Boolean algebras

In this appendix, X is always a (nonempty) set and S is a nonempty set of subsets of X.For any A⊆X, the symbol A⊥ denotes the set theoretical complement of A in X;thatis, A⊥ = X\A.

Definition A.1S is a (Boolean) ring if for every A, B ϵ S we have (A∪B) ϵ S and (A\B) ϵ S.

A ring is a set of sets that is closed with respect to set theoretical union and difference. If S is a ring, then ∅ ϵ S (because ∅ = A\A); however, X does not necessarily belong to S. If it does, then the (Boolean) ring is called Boolean algebra:

Definition A.2 A Boolean ring S is called Booelan algebra if X ϵ S.

A Boolean algebra S is thus closed with respect to the complement: if S is a Boolean algebra and AϵS, then A⊥ ϵ S.

One can define the notion of Boolean algebra directly: S is a Boolean algebra with respect to the set theoretical operations ∪, ∩, ⊥ if X ϵ S and if it holds that if A, B ϵ S, then (A∩B), (A∪B), and A⊥ are all in S.

The history of philosophy teaches us that metaphysical claims of sweeping generality are neither verifiable nor conclusively falsifiable. One can only aim at assessing their plausibility on the basis of the best available evidence provided by the sciences – both formal and empirical sciences. This is what has been done in this book in connection with the Principle of the Common Cause.

In Chapter 2, Reichenbach's notion of common cause and the related Common Cause Principle was formulated explicitly in terms of classical, Kolmogorovian probability measure spaces. The Definition 2.4 of common cause followed Reichenbach's original definition closely, and insisting on the quite obvious methodological principle that probabilistic concepts and statements (in particular claims about random events being correlated or probabilistically independent) are only meaningful within the context of a fixed probability measure space in terms of which some segment of reality is modeled, we specified the notions of common cause incomplete and common cause complete probability theories: a theory was defined to be common cause complete if it contains a proper common cause of every correlation it predicts, common cause incomplete otherwise. (There is a strong version of common cause completeness as well: a probability space was defined to be strongly common cause closed if it contains common causes of all admissible types – Definition 4.4.)

These explicitly defined notions form the basis on which one can start assessing the status of the Common Cause Principle in the spirit of empirical philosophy. Suppose we have an empirically confirmed scientific theory T describing some segment of reality using (possibly among other mathematical structures) a probability measure space (X, S, p).

Common cause partitions

Confronted with a common cause incomplete probability space (X, S, p)in which a direct causal influence between the correlated events is excluded, one can have in principle two strategies aiming at saving the Common Cause Principle: one may try to argue that S is not “rich enough” to contain a common cause, but there might exist a larger probability space (X′, S′, p′) that already contains a common cause of the correlation. As we have seen in Chapter 3 this strategy always works in the sense that it is always possible to enlarge (X, S, p) in such a way that the enlarged probability space already contains an event C that satisfies the Reichenbachian conditions.

Another natural idea is to suspect that the correlation between A and B is not due to a single factor but may be the cumulative result of a (possibly large) number of different “partial common causes,” none of which can in and by itself yield a complete common-cause-type explanation of the correlation, all of which, taken together, can however account for the entire correlation. In this chapter we elaborate this idea by formulating precisely a notion of the Reichenbachian Common Cause System (RCCS) and prove propositions on the existence and features of such systems.

As we have seen in Chapter 2, if the events A, B, C satisfy the Reichenbachian conditions (2.5)–(2.8) then there is a positive correlation between A and B (Proposition 2.5).

Common causes and common common causes

Proposition 3.9 tells us that every common cause incomplete classical probability space can be strongly common cause extended with respect to any (finite) set of correlations; Proposition 4.19 states that every classical probability space is even common cause completable. Note that what these propositions say is not that for a set of correlations between (Ai, Bi)(i = 1, 2 …n) there exists a single, common common cause C in the extension (or completion) for the whole set of correlations; in fact, the common causes Ci constructed explicitly in the proof of Proposition 3.9 are all different: Ci ≠ Cj (i ≠ j). This observation leads to the following question.

Let (Ai, Bi)(i = 1, … n)be a finite set of pairs of events in(X, S, p)that are correlated [Corrp(Ai, Bi) ＞ 0 for every i]. We say that C is a common common cause of these correlations if C is a Reichenbachian common cause of the correlated pair (Ai, Bi)for every i. Does every set of correlations in a classical probability space have a common common cause?

In view of the generality of this question one may surmise that the answer to it is “yes”; that is to say, one may conjecture that given any two correlations there can always exist a Reichenbachian common cause which is a common cause for both correlations, since, one may reason, we just have to refine our picture of the world by adding more and more events to the original event structure in a consistent manner, and finally we shall find a single common cause that explains both correlations.

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 ai (i = 1, 2 …n); in this case we write Sn.

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.

The Common Cause Principle in algebraic relativistic quantum field theory

Algebraic quantum field theory (AQFT) predicts correlations between projections A, B lying in von Neumann algebras A(V1), A(V2) associated with spacelike separated spacetime regions V1, V2 in Minkowski spacetime. These spacelike correlations predicted by AQFT lead naturally to the question of the status of Reichenbach's Common Cause Principle within AQFT. The aim of this chapter is to investigate the problem of status of Reichenbach's Common Cause Principle within AQFT.

Since the correlated projections belong to algebras associated with spacelike separated regions, a direct causal influence between them is excluded by the Special Theory of Relativity. Consequently, compliance of AQFT with Reichenbach's Common Cause Principle would mean that for every correlation between projections A and B lying in von Neumann algebras A(V1) and A(V2), respectively, associated with spacelike separated spacetime regions V1, V2, there must exist a projection C possessing the probabilistic properties that qualify it to be a Reichenbachian common cause of the correlation between A and B. However, since observables and hence also the projections in AQFT must be localized, one also has to specify the spacetime region V with which the von Neumann algebra A(V) is associated that contains the common cause C.

Intuitively, the region V should be disjoint from both V1 and V2 but should not be causally disjoint from them in order to leave room for a causal effect of C on the events A and B so that C can account for the correlation between A and B.

No correlation without causation. This is, in its most compact and general formulation, the essence of what has become Reichenbach's Common Cause Principle.

More explicitly, the Common Cause Principle says that every correlation is either due to a direct causal effect linking the correlated entities, or is brought about by a third factor, a so-called Reichenbachian common cause that stands in a well-defined probabilistic relation to the correlated events, a relation that explains the correlation in the sense of entailing it.

The Common Cause Principle is a nontrivial metaphysical claim about the causal structure of the World and entails that all correlations can (hence, should) be explained causally either by pointing at a causal connection between the correlated entities or by displaying a common cause of the correlation. Thus, the Common Cause Principle licenses one to infer causal connections from probabilistic relations; at the same time the principle does not address whether the causal connection holds between the correlated entities or between the common cause and the elements in the correlation.

While the technically explicit notion of common cause of a probabilistic correlation within the framework of classical Kolmogorovian probability theory is due to Reichenbach (1956), the Common Cause Principle was articulated explicitly only later, especially in the works by W. Salmon (see the “Notes and bibliographic remarks” to Chapter 2). The chief aim of this book is to investigate the Common Cause Principle; in particular, the problem of to what extent the Common Cause Principle can explain probabilistic correlations.

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 Corrp(A, B) ＞ 0 but S does not contain a common cause of the correlation Corrp(A, B).

Causal closedness and common cause closedness

Assuming that Reichenbach's Common Cause Principle is valid, one is led to the question of whether our probabilistic theories predicting probabilistic correlations can be causally rich enough to also contain the causes of all the correlations they predict. The aim of this chapter is to formulate precisely and investigate this question.

According to the Common Cause Principle, causal richness of a theory T would manifest in T's being causally closed (complete) in the sense of being capable of explaining the correlations by containing a common cause of every correlation between causally independent events A, B. This feature of a theory is formulated in the next two definitions of causal closedness. In both definitions (X, S, p)is a probability space and Rind is a two-place causal independence relation that is assumed to have been defined between elements of S. We treat the relation Rind as a variable in the problem of causal closedness; hence, at this point we leave open what properties Rind should have to be acceptable as a causal independence relation – later we will return to the issue of how to specify it.

Definition 4.1 The probability space (X, S, p) is called causally closed with respect to Rind, if for every correlation Corrp(A, B) ＞ 0 with A ϵ S and B ϵ S such that Rind (A, B) holds, there exists a common cause C of some type in S.

Main Claim

I argue in this chapter that Einstein and von Neumann meet in algebraic relativistic quantum field theory in the following metaphorical sense: algebraic quantum field theory was created in the late 1950s/early 1960s and was based on the theory of “rings of operators,” which von Neumann established in 1935–1940 (partly in collaboration with J. Murray). In the years 1936–1949, Einstein criticized standard, nonrelativistic quantum mechanics, arguing that it does not satisfy certain criteria that he regarded as necessary for any theory to be compatible with a field theoretical paradigm. I claim that algebraic quantum field theory (AQFT) does satisfy those criteria and hence that AQFT can be viewed as a theory in which the mathematical machinery created by von Neumann made it possible to express in a mathematically explicit manner the physical intuition about field theory formulated by Einstein.

The argument in favor of this claim has two components: