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 Reichenbach's definition. It is argued that these results very strongly restrict the possible ways of disproving Reichenbach's Common Cause Principle.

If {A(V)} is a net of local von Neumann algebras satisfying standard axioms of algebraic relativistic quantum field theory and V 1 and V 2 are spacelike separated spacetime regions, then the system (A(V 1), A(V 2), f) is said to satisfy the Weak Reichenbach’s Common Cause Principle iff for every pair of projections A ¥ A(V 1), B ¥ A(V 2) correlated in the normal state f there exists a projection C belonging to a von Neumann algebra associated with a spacetime region V contained in the union of the backward light cones of V 1 and V 2 and disjoint from both V 1 and V 2, a projection having the properties of a Reichenbachian common cause of the correlation between A and B. It is shown that if the net has the local primitive causality property then every local system (A(V 1), A(V 2), f) with a locally normal and locally faithful state f and suitable bounded V 1 and V 2 satisfies the

It is shown that in the Rovelli relational interpretation of quantum mechanics, in which the notion of absolute or observer independent state is rejected, the conclusion of the ordinary EPR argument turns out to be frame-dependent, provided the conditions of the original argument are suitably adapted to the new interpretation. The consequences of this result for the ‘peaceful coexistence ’ of quantum mechanics and special relativity are briefly discussed. 1 1

An overview is given of what mathematical physics can currently say about the vacuum state for relativistic quantum field theories on Minkowski space. Along with a review of classical results such as the Reeh–Schlieder Theorem and its immediate and controversial consequences, more recent results are discussed. These include the nature of vacuum correlations and the degree of entanglement of the vacuum, as well as the striking fact that the modular objects determined by the vacuum state and algebras of observables localized in certain regions of Minkowski space encode a remarkable range of physical information, from the dynamics and scattering behavior of the theory to the external symmetries and even the space–time itself. In addition, an intrinsic characterization of the vacuum state provided by modular objects is discussed. 1

. We provide an overview of the connections between Bell's inequalities and algebraic structure. 1. Introduction Motivated by the desire to bring into the realm of testable hypotheses at least some of the important matters concerning the interpretation of quantum mechanics evoked in the controversy surrounding the EinsteinPodolsky -Rosen paradox [18][5], Bell discovered the first example [3][4] of a family of inequalities which are now generally called Bell's inequalities. These inequalities provide an upper bound on the strength of correlations between systems which are no longer interacting but have interacted in the past. Stated briefly, Bell showed that if the correlation experiments can be modelled by a single classical probability measure, then the strength of the correlations must satisfy a bound which is violated by certain quantum mechanical predictions (and, as has been verified experimentally, by nature - for a review of this original application of Bell's inequalities and ...

A conjecture concerning vacuum correlations in axiomatic quantum field theory is proved. It is shown that this result can be applied both in the context of EPR-type experiments and Bell-type experiments. Key words: unified EPR, Bell, algebraic QFT. 1.

. Some algebraic invariants associated with Bell's inequalities are defined for inclusions of von Neumann algebras and studied in the context of general algebraic quantum theory. More special results are proven for quantum field theory, which establish that these invariants take infinitely many values. Sharp short-distance bounds on the Bell correlations are also demonstrated in the context of relativistic quantum field theory. I. Introduction Bell's inequalities have received a great deal of attention from many different points of view in the past twenty years. These inequalities limit the strength of correlations which can be modelled by a theory in the class of "classical and local" theories (cf. [8][19][20]). Our own initial interest was to show that the vacuum state in relativistic quantum field theory contains sufficiently strong quantum correlations between spacelike separated measurements that Bell's inequalities are (maximally 1 ) violated. We proved this to be true not onl...

The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types. The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras. However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable. The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions. In addition, differences between the probability theories in the type I, II and III settings are explained. A brief description is given of quantum systems

Abstract. We provide an overview of the connections between Bell’s inequalities and algebraic structure. 1.