Abstract
This paper examines the conceptual relations between the notions of determinism and locality. From a purely conceptual point of view, determinism does not appear to imply locality, nor (contrapositively) does nonlocality appear to imply indeterminism. The example of Newtonian mechanics strengthens this impression. It turns out, however, that in the context of quantum mechanics, a more complex connection between determinism and locality emerges. The connection becomes crucial when nonlocality is distinguished from no signaling. I argue that it is indeterminism that allows nonlocal theories such as quantum mechanics to comply with the no signaling constraint. I examine a number of interpretations of quantum mechanics, among them that of Schrödinger, Pitowsky and Popescu and Rohrlich, to support this claim.
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Notes
- 1.
I would like to thank Meir Hemmo, Hilary Putnam, Daniel Rohrlich, Orly Shenker and Mark Steiner for their critique of earlier versions of this paper.
- 2.
Such a general analysis involves, in my view, additional notions and constraints such as stability, conservation laws and extremum principles, which will not be considered in this paper.
- 3.
Clearly there is no single ‘standard’ interpretation, but the term is used here, as is common, to refer to descendents of the Copenhagen interpretation. Pitowsky’s interpretation, for instance, is based on the Birkhoff von Neumann axiomatization, the corner stone of the standard interpretation. Rival interpretations such as Bohm’s, GRW, modal interpretations and the many world interpretation deserve separate analyses.
- 4.
The concept of a closed system is of course an idealization that should be weakened to meet more realistic conditions. Further, the notion of the value of a parameter at a specific time also needs refinement, for some physical magnitudes such as velocity involve change over time, convergence to a limit, and so on.
- 5.
- 6.
Note that I here refer to locality, not Bell-locality; see below.
- 7.
See Frisch [2].
- 8.
They were not conceived as independent in Antiquity and the Middle Ages; see Glasner [3] Chap. 3. One of the reasons for the difference between ancients and moderns on this point is that the former were inclined to understand determinism in terms of the universality requirement—every event has a cause –rather than in terms of the same-cause-same effect requirement. On this construal, it is easier to appreciate why the contiguity of interaction appeared to exclude spontaneous occurrences.
- 9.
Again, by 'a grain of determinism' I do not mean the necessity of strict universal laws; probabilistic dependence would be sufficient to indicate nonlocality. Indeed this is what happens in some of the quantum mechanical cases.
- 10.
See, however Earman [4] and Norton [5] for counter examples to determinism in Newtonian mechanics. Despite these examples, Newtonian mechanics countenances numerous processes that are deterministic but nonlocal, attesting to the insufficiency of determinism for locality. It is also questionable as to whether STR is necessarily deterministic, but it is certainly compatible with determinism, which is all we need in order to demonstrate the feasibility of the first combination.
- 11.
Reichenbach [6] Chap. 19
- 12.
To test the existence of a common cause, one therefore compares the conditional probability of the joint event (on the common cause) with the product of the conditional probabilities of the individual events (on the common cause). When these probabilities are equal (unequal), one talks of factorizability (non-factorizability). See Chang and Cartwright [7] for an analysis of the relationship between factorizability and the existence of a common cause. They argue that since, in the probabilistic (indeterministic) case, factorizability is in general not a necessary condition for the existence of a common cause, the non-factorizability of quantum distributions does not exclude the possibility of common causes of the EPR correlations. They go on to propose such a common cause model, but their model requires discontinuous causal influences and is manifestly nonlocal.
- 13.
In the context of discussions of Bell’s inequalities, the assumption that a common cause (whether deterministic or stochastic) exists is sometimes referred to as locality, or Bell-locality. Note, however, that the Bell-locality is not identical to the requirement of locality characterized above, for it is committed not only to the continuity and finite speed of any causal interaction if it exists, but to the very existence of a cause—a ‘screening-off’ event.
- 14.
Bohmians reject this conclusion. See note no. 17 below.
- 15.
The notion of signaling has an anthropomorphic flavor, but I will not attempt to refine it. It should be noted that no signaling is not identical with Lorentz invariance; a theory can prohibit signaling while failing to be Lorents invariant. Note, further, that in the traditional understanding of the concept of locality the only possibilities were locality plus no signaling or nonlocality plus signaling. The distinction seeks to make room for a new possibility—nonlocality and no signaling which was previously seen as incoherent. (The fourth possibility, locality plus signaling, remains incoherent).
- 16.
- 17.
Bohmian QM seems to provide a counter-example, for despite being deterministic, it does not allow signaling. Recall, however, that in Bohmian QM the equilibrium state excludes knowledge of the predetermined states. In the absence of this information, the experimenter cannot use the correlations for signaling.
- 18.
Pitowsky’s formulation is slightly different from that of Birkhoff and von Neumann, but the difference is immaterial. Pitowsky makes significant progress, however, in his treatment of the representation theorem for the axiom system, in particular in his discussion of Solér’s theorem. The theorem, and the representation problem in general, is crucial for the application of Gleason’s theorem, but will not concern us here.
- 19.
In his book [16], Pitowsky studied the geometrical meaning of Boole’s classical conditions on probability. More details can be found in the introduction to this volume.
- 20.
In the literature, following in particular Jarrett [30], it is customary to distinguish outcome independence, violated in QM, from parameter independence, which is observed, a combination that makes possible the peaceful coexistence with STR. The non-contextuality of measurement amounts to parameter independence. See, however Redhead [8] and Maudlin [9], among others, for a detailed exposition and critical discussion of the distinction between outcome and parameter independence and its implications for the compatibility with STR.
- 21.
In the Clauser-Horn-Shimony-Holt inequality the classical limit reached by local realist considerations is −2 ≤ S ≤ 2. In QM this inequality can be violated, but as Boris Tsirelson has shown there is an upper bound to this violation: −2√2 ≤ S ≤ 2√2. Rohrlich and Popescu show that the Tsirelson bound can be violated without violation of STR, that is, without violation of the no signaling requirement.
- 22.
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Ben-Menahem, Y. (2012). Locality and Determinism: The Odd Couple. In: Ben-Menahem, Y., Hemmo, M. (eds) Probability in Physics. The Frontiers Collection. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21329-8_10
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