Aspects of Quantum Non-Locality II: Superluminal Causation and Relativity

Stud. Hist. Phil. Mod. Phys., Vol. 29, No. 4, 509 — 545, 1998  1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 1355-2198/98 $19.00#0.00

Aspects of Quantum Non-Locality II: Superluminal Causation and Relativity Joseph Berkovitz* In a preceding paper, I studied the significance of Jarrett’s and Shimony’s analyses of ‘factorisability’ into ‘parameter independence’ and ‘outcome independence’ for clarifying the nature of non-locality in quantum phenomena. I focused on four types of non-locality; superluminal signalling, action-at-a-distance, non-separability and holism. In this paper, I consider a fifth type of non-locality: superluminal causation according to ‘logically weak’ concepts of causation, where causal dependence requires neither action nor signalling. I conclude by considering the compatibility of non-factorisable theories with relativity theory. In this connection, I pay special attention to the difficulties that superluminal causation raises in relativistic spacetime. My main findings in this paper are: first, parameter-dependent and outcome-dependent theories both involve superluminal causal connections between outcomes and between settings and outcomes. Second, while relativistic deterministic parameter-dependent theories seem impossible on pain of causal paradoxes, relativistic indeterministic parameter-dependent theories are not subjected to the same challenge. Third, current relativistic non-factorisable theories seem to have some rather unattractive characteristics.  1998 Elsevier Science Ltd. All rights reserved.

1. Introduction Bell’s theorem asserts that granted very plausible assumptions, Bell-type experiments violate the so-called ‘factorisability’ condition. In Berkovitz (1998a) (henceforth, Part I), I considered the significance of Jarrett’s (1984, 1989) and Shimony’s (1993, Chapters 10 and 11) analyses of ‘factorisability’ for clarifying the nature of non-locality in quantum phenomena (as represented by (Received 26 April 1996; revised 19 February 1998) * Department of Philosophy, Logic and Scientific Method, The London School of Economics, Houghton Street, London WC2A 2AE, U.K. (e-mail: [email protected]). PII: S13 5 5- 2 1 98 (9 8) 0 0 02 4 - 0

509

510

Studies in History and Philosophy of Modern Physics

non-factorisable theories). In my considerations, I focused on the questions of superluminal (i.e. faster-than-light) signalling, action-at-a-distance, non-separability and holism. In this paper, I continue my investigation of quantum non-locality. In Section 2, I will study the significance of Shimony’s analysis of factorisability into ‘parameter independence’ and ‘outcome independence’ for clarifying the nature of causality in quantum phenomena. My focus will be on ‘logically weak’ notions of causation where a causal dependence between events requires neither signalling nor action between them. In Subsections 2.1—2.3, I will consider the view that while (a) outcome dependence involves superluminal causal dependence between distant outcomes, (b) parameter dependence involves superluminal causal dependence between a nearby setting and a distant outcome. (For simplicity’s sake, I will concentrate on ideal measurements. Also, for brevity’s sake, I will henceforth use the expression ‘the nature of causality in a certain theory’ to mean the nature of causality in worlds described by this theory.) Recently, Maudlin argued that this view is misleading since a different analysis of factorisability leads to opposite conclusions (1995, p. 94). In Subsections 2.1—2.2, I will consider Maudlin’s argument and argue that it is flawed. Then, in Subsection 2.3, I will show that claims (a) and (b) can be motivated by Lewis’ (1986) well developed and influential theory of causation. This may naturally suggest that outcome-dependent theories involve different superluminal Lewisian causal connections than parameter-dependent theories. However, as I will argue in Subsections 2.3 and 2.4, parameter-dependent theories also involve superluminal Lewisian causal dependence between distant outcomes, and outcome-dependent theories also involve superluminal Lewisian causal dependence between a nearby setting and a distant specific outcome. The reasoning of Subsections 2.3 and 2.4 is limited to non-relativistic theories since Lewis’ theory of causation presupposes a non-relativistic spacetime. Moreover, it relies on a controversial (‘non-backtracking’) resolution of the vagueness of counterfactuals. According to this resolution, the truth of counterfactuals—and thus of claims about causation—in a certain world (say, the actual situation) sometimes depends on the characteristics of worlds (counterfactual situations) that violate the laws of that world. After briefly considering these difficulties in Subsection 2.5, I will propose in Subsection 2.6 a theory of causation that overcomes them. I will then argue that

 Two remarks: (i) hereafter, by ‘quantum phenomena’, I will mean quantum phenomena as represented by non-factorisable theories, i.e. theories that violate factorisability; (ii) recall that factorisability, (2.1) below, asserts that (in Bell-type experiments) the probability of joint outcomes is different from the product of the probabilities of single outcomes.  Recall that parameter independence, (2.2) below, asserts that the probability of a distant outcome is independent of the nearby setting; and outcome independence, (2.3) below, asserts that the probability of a distant outcome is independent of the nearby outcome. For lack of space, I will not consider the significance of Jarrett’s analysis for clarifying the nature of causality in non-factorisable theories.

Aspects of Quantum Non-Locality

511

this theory, which is partially related to Mellor’s (1995), supports my conclusions in Subsections 2.3 and 2.4. In Section 3, I will turn to consider the question of the compatibility of non-factorisable theories with relativity. Subsections 3.1 and 3.2 will focus on parameter-dependent theories. In Subsection 3.1, I will argue that if relativistic parameter-dependent theories existed, they would involve backwards causation. Backwards causation is not a very attractive feature to have. However, one would not have to worry about this challenge if relativistic parameter-dependent theories were impossible. I will argue in Subsection 3.2 that a relativistic deterministic parameter-dependent theory seems impossible on pain of causal paradoxes. Arntzenius (1994, Section 5) argued that relativistic indeterministic parameter-dependent theories would similarly be impossible. I will argue that Arntzenius’s argument does not demonstrate that relativistic indeterministic parameter-dependent theories cannot exist. It rather seems to show that if such theories were possible, in certain causal loops causal consistency would require that the parameter dependence not be manifested. Finally, in Subsection 3.3, I will consider the prospects of (genuinely) relativistic non-factorisable theories. The prospects of relativistic ‘collapse’ theories are not very promising. It is commonly claimed that the prospects of relativistic ‘no-collapse’ theories are better. I will argue that while some versions of the so-called ‘modal interpretation’ of quantum mechanics are relativistic, they have some rather unattractive properties. Everett-type interpretations are also interesting candidates for relativistic ‘no-collapse’ interpretations, but they lie beyond the scope of the present discussion.

2. Superluminal Causation A common view has it that quantum phenomena require the existence of neither superluminal signalling nor action-at-a-distance. Since action and signalling are frequently considered as types of causation, one might be tempted to conclude that these phenomena do not necessarily require superluminal causation. But this conclusion is unwarranted since in various accounts of causation, action and signalling are not necessary conditions for causal dependence. Focusing on such accounts, the question arises then: do quantum phenomena require superluminal causal connections? A number of authors argue that the failure of factorisability indicates the existence of superluminal causation (cf. Bell, 1987, Chapters 7, 12, 16; Butterfield, 1994; Maudlin, 1994, Chapter 4; Berkovitz, 1995), whereas others reject this view (cf. Redhead, 1986, 1989; Cartwright, 1989, Chapter 6; Teller 1989; Chang and Cartweight, 1993; Dickson, 1996). But even among those who think that quantum phenomena require superluminal causation, there are some who question the significance of Shimony’s analysis of factorisability into parameter independence and outcome independence for clarifying the nature of superluminal

512

Studies in History and Philosophy of Modern Physics

causation in these phenomena. In particular, Maudlin recently argued that the interpretation of Shimony’s analysis of factorisability as introducing constraints on the causal role of settings, i.e. parameter independence, and constraints on the causal role of outcomes, i.e. outcome independence, leads to a reductio (1994, p. 95). In Subsection 2.2, I will consider Maudlin’s argument. But, first, to prepare the ground, I need to review the Bell-type experiment, the so-called ‘big-space’ approach to Bell’s theorem and Shimony’s analysis of factorisability. (As we shall see, this review will also be important for our considerations in Subsection 3.2.). 2.1. The big-space approach to Bell+s theorem Recall the Bell-type experiment (Part I, Section 2). Pairs of particles are emitted in opposite directions, ¸ (left) and R (right). When the particles are far apart (i.e. space-like separated), each of them encounters a measurement apparatus, which can be set to measure one of two different physical quantities: say l and l in the ¸-wing and r and r in the R-wing. In each run of the experiment, the measurements of any of these quantities can yield one of two different outcomes. Let x and x denote two opposite non-specific measurement outcomes in the ¸-wing, e.g. spin ‘up’ and spin ‘down’ without any specification of the quantity measured; and similarly for y and y in the R-wing. Let x and J (x) denote two opposite specific outcomes in a measurement of the quantity J l in the ¸-wing, say spin ‘up’ and spin ‘down’ in the l-direction. Similarly, let y P and (y) denote two opposite specific outcomes in the measurement of the P quantity r in the R-wing. It is commonly assumed that the pair’s state and the settings of the apparatuses jointly determine probabilities for single and joint specific outcomes. A theory for the Bell-type experiments is called ‘factorisable’ iff it satisfies the following condition: Factorisability. For any j, l, r, x and y , J P P(x & y /j & l & r)"P (x /j & l) ) P (y /j & r). (2.1) J P J P Bell’s theorem asserts that granted very plausible assumptions, factorisability is violated in Bell-type experiments (Part I, Sections 2 and 3). Shimony (1993, Chapters 10 and 11) analyses factorisability into two logically independent conditions: Parameter independence. The probability of a distant specific outcome is independent of the nearby setting. That is, for any j, l, l, r, r, x and y , if J P  In Part I, the distinction between specific and non-specific outcomes went unmentioned until the very end (in Subsection 6.3). However, since this distinction will play an important role in the discussion below, from now onward I will make it explicit.

Aspects of Quantum Non-Locality

P(x /j & l &r), P (x /j & l& r) and P(y /j & l & r) are non-zero, then: J J P P (x /j &l & r)"P (x /j & l & r)"P(x /j & l), J J J P(y /j & l & r)"P(y /j & l & r)"P(y /j & r); P P P and

513

(2.2)

Outcome independence. The probability of a distant specific outcome is independent of the nearby specific outcome. That is, for any j, l, r, x and y , if J P P(j & l & r & y ) and P (j & l & r & x ) are neither zero nor one, then: P J P(x /j & l & r & y )"P (x /j &l& r & (y) ), J P J P P(y /j & l & r & x )"P (y /j &l& r & (x) ). (2.3) P J P J Thus, in studying the nature of causality in quantum phenomena, attention has focused on the failures of outcome independence and parameter independence. Before turning to this study, let us reflect for a moment on the nature of the probabilities in (2.1)—(2.3). Note that by placing j, l and r after the conditionalisation stroke, one is in effect committed to the assumption that there is a ‘big’ probability space over j, l, r, x and y, whose events are all the various possible quintuples 1j, l, r, x, y2. In Part I, I used a different approach. In that different approach, where j, l and r appear in the subscript, there are many probability spaces, each labelled by either a triple 1j, l, r2, or a pair 1j, l2 (1j, r2). Each of the spaces labelled by 1j, l, r2 has four events: 1x , y 2, 1x , (y) 2, etc. And each J P J P of the probability spaces labelled by 1j, l2 (1j , r2) has two possible events x and J (x) (y and (y) ). J P P Mathematically, the two approaches can easily be related to each other (cf. Butterfield, 1989, p. 118; 1992a, Section 2). In particular, one may postulate that the conditional probabilities of the big-space approach, e.g. P (x /j, l,r), be equal J to the corresponding probabilities of the many-spaces approach, Pj, l, r (x ). Thus, J many believe that the differences between the two approaches are insiginificant for clarifying the metaphysical implications of quantum phenomena. In fact, there are authors who believe that the big-space approach is even preferable since it enables some probabilistic inferences that cannot be carried out in the many-spaces approach. But, conceptually, the two approaches are different. First, quantum theories assign definite values to the probabilities of the many-spaces approach, e.g. Pj,l, r (x ª ), but not to the corresponding probabilities of the big-space approach, J P (x /j, l, r). In these theories, the pair’s state, and the setting of the apparatuses J before the measurements occur jointly assign probabilities for outcomes. Thus, Pj, l, r (x ), i.e. the probability that j, l and r jointly assign to x , has a definite value. J J  For an anticipation of this analysis, see Van Fraassen (1982).  Here and henceforth, I will use ‘l’ and ‘r’ to denote certain measured quantities in some contexts and variables that range over measured quantities in others. Context will distinguish between these different uses.

514

Studies in History and Philosophy of Modern Physics

By contrast, quantum theories do not assign any definite value to probabilities of pair’s states, settings, outcomes and their conjunctions, at least if no further characteristics of the experiment are specified. Accordingly, these theories do not assign definite values to P (x & j & l & r), P (j & l & r) and (thus) J P (x /j & l & r). J Agreed, when all the relevant facts about the preparation of the Bell-type experiment are specified, these probabilities might have definite values. But, first, the assumption that such facts will always be sufficient for determining these probabilities is disputable. For instance, some authors believe that when the settings are decided by experimenters, they are good candidates for not having objective probabilities (cf. Butterfield, 1989, p. 118). So, according to this view, P (x /j & l & r) (interpreted as objective probability) will not necessarily have J a definite value. Second, some of the probabilities of the big-space approach, e.g. P (y/j & l), have no definite probabilities in the many-spaces approach; for j and l do not jointly assign a definite probability for y. Third, as we shall see in Subsection 3.2, the connection between the probabilities of the two approaches is not always straightforward. Fourth, as I will suggest in Subsection 3.2, the nature of the many-spaces probabilities is different from that of the big-space probabilities. The probabilities of the many-spaces approach are naturally interpreted as chances, whereas the probabilities of the big-space approach are not: they are better thought off as ‘long-run’ relative frequencies. But, as it turns out, in some set-ups, parameter dependence fails to be manifested in the relevant long-run relative frequencies. Thus, in drawing conclusions from the big-space probabilities, much care is needed. For these reasons, I prefer to use the many-spaces approach. Nevertheless, since (as we shall see below) the many-spaces approach would diminish the apparent plausibility of Maudlin’s argument, for the time being I will be using the big-space approach. 2.2. Maudlin+s reductio argument Maudlin (1994, p. 95) argues that Shimony’s analysis is not unique. Factorisability can also be analysed into the conjunction of two different logically independent conditions: M-parameter independence and M-outcome independence. M-parameter independence. The probability of a distant specific outcome given a nearby non-specific outcome is independent of the nearby setting. That is, for any j, l, r, x, x , y and y , if P(j & l & r & y) and P (j & l & r & x) are non-zero, J P then: (i) P (x / j & l & r & y)"P (x /j & l & y), J J (ii) P (y /j & l & r & x)"P(y /j & r & x). P P

(2.4)

Aspects of Quantum Non-Locality

515

M-outcome independence. The probability of a distant specific outcome is independent of the nearby non-specific outcome. That is, for any j, l, r, x, x , y and J y , if P (j & l & y) and P (j & r & x) are non-zero, then: P (i) P (x /j & l & y)"P (x /j & l), J J (ii) P (y /j & r & x)"P (y /j & r). (2.5) P P Maudlin calls (2.4) ‘parameter independence’ since it concerns conditionalising on a distant setting, and he calls (2.5) ‘outcome independence’ since it concerns conditionalising on a distant (non-specific) outcome. On the basis of this alternative analysis, Maudlin argues that the interpretation of Shimony’s analysis of factorisability as introducing constraints on the causal role of settings, i.e. parameter independence, and constraints on the causal role of outcomes, i.e. outcome independence, leads to a reductio. His argument runs as follows. Orthodox quantum mechanics (henceforth, QM) violates outcome independence and satisfies parameter independence. But, claims Maudlin, this theory violates M-parameter independence and satisfies M-outcome independence. Thus, if parameter dependence and M-parameter dependence each imply a causal dependence between a nearby setting and a distant outcome, outcome dependence and M-outcome dependence each imply a causal dependence between distant outcomes, and a lack of each of these probabilistic dependencies implies the corresponding causal independence, then the following absurdity ensues. According to Maudlin’s analysis of factorisability, QM involves superluminal causal dependence between a nearby setting and a distant outcome, and it does not involve a superluminal causal dependence between distant outcomes. On the other hand, according to Shimony’s analysis the same theory gives the opposite verdict. Maudlin’s argument relies on two premises. The first is that QM violates M-parameter independence and satisfies M-outcome independence, and the second is that a probabilistic dependence between two events is a necessary and sufficient condition for causal dependence between them. The first premise is false. As I have already noted in Subsection 2.1, the pair’s state and the distant setting do not (jointly) prescribe any definite probability for a nearby (specific or non-specific) outcome. Thus, unless we specify the nearby setting, or at least the experimental set-up that determines this setting, the left-hand side probabilities in (2.4) and (2.5). P(x /j & l & y) and P (y /j & r & x), J P have no definite values. Accordingly, M-parameter independence and M-outcome independence are neither true nor false in QM. This fact is more transparent in the many-spaces approach. In this approach, the corresponding probabilities, P (x /y) and Pj, r (y /x), do not exist since HJ J P Pj, l (y) and Pj,r(x) do not exist. In the big-space approach, it is easier to lose sight  In his notation, Maudlin does not make explicit the distinction between specific and non-specific outcomes, But, in presenting his analysis, he makes such a distinction. Anyway, the notation above gives more credibility to Maudlin’s suggestion that the violation of (2.4) would indicate parameter dependence and the violation of (2.5) would indicate outcome dependence.

516

Studies in History and Philosophy of Modern Physics

of this indefiniteness: for this approach is committed to the existence of P(x /j & l & y) and P (y /j & r & x). Accordingly, Maudlin’s argument gains J P some apparent plausibility by using this approach. So the prospects of M-parameter independence and M-outcome independence depend on the setting or the setting procedure of the nearby measurement apparatus. For instance, if the choice of the nearby setting is made randomly with a uniform distribution over all possible spin quantities, QM will satisfy M-outcome independence and violate M-parameter independence. On the other hand, if the nearby apparatus is set to measure one of two spin quantities in perpendicular directions, and the choice between these quantities is made randomly with certain chance, QM will violate both M-outcome independence and M-parameter independence. Yet, in some other set-ups, e.g. when the distant apparatus is set in advance to measure a certain fixed spin quantity, this theory will violate M-outcome independence and satisfy M-parameter independence. All these may not seem fatal to Maudlin’s argument. A similar argument could run with a weaker premise: for example, that QM sometimes violates M-parameter independence and satisfies M-outcome independence. But, since M-parameter independence and M-outcome independence depend on the particular setting procedure of the nearby apparatus, the violation or the lack of violation of these conditions are prima facie unreliable indicators of the nature of causality in non-factorisable theories. For at least according to current wisdom, the nature of causal dependencies between the distant wings is not supposed to depend on such facts about the setting procedure. Indeed, the sensitivity of M-outcome independence to the particular setting procedure in QM is not due to any exotic dependence between nearby setttings or setting procedures and distant outcomes. It is rather due to the fact that QM involves a probabilistic dependence between distant specific outcomes, henceforth specific outcome dependence. Outcome dependence is an instance of this condition: it asserts that the probability of a distant specific outcome, say spin ‘up’ in the l-direction, is dependent on whether the nearby outcome is ‘up’ or ‘down’ in the r-direction. Another instance of specific outcome dependence asserts that the probability of a distant specific outcome, say spin ‘up’ in the l-direction, is dependent on whether the nearby outcome is spin ‘up’ (‘down’) in the r-direction or spin ‘up’ (‘down’) in the r-direction. (For more about this issue, see Subsection 2.4.) The specific outcome dependence of QM could easily explain why the prospects of M-parameter independence and M-outcome independence depend on the setting procedure of the nearby measurement device. For instance, when the choice of the nearby setting is made randomly with a uniform distribution over all possible spin quantities, M-outcome independence obtains because the specific outcome dependence between distant outcomes is washed out. Here is why. Given the above setting procedure, the probability on the left-hand side of (2.5i) can be expanded as follows:

P P(xJ /j & l & r & y) ) f (r)dr ;

P(x / j & l & y)" J

(2.6)

Aspects of Quantum Non-Locality

517

where f ( ) ) is a uniform distribution over all possible spin quantities in the R-wing. (Note that the conditional probability in the right-hand side of (2.6) can be read as an implicit expression for the probability of a specific ¸-outcome given a specific R-outcome.) Now, in (measure) half of all the possible settings in the R-wing, a spin ‘up’ in the R-outcome would increase the probability of spin ‘up’ in the l-direction, whereas in the rest of them it would decrease this probability. Since the weighted average of the increase equals the weighted average of the decrease, (on the average) the probability of a spin ‘up’ in the l-direction would be independent of a non-specific outcome ‘up’ in the R-wing. Thus, M-outcome independence (for the ¸-wing, i.e. (2.5i) above) would obtain. In contrast to M-parameter dependence and M-outcome independence, the outcome independence, (or more generally) the specific outcome dependence and the parameter independence of QM are all independent of the setting procedure; they always hold (if the relevant settings and pair’s state obtain). Thus, prima facie, they are better indicators of the nature of causality in this theory. But, even if we ignored all these difficulties, Maudlin’s reductio argument would still be questionble since it relies on another disputable premise: namely, the assumption that probabilistic dependence between two events is a necessary and sufficient condition for causal dependence between them. For probabilistic dependence per se is generally a poor guide to causal dependence. Indeed, as we shall see in Subsection 2.4, Shimony’s and Maudlin’s different analyses of factorisability do not lead to any contradictions concerning causality. The question arises again then: can Shimony’s analysis be interpreted as analysing factorisability into constraints on the causal role of outcomes and on the causal role of settings? The answer to this question is not straightforward since there are many accounts of causation, and different accounts might give rise to different verdicts. Obviously, I cannot consider all the various accounts of causation. Rather, I will focus on a prevalent view that maintains that, in causation in the single case, causes influence the chance of their effects. My main motivation for this choice is that if Shimony’s and Jarrett’s analyses of factorisability were significant for clarifying the nature of causality in non-factorisable theories, such accounts of causation would be natural candidates for rendering this significance. I start with Lewis’s (1986) well developed and influential theory of causation. 2.3. Superluminal causation: *non-backtracking reasoning+ For Lewis, an event e depends causally on a distinct event c iff c raises the chance of e significantly above what it would have been without c in the circumstances (1986, pp. 175—184). Let N denote the counterfactual conditional as analysed by Lewis (1973; 1986, Chapters 16—17), c and e denote distinct particular events as characterised by Lewis (1986, Chapter 23) and O(c) denotes that c occurs. Then e depends causally on c iff for some p (0)p)1) the

518

Studies in History and Philosophy of Modern Physics

following counterfactuals obtain: (i) O(c) N ch [O(e)]"p,

(ii) O(c) N ch[O(e)]p,

(2.7)

where ‘’ means ‘lower by a large factor—not by a large difference’, and ch()) denotes the chance function just after c occurs or does not occur, as the case may be (1986, p. 177). (To avoid cumbersome notation, I will henceforth use ‘c’ to denote the event c in some contexts and its occurrence O(c) in others. Similarly for other events. Context will distinguish between these different uses.) I will follow Butterfield (1992a, p. 52; 1992b, pp. 30 and 32—33) and weaken Lewisian causal dependence. I will thus suppose that e causally depends on c just in case the chance of e would not have been the same if c had not occurred, i.e. if the following causal independence condition fails: p (0)p)1) such that (i) c N ch(e)"p and (ii)  c N ch(e)"p, (2.8) where ch ( ) ) denotes the chance function just after the time c occurs or does not occur. Butterfield’s main motivation for this modification is that if we deny superluminal causal dependence between events, we surely want there to be no superluminal causal influence, no matter how weak (1992a, p. 52; 1992b, p. 33). I argue elsewhere that Lewis’ (1986, Chapter 27) causal decision theory provides another motivation. For granted Lewis’ own definition of causal dependence, a follower of his causal decision theory will sometimes be better off if she decides to ignore the causal dependence between her optional actions and the things she cares about (Berkovitz, 1998b). Four key elements in Lewis’s analysis will be significant for our discussion below. First, the resolution of the vagueness of the counterfactuals in (2.7) and (2.8) is governed by the idea that counterfactuals are (generally) not backtrackers. Recall that a counterfactual, e.g., ‘if c had not occurred ...’, is called ‘non-backtracker’ if its supposition leads to possible worlds which match the actual world in matters of particular fact up to just before the time of c. (For more details, see Subsection 2.5.) Second, in Lewis’ theory, the socalled ‘cause’ is generally only one among all the causes of the effect (1986, p. 162). Third, for Lewis, causation is not equivalent to causal dependence. Causation is the ancestor of causal dependence: an actual event c causes a distinct actual event e if there is a finite chain of causal dependencies between actual events which starts with c and leads to e (ibid., p. 167). So while causation is transitive, causal dependence is not. In my discussion, I will focus on causal dependence. Fourth, Lewis’ account presupposes a non-relativistic spacetime; for recall that expressions like ‘just after’ have no precise meaning in relativistic spacetime. For the time being, I will be focusing on non-relativistic spacetime. I will argue below that Lewis’ account motivates the claims that (i) outcome dependence involves superluminal causal dependence between distant specific outcomes, and (ii) parameter dependence involves superluminal causal dependence between a nearby setting and a distant specific outcome.

Aspects of Quantum Non-Locality

519

Fig. 1. Straight lines indicate particle ‘trajectories’.

In my argument, I will basically follow Butterfield’s (1992a, Sections 3 and 7; 1992b) strategy. I will argue that granted certain natural premises, causal independence between distant specific outcomes implies outcome independence, and causal independence between a nearby setting and a distant specific outcome implies parameter independence. Thus, granted these premises, claims (i) and (ii) follow immediately. The argument is as follows. Consider Fig. 1. The specific ¸-outcome, x , will J be causally independent of the specific R-outcome, y , if for some q: P (i) y N ch (x )"q, (ii) (y) N ch(x )"q, (2.9) P J P J where ch ()) denotes the chance function just after the time of y , i.e. at t . And x P  J will be causally independent of the setting of the R-apparatus (or, more precisely, the state of the R-apparatus between t and t ), r, if for some p:   (i) r N ch (x )"p, (ii) r N ch(x )"p, (2.10) J J where ch()) denotes the chance function of x just after r, i.e. at t , and r denotes J  a setting of the R-apparatus to measure a different quantity. The inference from (2.9) and (2.10) to outcome independence and parameter independence requires an inference from counterfactuals with a chancy consequence to the corresponding conditional chances. But such an inference is not generally valid (cf. Lewis, 1986, p. 178; Butterfield, 1992c; Mellor, 1995, Section 3.2). However, the counterfactuals in (2.9) and (2.10) will imply the corresponding conditional chances, (2.11) and (2.12) below, if we make the following assumptions: Factors

the pair’s state (before both measurements occur), the setting of the apparatuses, and (possibly) the nearby specific outcomes are the only relevant factors for the chance of the distant specific outcome;

520

Studies in History and Philosophy of Modern Physics

Chances the probabilities in parameter dependent and outcome dependence denote chances; Evolution chances evolve by conditionalisation on intervening history: a latter chance distribution comes from an earlier one by conditionalising on the complete history of the interval in between; and Initial

all the worlds with the same laws have the same initial chance function.

(Recall that Chances and Factors are standard assumptions in discussions of the implications of Bell’s theorem for non-locality. Evolution is part of Lewis’ (1986, Chapter 19) theory of chance. And Initial seems reasonable, especially in light of Evolution: for granted Evolution, the failure of Initial would mean that two worlds with the same laws and with the same history up to a certain time could differ in their chance function at that time.) The inference from (2.9) and (2.10) to outcome independence and parameter independence runs as follows. Granted Factors, Evolution, Initial (and Lewis’ assumption that counterfactuals do not backtrack), the equality between the actual and the counterfactual chances in (2.9) and in (2.10) imply respectively (2.11) and (2.12) [if ch (r) and ch (r)O0 and ch ((y) )O0 or 1]: R R R P ch (x /y )"ch (x /(y) ) (2.11) R J P R J P (2.12) ch (x /r)"ch (x /r). R J R J And granted Factors, Evolution and Initial, (2.11) implies (2.13) and (2.12) implies (2.14) [if ch (j & l & r & y ), ch (j & l & r & (y) ), ch (j & l & r)O0]: R P R P R ch (x /j & l & r & y )"ch (x /j & l & r & (y) ), (2.13) R J P R J P (2.14) ch (x /j&l & r)"ch (x /j & l & r), R J R J where t is some time before the preparation of the experiment. But, assuming  Chances, (2.13) implies outcome independence and (2.14) implies parameter independence. Or contraposing, granted Factors, Chances, Evolution and Initial, outcome dependence implies causal dependence between distant specific outcomes and parameter dependence implies causal dependence between a nearby setting and a distant specific outcome. Since causation is the ancestor of causal dependence, each of these superluminal causal dependencies implies the existence of the corresponding superluminal causation. Following the reasoning above, it is natural to conclude that parameterdependent theories involve a different type of superluminal causation than outcome-dependent theories. But things are a bit more complicated.  If j, l and r are not assumed to be the only relevant factors for the chances of outcomes, Factors could be modified to accommodate any additional influences (for details, see Part I, Sections 3 and 4).  In fact, the implication from (2.10) to (2.12) relies also on another assumption: namely, that the R-apparatus is prepared to measure either r and r but actually measures r. Here, the idea is that granted such a preparation, the nearest r-worlds would be those in which the apparatus is set to measure r.

Aspects of Quantum Non-Locality

521

First, parameter-independent theories also involve superluminal Lewisian causal dependence between distant specific outcomes. Some theories, e.g. nonrelativistic indeterministic parameter-dependent theories (that embody no backwards causation and respect the perfect anti-correlations of the singlet state), involve such a causal connection since they violate outcome independence. Other parameter-dependent theories involve superluminal causal dependence between specific outcomes despite the fact that they do not violate outcome independence. Consider, for instance, Bohm’s theory (cf. Bohm, 1952; Bell, 1987, Chapter 17; Du¨rr, Goldstein and Zanghi, 1992; Bohm and Hiley, 1993; Holland, 1993; Cushing, 1994). In this theory, the wave function of systems, their positions and the settings of measurement apparatuses jointly determine outcomes of measurements of these systems (as long as these outcomes get recorded in the positions of particles). Consider a Bell-type experiment for spin in which the particle pair is prepared in the singlet state, the R-measurement occurs shortly before the ¸-measurement and the ¸- and the R-apparatus are set to measure the same spin quantity. If the R-outcome had been different, the chance of the actual ¸-outcome would have been much lower, in fact zero. Accordingly, there is superluminal Lewisian causal dependence between specific outcomes. More generally, other deterministic parameter-dependent theories that respect the perfect anti-correlation of the singlet state would also involve such a causal connection. Second, as I will argue below, outcome-dependent theories also involve superluminal Lewisian causal dependence between a nearby setting and a distant specific outcome. 2.4. Specific outcome dependence A little reflection on the various ways in which factorisability fails in quantum theories shows that these theories violate the following condition, which may be interpreted as a generalisation of outcome independence: Specific outcome independence. The probability of the distant specific outcome is independent of the nearby specific outcome. That is, for any j, l, r, r, x , J y and y : P PY (i) P (x /j & l & r & y )"P(x /j & l & r & (y) ), (2.15) J P J P (ii) P (x /j & l & r & y )"P(x /j & l & r & y ), (2.15) J P J PY (iii) P (x /j & l & r & y )"P(x / j & l & r & (y) ) (2.15) J P J PY  Here is why. Consider a Bell-type experiment for spin in which the distant measurement occurs first and the pair’s state is prepared in the singlet state. If backwards causation does not exist, the chance of a distant specific outcome will be independent of the nearby setting. Now, granted indeterminism, there will be pair’s states and distant settings for which the chance of the distant specific outcome will be strictly between zero and one. Thus, if the nearby setting is the same as the distant one, the nearby specific outcome will have to be opposite to the distant specific outcome. Accordingly, outcome dependence must obtain.

522

Studies in History and Philosophy of Modern Physics

(if the relevant probabilities are defined); and similarly for the corresponding probabilities in the R-wing. The specific outcome independence in (2.15i) is the familiar outcome independence: it asserts that a distant specific outcome, say spin ‘up’ in the l-direction, is independent of whether the nearby specific outcome is spin ‘up’ or spin ‘down’ in the r-direction. The specific outcome independence conditions in (2.15ii) and (2.15iii) are less familiar. (2.15ii) asserts that a distant specific outcome, say spin ‘up’ in the l-direction, is independent of whether the nearby specific outcome is spin ‘up’ in the r-direction or spin ‘up’ in the r-direction. And (2.15iii) asserts that a distant specific outcome, say spin ‘up’ in the l-direction, is independent of whether the nearby specific outcome is spin ‘up’ in the r-direction or spin ‘down’ in the r-direction. Outcome-dependent and indeterministic parameter-dependent theories violate all the three types of specific outcome independence. Deterministic parameter-dependent theories violate only (2.15ii) and (2.15iii); for (2.15i) is undefined in these theories (since either P(j & l & r & y ) or P(j & l & r & (y) ) P P equals zero). Recall (Subsection 2.3), however, that Bohm’s theory and other theories that respect the perfect anti-correlations of the singlet state violate a counterfactual version of (2.15i): if the specific outcome had been (y) instead P of y , the chance of the actual specific outcome, x , would have been zero instead P J of one. Following Subsection 2.3, I will now argue that due to specific outcome dependence, outcome-dependent theories involve superluminal Lewisian causal dependence between a nearby setting and a distant specific outcome. Consider again Fig. 1. Let r denote the actual setting of the R-apparatus during the R-measurement. A specific ¸-outcome, x , will be causally independent of r if, for J some p, the following counterfactuals obtain: (i) r N ch (x )"p, (ii) r N ch (x )"p, (2.16) J J i.e. if the actual and the counterfactual chances of x at t, are the same. Suppose, J as before, that in the nearest r-worlds, the setting of the R-apparatus is r. Then, granted Factors, Evolution and Initial, (2.16) will imply (2.17) or (2.18), or both (depending on whether the R-outcome is ‘up’ or ‘down’, or both, in the nearest r-worlds): ch (x /j & l & r & y )"ch (x /j & l & r& y ), (2.17) R J P R J PY ch (x /j & l & r & y )"ch (x /j & l & r & (y) ). (2.18) P R J PY R J Since these equalities hold for any j, l, r, r, x , y and y , (2.17) and (2.18) imply J P PY the specific outcome dependence of (2.15ii) and (2.15iii), respectively. Thus,

 Since the measured quantity in the nearest r-worlds is r, rather than r, it is not clear that the non-specific R-outcome in the actual world should carry any weight in determining the nearest r-worlds.

Aspects of Quantum Non-Locality

523

granted Factors, Evolution and Initial, the specific outcome dependence of outcome-dependent theories implies superluminal Lewisian causal dependence between a nearby setting and a distant specific outcome. So outcome-dependent and parameter-dependent theories both involve superluminal Lewisian causal dependence between distant specific outcomes and between nearby settings and distant specific outcomes. This, I submit, is a consequence of the fact that quantum phenomena, as represented by (nonrelativistic) non-factorisable theories, require that the chance of a distant specific outcome be dependent on the (earlier) nearby specific outcome. Finally, recall (Subsection 2.2) our second objection to Maudlin’s reductio argument: namely, that probabilistic dependence is generally a poor guide to causal dependence. Indeed, following our argument above, it is easily seen that the parameter independence of QM does not imply a lack of Lewisian causal dependence between a nearby setting and a distant setting: for the parameter independence of QM is compatible with its specific outcome dependence, and (granted some natural assumptions) this dependence implies Lewisian causal dependence between a nearby setting and a distant specific outcome. Second, the fact that QM sometimes satisfies M-outcome independence does not imply that this theory sometimes lacks a Lewisian causal dependence between distant specific outcomes; for M-outcome independence is compatible with the specific outcome dependence of QM, and (granted some natural assumptions) this dependence implies Lewisian causal dependence between distant specific outcomes. 2.5. Some reservations The scope of the reasoning above is limited to non-relativistic theories. For recall that Lewis’ theory presupposes a non-relativistic spacetime. On this account, an event x depends causally on a distinct event y iff the actual chance of x just after the event y is not the same as it would have been at the same time if y had not occurred. However, in general, expressions like ‘just after’ and ‘at the same time’ have no definite meaning in relativistic spacetime. While it is easy to generalise Lewis’ theory of causation to relativistic spacetime—for instance, we may postulate that x causally depends on y iff the chance of x on at least one space-like hyperplane that lies just after y is not the same as it would have been if y had not occurred—it is more difficult to say how our discussion in Subsections 2.3 and 2.4 would impinge on relativistic non-factorisable theories. But, prima facie, it is plausible to assume that such theories, if they existed, would involve specific outcome dependence. If so, relativistic nonfactorisable theories would also involve causal dependence between both distant specific outcomes and nearby settings and distant specific outcomes; for as is easily shown, specific outcome dependence would imply such causal dependencies according to the above slight modification of Lewis’ theory. In any case, it may be objected to our reasoning above that Lewis’ theory relies on a controversial resolution of the vagueness of counterfactuals.

524

Studies in History and Philosophy of Modern Physics

According to this resolution, the truth of counterfactuals, and accordingly the truth of claims about causation, generally depends on possible worlds that violate the laws of the actual world. For instance, recall that in Bohm’s theory, if the R-measurement in Bell-type experiment occurs first, the wave function of the particle pair, the position of the R-particle and the state of the R-apparatus just before the R-measurement jointly determine the R-oucome. So unless the laws of Bohm’s theory are violated, the R-outcome could not be different if its past were the same. But, the question arises, how could worlds that violate the laws of a certain world be relevant for causation in that world? In resolving the truth value of counterfactuals, the non-backtracking strategy, as advocated by Lewis (1986, pp. 47—48), is mainly governed by the following system of priorities: (2a) it is of the first importance to avoid a big, widespread, diverse violation of law; (2b) it is of the second importance to maximise the spatio-temporal region throughout which perfect match of particular fact prevails; and (2c) it is of the third importance to avoid even small, localised, simple violations of law. By contrast, the backtracking strategy of resolving the vagueness of counterfactuals maintains that avoiding any violation of law is of primary importance. Thus, in contrast to non-backtracking counterfactuals, in backtracking counterfactuals the supposition ‘if y had not occurred’ leads to possible worlds which do not in general match the actual world in matters of particular fact up to just before the time of y. On the usual understanding of the backtracking strategy, a backtracking counterfactual dependence of event x on a distinct event y does not necessarily express a causal dependence between them. The idea is that if the past in the nearest worlds could be different, such a dependence might rather be due to a common cause c: if y had not occurred (c would not have occurred, and thus) x would not have occurred. Moreover, even when x is counterfactually dependent on y due to a direct cause, it does not follow that x depends causally on y; it might rather be y which is causally dependent on x. Thus, a number of authors believe that if causal dependence is to be explicated in terms of counterfactuals, the non-backtracking strategy is indispensable.

 On the basis of such objections Dickson (1996) argues that it is not at all clear that Bohm’s theory involves superluminal causal dependence between distant outcomes.  Maudlin (1994, pp. 130—139) argues that backtracking counterfactuals could nevertheless be used to demonstrate the existence of superluminal causation in non-factorisable theories, His reasoning is basically the following. Let two events a and b be causally implicated with each other if they either share a common cause or are connected to each other by a chain of causal dependencies. Maudlin assumes that a backtracking counterfactual dependence between two events entails that they are causally implicated with each other. Granted this assumption, he argues that two distant (space-like separated) events, a and b, are causally implicated with each other if the following backtracking

Aspects of Quantum Non-Locality

525

2.6. Superluminal causation: *backtracking reasoning+ I will aruge below that granted an adequate similarity relation, backtracking counterfactuals with a chancy consequence could express a causal dependence between events. On the basis of this similarity relation, I will propose a theory of causation that overcomes the above limitations of Lewis’ theory. The theory I have in mind is related to Mellor’s (1995) theory, but differs from it in some significant respects. It assumes a certain metaphysical framework described below. First, it assumes a certain theory of chance, which is similar to Mellor’s (1995, Chapters 3 and 4). According to this theory, every chance that a certain event will occur (or that a certain fact will be true) is a property of (or a fact about) the ‘chance set-up’ or the ‘circumstances’ in which this event occurs or does not occur (as the case may be). These circumstances and the laws jointly make these chances true, or in other words they are their truth-makers. Whether or not the notions of ‘being a property of’, ‘making true’ and ‘being a truth-maker’ can be explicated in non-causal terms is an intricate matter that I will not tackle here. In any case, I believe that in many situations, including those under consideration, it is possible to identify the circumstances of which chances of events are properties. If chances are properties of circumstances, almost all contingent events have more than one actual chance of occurrence since they typically have different chances at different times. More generally, different chances of an event are properties of different circumstances, which may occur at different times. Chances of events are not just any properties of circumstances, they are properties of circumstances that satisfy the axioms of the calculus of probability and some other certain conditions. For instance, if the chance of e is one, then it is a fact that e occurs, and if one’s evidence about e is that its chance is p then one’s credence in e should also be p (Mellor, 1995, p. 44). Chances are also supposed to be related to frequencies. But the exact nature of the connection between them is a matter of controversy. Fortunately, this issue

 (continued) counterfactual holds: a (b) would not have occurred if b (a) had not occurred and everything in a’s (b’s) past light cone had been the same (ibid., p. 130). Then he demonstrates that distant events in non-factorisable theories would be causally implicated with each other. Accordingly, he concludes that such theories would involve superluminal causation; for even if the causal implication between distant events were due to a common cause, such a cause would have to lie outside the past light cone of (at least) one of these events. Unfortunately, Maudlin’s reasoning will be of no avail to our consideration below. For our aim here is not only to argue for the existence of superluminal causation in non-factorisable theories, but also to analyse its exact nature. Also, it is not clear that this account is entirely ‘free’ of non-backtracking reasoning: for it is not clear that the supposition that ‘b does not occur’ would always be compatible with the supposition that everything in a’s past light cone remains the same’.  For other accounts of chance, see for example Popper (1957, 1990), Hacking (1965), Levi (1980, Chapters 11—17) and Lewis (1986, Chapter 19).

526

Studies in History and Philosophy of Modern Physics

will not be central to our consideration in this section (though it will be significant for our discussion in Subsection 3.2). Second, in connecting causes and effects, I will make the following assumptions. Causes typically cause their effects in the presence of certain relevant circumstances. Intuitively, these circumstances are supposed to consist of all the other partial causes of the effect. When a cause c causes an effect e in the relevant circumstances S, e has some actual chance which is a property of c and S. A cause influences the chance of its effects, and this change is a property of the laws and the relevant circumstances in which the cause causes these effects. That is, for any cause c, effect e and relevant circumstances S in which c causes e, there is an actual chance that e has with c and there typically are counterfactual chances +p, that e might have had without c which are all properties of S. When the chances +p, exist, at least some of them differ from p. So the relevant circumstances S and the laws jointly make true the following condition: (i) c N ch(e)"p, (ii)  (c N ch(e)"p).

(2.19)

There are philosophers who argue that causes and the relevant circumstances in which causes cause their effects always precede these effects and are local and contiguous to them (cf. Mellor, 1995, Sections 2.1—2.2, Chapter 17). However, this view is controversial, especially in the context of quantum phenomena. For as we have seen in Part I, in various non-factorisable theories, the chances of outcomes in Bell-type experiments depend both on distant occurrences and holistic properties which are not localised. Anyway, in resolving the vagueness of the counterfactuals in (2.19), I will assume that the similarity relation is mainly governed by the following system of priorities: (2d) it is of the first importance to avoid any violation of law; (2e) it is of the second importance to match the relevant circumstances S; and (2m) it is of the third importance to maximise similarity in matters of particular facts, while optimising the range of the ways in which the counterfactual supposition is realised in the nearest worlds. This system of priorities can be motivated as follows. First, recall that according to the backtracking strategy of resolving the vagueness of counterfactuals, avoiding any violation of laws is of first priority. Second, matching the relevant circumstances is particularly desirable. For recall that the chances that e has with c and the chances that e might have had without it (if they exist) are all assumed to be properties of the circumstances S and the laws, and the nearestworlds semantics is supposed to cash out this very fact. Third, while maximising  When there are different ways in which c could fail to occur, i.e. various nearest c-worlds, e might have different chances p in different nearest c-worlds.  The assumption that causes change the chance of their effects is not without difficulties, as cases of causal pre-emption and causal overdetermination demonstrate (see, for example, Lewis (1986, Chapters 21 and 23) and Menzies (1989; 1996, Sections 1 and 2)). For lack of space, I cannot consider how the theory above fares with respect to these difficulties.

Aspects of Quantum Non-Locality

527

similarity of particular facts is important, it is also very important to secure the range of possible ways in which the counterfactual supposition is realised in the nearest worlds. As Lewis (1986, p. 211) rightly emphasises, ‘a similarity theory needn’t suppose that just any sort of similarity we can think of has nonzero weight’. Under too strict a similarity relation, a counterfactual dependence (backtracking or non-backtracking) would not properly express causal dependence. Finally, I will assume that e causally depends on c if for some actual circumstances S: (I) there exists some chance p that e actually has which is a property of c and S; (II) there exists no other actual chance of e which is a property of c and actual circumstances S that include S; and (III) if there exist chances +p, that e might have had without c which are properties of S, some of them will differ from p, i.e. (2.19) holds. Intuitively, conditions (I)—(II) are supposed to single out all the relevant circumstances in which the alleged cause c causes its alleged effect e. Condition (III) is then supposed to ‘certify’ c as a (partial) cause of e. Two remarks. First, note that this theory has no special difficulties with relatively. Since chances are properties of circumstances, they are (essentially) time-independent. In particular, the relevant chances in (I)—(III) are frameindependent. Accordingly, causal dependence is frame-independent. Secondly, similarly to Lewis’ theory, we could define causation as an ancestor of causal dependence (see Subsection 2.3). So much for the presentation of the theory. I will now argue that this theory supports our conclusions in Subsections 2.3 and 2.4: namely, that parameterdependent and outcome-dependent theories both involve superluminal causal dependence between distant specific outcomes and between nearby settings and distant outcomes. I will focus on the apparently more controversial claims: namely, that (i) parameter-dependent theories involve superluminal causal dependence between specific outcomes; and (ii) outcome-dependent theories involve superluminal causal dependence between a nearby setting and a distant specific outcome. Turning to the first claim, consider for example Bell’s (1987, Chapter 17) ‘minimal’ Bohm theory in the context of the Bell-type experiment. Suppose that the R-measurement occurs shortly before the ¸-measurement and the ¸and the R-apparatus are set to measure the same spin quantity. Let x and y be J P the actual specific ¸-outcome and the actual specific R-outcome, respectively. The chance of x shortly after the R-measurement and just before the J  For simplicity’s sake, from now onward, when I say that a chance is a property of certain circumstances, I will mean that it is a property of these circumstances and the laws.  See also Du¨rr, Goldstein and Zanghı` (1992) and Cushing (1994).

528

Studies in History and Philosophy of Modern Physics

¸-measurement is a property of the value of the guiding field at the positions of the particles at that time, say t. Thus, substituting the conjunction of the value of the guiding field and the position of the ¸-particle at t for S, y for c and x for e, P J it is not difficult to show that conditions (I)—(III) are satisfied. Condition (I) holds since the chance of x at t is a property of S and y for S and y jointly determine J P P the value of the guiding field at the particles’ positions at t. Condition (II) obtains since there is no other actual chance of x which is a property of y and J P actual circumstances S that include S. Finally, condition (III) holds since the chance that x would have had if (y) had occurred, which is a property of S, is J P zero; for in the nearest (y) -worlds, the chance of x which is a property of P J (y) and S (i.e. the value of the guiding field at the particles’ counterfactual P positions at t) is zero. Two remarks. First, note that in contrast to Subsection 2.3, the reasoning above requires no violation of laws. Indeed, granted the new similarity relation (see (2d)—(2m) above), the counterfactual supposition that the R-outcome had been (y) instead of y (say, spin ‘down’ instead of spin ‘up’ in the r-direction), P P does not lead to worlds that violate Bohm’s theory. According to this new relation, the specific R-outcome in the nearest (y) -worlds differs from the P actual one not because of a violation of laws, but rather due to the fact that the position of the R-particle before the R-measurement occurs is different in these worlds. Second, note that this reasoning fits neatly with Bell’s (1987, p. 101) view that in physical theories [w]e can calculate the consequence of changing free elements in a theory [2] and so can explore the causal structure of the theory [2]. A respectable class of theories, including contemporary quantum theory as it is practised, have ‘free’ ‘external’ variables, in addition to those internal to and conditioned by the theory.

For, using Bell’s terminology, we may say that the position of the R-particle shortly before the R-measurement occurs is a free external variable in the context of exploring the causal connection between the distant specific outcomes. Bohm’s theory is deterministic. But, as I mentioned in Subsection 2.3, indeterministic parameter-dependent theories violate outcome independence. Thus, similary to outcome-dependent theories, they would involve causal dependence between distant specific outcomes. Turning to outcome-dependent theories, specific-outcome dependence motivates the claim that these theories involve superluminal causal dependence between a nearby setting and a distant specific outcome. For suppose that the actual specific outcomes are x and y , and the setting of the R-apparatus during J P the R-measurement is r. Then, substituting j & l & y (i.e. the pair’s state, the ¸-setting and the non-specific R-outcome) for S, x for e and r for c, Factors and J Chances (see Subsection 2.3) and specific outcome dependence jointly motivative conditions (I)—(III) above.

Aspects of Quantum Non-Locality

529

Here is why. Granted Chances, the specific outcome dependence in (2.15ii), as expressed in terms of the many-spaces approach, Pj, l, r (x /y )OPj, l, r (x /yr ) [if Pj, l, r (y ), Pj, l, r (y ) O0], (2.20) J P J P PY can plausibly be interpreted as asserting that there are conditional chances of x J given y which are properties of j & l & r. Although conditional chances do not P generally imply the corresponding counterfactuals with a chancy consequence (see Subsection 2.3), Factors and (2.20) jointly motivate the assumption that there exists an actual chance of x which is a property of j & l & r & y. Now, J Factors also motivates the assumption that there exists no other actual chance which is a property of actual circumstances that include j & l & r & y; for Factors implies that there could be no other factors which are relevant for chances of x . J Accordingly, conditions (I) and (II) obtain. Finally, Factors, Chances and (2.20) jointly motivate the assumption that if the R-setting had been r, x would have J had a different chance which is a property of j & l & y; for granted these assumptions, the chance that x has in the nearest r-worlds, which is a property J of j & l & r & y, differs from its corresponding chance in the actual world. Thus, condition (III) also obtains. 2.7. Concluding remarks I argued above that (according to two accounts of causation) parameterdependent and outcome-dependent theories both involve causal dependence between distant specific outcomes and between a nearby setting and a distant specific outcome. I also argued that these causal dependencies are due to the specific outcome dependence of these theories. The main difference between outcome-dependent and parameter-dependent theories is that in outcome-dependent theories the dependence of the distant specific outcome on the nearby setting is washed out (i.e. Pj, l, r (x )"Pj, l, r (x /y ) ) Pj, l, r (y )#Pj, l, r (x / J J P P J (y) ) ) Pj, l, r ((y) )"Pj, l, r (x /y ) ) Pj, l, r (y )#Pj, l, r (x /(y) ) ) Pj, l, rY ((y) )" P J PY PY J PY PY P Pj, l, r (x ), whereas in parameter-dependent theories it is not (i.e. J Pj, l, r (x )OPj, l,r (x )). J J This is not to deny, of course, that the ‘mechanism’ by which causal connections are realised varies from one theory to another (even among theories of the same type, e.g. outcome-dependent). For recall that here we are dealing with ‘logically weak’ concepts of causation; different types of mechanism, or more generally different ontologies, could realise a similar type of causal dependencies. In a sense, this is to be expected: investigations that are based on general structural probabilistic dependencies are bound to be limited in this way. (For a detailed consideration of differences and similarities in the ontology of nonfactorisable theories, see Part I and Section 3 below.)

 As I have said in Subsection 2.1, I prefer to work in the many-spaces approach. Thus, unless required otherwise, I will henceforth use this approach.

530

Studies in History and Philosophy of Modern Physics

I now turn to consider the question of the compatibility of non-factorisable theories with relativity.

3. Non-factorisable Theories and Relativity A popular view has it that special relativity prohibits any superluminal influence. A different prevalent view maintains that this theory prohibits only some types of superluminal influence, but there is a controversy over what exactly these types are. Many believe that relativity excludes superluminal signalling. Alternatively, others believe that this theory also excludes superluminal transport of matter—energy and/or action-at-a-distance. On the other hand, there is the view that relativity per se prohibits only superluminal influences that are incompatible with the relativistic spacetime (i.e. Minkowski spacetime), and that this prohibition is compatible with all the superluminal influences mentioned above (cf. Maudlin, 1994; 1996, Section 2). It is commonly agreed that relativity requires that theories be Lorentz invariant. If this requirement is to reflect the structure of the relativistic spacetime, Lorentz invariance must hold at the level of individual processes, and not only at the level of ensembles or observed phenomena. However, such an invariance is not sufficient for compatibility with the relativistic spacetime; for Lorentz invariance at the level of individual systems is also satisfied by some theories that postulate a preferred frame (cf. Bell, 1987, Chapter 9). So compatibility with the relativistic spacetime requires more than Lorentz invariance. Maudlin (1996, Section 2) suggests that a theory is genuinely relativistic if it can be formulated without ascribing to spacetime any more or different intrinsic structure than the relativistic metrics. Following up this proposal, I will consider below the prospects of relativistic non-factorisable theories. I start with parameter-dependent theories. 3.1. Relativistic parameter-independent theories: the challenge of backward causation All current parameter-dependent theories are not (genuinely) relativistic, and many believe that the prospects of relativistic parameter-dependent theories are not promising. In this subsection and in the following one, I will consider two challenges for such theories. The first challenge, due to Butterfield et al. (1993, Section 4) and Ghirardi et al. (1993, Section 5), is intended to show that if relativistic parameter-dependent theories existed, they would involve backwards causation.

 As Maudlin notes, it is not clear that a general criterion for identifying a structure of spacetime as intrinsic could be found (1996, pp. 292—293). Fortunately, our discussion below will not require such a criterion.

531

Aspects of Quantum Non-Locality

The argument runs as follows. Consider, with reference to a Bell-type experiment, the non-relativistic limit of an allegedly relativistic parameter-dependent theory in a given reference frame F. Suppose that in this frame, the R-measurement takes place at an earlier time than the ¸-measurement. Let P$j, l, r (x ). J P$j, l, r (y ), P$j, l, r (y ) and P$j, l, r (x ) be probability functions prescribed respectively P P J by j&l&r, j&l&r and j&l&r in F. Since the theory involves parameter dependence, there are some js with non-zero probability of occurrence such that for some l, r, r and x : J

P$j, l, r (x )OP$j, l, r (x ), J J

(3.1)

while the absence of backwards causation would imply that for all j, l, r, r and y : P

P$j, l, r (y )OP$j, l, r (y ). P P

(3.2)

Now, when the R- and the ¸-measurement are very far apart from each other, there exist reference frames, say FM, moving even with a very small velocity with respect to F, for which the temporal order of the spacetime regions of the ¸- and the R-measurement is inverted. Since the probability of an outcome is an objective fact, in the sense that it cannot depend on a reference frame, it must also hold for FM that: FM PFM jM, lM, rM (xlM)OPjM, lM, rM (x M), J

(3.3)

where jM, lM, rM and x M have the obvious meanings in the reference frame FM. J Since for FM the ¸-measurement occurs before the R-measurement, (3.3) implies that a later setting would influence the probability of an earlier outcome. But if the theory above were invariant for the transformation under consideration, an analogous situation would occur also for the reference frame F, in contradiction to (3.2). So a relativistic parameter-dependent theory would involve frame-dependent backwards causation: in some reference frames, the objective probability (chance) of a distant outcome at a certain time would depend on the setting of the nearby apparatus at a later time. In fact, as I will now argue, such a theory would also involve a frame-independent backwards causation. Here is why. Consider the following pair of parallel Bell-type experiments, described in Fig. 2. Due to the relativistic parameter dependence, the R-setting in experiment 1 would influence the (chance of the) ¸-outcome in that experiment, and the ¸-setting of experiment 2 would influence the (chance of the) R-outcome in that experiment in all reference frames. And due to local influences, the ¸-outcome of experiment 1 would influence the setting of the ¸apparatus of experiment 2 and the R-outcome of experiment 2 would influence the setting of the R-apparatus of experiment 1 in all reference frames. Accordingly, by the transitivity of causation, there would be a frame-independent causal connection from the R-apparatus of experiment 1 to the earlier R-outcome of experiment 2 and from the ¸-apparatus of experiment 2 to the earlier ¸-outcome of experiment 1 (see Fig. 3).

532

Studies in History and Philosophy of Modern Physics

Fig. 2. Dotted lines indicate the ‘trajectories’ of particles; arrows indicate causal connections and the direction of time; ellipses indicate specific outcomes; rectangles indicate measurement apparatuses; Square brackets indicate either alternative outcomes or alternative settings.

Fig. 3. A ‘map’ of the causal dependencies of Fig. 2’s set-up; arrows indicate invariant causal dependencies, vertical arrows indicate both influences on settings and the direction of time; arrows with ‘p-d’ indicate causal dependencies due to parameter dependence.

3.2. Relativistic parameter-dependent theory: the challenge of causal paradoxes A relativistic parameter-dependent theory, if it existed, would not only involve backwards causation but also causal loops (see Figs 2 and 3). Causal loops per se are not impossible: only inconsistent loops are impossible. However, a relativistic deterministic parameter-dependent theory could not exist on pain of inconsistent causal loops. (More precisely, such a theory would be impossible if some prima facie very plausible assumptions about the nature of physical reality obtain; for example, the assumption that the arrangement of the settings of the measurement apparatuses in Fig. 2’s set-up is not physically impossible.) Here is why. Consider again Fig. 2. Let ¸2 and R1 be respectively the ¸-apparatus of experiment 2 and the R-apparatus of experiment 1. Suppose that

533

Aspects of Quantum Non-Locality

the ¸-outcome of experiment 1, x , influences the setting of the ¸-apparatus of J experiment 2 in the following way: ¸ -Setting. If x is ‘#1’ (say, spin ‘up’), ¸ is set to measure l>; and if x is ‘!1’  J  J (i.e. (x) occurs), ¸ is set to measure l\; where l> and l\ are two different J  quantities. Suppose also that the R-outcome of experiment 2, y , influences the setting of the P R-apparatus of experiment 1 in a similar way: R -Setting. If y is ‘#1’, R is set to measure r\; and if y is ‘!1’ (i.e. (y)  P  P P occurs), R is set to measure r>; where r> and r\ are two different quantities.  Suppose further that the ¸-apparatus of experiment 1 and the R-apparatus of experiment 2 are set to measure the same quantity and the particle pair in each of these experiments is prepared in the same state. Finally, suppose that in this state and in these settings, the following dependencies would obtain invariantly in all reference frames:





‘#1’ if R "r>, ‘#1’ if ¸ "l>,   x" y" (3.4) J P ‘!1’ if R "r\, ‘!1’ if ¸ "l\.   It is then not difficult to show that the causal loop of Fig. 2 (see Fig. 3) would be inconsistent. If R were set to measure r>, x would have to be ‘#1’. Thus, ¸  J  would be set to measure l> and so y would have to be ‘#1’. But, if y were ‘#1’, P P the setting of R could not be r>. On the other hand, if R were set to measure   r\, x would have to be ‘!1’. Thus, ¸ would be set to measure l\ and so y J  P would have to be ‘!1’. But, if y were ‘!1’, the setting of R could not P  be r\. So a relativistic deterministic parameter-dependent theory seems to be impossible. One may hope to avoid inconsistency by retreating to an indeterministic parameter dependence. Arntzenius (1994, Section 5) argues that inconsistency would also be inherent in relativistic indeterministic parameterdependent theories. His argument proceeds by reductio. First, he supposes, with reference to the set-up described in Fig. 2, that a relativistic indeterministic parameter-dependent theory, if it existed, would imply that (in all reference frames): for some j, l>, l\, r>, r\, x , y and p and p , p Op : J P     (i) P(x /r>)"p , P(x /r\)"p , J  J  (ii) P(y /l>)"p , P(x /l\)"p . P  J 

(3.5)

 Arntzenius talks about a relativistic probabilitistic parameter-dependent theory. But since the pair’s state and the settings are assumed to be the only relevant factors for the chance of outcomes, a probabilistic parameter dependence would imply indeterminism.  This is Arntzenius’s notation.

534

Studies in History and Philosophy of Modern Physics

Then, he argues that due to the deterministic causal relations between the ¸-outcome of experiment 1 and the ¸-setting of experiment 2 and the Routcome of experiment 2 and the R-setting of experiment 1 (see ¸ -Setting and  R -Setting above), (3.5) would imply that:  (i) P (x /( y) )"p , P(x /y )"p , J P  J P  (ii) P (y / x )"p , P(y /(x) )"p . P J  P J 

(3.6)

Finally, he demonstrates that (3.6) cannot hold since consistency in the causal loop of Fig. 2 requires that for any l>, l\, r> and r\: (i) P(x /r>)"P (x /r\), (ii) P (y /l>)"P(y /l\). J J P P

(3.7)

Thus, he concludes that a relativistic theory that exhibits an indeterministic parameter dependence is also impossible. The claim that (3.7) implies the impossibility of relativistic indeterministic parameter-dependent theories is natural, especially in the context of the bigspace approach (see Subsection 2.1 above). For recalling that the particle pairs are prepared in the state j and the ¸- apparatus of experiment 1 and the R-apparatus of experiment 2 are set respectively to measure l and r, (3.7) implies that for any j, l>, l\, r>, r\, x , y : J P (i) P(x /j & l & r>)"P(x /j & l & r\), J J (ii) P(y /j & l> & r)"P (y /j&l\&r), P P

(3.8)

i.e. parameter independence in the big-space approach. Nevertheless, the claim that (3.7) implies the impossibility of an indeterministic relativistic parameterdependent theory is questionable. Consider a ‘simple’ Bell-type experiment (see Subsection 2.1) in which the R-measurement occurs before the ¸-measurement. Suppose that the R-apparatus is set to measure either r> or r\ and the ¸-apparatus is set to measure l. Consider any non-relativistic theory that assigns the following probabilities for some j, l, r>, r\, x , y and y : J P> P\ (i) Pj, l,r> (x /y >)"0, J P

(iv) Pj, l,r\ (x /y \)"1, J P

(ii) Pj, l,r> (x /(y) >)"1, (v) Pj, l,r\ (x /(y) \)"0, J P J P (iii) Pj, l,r> (y >)"0.3, P

(3.9)

(vi) Pj, l,r\ (y \)"0.6. P

These probabilities (as well as any other probabilities of the many-spaces approach) can naturally be interpreted as chances a` la Subsection 2.6’s account, where the letters in the subscript stand for the circumstances of which these chances are properties. For instance, Pj, l,r> (y >) can be interpreted as the chance P of y > which is a property of j & l & r>. P

Aspects of Quantum Non-Locality

535

Suppose further that there exists no backwards causation. Then, expanding Pj,l,r> (x ) and Pj,l,r\ (x ) in terms of the chances in (3.9), it is easily seen that this J J theory is parameter-dependent: 0.7"Pj,l, r> (x )OPj,l,r\ (x )"0.6. (3.10) J J What are the values of the corresponding probabilities of the big-space approach, i.e. P(x /j & l & r>) and P(x /j & l & r\)? A common view has it that J J they have to be equal to the probabilities in (3.10). Arntzenius himself believes that whenever the probabilities of the many-spaces approach are well defined and objective, the corresponding probabilities in the big-space approach should be the same (personal communication). Suppose then that (3.10) implies: 0.7"P(x /j & l & r>)OP(x /j & l & r\)"0.6. (3.11) J J Consider, now, the same non-relativistic theory but in a different ‘non-simple’ set-up. Suppose that the R-measurement occurs before the ¸-measurement. Suppose further that the specific R-outcomes mentioned in (3.9) influence the ¸-setting as follows: if the R-outcome is either (y) or y , the ¸-apparatus is P> P\ switched off; otherwise, this apparatus is set to measure l. It is not difficult to show that in this set-up, the following probabilities would obtain: P(x /j & l & r>)"P(x /j & l & r\)"0; (3.12) J J for as is easily seen, the ¸-apparatus is switched off whenever x has any chance J of occurring. If, as Arntzenius argues, (3.7) implies the impossibility of relativistic parameter-dependent theories, (3.12) would similarly imply the impossibility of the non-relativistic parameter-dependent theory above. In fact, a similar reasoning would imply the impossibility of any non-relativistic parameter-dependent theory (including Bohm’s)! So something must be wrong with the assumption that (3.12) implies the impossibility of the non-relativistic parameter-dependent theory above. In his argument, Arntzenius in effect presupposes that in relativistic parameter-dependent theories, the (probabilistic) dependence of a distant outcome on a nearby setting would always be manifested (if the relevant pair’s state and settings obtain). A similar reasoning would dictate that in non-relativistic parameterdependent theories (which involve no backwards causation), the (probabilistic) dependence of a distant outcome on an earlier nearby setting would always be manifested. But, as the ‘non-simple’ set-up above seems to demonstrate, this assumption is disputable. Although parameter dependence is an invariant characteristic of parameterdependent theories, it is not always manifested. (In a sense, this is less surprising if we make explicit the fact that the parameter dependence in (3.10) is a property of the pair’s state, the settings and the set-up of the ‘simple’ Bell-type experiment.) Indeed, in the ‘non-simple’ set-up, the parameter dependene fails to be manifested just because the ¸-apparatus is turned off whenever x has any J chance of occurring.

536

Studies in History and Philosophy of Modern Physics

Similarly to the non-relativistic theory above, the fact that parameter dependence is not manifested in Fig. 2’s set-up—i.e. the fact that (3.7) obtains—does not imply the impossibility of a relativistic indeterministic parameter-dependent theory. However, due to complications coming from the causal loop, it is much more difficult to account for why the parameter dependence fails to be manifested in this set-up. Thus, for lack of space, my argument below will be more suggestive in nature. The argument will have two parts. In the first one, I will argue that while the probabilities of the many-spaces approach can naturally be interpreted as chances a` la Subsection 2.6, the probabilities in (3.5)—(3.7)—i.e. P(x /r>), J P(x /r\), P(y /l>), P(x /l\), P(x /(y) ), P(x /y ), P(y /x ) and P(y /(x) )— J P J J P J P P J P J cannot; they are better thought of as ‘long-run’ relative frequencies. Then, in the second part, I will explain why parameter dependence fails to be manifested in the relevant long-run relative frequencies of Fig. 2’s set-up. In order for the probabilities in (3.5)—(3.7) to be chances (a` la Subsection 2.6), their values have to be properties of some facts about Fig. 2’s set-up; facts that would make these values true. Now, the values of different probabilities, e.g. P(x /r>) and P(x /y ), would be properties of different facts. For instance, the J J P value of P(x /r>) is a property of j, l, (y) and the fact that (y) determines the J P P setting of the R-apparatus of experiment 1 to be r>, whereas the value of P(x /y ) J P is not. This means that if these probabilities were chances, they would have to ‘live’ in different spaces. This becomes clear when the facts of which these chances are properties appear explicitly in the subscript (as in the probabilities in (3.9) and (3.10) above). However, some reflection on the details of Arntzenius’s argument for (3.7) (details which for a lack of space I declined to present above) can show that the derivation of this equality requires that the probabilities in (3.5)—(3.7) all live in the same space. Thus, if the argument for (3.7) is to go through, the probabilities cannot be interpreted as chances (at least if chances are interpreted as in Subsection 2.6). On the other hand, if these probabilities were interpreted as ‘long-run’ relative frequencies, they could all be assumed to be in the same space. (The exact characterisation of the term ‘long-run relative frequency’ is an intricate matter that I cannot discuss here. Thus, in what follows, I will assume an intuitive understanding of this term. For instance, intuitively, the long-run relative frequency of x in Fig. 2’s set-up is its relative frequency in a ‘long’ series of J ‘typical’ runs of this set-up.) On such interpretation, P(x /j & l & r>) would be the ratio of the long-run J relative frequencies of x &j&l&r> and of j&l&r> (in runs of Fig. 2’s experiJ ment). So this conditional probability would in effect measure the long-run relative frequency of x in the references class of j, l and r>. Similarly for J P(x /j & l & r\). J In general, such relative frequencies would be determined by the theory and the particular experimental set-up. For example, in the ‘simple’ Bell-type experiment, the long-run relative frequency of x in the reference class of j, l and r>, J

Aspects of Quantum Non-Locality

537

i.e. P(x /j & l & r>), would be expected to be close to Pj,l,r> (x ) (of (3.9) above). On J J the other hand, in the ‘non-simple’ set-up, due to the causal connection between the R-outcome and the ¸-setting, the long-run relative frequency of x in the J same reference class would be zero. Granted the above interpretation of the probabilities of the many-spaces and the big-space approaches, I now turn to explain why parameter dependence fails to be manifested in Fig. 2’s set-up. The reasoning is as follows. Suppose that for some j, l, r>,r\ and x , Pj,l,r> (x )'Pj,l,r\ (x ); where these chances are strictly J J J between zero and one. Then, due to the symmetry of Fig. 2’s set-up (see (3.5) above), namely the fact that Pj,l>,r (y )"Pj,l, r> (x ) and Pj ,l\,r (y )"Pj ,l,r\ (x ), we P J P J will also have: Pj,l>,r (y )'Pj,l\,r (y ). P P If the two experiments in Fig. 2’s set-up were separated from each other, Pj,l,r> (x )'Pj,l,r\ (x ) would imply that the long-run relative frequency of x in J J J the reference class of j, l and r> would be higher than its long-run relative frequency in the reference class of j, l and r\. In other words, P(x /j&l&r>) J would be greater than P(x /j&l&r\). J But , in Fig. 2’s set up, Pj,l>,r (y )'Pj,l\,r (y ) ‘pulls’ these relative frequencies in P P the opposite direction. For given the strict correlation between l> and x and J between l\ and (!x) , y would have a greater chance to occur with x than with J P J (x) . Thus, the long-run relative frequency of y in the reference class of j, l and J P x should be higher than its long-run relative frequency in the reference class of j, J l and (x) , i.e. P(y /j&l&x ) should be greater than P (y /j&l&(x) ). HowJ P J P J ever, granted the strict correlation between y and r\ and (y) and r>, P P P(y /j&l&x )'P(y /j&l&(x) ) would imply that P(x /j&l&r>) should be P J P J J lower than P(x /j&l&r\). J In short, in Fig. 2’s set-up, a relativistic indeterministic parameter dependence would give rise to opposite ‘forces’ that would pull the long-run frequencies of x J in the reference classes of j&l&r> and j&l&r\ in opposite direction, Pj,l,r> (x )'Pj,l,r\ (x ) would push the long-run relative frequency of x in the J J J reference class of j & l & r> to be higher than its relative frquency in the reference class of j & l & r\, i.e. in the direction of P(x /j & l & r>)'P(x /j & l & r\); J J whereas Pj,l>,r (y )'Pj,l\,r (y ) would push these frequencies in the opposite P P direction, i.e. in the direction of P(x /j & l & x>)(P(x /j & l & r\). Arntzenius’s J J argument seems to demonstrate that causal consistency in this set-up would obtain if these opposite forces balance each other, so that the long-run relative  Although my argument below relies on this interpretation, I believe that the claim that parameter dependence could exist yet fail to be manifested does not depend on this interpretation.  Arntzenius expresses both the parameter dependence and the symmetry of Fig. 2’s set-up in terms of big-space probabilities. But recall that he agrees that these probabilities are equal to the corresponding probabilities of the many-spaces approach.  For by Bayes theorem, P(y /j & l & x )"P(x /j & l & y ) ) P(y /j & l)]/P(x /j & l) and P(y / P J J P P J P j & l & (x) )"P((x) /j & l & y ) ) P(y /j & l)/P((x) /j & l). Thus, by the probability calculus, J J P P J P(y /j & l & x )'P(y /j & l & (x) ) implies that P(x /j & l & y )'P(x /j & l), which in turns imP J P J J P J plies that P(x /j & l & y )'P(x /j & l & (y) ). But due to the strict correlation between y and r\ J P J P P and (y) and r>, this latter inequality implies that P(x /j & l & r>)(P(x /j & l & r\). P J J

538

Studies in History and Philosophy of Modern Physics

frequencies of x in the reference classes of j & l & r> and of j & l & r\ would be J the same, i.e. P(x /j&l&r>)"P(x /j&l&r\) and (thus) P(x /r>)"P(x /r\). J J J J In the argument above, I assumed that Pj,l,r> (x )'Pj,l,r\ (x ). But, as is not J J difficult to show, a similar reasoning applies to the assumption that Pj,l,r> (x )(Pj,l,r\ (x ). Moreover, because of the symmetry of Fig. 2’s set-up, the J J same type of reasoning applies also to the corresponding probabilities of the R-outcome of experiment 2, i.e. for P(y /l>) and P(y /l\). P P To sum up: (3.7) — i.e. P(x /r>)"P(x /r\) and P(y /l>)"P(y /l\) — does not J J P P demonstrate the inconsistency of a relativistic indeterministic parameter-dependent theory. On the contrary, it shows that if such a theory were possible, causal consistency in Fig. 2’s set-up would require that (3.7) obtains. 3.3. The prospects of relativistic non-factorisable quantum theories I now turn to the question of the compatibility of non-factorisable theories with relativity. I start with the so-called ‘collapse’ theories. In recent years, there have been various attempts to account for the infamous state reduction of QM as a real dynamical process (see, for instance, Ghirardi, Rimini and Weber, 1986; Bell, 1987, Chapter 22; Pearle, 1989; Ghirardi, Rimini and Pearle, 1990; Butterfield et al., 1993, and Ghirardi et al., 1993). Current dynamical reduction models are not genuinely relativistic. And so far, attempts to generalise these models have met serious difficulties. (For some recent attempts to solve these difficulties, see for example Ghirardi (1996) and Pearle (1996) and references therein.) A different attempt to construct a relativistic collapse theory by postulating a radical hyperplane dependence has been pursued by Fleming (1989, 1992) (cf. Fleming and Bennett, 1989). In standard non-relativistic collapse theories, the state reduction is instantaneous. An analogue instantaneous collapse would be incompatible with the relativistic spacetime since the state reduction can be instantaneous in at most one reference frame. In the hyperplane-dependent theory, this difficulty is circumvented by postulating that in every reference frame there is a reduction which is instantaneous in that frame; for state reductions occur on any family of flat parallel space-like hyperplanes that intersects the spacetime region of at least one of the measurements. Since all these reductions are supposed to be real objective processes (Fleming, 1992, p. 109), each reference frame reproduces the predictions of QM. The hyperplane-dependent theory is genuinely relativistic. However, as Maudlin (1996, Section 5) has pointed out, this theory involves mysterious correlations between various state reductions. If all families of flat parallel space-like hyperplanes are to tell the same story about measurement outcomes in Bell-type experiments, the state reductions in all the different familes have to be correlated with each other. But, since in the hyperplane-dependent theory the wave functions on flat space-like hyperplanes are supposed to provide the  Fleming (1995) argues that his theory is just as radical as other relativistic quantum theories since hyperplane dependence would also be inherent in such theories.

Aspects of Quantum Non-Locality

539

complete description of physical reality, there seems to be nothing in the ontology of this theory that can account for these correlations. The prospects of relativistic collapse theories are not promising. It is frequently claimed that the prospects of relativistic no-collapse theories are better. In these theories, the wave function does not (generally) provide the complete description of physical reality: there exist additional variables which are supposed to represent real physical quantities. Since the wave function has a covariant dynamics, the question of the compatibility with relativity turns on the dynamics of these additional variables (cf. Maudlin, 1996, p. 302; Dickson and Clifton, 1998). If such a dynamics is to be satisfactory, it would probably have to meet some constraints. In the context of the so-called ‘modal interpretations’, Dickson and Clifton (1998, Section 4) argue that that Stability, i.e. the requirement that a ‘freely’ evolving system does not undergo transitions (or, more precisely, that its definite properties follow the unitary evolution of the system) is one of them. Almost all the so-called ‘modal interpretations’ satisfy Stability. (Van Fraassen’s (1991) modal interpretation is agnostic about Stability since it is agnostic about dynamics in general.) Moreover, as Dickson and Clifton (ibid., Section 4) stress, there are good reasons to require Stability. But, argue Dickson and Clifton, if Stability obtains, the modal interpretations proposed by Van Fraassen (1979, 1991), Kochen (1985), Dieks (1988, 1989), Healey (1989), Bub (1992, 1994), Vermaas and Dieks (1995) and Bacciagaluppi and Dickson (1996) cannot be fundamentally Lorentz invariant. Thus, they conclude that the prospects of a relativistic modal interpretation with satisfactory dynamics are dim. In their argument, Dickson and Clifton concentrate on Vermaas—Dieks theory. They show that when supplemented with Stability, this theory cannot be Lorentz invariant (at the level of individual processes). Then they consider how this argument impinges on other modal interpretations. When supplemented with Stability, Vermaas—Dieks theory entails that in any reference frame in which the nearby measurement in a Bell-type experiment occurs before the distant one, the probability of definite properties of the nearby particle after the nearby measurement depends on its definite properties before that measurement (henceforth, Influence). And similarly to almost all other modal interpretations, in this theory distant systems in a Bell-type experiment possess definite properties and have joint probabilities for these properties throughout the experiment (henceforth, Joint Probabilities).  Indeed, in some no-collapse theories, e.g. the so-called ‘many-minds’ theory, the wave function does provide the complete description of physical reality (cf. Albert and Loewer, 1988). However, the many-minds theory does not violate factorisability, at least not if outcomes are real physical events; for recall that in this theory, outcomes are not real physical events.  In fact, Dickson and Clifton express some doubt about the applicability of their argument to Kochen’s and Bub’s modal interpretations (1998, note 3 and Appendix 1).  That this is the case should be clear from Dickson and Clifton’s derivation of transition probabilities in their equations (32)—(37).

540

Studies in History and Philosophy of Modern Physics

Now, as is not difficult to show, Dickson and Clifton’s no-go theorem for relativistic dynamics in modal interpretations does not apply to theories that violate Stability, Joint Probabilities or Influence. In fact, a violation of each of these conditions might furnish a way to a relativistic modal interpretation. Kochen’s (1985) modal interpretation furnishes the way to a relativistic modal interpretation by violating Joint Probabilities. In this theory, the range of possible determinate properties of a system S and their probabilities are assigned as follows. First, the universe is divided into two ‘systems’, a system S and the rest of the universe. Then the possible determinate properties of S and their probabilities are derived from S’s (reduced) density operator, i.e. from the partial trace of the state of the universe over the rest of the universe. Similarly to some other modal interpretations, P is a possible determinate property of S iff it corresponds to one of the spectral projections of S’s (reduced) density operator (or rather it is a projection in the Boolean algebra it generates). But, in contrast to other modal interpretations, the determinate properties of (sub-)systems are not generally intrinsic to them. Properties of (sub-)systems are generally only ‘related to’, the rest of the universe. For instance (in a toy universe that consists of a Bell-type experiment before the measurements occur, say a universe with a particle pair in a superposition of spin and two measurement apparatuses in a ‘ready state’), the ¸-particle may have l-spin ‘up’ relative to the R-particle and the apparatuses. Yet, as a subsystem of the pair, this particle may have no definite spin property relative to the apparatuses. So in Kochen’s theory, properties are generally relational! Moreover, in this theory, joint probabilities of determinate properties of (sub-)systems exist only when either these properties (e.g. l-spin ‘up’ in the ¸-particle and r-spin ‘up’ in the R-particle) are related to the same context (e.g. the measurement apparatuses), or the (sub-)systems have these properties relative to some context. Now, before the measurements occur, particles in Bell-type experiments do not have any (relevant) determinate properties (e.g. spin properties) that can be related to the same context, nor do they have them as subsystems of the pair. Accordingly, Joint Probabilities fails in a way that blocks Dickson and Clifton’s no-go theorem (since the joint probabilities that are required in their theorem do not exist), and moreover clears the way for developing a Lorentzinvariant dynamics for determinate properties that satisfies both Stability and Influence. Thus, at the price of radical relationalism, Kochen’s theory would be relativistic.

 After the measurements occur the particles have both determinate spin properties and joint probabilities for these properties. In fact, the particles (as subsystems of the pair) both have spin properties relative to the apparatuses.  Actually, it is not only the radical relationlism that would furnish the way to relativistic modal interpretation, but also the assignment rule for joint probabilities of determinate properties: namely, the rule that postulates that joint probabilities for determinate properties exist only when these properties belong to the same context. For this relationalism per se does not imply that a rule for joint probabilities of determinate properties which are related to different contexts could not exist.

Aspects of Quantum Non-Locality

541

The second route to try to achieve a relativistic modal interpretation is to abandon Influence. Dieks’ (1998) modal interpretation provides the following recipe. First, postulate that the set of the determinate properties that could be possessed by system is derivable from its (reduced) density operator. Second, postulate that Stability obtains. Finally (in contrast to Vermaas—Dieks theory), postulate that when an interaction occurs, the probability of a transition to new determinate properties depends solely on the total quantum state; so this probability does not depend on the possessed properties of the systems involved (so as to abandon the Vermaas—Dieks rule for assigning joint probabilities to determinate properties). Accordingly, Influence fails: the ¸- and the R-particle might each possess a spin ‘up’ in the l-direction all the way up to an (ideal) l-spin measurement, yet the ¸- and the R-outcome turn ‘down’ (in the l-direction). As a result, the same possessed properties and the same transition probabilities are found in all reference frames. Thus, at the price of postulating the existence of possessed properties that basically influence nothing (except for their own evolution when no interactions occur), Dieks’ theory is relativistic. A third way to try to achieve a relativistic modal interpretation is to abandon Stablity. For instance, the transition probabilities from (determinate properties of a system at) an earlier time to (possible determinate properties of that system at) a later time can be given by the so-called ‘single-time’ probabilities at the later time. Granted such a rule, the dynamics of the determinate properties would be relativistic. However, a violation of Stability would probably be at least as costly as a violation of Influence. So while some modal interpretations are genuinely relativistic, they have some rather unattractive properties. We are left with the question: can a relativistic no-collapse theory escape this destiny?

4. Summary of Conclusions In this paper and in a preceding one (Part I), I have considered the nature of non-locality in the so-called ‘non-factorisable’ theories. Here follows a summary of my main conclusions: (i) A failure of locality* is a necessary condition for superluminal signalling, whereas a failure of completeness, parameter independence or outcome independence is not. Superluminal signalling also requires that nonlocality* be due to a controllable factor and that the distribution of pairs’ states be controllable (Part I, Subsections 4.1—4.3.3).  Recall (Part I, Subsections 3.2 and 4.2) that completeness asserts that the pair’s state, the settings and some other relevant physical factors jointly render the distant outcomes probabilistic independent; and locality* asserts that the probability of a distant specific outcome is independent of the setting and any other physical factor (e.g. the apparatus microstate) in the nearby wing.

542

Studies in History and Philosophy of Modern Physics

(ii) A number of non-factorisable theories apparently satisfy locality.* But this appearance is largely misleading since discussions of these theories make unrealistic assumptions about measurement processes: when measurements are modelled realistically, locality* might fail. Thus, the question of signalling is likely to arise also to apparently local* theories (Part I, Subsection 4.3.4). (iii) The question of the controllability of the non-locality* has implications for the compatibility of quantum theories with relativity, even when superluminal signalling is impossible in principle (because e.g. the distributions of states of pairs are uncontrollable in principle). Indeed, the controllability of parameter dependence seems to imply that relativistic deterministic parameter-dependent theories could not be possible on pain of causal paradoxes (Subsection 3.2 above). (iv) Arntzenius (1994, Section 5) has argued that relativistic indeterministic parameter-dependent theories would be similarly impossible. I have argued that Arntzenius’s argument does not demonstrate the impossibility of such theories. It rather shows that if such theories were possible, causal consistency would sometimes require that the parameter dependence fails to be manifested in the long-run relative frequencies of specific outcomes (Subsection 3.2 above). (v) Relativistic parameter-dependent theories, if they existed, would involve frame-independent backwards causation (Subsection 3.1. above). (vi) The view that parameter dependence (non-locality*) is due to action-at-adistance, whereas outcome dependence (completeness) is due to some type of holism and/or non-separability is unfounded. First, all current parameter-dependent theories involve some type of holism or non-separability. Second, parameter-dependent theories do not necessarily involve actionat-a-distance. Third, outcome-dependent theories might also involve action-at-a-distance (Part I, Sections 5 and 6). (vii) The view that action-at-a-distance does not make sense in non-separable or holistic theories is unfounded. On the contrary, action-at-a-distance makes better sense in such theories (Part I, Subsection 5.3). (viii) Parameter-dependent and outcome-dependent theories both involve superluminal causal dependence between distant specific outcomes and between nearby settings and distant specific outcomes. These causal dependences are due to the fact that both of these theories involve specific outcome dependence (which is a generalisation of the familiar outcome dependence). This is not to deny, of course, that the ‘mechanism’ by which causal connections are realised varies from one theory to another. For, here, by ‘causation’ we mean ‘logically weak’ concepts of causation; different types of mechanisms, or more generally different ontologies, could realise a similar type of causal dependence. (ix) The prospects of relativistic ‘collapse’ theories are not promising. It is commonly claimed that the prospects of relativistic no-collapse theories are better. However, while some versions of the so-called ‘modal

543

Aspects of Quantum Non-Locality

interpretation’ of quantum mechanics are genuinely relativistic, these theories have some rather unattractive properties (Subsection 3.3 above). Acknowledgements—For discussions of and comments on parts of an earlier version, I would like to thank Gordon Fleming, Meir Hemmo, Martin Jones, Isaac Levi, Itamar Pitowsky, Michael Redhead, Katinka Ridderbos, anonymous referees and audiences at the Universities of Cambridge, Northwestern and Pittsburgh. I am very grateful to Frank Arntzenius, Jeremy Butterfield and especially Guido Bacciagaluppi, Rob Clifton and Fred Kronz. For financial support during my research on this paper, I would like to thank the British Council, Vatat Foundation, University of Haifa and the Centre for Philosophy of Science, University of Pittsburgh.

References Albert, D. and Loewer, B. (1988) ‘Interpreting the Many-Worlds Interpretation’, Synthese 77, 195—213. Arntzenius, F. (1994) ‘Spacelike Connections’, British Journal for the Philosophy of Science 45, 201—217. Bacciagaluppi, G. and Dickson, M. (1997) ‘Dynamics for Density-Operator Interpretations of Quantum Theory’, quant-ph/9711048. Bell, J. S. (1987) Speakable and ºnspeakable in Quantum Mechanics (Cambridge: Cambridge University Press). Berkovitz, J. (1995) ‘What Econometrics Cannot Teach Quantum Mechanics’, Studies in History and Philosophy of Modern Physics 26, 163—200. Berkovitz, J. (1998a), Aspects of Quantum Non-Locality. I: Superluminal Signalling, Action-at-a-Distance, Non-Separability and Holism’, Studies in History and Philosophy of Modern Physics 29, 183—222. Berkovitz, J. (1998b) ‘The Notion of Causation in Causal Decision Theories’, in preparation. Bohm, D. (1952) ‘A Suggested Interpretation of Quantum Theory in Terms of Hidden Variables, Part I and II’, Physical Review 85, 166—179 and 180—193. Bohm, D. and Hiley, B. (1993) ¹he ºndivided ºniverse: An Ontological Interpretation of Quantum Mechanics (London: Routledge). Bub, J. (1992) ‘Quantum Mechanics Without the Projection Postulate’, Foundations of Physics 22, 737—754. Bub, J. (1994) ‘On the Structure of Quantal Proposition Systems’, Foundations of Physics 24, 1261—1280. Butterfield, J. N. (1989) ‘A Space-Time Approach to the Bell Inequality’, in Cushing and McMullin (1989), pp. 114—144. Butterfield, J. N. (1992a) ‘Bell’s Theorem: What It Takes’, British Journal for the Philosophy of Science 42, 41—83. Butterfield, J. N. (1992b) ‘David Lewis Meets John Bell’, Philosophy of Science 59, 26—43. Butterfield, J. N. (1992c) ‘Probabilities and Conditionals: Distinction by Example’, Aristotelian Society 92, 251—272 Butterfield, J. N. (1994) ‘Outcome Dependence and Stochastic Einstein Locality’, in D. Prawitz and D. Westersta¨hl (eds), ¸ogic and Philosophy of Science in ºppsala (Dordrecht: Kluwer), pp. 385—424. Butterfield, J. N., Fleming, G. N., Ghirardi, G. C. and Grassi, R. (1993) ‘Parameter Dependence in Dynamical Models for State Vector Reduction’, International Journal of ¹heoretical Physics 32, 2287—2303. Cartwright, N. (1989) Nature’s Capacities and ¹heir Measurements (Oxford: Clarendon Press). Chang, H. and Cartwright, N. (1993) ‘Causality and Realism in the EPR Experiment’, Erkenntnis 38, 169—190.

544

Studies in History and Philosophy of Modern Physics

Cushing, J. (1994) Quantum Mechanics: Historical Contingency and the Copenhagen Hegemony (Chicago: Chicago University Press). Cushing, J. and McMullin, F. (eds) (1989) Philosophical Consequences of Quantum ¹heories: Reflections on Bell’s ¹heorem (Notre Dame: University of Notre Dame Press). Cushing, J., Fine, A. and Goldstein, S. (eds) (1996) Bohmian Mechanics and Quantum ¹heory: An Appraisal (Dordrecht: Kluwer). Dickson, M. (1996) ‘Is Bohm’s Theory Local?’, in Cushing et al. (1996), pp. 321—330. Dickson, M. and Clifton, R. (1998) ‘Lorentz Invariance in Modal Interpretations’, forthcoming in Dieks and Vermaas (1998). Dieks, D. (1988) ‘The Formalism of Quantum Mechanics: An Objective Description of Reality?’, Annalen der Physik 7, 174—190. Dieks, D. (1989) ‘Quantum Mechanics Without the Projection Postulate and Its Realistic Interpretation’, Foundations of Physics 19, 1397—1423. Dieks, D. (1998) ‘Locality and Lorentz-Covariance in the Modal Interpretation of Quantum Mechanics’, forthcoming in Dieks and Vermaas (1998). Dieks, D. and Vermaas, P. E. (eds) (1998) ¹he Modal Interpretation of Quantum Mechanics (Dordrecht: Kluwer) forthcoming. Du¨rr, D., Goldstein, S. and Zanghı` , N. (1992) ‘Quantum Equilibrium and the Origin of Absolute Uncertainty’, Journal of Statistical Physics 67, 843—907. Fleming, G. N. (1989) ‘Lorentz Invariant State Reduction and Localisation’, in A. Fine and J. Leplin (eds), PSA 1988, Vol. 2 (East Lansing: Philosophy of Science Association), pp. 112—126. Fleming, G. N. (1992) ‘The Objectivity and Invariance of Quantum Predictions’, in D. M. Hull, M. Forbes and K. Okruhlik (eds), PSA 1992, Vol. 1 (East Lansing: Philosophy of Science Association), pp. 104—113. Fleming, G. N. (1995) ‘Just How Radical is Hyperplane Dependence?’, in R. Clifton (ed.), Perspectives on Quantum Reality (Dordrecht: Kluwer), pp. 11—28. Fleming, G. N. and Bennett, H. (1989) ‘Hyperplane Dependence in Relativistic Quantum Mechanics’, Foundations of Physics 19, 231—267. Ghirardi, G. C. (1996) ‘Properties and Events in Relativistic Context: Revisiting the Dynamical Reduction Program’, Foundations of Physics ¸etters 9, 313—355. Ghirardi, G. C., Rimini, A. and Weber, T. (1986) ‘Unified Dynamics for Microscopic and Macroscopic Systems’, Physical Review D 34, 470—491. Ghirardi, G. C., Pearle, P. and Rimini, A. (1990) ‘Markov Processes in Hilbert Space and Continuous Spontaneous Localisation of Systems of Identical Particles’, Physical Review A 42, 78—89. Ghirardi, G. C., Grassi, R., Fleming, G. N. and Butterfield, J. N. (1993) ‘Parameter Dependence in Dynamical Models for State Reduction’, Foundations of Physics 23, 341—364. Hacking, I. (1965) ¹he ¸ogic of Statistical Inference (Cambridge: Cambridge University Press). Healey, R. (1989) ¹he Philosophy of Quantum Mechanics: An Interactive Interpretation (Cambridge: Cambridge University Press). Holland, P. R. (1993) ¹he Quantum ¹heory of Motion (Cambridge: Cambridge University Press). Jarrett, J. (1984) ‘On the Physical Significance of the Locality Conditions in the Bell Arguments’, NouL s 18, 569—589. Jarrett, J. (1989) ‘Bell’s Theorem: A Guide to Implications’, in Cushing and McMullin (1989), pp. 60—79. Kochen, S. (1985) ‘A New Interpretation of Quantum Mechanics’, in P. Lahti and P. Mittelstaedt (eds), Symposium of the Foundations of Modern Physics (Singapore: World Scientific), pp. 151—170.

Aspects of Quantum Non-Locality

545

Levi, I. (1980) ¹he Enterprise of Knowledge (Cambridge, Mass.: MIT Press). Lewis, D. (1973) Counterfactuals (Oxford: Blackwell). Lewis, D. (1986) Philosophical Papers, Vol. 2 (Oxford: Oxford University Press). Maudlin, T. (1994) Quantum Nonlocality and Relativity (Oxford: Blackwell). Maudlin, T. (1996) ‘Space-Time in the Quantum World’, in Cushing et al. (1996). pp. 285—307. Mellor, D. H. (1995) ¹he Facts of Causation (London: Routledge). Menzies, P. (1989) ‘Probabilistic Causation and Causal Process: A Critique of Lewis’, Philosophy of Science 59, 642—663. Menzies, P. (1996) ‘Probabilistic Causation and the Pre-emption Problem’, Mind 105, 85—117. Pearle, P. (1989) ‘Combining Stochastic Dynamical State-Vector Reduction with Spontaneous Localisation’, Physical Review A 39, 2277—2289. Popper, K. (1957) The Propensity Interpretation of the Calculus of Probability, and the Quantum Theory’, in S. Korner (ed.), Observation and Interpretation (London: Butterworths), pp. 65—70. Popper, K. (1990) A ¼orld of Propensities (Bristol: Thoemmes) Redhead, M. L. G. (1986) ‘Relativity and Quantum Mechanics: Conflict or Peaceful Coexistence?’, Annals of the New ½ork Academy of Science 480, 14—20. Redhead, M. L. G. (1989) ‘Nonfactorizability, Stochastic Causality, and Passion-at-aDistance’, in Cushing and McMullin (1989), pp. 145—153. Shimony, A. (1993) Search for a Naturalistic ¼orld »iew, Vol. 2 (Cambridge: Cambridge University Press). Teller, P. (1989) ‘Relativity, Relational Holism and Quantum Mechanics’, in Cushing and McMullin (1989), pp. 208—223. Vermaas, P. E. and Dieks, D. (1995) ‘The Modal Interpretation of Quantum Mechanics and Its Generalisation to Density Operators’, Foundations of Physics 25, 145—158. Fraassen, B. C. van (1979) ‘Hidden Variables and the Modal Interpretation of Quantum Mechanics’, Synthese 42, 155—165. Fraassen, B. C. van (1982) ‘The Charybids of Realism’, Synthese 52, 25—38, page reference to reprint in Cushing and McMullin (1989), pp. 97—113. Fraassen, B. C. van (1991) Quantum Mechanics: An Empiricist »iew (Oxford: Clarendon Press).