Results 1 
5 of
5
On Reichenbach's common cause principle and Reichenbach's notion of common cause
"... 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 correlation ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
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.
Local primitive causality and the common cause principle in quantum field theory
 FOUND. PHYS
, 2002
"... 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 pai ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
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
Legacy of john von neumann in the theory of operator algebras
 In The Neumann Compendium
"... After some earlier work on single operators, von Neumann turned to families of operators in [1]. He initiated the study of rings of operators which are commonly called von Neumann algebras today. The papers which constitute the series “Rings of operators ” opened a new field in mathematics and influ ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
After some earlier work on single operators, von Neumann turned to families of operators in [1]. He initiated the study of rings of operators which are commonly called von Neumann algebras today. The papers which constitute the series “Rings of operators ” opened a new field in mathematics and influenced research for half a century (or even longer). In the standard theory of modern operator algebras, many concepts and ideas have their origin in von Neumann’s work. Since its inception, operator algebra theory has been in intimate relation to physics. The mathematical formalismofquantumtheorywasoneofthemotivationsleadingnaturallytoalgebras of Hilbert space operators. After decades of relative isolation, again physics fertilized the operator algebra theory by mathematical questions of quantum statistical mechanics and quantum field theory. The objective of the present article is twofold. On the one hand, to sketch the early development of von Neumann algebras, to show how the fundamental classification of algebras emerged from the lattice of projections. These old ideas of von Neumann and Murray revived much later in connection with Jordan operator algebras
When can statistical theories be causally closed?
"... The notion of common cause closedness of a classical, Kolmogorovian probability space with respect to a causal independence relation between the random events is defined, and propositions are presented that characterize common cause closedness for specific probability spaces. It is proved in partic ..."
Abstract
 Add to MetaCart
(Show Context)
The notion of common cause closedness of a classical, Kolmogorovian probability space with respect to a causal independence relation between the random events is defined, and propositions are presented that characterize common cause closedness for specific probability spaces. It is proved in particular that no probability space with a finite number of random events can contain common causes of all the correlations it predicts; however, it is demonstrated that probability spaces even with a finite number of random events can be common cause closed with respect to a causal independence relation that is stronger than logical independence. Furthermore it is shown that infinite, atomless probability spaces are always common cause closed in the strongest possible sense. Open problems concerning common cause closedness are formulated and the results are interpreted from the perspective of Reichenbach’s Common Cause Principle. 1 The problem and informal review of results Let T be a theory formal part of which contains classical probability theory (S, p), where S is a Boolean algebra of sets representing random events (with Boolean operations ∪,∩,⊥) and