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# COMMON CAUSE PRINCIPLE

No correlation without causation. This is the most compact formulation of Reichenbach's Common Cause Principle (RCCP). More explicitly RCCP is the claim that if two events A and B are correlated, then either A and B stand in a causal relation, Rcause(A, B), or, if A and B are causally independent, Rind(A, B), then there is a third event C, a so-called Reichenbachian common cause that brings about the correlation by being related to A and B in a specific manner spelled out in the following definition, first given by Reichenbach (1956): Event C is called a (Reichenbachian) common cause of the correlation
(1)      p (A B ) p (A )p (B ) > 0
if the following conditions hold:
(2)      p (A B |C ) = p (A |C )p (B |C )
(3)      p (A B |C ) = p (A |C )p (B |C )
(4)      p (A |C ) > p (A |C )
(5)      p (B |C ) > p (B |C )
Here A, B, and C are assumed to be elements in a Boolean algebra 𝑠 and they are to be interpreted as representatives of random events. p (A |C ) = p (A C )/p (C ) and so on denote the conditional probability of A on condition C, C denotes the complement of C, and it is assumed that none of the probabilities involved is equal to zero.

RCCP is a metaphysical claim about the causal structure of the world, and it has been debated extensively in the philosophical literature whether RCCP is a valid principle. How could RCCP fail? The first step in any attempt to falsify RCCP is to display common cause incomplete probability spaces, that is, probability spaces that contain at least one correlation that does not have a common cause in the given probability space. Common cause incomplete probability spaces exist; however, the mere existence of such probability spaces does not entail that RCCP is not valid because RCCP is not the claim that given a correlated pair (A, B ) of events in 𝑠 there has to exist a common cause C that belongs to 𝑠: RCCP is a pure existence claim, not requiring the common cause to belong to the specific set of events 𝑠. If, however, one wishes to maintain the validity of RCCP against the threat posed by the existence of common cause incomplete probability spaces, one has to be able to claim that the probability space (𝑠, p ) is consistently extendable into a larger probability space (𝑠, p ) that does contain a common cause of the given correlation. If this can be done, one calls (𝑠, p ) "common cause completable" (with respect to the given correlation). It can be shown that every common cause incomplete probability space is common cause completable with respect to any finite set of correlations in it. (It is an open problem whether common cause extendability with respect to an infinite number of correlated events also holds.)

In view of common cause completability of probability spaces, one can always defend RCCP against attempts of falsification by referring to "hidden" common causes"hidden" in the sense of not being accounted for in the set of events 𝑠. Thus any successful falsification of RCCP must require some properties of the common cause in addition to those required by Reichenbach. One such possible requirement is that different correlations have a common common cause. One can show that different correlations cannot in general have a common common causenot even in case of two correlations.

Assuming that RCCP is valid, one is led to the question of whether our theories predicting probabilistic correlations can be causally rich enough to contain also the causes of the correlations. According to RCCP, causal richness of a theory (𝑠, p ) would manifest in the theory's being causally closed: (𝑠, p ) is called common cause closed with respect to R ind , if for every pair (A, B ) of correlated events such that R ind (A, B ) holds, there exists a common cause C in 𝑠 of the correlation.

Whether a probabilistic theory is common cause closed with respect to the causal independence relation R ind depends on how R ind is specified. The weaker R ind (i.e., the more pairs of random events are causally independent) the stronger the notion of common cause closedness with respect to R ind and the more difficult it is for 𝑠 to be common cause closed with respect to R ind . For instance no probability space with a finite set of random events can be common cause closed with respect to the weakest R ind (i.e., if R ind (A, B ) holds for all A and B ). However if R ind (A, B ) is strong enough to imply that the presence of A implies neither the presence of B nor the presence of B (and conversely, replacing A with B ) then finite probability spaces can be common cause closed (with respect to R ind )though they are not necessarily so. EPR correlations predicted by quantum mechanics are generally viewed as ones that might not admit a common cause type explanationif the common causes are required also to conform to relativistic causality (such common causes are called "local").

Proving the impossibility of local common causes of EPR correlations involves two difficulties: First one has to link RCCP to quantum mechanics, which is non-trivial task since Reichenbach's notion of common cause was defined in terms of classical probability theory, not in terms of quantum mechanics. Second one has to formulate "locality" of common causes. One can approach the first problem in two ways: (i) reformulating Reichenbach's notion of common cause in terms of non-classical (quantum) probability spaces; (ii) representing quantum probabilities and quantum correlations in terms of classical probability theory.

Reichenbach's notion of common cause can be reformulated in terms of non-classical probability theory, where 𝑠 is replaced by the lattice of projections of avon Neumann algebra and p by a state on the von Neu-mann algebra. The notions of common cause and ofcommon cause completability can be adapted to the non-commutative case, and it can be shown that every non-commutative probability space also is common cause completable. Relativistic causality can also be formulated in terms of non-commutative probability spacesthe resulting theory is known as local algebraic quantum field theory. Locality of common causes of EPR correlations predicted by local quantum field theory can be defined by requiring the common causes to belong to a spacetime region located within the intersection of backward light cones of the spacelike separated regions containing the correlated observables. Whether such localized common causes exist in quantum field theory is an open problem, only partial results are known.

One can also take approach (ii) and formulate locality conditions for the hypothetical common causes of EPR correlations predicted by non-local, non-relativistic quantum mechanicsnow represented in classical probability theory. These locality conditions express two sorts of independence: (i) the statistical independence of the random events of choosing measurements in the two wings of a typical correlation experiment and (ii) the statistical independence between choosing measurements in any wing and presence of any combination of the hypothetical common causes of spin correlations in different directions. Again it is an open question whether common causes satisfying these locality conditions can exist. It is known however that the EPR correlations between outcomes of spin measurements in different directions cannot have a common common cause because the assumption of common common causes of EPR correlations in different directions implies Bell's inequality.

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