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arXiv:quant-ph/9805066 · The British Journal for the Philosophy of Science

On Reichenbach's Common Cause Principle and Reichenbach's Notion of Common Cause

Gábor Hofer‐Szabó, Miklós Rédei, László E. Szabó

Published Sep 1, 1999• 105 citations
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Abstract

It is shown that, given any finite set of pairs of random events in a Boolean algebra which are correlated with respect to a fixed probability measure on the algebra, the algebra can be extended in such a way that the extension contains events that can be regarded as common causes of the correlations in the sense of Reichenbach's definition of common cause. It is shown, further, that, given any quantum probability space and any set of commuting events in it which are correlated with respect to a fixed quantum state, the quantum probability space can be extended in such a way that the extension contains common causes of all the selected correlations, where common cause is again taken in the sense of Reichenbachs definition. It is argued that these results very strongly restrict the possible ways of disproving Reichenbach's Common Cause Principle.

DOI
10.1093/bjps/50.3.377

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It is shown that, given any finite set of pairs of random events in a Boolean algebra which are correlated with respect to a fixed probability measure on the algebra, the algebra can be extended in such a way that the extension contains events that can be regarded as common causes of the correlations in the sense of Reichenbach's definition of common cause. It is shown, further, that, given any quantum probability space and any set of commuting events in it which are correlated with respect to a fixed quantum state, the quantum probability space can be extended in such a way that the extension contains common causes of all the selected correlations, where common cause is again taken in the sense of Reichenbachs definition. It is argued that these results very strongly restrict the possible ways of disproving Reichenbach's Common Cause Principle.
Reichenbach's notion of common cause In what follows ( X, S, p ) denotes a classical Kolmogorovian probability space with Boolean algebra S of subsets of a set X (with respect to the set theoretic operations ∩, ∪ and A ⊥ = X\A as Boolean algebra operations) and with the probability measure p on S . (See the Appendix for a concise review of the basic concepts of measure theoretic probability theory.) Elements of S are called random events (elements of X are sometimes called (random) elementary events). It is common to assume in probability theory that p is a countably additive (also called σ-additive) and not just a finitely additive measure but the assumption of countable additivity is somewhat controversial in the philosophical literature. The distinction between countable and finite additivity will not play any role in Chapters 3–5, 7 and 9: the results presented are valid under the assumption of either finite or countable additivity. Countable additivity will play a role in Chapters 6 and 8, however, where the problem of correlations in nonclassical (quantum) probability spaces will be investigated, and where the quantum counterpart of p will be assumed to be countably additive (“normal” in the terminology of the theory von Neumann algebras). If S has a finite number of elements, then it is the power set P(X) of a set X having n ＜ ∞ elements denoted by a i ( i = 1, 2 … n ); in this case we write S n .
Given a probabilistic correlation between two events, this correlation might be explained in terms of a common cause. In [1] Reichenbach defines the notion of common cause and shows that the definition is consistent with the explicable correlation, i.e. if two events have a common cause then they do correlate. In this paper we generalize the notion of common cause to Hilbert lattices in two different ways according to the two different definitions of conditional probability in the quantum case, and show that Reichenbach's theorem does not hold in either case. There will be given counter-examples when a common cause 'causes' correlation, anticorrelation and independence, respectively.
Reichenbach's Common Cause Principle is the claim that if two events are correlated, then either there is a causal connection between the correlated events that is responsible for the correlation or there is a third event, a so called (Reichenbachian) common cause, which brings about the correlation. The paper reviews some results concerning Reichenbach's notion of common cause, results that are directly relevant to the problem of how one can falsifY Reichenbach's Common Cause Principle. Special emphasis will be put on the question of whether EPR-type correlations can have an explanation in terms of Reichenbachian common causes. Most of the results to be recalled indicate that falsifYing Reichenbach's Common Cause Principle is much more tricky than one may have thought and that, contrary to some claims in the literature, there is no conclusive proof yet that the EPR correlations predicted by ordinary, non-relativistic quantum mechanics cannot have a common cause; furthermore, recent results about possible Reichenbachian common causes of correlations predicted by quantum field states between spacelike separated local observable algebras in algebraic quantum field theory strongly indicate that there may very well exist Reichenbachian common causes of superluminal correlations predicted by quantum field theory.
Common cause (in)completeness and extendability In principle, there are two ways to interpret Reichenbach's Common Cause Principle in general, each determined by how one views the status of the Principle with respect to the conditions of its validity: the Principle can be viewed as a falsifiable or a nonfalsifiable principle. In the falsificationist interpretation, the Common Cause Principle is a claim that can possibly and conclusively be shown not to hold for some empirically given events and their correlations; in the nonfalsificationaist interpretation, the Common Cause Principle cannot be falsified conclusively – whatever the actual circumstances. Is the Common Cause Principle falsifiable or nonfalsifiable? We shall argue that the Common Cause Principle, as formulated in Chapter 2, is not falsifiable conclusively, but the argument cannot be trivial, since the Common Cause Principle is certainly not trivially nonfalsifiable: it is not true that every classical probability space ( X, S, p ) is provably common cause complete in the sense that for any A, B ϵ S that are correlated in p there exists a C ϵ S that is a ( proper ) common cause of the correlation between A and B . There exists common cause incomplete probability spaces (for instance the probability space described in Figure 3.1 is common cause incomplete). This makes the following definition nonempty: Definition 3.1 The probability space ( X, S, p )is common cause incomplete if there exist events A, B ϵ S such that Corr p ( A, B ) ＞ 0 but S does not contain a common cause of the correlation Corr p ( A, B ).
Einstein-Podolsky-Rosen (EPR) correlations Explaining the correlations predicted by quantum mechanics in the case of joint quantum systems of the EPR type has proved to be a major challenge for the Common Cause Principle. The EPR correlations have been confirmed by a large number of sophisticated experiments carried out since the early 1980s, and, since quantum theory does not give us any hints as to where to look for possible common causes that could explain the correlations in question, the suspicion arose that common causes of EPR correlations might not exist at all. If this were indeed the case, then the EPR correlations would constitute strong empirical evidence against Reichenbach's Common Cause Principle. But how could this be possible in view of the results presented in the previous chapters? As we have seen (Chapter 3), every common cause incomplete probability space is common cause extendible (even strongly common cause extendible with respect to any finite set of correlations; Proposition 3.9); therefore every correlation can, in principle, be explained by a (possibly hidden) common cause, including the (finite number of) EPR correlations. Could the EPR correlations (not) be explained by referring to such hidden common causes? To answer this question, one has to realize first that it is not quite obvious how to link the Common Cause Principle to quantum mechanics because Reichenbach's notion of common cause was defined in terms of classical probability theory (Chapter 2), not in terms of quantum probability theory.
A condition is formulated in terms of the probabilities of two pairs of correlated events in a classical probability space which is necessary for the two correlations to have a single (Reichenbachian) common-cause and it is shown that there exists pairs of correlated events, probabilities of which violate the necessary condition. It is concluded that different correlations do not in general have a common common-cause. It is also shown that this conclusion remains valid even if one weakens slightly Reichenbach's definition of common-cause. The significance of the difference between common-causes and common common-causes is emphasized from the perspective of Reichenbach's Common Cause Principle.