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.

Fundamentals of quantum mechanics

Quantum mechanics remains one of the most successful, yet enigmatic, formulations of physics. Uncertainty and measurement constraints are incorporated at the core of this fundamental description of micro-physical dynamics. Quantum mechanics successfully describes the observed structures in chemistry and materials science. In particular, the impenetrability of matter due to Pauli exclusion is a consequence of incorporating special relativity into quantum mechanics. Microscopic causality, or the requirement that communications cannot propagate at greater than the speed of light, relates a particle's spin to its quantum statistics but does not exclude the space-like correlations associated with a coherent quantum state.

This section will examine some of the foundations of quantum physics. An emphasis will be placed upon how the microscopic fundamentals of quantum mechanics relate to the geometric fundamentals of gravitational mechanics.

Quantum formalism

There are several interpretations of quantum mechanics offering models of the underlying (often hidden) dynamics that generate the observed quantum phenomenology (for example, the Copenhagen interpretation, or the many-worlds interpretation). Since any interpretation consistent with quantum physics cannot be proved or disproved by experiment, only those aspects of quantum formalism directly connected to experimental observables will be developed, with minimal interpretation imposed. To establish the conventions utilized in what follows, a brief overview of the formalism that describes those calculable observables modeled by quantum physics will be given.

This book summarizes and develops further in some respects the results of research the authors have undertaken in the past several years on the problem of explaining probabilistic correlations in terms of (Reichenbachian) Common Causes. The results have been published by the authors of this book in a number of papers, partly in collaborations with each other and with other colleagues; these papers form the basis of the present book. We wish to thank especially Balázs Gyenis, Zalán Gyenis, Inaki San Pedro, Stephen J. Summers, and Péter Vecsernyés for the cooperation on the topic of the book and in particular for allowing us to use material in joint publications.

In our work we also have benefited greatly from collaborations and informal discussions with a number of other colleagues. These include Nuel Belnap, Arthur Fine, Jeremy Butterfield, Rob Clifton, Gerd Grasshoff, David Malament, Tomasz Placek, Samuel Portman, Elliott Sober, Leszek Wronski, and Adrien Wüthrich – we thank them all for their interest in our work and for their readiness to share with us their insights and views.

The research that the book is based on was partially supported by several small grants from the Hungarian National Science Found (OTKA). The final writing was facilitated by the OTKA grant with contract number K100715.

The fundamental “players” in the cosmological arena are microscopic particles and the interactions by which they exchange well-defined quantum numbers. Many of the critical properties of micro-physics can be determined by their behaviors in nearly flat space-time, as described by Minkowski. There are several requirements that a successful model of fundamental processes should fulfill. Among these characteristics are:

1. • Describes quantum phenomenology: Quantum mechanics successfully describes the subtle behaviors of matter and energies undergoing microscopic exchanges. Quantum behaviors inherently have aspects beyond measurement.

2. • Conservation properties: Most particles have internal quantum numbers, like charge, lepton number, baryon number, etc., that are carried undiminished throughout complicated interactions with other particles. In addition, they carry properties like mass and spin, whose kinematic transformations under space-time transformations are well-defined, and which satisfy composite conservation laws (energy, momentum, angular momentum) for sufficiently isolated homogeneous and isotropic systems.

3. • Unitarity: Despite being unable to follow quantum coherent particle properties while that coherence is maintained, the evolution of those properties is described in a manner that ensures conservation of probability. This means that these properties do not just “pop” into or out of existence between the detections and interrelations of the particles.

4. • Cluster decomposability and classical correspondence: Classical physics is quite successful in describing much of common phenomenology. Classical models exploit those characteristics of a system that can be isolated, studied, and parameterized independent of observation.

5. […]

The incorporation of quantum mechanics into gravitational dynamics introduces perplexing issues into modern physics. In contrast to other interactions like electromagnetism, the classical trajectory of a gravitating system is independent of the mass coupling to the gravitational field. As previously discussed, this allows the gravitation of arbitrary test particles to be described in terms of local geometry only, the basis of general relativity. Thus, the geometrodynamics of classical general relativity are most directly expressed using localized geodesics. However, quantum dynamics incorporate measurement constraints that disallow complete localization of physical systems. A coherent quantum system is not represented by a path or a classical trajectory; rather, it self-interferes throughout regions. This complicates the use of classical formulations in describing inherently quantum processes.

In addition, the equations of general relativity are complex and non-linear in the interrelations between sources and geometry, which makes solutions of even classical systems complicated. The key to describing complex systems is to determine the most useful set of parameters and coordinates that give concise predictive explanations of those systems. This chapter will develop tools for examining quantum behaviors in gravitating systems.

Quantum coherence and gravity

The behaviors of quantum objects in Minkowski space-time are well understood, despite a lack of consensus on the various interpretations (Copenhagen, many worlds, etc.) of the underlying fundamentals of the quantum world, or concerns of the completeness of quantum theory. There have also been tests of systems modeled by equations that involve both Newton's gravitational constant GN and Planck's constant ħ, as described in Section 2.3.1.

One of the principles of modern physics that is most adhered to is the expectation that models constructed to describe the phenomena of the physical universe should not depend upon any absolute frame of reference. The discovery of the CMB radiation perhaps demonstrates a counter example to this supposition, due to the preferred frame at rest relative to the energy content of the universe during its initial phase of expansion, as will be discussed in the next chapter. However, for most phenomena, the co-variance of the laws modeling those phenomena is consistent with the expectation of independence of the fundamental physics from the particular frame of reference utilized by the observer. This principle is embodied in the concept of complementarity [23] in the description of black holes. In its most direct expression, complementarity simply states that no observer should ever witness a violation of a law of nature. In particular, one expects that for a freely falling observer, there should be no local effects of gravitation as espoused by the principles of equivalence and relativity.

In this treatment, a horizon will always be a light-like surface that globally separates causally disconnected regions of space-time. Since light itself is characterized by both classical and quantum properties, geometries with horizons offer insights into the subtle relationships between general relativity and quantum physics. Generally, horizons can be only globally (not locally) defined, which means that local experiments performed by freely falling, inertial observers cannot detect the presence of such horizons.

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.

This book is the result of decades of efforts by the author to understand physics at its most basic foundations, and to construct models that can address some of the unanswered questions about gravity, quantum mechanics, and the spectrum of fundamental particles. It is felt that foundational explorations should examine the conceptual and philosophical basis of a discipline. As such, less emphasis was placed upon the mathematics of complex calculations, while more emphasis was placed upon the consistencies and critiques of basic premises, in the preparation of this book. As a scientist, there is often a tendency to be drawn towards mathematical and formal pursuits, simply because of the beauty and elegance of mathematics. Such approaches sometimes bypass difficult conceptual approaches and thought experiments, or develop speculative formulations just because of their elegance. This book attempts to establish a balance towards the exploration of basic concepts and puzzles.

The prior book on black holes by Lenny Susskind and myself serves as an introduction to quantum physics and relativity for static geometries, as well as information in horizon physics. However, some of my more recent explorations indicate that qualitative modifications in descriptions of the physics occur once dynamics has been incorporated. This then calls into question any intuitions from static geometries that have been used to imply that those very geometries cannot be static. This book incorporates dynamic, spatially coherent geometries, as well as expanding upon the well-established foundations of the prior work.

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.

Groups and special relativity

Fundamentals of group theory

Properties of groups

A group is a set of elements that have the following properties:

1. • Contains the identity transformation 1;

2. • If E and E′ are elements in the group, then there exists a group operation (generically called “multiplication”) that always produces an element of the group, E″ = E′ · E (closure);

3. • For every transformation element E, there exists in the group an inverse element E-1, where E-1 • E = 1;

4. • The group operation is associative, E′ · (E′ · E) = (E″ · E′) · E.

A particular type of group satisfies an additional property. For an abelian group, the group operation yields the same answer regardless of the order, E′ · E = E · E′. Often, a subset of the elements within a group satisfies all four group properties. This subset is referred to as a subgroup.

Generally, two different groups are isomorphic if there is a one-to-one relationships between the elements of the groups with regards to group operations. More generally, if a set of elements in one group are in direct relationship with one element in another, the groups are homomorphic.

A particular class of groups is quite useful for describing transformations in quantum physics. Invertible N × N matrices (i.e., matrices with non-vanishing determinants) form the set of linear groups.

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.