a more

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the question of whether quantum non-locality can be studied in a moregeneral way, namely by analyzing the types of non-locality involved inviolations of PI and in violations of OI, independently of how theseviolations are realized.

8.1 Non-separability, holism and action at a distance

It is frequently argued or maintained that violations of OI involve statenon-separability and/or some type of holism, whereas violations of PIinvolve action at a distance. For notable examples, Howard (1989) arguesthat spatiotemporal separability (see section 4.3) implies OI, andaccordingly a violation of it implies spatiotemporal non-separability;Teller (1989) argues that particularism (see section 4.3) implies OI, andthus a violation of it implies relational holism; and Jarrett (1984, 1989)argues that a violation of PI involves some type of action at a distance.These views are controversial, however.

First, as we have seen in section 5, in quantum theories the violation ofeither of these conditions involves some type of non-separability and/orholism.

Second, the explicit attempts to derive OI from separability orparticularism seem to rely (implicitly) on some locality conditions.Maudlin (1998, p. 98) and Berkovitz (1998a, section 6.1) argue thatHoward's precise formulation of spatiotemporal separability embodiesboth separability and locality conditions, and Berkovitz (1998a, section6.2) argues that Teller's derivation of OI from particularism implicitlyrelies on locality conditions. Thus, the violation of OI per se does notimply non-separability or holism.

Third, a factorizable model, i.e., model that satisfies OI, may be non-separable (Berkovitz 1995b, section 6.5). Thus, OI cannot be simplyidentified with OI.



Fourth, Howard's spatiotemporal separability condition (see section 4.3)requires that states of composite systems be determined by the states oftheir subsystems. In particular, spatiotemporal separability requires thatjoint probabilities of outcomes be determined as some function of thesingle probabilities of these outcomes. Winsberg and Fine (2003) objectthat as a separability condition, OI arbitrarily restricts this function to be aproduct function. And they argue that on a weakened formalization ofseparability, a violation of OI is compatible with separability. Fogel(2004) agrees that Winsberg and Fine's weakened formalization ofseparability is correct, but argues that, when supplemented by a certain‘isotropy’ condition, OI implies this weakened separability condition.Fogel believes that his suggested ‘isotropy’ condition is very plausible,but, as he acknowledges, this condition involves a nontrivial measurementcontext-independence.[33]

Fifth, as the analysis in section 5 demonstrates, violations of OI mightinvolve action at a distance. Also, while the minimal Bohm theoryviolates PI and arguably some modal interpretations do not, the type ofaction at a distance they postulate, namely action* at a distance (seesection 5.2), is similar: In both cases, an earlier spin-measurement in(say) the L-wing does not induce any immediate change in the intrinsicproperties of the R-particle. The L-measurement only causes animmediate change in the dispositions of the R-particle—a change thatmay influence the behavior of the R-particle in future spin-measurementsin the R-wing. But, this change of dispositions does not involve anychange of local properties in the R-wing, as these dispositions arerelational (rather than intrinsic) properties of the R-particle. Furthermore,the action at a distance predicated by the minimal Bohm theory is weakerthan the one predicated by orthodox collapse quantum mechanics and theGRW/Pearle collapse models; for in contrast to the minimal Bohmtheory, in these theories the measurement on the L-particle induces achange in the intrinsic properties of the R-particle, independently of

Joseph Berkovitz


change in the intrinsic properties of the R-particle, independently ofwhether or not the R-particle undergoes a measurement. Thus, if the R-particle comes to possess (momentarily) a definite position, the EPR/Bexperiment as described by these theories involves action at a distance —a stronger kind of action than the action* at a distance predicated by theminimal Bohm theory.

8.2 Superluminal signaling

It was also argued, notably by Jarrett 1984 and 1989 and Shimony 1984,that in contrast to violations of OI, violations of PI may give rise (at leastin principle) to superluminal signaling. Indeed, as is not difficult to seefrom section 7.1, in theories that satisfy λ-independence there is anasymmetry between failures of PI and failures of OI with respect tosuperluminal signaling: whereas λ-distribution and the failure of PI aresufficient conditions for the in-principle possibility of superluminalsignaling, λ-distribution and the failure of OI are not. Thus, the prospectsof superluminal signaling look better in parameter-dependent theories,i.e., theories that violate PI. Yet, as we have seen in section 7.2.1, if theBohmian quantum-equilbrium distribution obtains, then Bohm's theory,the paradigm of parameter dependent theories, prohibits superluminalsignaling. And if this distribution is obtained as a matter of law, thenBohm's theory prohibits superluminal signaling even in theory.Furthermore, as we remarked in section 7.1, if the in-principle possibilityof violating λ-independence is not excluded, superluminal signaling mayexist in theories that satisfy PI and violate OI. In fact, as section 7.4 seemsto suggest, the possibility of superluminal signaling in theories that satisfyPI but violate OI cannot be discounted even when λ-independence isimpossible.

8.3 Relativity



Jarrett (1984, 1989), Ballentine and Jarrett (1997) and Shimony (1984)hold that superluminal signaling is incompatible with relativity theory.Accordingly, they conclude that violations of PI are incompatible withrelativity theory, whereas violations of OI may be compatible with thistheory. Furthermore, Sutherland (1985, 1989) argues that deterministic,relativistic parameter-dependent theories (i.e., relativistic, deterministictheories that violate PI) would plausibly require retro-causal influences,and in certain experimental circumstances this type of influences wouldgive rise to causal paradoxes, i.e., inconsistent closed causal loops (whereeffects undermine their very causes). And Arntzenius (1994) argues thatall relativistic parameter-dependent theories are impossible on pain ofcausal paradoxes. That is, he argues that in certain experimentalcircumstances any relativistic, parameter-dependent theory would giverise to closed causal loops in which violations of PI could not obtain.

It is noteworthy that the view that relativity per se is incompatible withsuperluminal signaling is disputable (for more details, see section 10).Anyway, recalling (section 8.2), if λ-distribution is excluded as a matterof law, it will be impossible even in theory to exploit the violation of PIto give rise to superluminal signaling, in which case the possibility ofrelativistic parameter-dependent theories could not be discounted on thebasis of superluminal signaling.

Furthermore, as mentioned in section 7.1 and 7.4, the in-principlepossibility of superluminal signaling in theories that satisfy PI and violateOI cannot be excluded a priori. Thus, if relativity theory excludessuperluminal signaling, the argument from superluminal signaling mayalso be applied to exclude the possibility of some relativistic outcome-dependent theories.

Finally, Berkovitz (2002) argues that Arntzenius's argument for theimpossibility of relativistic theories that violate PI is based onassumptions about probabilities that are common in linear causal

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assumptions about probabilities that are common in linear causalsituations but are unwarranted in causal loops, and that the real challengefor these theories is that in such loops their predictive power isundermined (for more details, see section 10.3).

8.4 Superluminal causation

In various counterfactual and probabilistic accounts of causationviolations of PI entail superluminal causation between the setting of thenearby measurement apparatus and the distant measurement outcome,whereas violations of OI entail superluminal causation between thedistant measurement outcomes (see Butterfield 1992b, 1994, Berkovitz1998b, section 2). Thus, it seems that theories that violate PI postulate adifferent type of superluminal causation than theories that violate OI. Yet,as Berkovitz (1998b, section 2.4) argues, the violation of PI in Bohm'stheory does involve some type of outcome dependence, which may beinterpreted as a generalization of the violation of OI. In this theory, thespecific R-measurement outcome in the EPR/B experiment depends onthe specific L-measurement outcome: For any three different directions x,y, z, if the probabilities of x-spin ‘up’ and y-spin ‘up’ are non-zero, theprobability of R-outcome z-spin ‘up’ will generally depend on whetherthe L-outcome is x-spin ‘up’ or y-spin ‘up’. Yet, due to the determinismthat Bohm's theory postulates, OI trivially obtains. Put it another way, OIdoes not reflect all the types of outcome independence that may existbetween distant outcomes. Accordingly, the fact that a theory satisfies OIdoes not entail that it does not involve some other type of outcomedependence. Indeed, in all the current quantum theories that violatefactorizability there are correlations between distant specific measurementoutcomes — correlations that may well be interpreted as an indication ofcounterfactual superluminal causation between these outcomes.

8.5 On the origin and nature of parameter dependence



Parameter dependence (PI) postulates that in the EPR/B experiment theprobability of the later, distant measurement outcome depends on thesetting of the apparatus of the nearby, earlier measurement. It may betempting to assume that this dependence is due to a direct influence of thenearby setting on the (probability of the) distant outcome. But a littlereflection on the failure of PI in Bohm's theory, which is the paradigm forparameter dependence, demonstrates that the setting of the nearbyapparatus per se has no influence on the distant measurement outcome.Rather, it is because the setting of the nearby measurmenent apparatusinfluences the nearby measurement outcome and the nearby outcomeinfluences the distant outcome that the setting of the nearby apparatus canhave an influence on the distant outcome. For, as is not difficult to seefrom the analysis of the nature of non-locality in the minimal Bohmtheory (see section 5.3.1), the setting of the apparatus of the nearby(earlier) measurement in the EPR/B experiment influences the outcomethe nearby measurement, and this outcome influences the guiding field ofthe distant particle and accordingly the outcome of a measurement on thatparticle.

While the influence of the nearby setting on the nearby outcome isnecessary for parameter dependence, it is not sufficient for it. In all thecurrent quantum theories, the probabilities of joint outcomes in the EPR/Bexperiment depend on the settings of both measurement apparatuses: Theprobability that the L-outcome is l-spin ‘up’ and the R-outcome is r-spin‘up’ and the probability that the L-outcome is l-spin ‘up’ and the R-outcome is r-spin ‘down’ both depend on (l − r), i.e., the distancebetween the angles l and r. In theories in which the sum of these jointprobabilities is invariant with respect to the value of (l − r), parameterindependence obtains: for all pairs' states λ, L-setting l, and R-settings rand r′, L-outcome xl, and R-outcomes yr and yr′ :

Joseph Berkovitz


(PI) Pλ l r(xl & yr) + Pλ l r(xl & ¬yr) = Pλ l r′ (xl & yr′ ) + Pλ l r′ (xl & ¬yr′ ).

Parameter dependence is a violation of this invariance condition.

9. Can there be ‘local’ quantum theories?

The focus of this entry has been on exploring the nature of the non-localinfluences in the quantum realm as depicted by quantum theories thatviolate factorizability, i.e., theories in which the joint probability of thedistant outcomes in the EPR/B experiment do not factorize into theproduct of the single probabilities of these outcomes. The motivation forthis focus was that, granted plausible assumptions, factorizability mustfail (see section 2), and its failure implies some type of non-locality (seesections 2-8). But if any of these plausible assumptions failed, it may bepossible to account for the EPR/B experiment (and more generally for allother quantum phenomena) without postulating any non-local influences.Let us then consider the main arguments for the view that quantumphenomena need not involve non-locality.

In arguments for the failure of factorizability, it is presupposed that thedistant measurement outcomes in the EPR/B experiment are real physicalevents. Recall (section 5.3.3) that in Albert and Loewer's (1988) many-minds interpretation this is not the case. In this interpretation, definitemeasurement outcomes are (typically) not physical events. In particular,the pointers of the measurement apparatuses in the EPR/B experiment donot display any definite outcomes. Measurement outcomes in the EPR/Bexperiment exist only as (non-physical) mental states in observers' minds(which are postulated to be non-physical entities). So sacrificing some ofour most fundamental presuppositions about the physical reality andassuming a controversial mind-body dualism, the many-mindsinterpretation of quantum mechanics does not postulate any action at adistance or superluminal causation between the distant wings of the



distance or superluminal causation between the distant wings of theEPR/B experiment. Yet, as quantum-mechanical states of systems areassumed to reflect their physical states, the many-minds theory doespostulate some type of non-locality, namely state non-separability andproperty and relational holism.

Another way to get around Bell's argument for non-locality in the EPR/Bexperiment is to construct a model of this experiment that satisfiesfactorizability but violates λ-independence (i.e., the assumption that thedistribution of all the possible pairs' states in the EPR/B experiment isindependent of the measured quantities). In section 2, we mentioned twopossible causal explanations for the failure of λ-independence. The first isto postulate that pairs' states and apparatus settings share a commoncause, which correlates certain types of pairs' states with certain types ofsettings (e.g. states of type λ1 are correlated with settings of type l and r,whereas states of type λ2 are correlated with settings of type l′ and r′,etc.). As we noted, thinking about all the various ways one can measureproperties, this explanation seems conspiratorial. Furthermore, it runscounter to one of the most fundamental presuppositions of empiricalscience, namely that in experiments preparations of sources and settingsof measurement apparatuses are typically independent of each other. Thesecond possible explanation is to postulate causation from themeasurement events backward to the source at the emission time. (Foradvocates of this way out of non-locality, see Costa de Beauregard 1977,1979, 1985, Sutherland 1983, 1998, 2006 and Price 1984, 1994, 1996,chapters 3, 8 and 9.) Maudlin (1994, p. 197-201) argues that theories thatpostulate such causal mechanism are inconsistent. Berkovitz (2002,section 5) argues that Maudlin's line of reasoning is based on unwarrantedpremises. Yet, as we shall see in section 10.3, this way out of non-localityfaces some challenges. Furthermore, while a violation of λ-independenceprovides a way out of Bell's theorem, it does not necessarily implylocality; for the violation of λ-independence is compatible with the failureof factorizability.

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of factorizability.

A third way around non-locality is to ‘exploit’ the inefficiency ofmeasurement devices or (more generally) measurement set-ups. In anyactual EPR/B experiment, many of the particle pairs emitted from thesource fail to be detected, so that only a sample of the particle pairs isobserved. Assuming that the observed samples are not biased, it is nowgenerally agreed that the statistical predictions of orthodox quantummechanics have been vindicated (for a review of these experiments, seeRedhead 1987, section 4.5). But if this assumption is abandoned, there areperfectly local causal explanations for the actual experimental results(Clauser and Horne 1974, Fine 1982b, 1989a). Many believe that this wayout of non-locality is ad hoc, at least in light of our current knowledge.Moreover, this strategy would fail if the efficiency of measurementdevices exceeded a certain threshold (for more details, see Fine 1989a,Maudlin 1994, chapter 6, Larsson and Semitecolos 2000 and Larsson2002).

Finally, there are those who question the assumption that factorizability isa locality condition (Fine 1981, 1986, pp. 59-60, 1989b, Cartwright 1989,chaps. 3 and 6, Chang and Cartwright 1993). Accordingly, they deny thatnon-factorizability implies non-locality. The main thrust of this line ofreasoning is that the principle of the common cause is not generally valid.Some, notably Cartwright (1989) and Chang and Cartwright (1993),challenge the assumption that common causes always screen off thecorrelation between their effects, and accordingly they question the ideathat non-factorizability implies non-locality. Others, notably Fine, denythat correlations must have causal explanation.

While these arguments challenge the view that the quantum realm asdepicted by non-factorizable models for the EPR/B experiment mustinvolve non-locality, they do not show that viable local, non-factorizablemodels of the EPR/B experiment (i.e., viable models which do not



models of the EPR/B experiment (i.e., viable models which do notpostulate any non-locality) are possible. Indeed, so far none of theattempts to construct local, non-factorisable models for EPR/Bexperiments has been successful.

10. Can quantum non-locality be reconciled withrelativity?

The question of the compatibility of quantum mechanics with the specialtheory of relativity is very difficult to resolve. (The question of thecompatibility of quantum mechanics with the general theory of relativityis even more involved.) The answer to this question depends on theinterpretation of special relativity and the nature of the exact constraints itimposes on influences between events.

A popular view has it that special relativity prohibits any superluminalinfluences, whereas theories that violate factorizability seem to involvesuch influences. Accordingly, it is held that quantum mechanics isincompatible with relativity. Another common view has it that specialrelativity prohibits only certain types of superluminal influence. Manybelieve that relativity prohibits superluminal signaling of information.Some also believe that this theory prohibits superluminal transport ofmatter-energy and/or action-at-a-distance. On the other hand, there is theview that relativity per se prohibits only superluminal influences that areincompatible with the special-relativistic space-time, the so-called‘Minkowski space-time,’ and that this prohibition is compatible withcertain types of superluminal influences and superluminal signaling (for acomprehensive discussion of this issue, see Maudlin, 1994, 1996, section2).[34]

It is commonly agreed that relativity requires that the descriptions ofphysical reality (i.e., the states of systems, their properties, dynamicallaws, etc.) in different coordinate systems should be compatible with each

Joseph Berkovitz


laws, etc.) in different coordinate systems should be compatible with eachother. In particular, descriptions of the state of systems in differentfoliations of spacetime into parallel spacelike hyperplanes, whichcorrespond to different inertial reference frames, are to be related to eachother by the Lorentz transformations. If this requirement is to reflect thestructure of the Minkowski spacetime, these transformations must hold atthe level of individual processes, and not only at the level of ensembles ofprocesses (i.e., at the statistical level) or observed phenomena. Indeed,Bohm's theory, which is manifestly non-relativistic, satisfies therequirement that the Lorentz transformations obtain at the level of theobserved phenomena.

However, satisfying the Lorentz transformations at the level of individualprocesses is not sufficient for compatibility with Minkowski spacetime;for the Lorentz transformations may also be satisfied at the level ofindividual processes in theories that postulate a preferred inertialreference frame (Bell 1976). Maudlin (1996, section 2) suggests that atheory is genuinely relativistic (both in spirit and letter) if it can beformulated without ascribing to spacetime any more, or different intrinsicstructure than the relativistic metrics.[35] The question of thecompatibility of relativity with quantum mechanics may be presented asfollows: Could a quantum theory that does not encounter the measurementproblem be relativistic in that sense?


The main problem in reconciling collapse theories with special relativityis that it seems very difficult to make state collapse (modeled as a realphysical process) compatible with the structure of the Minkowskispacetime. In non-relativistic quantum mechanics, the earlier L-measurement in the EPR/B experiment induces a collapse of theentangled state of the particle pair and the L-measurement apparatus.Assuming (for the sake of simplicity) that measurement events occur



Assuming (for the sake of simplicity) that measurement events occurinstantaneously, state collapse occurs along a single spacelike hyperplanethat intersects the spacetime region of the L-measurement event—thehyperplane that represents the (absolute) time of the collapse. But thistype of collapse dynamics would involve a preferred foliation ofspacetime, in violation of the spirit, if not the letter of the Minkowskispacetime.

The current dynamical collapse models are not genuinely relativistic, andattempts to generalize them to the special relativistic domain haveencountered difficulties (see, for example, the entry on collapse theories,Ghirardi 1996, Pearle 1996, and references therein). A more recentattempt to address these difficulties due to Tumulka (2004) seems morepromising.

In an attempt to reconcile state collapse with special relativity, Fleming(1989, 1992, 1996) and Fleming and Bennett (1989) suggested radicalhyperplane dependence. In their theory, state collapse occurs along aninfinite number of spacelike hyperplanes that intersect the spacetimeregion of the measurements. That is, in the EPR/B experiment a collapseoccurs along all the hyperplanes of simultaneity that intersect thespacetime region of the L-measurement. Similarly, a collapse occursalong all the hyperplanes of simultaneity that intersect the distant (space-like separated) spacetime region of the R-measurement. Accordingly, thehyperplane-dependent theory does not pick out any reference frame aspreferred, and the dynamics of the quantum states of systems and theirproperties can be reconciled with the Minkowski spacetime. Further, sinceall the multiple collapses are supposed to be real (Fleming 1992, p. 109),the predictions of orthodox quantum mechanics are reproduced in eachreference frame.

The hyperplane-dependent theory is genuinely relativistic. But the theorydoes not offer any mechanism for state collapses, and it does not explain

Joseph Berkovitz


does not offer any mechanism for state collapses, and it does not explainhow the multiple collapses are related to each other and how ourexperience is accounted for in light of this multiplicity.

Myrvold (2002b) argues that state collapses can be reconciled withMinkowski spacetime even without postulating multiple differentcollapses corresponding to different reference frames. That is, he argueswith respect to the EPR/B experiment that the collapses induced by the L-and the R-measurement are local events in the L- and the R-wingrespectively, and that the supposedly different collapses (corresponding todifferent reference frames) postulated by the hyperplane-dependent theoryare only different descriptions of the same local collapse events. Focusingon the state of the particle pair, the main idea is that the collapse event inthe L-wing is modeled by a (one parameter) family of operators (theidentity operator before the L-measurement and a projection to thecollapsed state after the L-measurement), and it is local in the sense that itis a projection on the Hilbert space of the L-particle; and similarly,mutatis mutandis, for the R-particle. Yet, if the quantum state of theparticle pair represents their complete state (as the case is in the orthodoxtheory and the GRW/Pearle collapse models), these collapse events seemnon-local. While the collapse in the L-wing may be said to be local in theabove technical sense, it is by definition a change of local as well asdistant (spacelike) properties. The operator that models the collapse in theL-wing transforms the entangled state of the particle pair—a state inwhich the particles have no definite spins—into a product of non-entangled states in which both particles have definite spins, andaccordingly it causes a change of intrinsic properties in both the L- andthe R-wing.

In any case, Myrvold's proposal demonstrates that even if state collapsesare not hyperplane dependent, they need not be incompatible withrelativity theory.




Recall (section 5.3) that in no-collapse theories, quantum-mechanicalstates always evolve according to a unitary and linear equation of motion(the Schrödinger equation in the non-relativistic case), and accordinglythey never collapse. Since the wave function has a covariant dynamics,the question of the compatibility with relativity turns on the dynamics ofthe additional properties —the so-called ‘hidden variables’— that no-collapse theories typically postulate. In Albert and Loewer's many-mindstheory (see section 5.3.3), the wave function has covariant dynamics, andno additional physical properties are postulated. Accordingly, the theoryis genuinely relativistic. Yet, as the compatibility with relativity isachieved at the cost of postulating that outcomes of measurements (and,typically, any other perceived properties) are mental rather than physicalproperties, many find this way of reconciling quantum mechanics withrelativity unsatisfactory.

Other Everett-like interpretations attempt to reconcile quantum mechanicswith the special theory of relativity without postulating such acontroversial mind-body dualism. Similarly to the many-mindsinterpretation of Albert and Loewer, and contrary to Bohm's theory andmodal interpretations, on the face of it these interpretations do notpostulate the existence of ‘hidden variables.’ But (recalling section 5.3.3)these Everett-like interpretations face the challenge of making sense ofour experience and the probabilities of outcomes, and critics of theseinterpretations argue that this challenge cannot be met without addingsome extra structure to the Everett interpretation (see Albert and Loewer1988, Albert 1992, pp. 114-5, Albert and Loewer 1996, Price 1996, pp.226-227, and Barrett 1999, pp. 163-173); a structure that may render theseinterpretations incompatible with relativity. Supporters of the Everettinterpretation disagree. Recently, Deutsch (1999), Wallace (2002, 2003,2005a,b) and Greaves (2004) have suggested that Everettians can makesense of the quantum-mechanical probabilities by appealing to decision-

Joseph Berkovitz


sense of the quantum-mechanical probabilities by appealing to decision-theoretical considerations. But this line of reasoning has been disputed(see Barnum et al. 2000, Lewis 2003b, Hemmo and Pitowsky 2005 andPrice 2006).

Modal interpretations constitute another class of no-collapseinterpretations of quantum mechanics that were developed to reconcilequantum mechanics with relativity (and to solve the measurementproblem). Yet, as the no-go theorems by Dickson and Clifton (1998),Arntzenius (1998) and Myrvold (2002) demonstrate, the earlier versionsof the modal interpretation are not genuinely compatible with relativitytheory. Further, Earman and Ruetsche (2005) argue that a quantum-fieldversion of the modal interpretation (which is set in the context ofrelativistic quantum-field theory), like the one proposed by Clifton(2000), would be subject to serious challenges. Berkovitz and Hemmo(2006a,b) develop a relational modal interpretation that escapes all theabove no-go theorems and to that extent seems to provide better prospectsfor reconciling quantum mechanics with special relativity.

10.3 Quantum causal loops and relativity

Recall (section 8) that many believe that parameter-dependent theories(i.e., theories that violate parameter independence) are more difficult oreven impossible to reconcile with relativity. Recall also that one of thelines of argument for the impossibility of relativistic parameter-dependenttheories is that such theories would give rise to causal paradoxes. In ourdiscussion, we focused on EPR/B experiments in which the measurementsare distant (spacelike separated). In a relativistic parameter-dependenttheory, the setting of the nearby measurement apparatus in the EPR/Bexperiment would influence the probability of the distant (spacelikeseparated) measurement outcome. Sutherland (1985, 1989) argues that itis plausible to suppose that the realization of parameter dependence wouldbe the same in EPR/B experiments in which the measurements are not



be the same in EPR/B experiments in which the measurements are notdistant from each other (i.e., when the measurements are timelikeseparated). If so, relativistic parameter-dependent theories would involvebackward causal influences. But, he argues, in deterministic, relativisticparameter-dependent theories these influences would give rise to causalparadoxes, i.e., inconsistent closed causal loops.

Furthermore, Arntzenius (1994) argues that all relativistic parameter-dependent theories are impossible on pain of causal paradoxes. In hisargument, he considers the probabilities of measurement outcomes in asetup in which two EPR/B experiments are causally connected to eachother, so that the L-measurement outcome of the first EPR/B experimentdetermines the setting of the L-apparatus of the second EPR/B experimentand the R-measurement outcome of the second EPR/B experimentdetermines the setting of the R-apparatus of the first EPR/B experiment.And he argues that in this experiment, relativistic parameter-dependenttheories (deterministic or indeterministic) would give rise to closed causalloops in which parameter dependence would be impossible. Thus, heconcludes that relativistic, parameter-dependent theories are impossible.(Stairs (1989) anticipates the argument that the above experimental setupmay give rise to causal paradoxes in relativistic, parameter-dependenttheories, but he stops short of arguing that such theories are impossible.)

Berkovitz (1998b, section 3.2, 2002, section 4) argues that Arntzenius'sline of reasoning fails because it is based on untenable assumptions aboutthe nature of probabilities in closed causal loops—assumptions that arevery natural in linear causal situations (where effects do not cause theircauses), but untenable in causal loops. (For an analysis of the nature ofprobabilities in causal loops, see Berkovitz 2001 and 2002, section 2.)Thus, he concludes that the consistency of relativistic parameter-dependent theories cannot be excluded on the grounds of causalparadoxes. He also argues that the real challenge for relativisticparameter-dependent theories is concerned with their predictive power. In

Joseph Berkovitz


parameter-dependent theories is concerned with their predictive power. Inthe causal loops predicted by relativistic parameter-dependent theories inArntzenius's suggested experiment, there is no known way to compute thefrequency of events from the probabilities that the theories prescribe.Accordingly, such theories would fail to predict any definite statistics ofmeasurement outcomes for that experiment. This lack of predictabilitymay also present some new opportunities. Due to this unpredictability,there may be an empirical way for arbitrating between these theories andquantum theories that do not predicate the existence such causal loops inArntzenius's experiment.

Another attempt to demonstrate the impossibility of certain relativisticquantum theories on the grounds of causal paradoxes is advanced byMaudlin (1994, pp. 195-201). (Maudlin does not present his argument inthese terms, but the argument is in effect based on such grounds.) Recall(sections 2 and 9) that a way to try to reconcile quantum mechanics withrelativity is to account for the curious correlations between distantsystems by local backward influences rather than non-local influences. Inparticular, one may postulate that the correlations between the distantmeasurement outcomes in the EPR/B experiment are due to localinfluences from the measurement events backward to the state of theparticle pair at the source. In such models of the EPR/B experiment,influences on events are always confined to events that occur in their pastor future light cones, and no non-locality is postulated. Maudlin arguesthat theories that postulate such backward causation will be inconsistent.More particularly, he argues that a plausible reading of Cramer's (1980,1986) transactional interpretation, and any other theory that similarlyattempts to account for the EPR/B correlations by postulating causationfrom the measurement events backward to the source, will beinconsistent.

Berkovitz (2002, section 5) argues that Maudlin's argument is, in effect,that if such retro-causal theories were true, they would involve closed



that if such retro-causal theories were true, they would involve closedcausal loops in which the probabilities of outcomes that these theoriesassign will certainly deviate from the statistics of these outcomes. And,similarly to Arntzenius's argument, Maudlin's argument also rests onuntenable assumptions about the nature of probabilities in causal loops(for a further discussion of Maudlin's and Berkovitz's arguments and,more generally, the prospects of Cramer's theory, see Kastner 2004).Furthermore, Berkovitz (2002, sections 2 and 5.4) argues that, similarly torelativistic parameter-dependent theories, the main challenge for theoriesthat postulate retro-causality is not causal paradoxes, but rather the factthat their predictive power may be undermined. That is, the probabilitiesassigned by such theories may fail to predict the frequency of events inthe loops they predicate. In particular, the local retro-causal theories thatMaudlin considers fail to assign any definite predictions for the frequencyof measurement outcomes in certain experiments. Yet, some othertheories that predicate the existence of causal loops, such as Sutherland's(2006) local time-symmetric Bohmian interpretation of quantummechanics, seem not to suffer from this problem.

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Other Internet Resources

Donald, M. (1999), Progress in a many-minds interpretation ofquantum theory.Fogel, B. (2004), The Structure of Quantum Holism. (PDF)Larsson, J-A and Semitecolos, J. (2000), Strict detector-efficiencybounds for n-site Clauser-Horne inequalities.Larsson, J. A. (2002), A Kochen-Specker inequality.Lewis, P. J. (2003b), Deutsch on Quantum Decision Theory.––– (2004), Interpreting spontaneous collapse theories (DOC file).Sutherland, R. I. (2006), Causally symmetric Bohm model (PDF).Tumulka, R. (2004), A relativistic version of the Ghirardi-Rimini-Weber model (PDF).

Joseph Berkovitz


Valentini, A. (2002), Signal-locality in hidden-variables theories(PDF).Valentini, A. and Westman, H. (2004), Dynamical origin of quantumprobabilities (PDF).Wallace, D. (2002), Quantum probability and decision theory,revisited, also available from PhilSci Archive.––– (2005a), Epistemology quantised: circumstances in which weshould come to believe in the Everett interpretation (PDF).––– (2005b), Quantum probability from subjective likelihood:improving on Deutsch's proof of the probability rule, also availablefrom PhilSci Archive.––– (2005c), Everett and structure.

Related Entries

Bell's Theorem | causation: causal processes | intrinsic vs. extrinsicproperties | physics: holism and nonseparability | physics: Reichenbach'scommon cause principle | quantum mechanics | quantum mechanics:Bohmian mechanics | quantum mechanics: collapse theories | quantummechanics: Everett's relative-state formulation of | quantum mechanics:Kochen-Specker theorem | quantum mechanics: many-worldsinterpretation of | quantum mechanics: modal interpretations of | quantumtheory: the Einstein-Podolsky-Rosen argument in | supervenience |Uncertainty Principle

Acknowledgments

For comments on earlier versions of this entry, I am very grateful toGuido Bacciagaluppi.

Notes to Action at a Distance in QuantumMechanics



1. Intuitively, the spin-component of a particle in a certain direction canbe thought of as its intrinsic angular momentum along that direction. But,as we shall see in section 5, the nature of spin properties depends on theinterpretation of quantum mechanics. In any case, the exact nature of thisquantity will not be essential for what follows in sections 1-4. Theimportant thing is that in various quantum states the properties of distantphysical systems may be curiously correlated.

2. Recall Bell's (1981) example of Bertlmann's socks. ‘Dr. Bertlmannlikes to wear two socks of different colours. Which colour he will have ona given foot on a given day is quite unpredictable. But when you see thatthe first sock is pink you can already be sure that the second sock will notbe pink. Observation of the first, and experience of Bertlmann, givesimmediate information about the second.’

3. Two comments:

(i) In some Bell-type models of the EPR/B experiment, it is assumed thatin addition to the pair's state and the settings of the measurementapparatuses, there are other factors that may be relevant for theprobabilities of measurement outcomes. In particular, in his presentationof stochastic, local models of the EPR/B experiment Bell (1971, p. 37)assumes that the setting of the apparatuses need not specify their entirerelevant states. The outcomes may also be influenced by some otheraspects of the apparatus microstates, which may be different for the samesettings (see also Jarrett 1984). More generally, in addition to the state ofthe L- (R-) particle and the setting of the L- (R-) measurement apparatus,there may be some other (local) physical quantities that are relevant forthe probability of the L- (R-) measurement outcome. That is, letting α andβ denote all the relevant local physical quantities (other than the settings)that are relevant for the probability of the L- and the R-outcomerespectively, in such models the single and joint probabilities of outcomeswill be: Pλ l α(xl), Pλ r β(xl) and Pλ l r α β(xl & yr). We shall refer to this

Joseph Berkovitz


will be: Pλ l α(xl), Pλ r β(xl) and Pλ l r α β(xl & yr). We shall refer to thistype of models in section 7. But, for the sake of simplicity, in the rest ofthis entry we shall focus on the simpler models above.

(ii) There are two different approaches to modeling the probabilities inBell-type models of the EPR/B experiment: The many-spaces and thebig-space approaches (see Butterfield 1989, 1992a). In the many-spaceapproach, which we use in this review, each triple of pair's state, L- andR-setting labels a different probability space of measurement outcomes.For example, letting l and l′ be different L-apparatus settings, theprobability Pλ l r(xl & yr) belongs to one probability space, whereas theprobability Pλ l′ r(xl′ & yr) belongs to another. By contrast, in the big-space approach, all the probabilities of a Bell-type model belong to onebig probability space. In this approach the probabilities of outcomes areexpressed in terms of conditional probabilities. For example, theprobabilities P(xl & yr / λ & l & r) and P(xl′ & yr / λ & l′ & r) correspondto Pλ l r(xl & yr) and Pλ l′ r(xl′ & yr), respectively. (Note that in contrast tothe above notation, in the literature probabilities of spin-measurementoutcomes in the big-space approach are frequently expressed asconditional probabilities of non-specific spin-outcomes, i.e. the non-specific outcomes ‘up’ or ‘down’, given certain settings: P(x & y / λ & l& r), where ‘x’ and ‘y’ denote non-specific outcomes.) Mathematically,the two approaches can easily be related to each other. In particular, onecan construct a big-probability space in which the conditionalprobabilities of outcomes, given a pair's state and apparatus settings, areequal to the corresponding unconditional probabilities in the many-spacesapproaches: P(xl & yr / λ & l & r) = Pλ l r(xl & yr), P(xl′ & yr / λ & l′ &r) = Pλ l′ r(xl′ & yr), etc. But, conceptually the two approaches aredifferent. First, in contrast to the many-spaces approach, in the big-spaceapproach it is presupposed that settings always have definite probabilities(Butterfield 1989, p. 118, 1992a, section 2). Secondly, some of theprobabilities of the big-space approach, e.g. P(xl / λ & r), have no



probabilities of the big-space approach, e.g. P(xl / λ & r), have nocorrespondence in the many-spaces approach. Third, as we shall see inthe next section, factorizability can be analyzed into two conditions:parameter independence and outcome independence. Berkovitz (2002)argues that the meaning of parameter independence need not be the samein the two different approaches. That is, in some circumstances theparameter independence of the big-space approach expresses differentproperties than the parameter independence of the many-spaces approach.Indeed, in these circumstances the parameter independence of the many-spaces approach fails, whereas the parameter independence of the big-space approach holds. For arguments for the superiority of the many-spaces approach over the big-space approach, see Butterfield (1989, p.118), and Berkovitz (2002, section 4.2).

4. Or when the range of the values of λ is discrete,

5. For a dissenting view, see Fine (1981, 1982a), Cartwright (1989) andChang and Cartwright (1993). We shall discuss this view in section 9.

6. While the fullest analysis of factorizability is due to Jarrett, precursorsare Suppes and Zanotti (1976) and van Fraassen (1982).

7. See van Fraassen (1982) and Jarrett (1984, 1989).

8. As we shall see below, in the literature the term ‘interpretation’ isfrequently used to refer to alternative quantum theories. The question ofwhether this use is justified and the criteria for distinguishing between aninterpretation of orthodox quantum mechanics and an alternative quantumtheory will be insubstantial for the considerations below.

Pψ l r(xl & yr) = ∑λ Pλ l r(xl & yr) · ρψ l r(λ),Pψ l (xl) = ∑λ Pλ l (xl) · ρψ l(λ), and Pψ r(yr) = ∑λ Pλ r(yr) · ρψ r(λ).

Joseph Berkovitz


9. For a history of the notion of action at a distance, see Hesse (1969).

10. In reality, the position of different particles will be different: |upi>pi(|downi>pi). But this is immaterial for the analysis below.

11. This is a variant of the so-called ‘tails problem’ (see the entry oncollapse theories, section 12, and Albert 1992, chapter 5).

12. For a recent interesting discussion of Newton's view of action at adistance, see Henry (1994), and references therein.

13. For the Clarke-Leibniz correspondence, see Alexander (1956).

14. Of course, here ‘field’ is not intended to mean a field in the sense ofquantum field theory.

15. For discussions of this version of Bohm's theory, see for exampleDürr, Goldstein and Zanghì (1992a), Albert (1992) and Cushing (1994).

16. The wave function propagates according to the Schrödinger equationin the ‘configuration’ space of the particles, which for an N-particlesystem is a 3N-dimensional space, coordinatized by the 3N positioncoordinates of the particles. For more details, see the entry on Bohmianmechanics.

17. Here follows a more technical account of the above experimentaccording to the minimal Bohm theory. Let the wave function, i.e. thestate of the guiding field, before any measurement occurs be:

where f1(z1) and f2(z2) are non-overlapping Gaussian wavepackets; z1and z2 are respectively the positions of particle 1 and particle 2 along thez-direction; and |z-up> and |z-down> are z-spin eigenstates (i.e. z-spin‘up’ and z-spin ‘down,’ respectively). Suppose that we perform a z-spin

ψ = 1/√2f1(z1) f2(z2) ( |z-up>1 |z-down>2 − |z-down>1 |z-up>2),



‘up’ and z-spin ‘down,’ respectively). Suppose that we perform a z-spinmeasurement on particle 1, and switch off the R-apparatus. Then(suppressing for simplicity's sake the free time evolution of the twowavepackets as they move towards their respective Stern-Gerlach devicesand the states of the Stern-Gerlach devices), the state of the guiding fieldduring the L-measurement will be:

where g1 is the coupling constant for the spin measurement on particle 1(coupling the position and the spin degrees of freedoms that are related tothe guidance of particle 1); and T is the duration of the measurement.Since the guiding field of the particle pair factorizes into f2(z2) and ( f1(z1+ g1T) |z-up>1 |z-down>2 − f1(z1 − g1T) |z-down>1 |z-up>2), it followsfrom the guiding equation that particle 2's velocity along the z-axis doesnot depend on particle 1's position.

18. See Bohm, Schiller and Tiomno (1955), Dewdney, Holland andKyprianidis (1987), Bohm and Hiley (1993, chapter 10), and Holland(1993).

19. While in the minimal and the non-minimal Bohm theories, the wavefunction is interpreted as a field, Dürr, Goldstein and Zanghì (1997,section 12) propose that the wave function should be interpreted as aparameter of a physical law. This, they argue, may explain why there isno action of configurations of particles on wave functions.

20. For discussions of the above experiment in the non-minimal theory,see Dewdney, Holland and Kyprianidis (1987), Bohm and Hiley (1993,section 10.6), and Holland (1993, section 11.3).

21. For discussions of the prospects of relativistic modal interpretations,see Dickson and Clifton (1998), Arntzenius (1998), Myrvold (2002a),Earman and Ruetsche (2005) and Berkovitz and Hemmo (2005, 2006a,b).

1/√2 f2(z2) ( f1(z1 + g1T) |z-up>1 |z-down>2 − f1(z1 − g1T) |z-down>1 |z-up>2)

Joseph Berkovitz


Earman and Ruetsche (2005) and Berkovitz and Hemmo (2005, 2006a,b).We shall discuss this issue at the end of this section and in section 10.2.

22. If some of the ci are degenerate, the Schmidt biorthogonaldecomposition is not unique, and the properties assigned by the aboverule are projections onto multi-dimensional subspaces.

23. Note the difference between |ψ9> and the singlet state |ψ3>. In |ψ9>,the coefficients of the two branches of the superposition are unequal. Andwhile EPR/B-like experiments can be prepared with both the state |ψ3>and the state |ψ9>, the difference between these states is significant forthe above interpretation. For unlike |ψ3>, |ψ9> has a unique factorization.Accordingly, the L- and the R-particle each have definite spin propertiesin the state |ψ9> but not in |ψ3>.

24. In fact, as we shall see later in this section (in the discussion of‘property composition’), this claim needs some qualification.

25. In the original modal interpretations, the question of the relationbetween the dynamics of properties of systems and the dynamics of theproperties of their subsystems has been largely overlooked. For adiscussion of this issue, see Vermaas (1997, 1999), Berkovitz and Hemmo(2005, 2006a,b).

26. Similarly to any other physical object, the brain of a human observerhas many different sets of relational properties, i.e. sets of properties thatare related to different systems. Brain properties that are defined relativeto different systems are generally different. Thus, the question arises as towhich of these different brain properties are correlated to our beliefsabout the properties of physical systems that figure in our experience. Fora discussion of this question, see Berkovitz and Hemmo (2005, section 6,2006b, section 6).



27. The challenge is to explicate the nature of such holistic properties andto relate them to our experience.

28. That is, the property of a system is given by the spectraldecomposition of its so-called ‘reduced state’ (a statistical operatorobtained by a partial tracing). For example, the reduced state of the L-particle in the state |ψ9> is obtained by a partial tracing of |ψ9> over theHilbert space of the R-particle.

30. The above formulation of screening off is motivated by the fact thatwe work in the framework of the many-spaces approach to theprobabilities of outcomes in Bell-type models of the EPR/B experiment.In the literature, the formulation of screening off is slightly different:

30. In his celebrated theorem, Bell did not mention Reichenbach'sprinciple or FactorUCP. But it is reasonable to assume that he had inmind some similar principles.

31. There are some obvious candidates for superluminal signaling. First,the potentials in the Schrödinger equation are Newtonian. Therefore, ifone is allowed to vary the potential somewhere, this will be feltinstantaneously throughout space. But, in the context of this entry suchsuperluminal signaling is less interesting because it will be due toNewtonian effects rather than quantum effects. Second, wave functionscan spread instantaneously: If you have a particle confined to a box (sothat its wave function is zero outside the box) and open the box, the wavefunction will instantaneously be non-zero everywhere, and superluminalsignaling will be possible. It is noteworthy, however, that the preparationof such state requires the existence of an infinite potential barrier—a statethat is impossible. In any case, in what follows we shall focus on the

P(x/y & CC(x,y)) = P(x / CC(x,y)) P(y / CC(x,y)) ≠ 0

P(y/x & CC(x,y)) = P(y / CC(x,y)) P(x / CC(x,y)) ≠ 0

Joseph Berkovitz


that is impossible. In any case, in what follows we shall focus on thequestion of whether the non-locality in the EPR/B experiment, as depictedby various interpretations of quantum mechanics, can be exploited to giverise to superluminal signaling.

32. Note that according to this suggestion, the statistical predictions ofBohm's theory slightly deviate from the statistical predictions of orthodoxquantum mechanics.

33. It is noteworthy, however, that while separability does not imply OI,the prospects of separable models that violate OI are dim. To see why, letus consider Maudlin's (1994, p. 98) criticism of Howard's claim that OIfollows from spatiotemporal separability. Maudlin invites us to considerthe following model for the EPR/B experiment. Suppose that each particlehad some means of superluminal communication, which may be realizedby a tachyon. Suppose also that each of the particles carries the sameinstructions: If it arrives at a measurement apparatus without havingreceived a message from its partner, and the measurement apparatus is ina state of being ready to measure z-spin, its state of not having definite z-spin evolves with equal chance to either the state of having z-spin ‘up’ orthe state of having z-spin ‘down.’ It then communicates to its partner thesetting of its measurement apparatus and the outcome of themeasurement. If it receives a message from its partner, its state ismodified accordingly, so that the new chance of spin outcomes agreeswith the predictions of orthodox quantum mechanics. Such a model willinvolve a violation of OI, but by construction it is separable: The particlesand the tachyons have separable states at all times, and the joint state ofany two systems is just the product of their individual states. Yet, themodel will be separable in the intended sense only if the above set ofcommunication instructions could be encoded into the qualitative,intrinsic properties of each of the particles, and each of the particles couldkeep an open line of communication with its partner and no otherparticles. But a little reflection on the grave difficulties involved with that



particles. But a little reflection on the grave difficulties involved with thattask suggests that the physical feasibility and plausibility of any suchseparable model of the EPR/B experiment will be highly questionable.

34. Friedman (1983, sections 4.6-4.7) holds that special relativity per sedoes not prohibit superluminal signaling, but that such signaling will leadto paradoxes of time travel. Maudlin (1994, pp. 112-116) argues thatsuperluminal signaling need not imply such paradoxes, as the conditionsfor them are much more complex than merely the existence ofsuperluminal signaling.

35. As Maudlin (1996, pp. 292-293) notes, it is not clear that a generalcriterion for identifying a structure of spacetime as intrinsic could befound.

Copyright © 2009 by the author Joseph Berkovitz

Joseph Berkovitz


Action and Distance in Quantum Mechanics

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