## Abstract

There is widespread belief in a tension between quantum theory and special relativity, motivated by the idea that quantum theory violates J. S. Bell’s criterion of local causality, which is meant to implement the causal structure of relativistic space-time. This paper argues that if one takes the essential intuitive idea behind local causality to be that probabilities in a locally causal theory depend only on what occurs in the backward light cone and if one regards objective probability as what imposes constraints on rational credence along the lines of David Lewis’ *Principal Principle*, then one arrives at the view that whether or not Bell’s criterion holds is irrelevant for whether or not local causality holds. The assumptions on which this argument rests are highlighted, and those that may seem controversial are motivated.

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## Notes

- 1.
See (Friederich 2011), (Healey 2012), (Friederich 2013), and (Friederich 2014) for recent explorations and defences of the idea that quantum states are in some sense relative to the agents who assign them and do not straightforwardly correspond to objective features of the systems they are assigned to.

- 2.
- 3.
See Section 7 in (Shimony 2004). An influential argument for peaceful coexistence due to Jarrett (1984) has been thoroughly criticised in the past few years, for example by (Maudlin (2011), pp. 85–90). Others have argued that it rests on serious misunderstandings of the points Bell really wanted to make, see (Norsen 2009), (Norsen 2011) and (Näger 2013).

- 4.
(Maudlin (2011), pp. 141–144), elaborates on this point in great detail. While Maudlin thinks that the quantum correlations require superluminal causation, he does not regard this as by itself raising any serious problems of incompatibility between quantum theory and special relativity. What he does regard as problematic, however, is quantum theory’s violation of the probabilistic criterion of local causality discussed further below.

- 5.
He even gives an enthusiastic recommendation of backward causation as the clue for resolving the foundational problems of quantum theory, see Chapts. 8 and 9 of (Price 1996).

- 6.
Contrary to Fig. 1, in the figure displayed in Bell’s work the space-time region 3 is only a subregion of the backward light cone of region 1. Crucially, as Bell emphasises, it is one which completely shields region 1 from the overlap of the backward light cones of regions 1 and 2 (as the full backward light cone of region 1 trivially does). For the purposes of this paper I identify region 3 with the full backward light cone of region 1. The main advantages of this choice are, first, that it avoids any potential difficulties arising from hypothetical non-Markovian local theories that might incorrectly be diagnosed as

*not*locally causal even though one sees that they are locally causal if one chooses a sufficiently large region 3 (Goldstein et al. 2011, Section “Bell’s definiton of locality”) and, second, that it facilitates our discussion in the following sections. For a potential disadvantage see (Goldstein et al. 2011, note 34). - 7.
The present formulation of (BPLC) ignores variables \(a\) and \(b\) for the measurement settings in the regions 1 and 2. The full formulation reads \(Pr_a(A|E)=Pr_{a,b}(A|E,B)\) (Seevinck and Uffink 2011, Eq. (21)). I try to keep the present discussion as simple as possible by ignoring the measurement settings in my notation for the sake of simplicity. The role of the measurement settings can no longer be ignored when discussing local commutativity, as will be done in Sect. 5, see Eq. (5). For a useful and detailed investigation of criteria related to (BPLC), which focuses on very different aspects than the present discussion, see (Butterfield 2007).

- 8.
Healey suggests this perspective on local causality: according to his preferred reading “quantum theory cannot fail to be a locally causal theory—not because it satisfies Local Causality but because that condition is simply inapplicable to quantum theory” (Healey 2014, p. 6). Healey adds that not much hinges on whether one regards Bell’s criterion as applicable to quantum theory or not: “[I]n the end it is unimportant whether one understands the condition of Local Causality to be violated by quantum theory (including relativistic quantum field theory) or simply inapplicable to that theory. [...] [O]n neither understanding does quantum theory conflict with the intuitive principle on which Bell based that condition” (Healey 2014, p. 6). The conclusion of the present paper is in marked agreement with this last point, even though Healey reaches it along very different lines than the present paper.

- 9.
I would like to thank an anonymous referee for suggesting to highlight this point.

- 10.
See (Lewis (1980), pp. 87–89) for details.

- 11.
In what follows I will use the letter “\(A\)” to denote both the proposition concerning some chancy fact as well as the “chancy” event itself. So, the expression “\(Pr(A)\)” can either be read as “the chance of (the proposition) \(A\) being true” or as “the chance of (the event) \(A\) to occur”.

- 12.
Ibid. pp. 92–96.

- 13.
See (Lewis 1980, p. 97) as “the Principal Principle reformulated”.

- 14.
As persuasively argued by Cusbert (2013), there are strong reasons for conceiving of chances as relative to causal histories rather than times. It seems natural to suppose that in typical “chance theories” in the special relativistic space-time framework causal histories will approximately coincide for all points in a finite and not overly large space-time region, whereas they will differ for distant points on the same space-like hypersurface. So, if Cusbert is right, this provides us with further reasons for expecting objective probabilities to be relative to “local” space-time regions rather than space-like hypersurfaces.

- 15.
This allows me to ignore a further complication that Hall criterion successfully addresses, namely, the difficulty that, according to Lewis’ criterion (TPP), a pattern of events which according to the chance theory \(T\) can occur with nonzero probability may

*undermine*\(T\) as the correct chance theory of the world. The problem is solved by replacing the original principle (TPP) with the modified “New Principle” that reduces to the original one if there is only one candidate chance theory \(T\). (Lewis 1994, p. 487 and Hall 1994, p. 511). - 16.
Hall’s account of admissible evidence does in fact have subtly different implications from Lewis’, but these can be neglected for our present purposes: Lewisian chances come apart from Hallian chances if agents have evidence that allows them to have more informed rational credences than those obtained on grounds of the chance theory that they use. To see what this means in practice, assume that, according to the chance theory \(T\) that is used, only evidence about the past is admissible. Then, according to Lewis, if the agent acquires evidence about the future due to reliable oracles and the like (referred to as “crystal balls” by (Hall (1994), p. 508)), this evidence is inadmissible. Nevertheless, in order for her credences to be rational, she may need to take such evidence into account, so rational credences will no longer coincide with the chances. According to Hall, in contrast, any evidence that an agent has automatically counts as admissible, even if obtained by the help of crystal balls. As a consequence, on the Hallian account the rational credences coincide in all circumstances with the chances, but the chances are not in all cases those derived from the chance theory \(T\), according to which only evidence about the past is admissible. For the sake of simplicity, I assume for our present purposes that there are no “crystal balls” around or, to consider the more general case, no exemptions from the chance theory \(T\)—quantum theory—that our agents use.

- 17.
In what follows, I assume that the admissible evidence for the agent in region 1 does not include the event \(A\) itself, for otherwise the associated probability \(Pr_1(A)\) would be trivial (i.e. \(0\) or \(1\)). The problem we encounter here is essentially that of specifying the probability of an event with respect to the very moment where it occurs (or fails to occur). Since this problem is in no way specific to the present investigation, I feel free to ignore it here.

- 18.
- 19.
- 20.
She may also derive it from what occurs

*now*where she is together with evidence about her backward light cone. Evidence about what occurs in the backward light cone may include, for example, memories of agreements that were made as to which obervables are to be measured in the future at space-like separation from her. - 21.
There are further potentially interesting ramifications of the considerations offered here. For example, if we interpret the present argument to the end that (IPLC) is not violated in quantum theory as suggesting that (ILC) is not violated as well and that, hence, the quantum correlations are non-causal (which admittedly would require further argument), then, as persuasively argued by Cavalcanti (2010), this may spell trouble for

*causal*(as opposed to standard*evidential*) decision theory in that it recommends losing betting strategies in gambles that refer to EPRB-type setups. - 22.
A remaining worry that one might have is how nature manages to “perform the trick” (see the abstract of Gisin 2009) of producing correlations that violate (BPLC) without availing herself of non-local influences that are incompatible with the spirit, if not the letter, of relativity theory. In (Friederich 2014) I argue that this worry rests on a misguided picture of the role of time in the “coming about” of the quantum correlations. Once the block universe view is accepted as the mature perspective on the role and status of time the question of how nature might be able to produce certain correlations is no longer well posed.

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## Acknowledgments

I would like to thank Andreas Bartels, Jeremy Butterfield, Michael Esfeld, Thorben Petersen, two anonymous referees, and the members of the Göttinger Philosophisches Reflektorium for useful comments on earlier versions. I am grateful to discussions with conference audiences in Hanover and Munich, where the material developed here was presented.

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The work was carried out at the University of Göttingen.

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Friederich, S. Re-thinking local causality.
*Synthese* **192, **221–240 (2015). https://doi.org/10.1007/s11229-014-0563-6

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### Keywords

- Quantum theory
- Special relativity
- Causality
- Probability
- Principal Principle