14 research outputs found
Characterizing common cause closedness of quantum probability theories
We prove new results on common cause closedness of quantum probability
spaces, where by a quantum probability space is meant the projection lattice of
a non-commutative von Neumann algebra together with a countably additive
probability measure on the lattice. Common cause closedness is the feature that
for every correlation between a pair of commuting projections there exists in
the lattice a third projection commuting with both of the correlated
projections and which is a Reichenbachian common cause of the correlation. The
main result we prove is that a quantum probability space is common cause closed
if and only if it has at most one measure theoretic atom. This result improves
earlier ones published in Z. GyenisZ and M. Redei Erkenntnis 79 (2014) 435-451.
The result is discussed from the perspective of status of the Common Cause
Principle. Open problems on common cause closedness of general probability
spaces are formulated, where is an
orthomodular bounded lattice and is a probability measure on
.Comment: Submitted for publicatio
Quantum Probability Theory
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 for
which probability theory based on type I algebras is known to be insufficient.
These illustrate the physical significance of the previously mentioned
differences.Comment: 28 pages, LaTeX, typos removed and some minor modifications for
clarity and accuracy made. This is the version to appear in Studies in the
History and Philosophy of Modern Physic
Local Primitive Causality and the Common Cause Principle in Quantum Field Theory
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),\phi) is said to
satisfy the Weak Reichenbach's Common Cause Principle iff for every pair of
projections A \in A(V_1), B \in A(V_2) correlated in the normal state \phi
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),\phi) with a locally normal and locally faithful
state \phi and open bounded V_1 and V_2 satisfies the Weak Reichenbach's Common
Cause Principle.Comment: 14 pages, Late
On the history of the isomorphism problem of dynamical systems with special regard to von Neumann's contribution
This paper reviews some major episodes in the history of the spatial
isomorphism problem of dynamical systems theory (ergodic theory). In
particular, by analysing, both systematically and in historical context, a
hitherto unpublished letter written in 1941 by John von Neumann to Stanislaw
Ulam, this paper clarifies von Neumann's contribution to discovering the
relationship between spatial isomorphism and spectral isomorphism. The main
message of the paper is that von Neumann's argument described in his letter to
Ulam is the very first proof that spatial isomorphism and spectral isomorphism
are not equivalent because spectral isomorphism is weaker than spatial
isomorphism: von Neumann shows that spectrally isomorphic ergodic dynamical
systems with mixed spectra need not be spatially isomorphic.Comment: Forthcoming in: Archive for History of Exact Sciences. The final
publication will be available at
http://www.springer.com/mathematics/history+of+mathematics/journal/40
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 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