Local Primitive Causality and the Common Cause Principle in Quantum Field Theory

Miklós Rédei, Stephen J. Summers
August 06, 2001
If $\{\mathcal{A}(V)\}$ 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 $(\mathcal{A}(V_1),\mathcal{A}(V_2),\phi)$ is said to satisfy the Weak Reichenbach's Common Cause Principle iff for every pair of projections A \in $\mathcal{A}(V_1)$, $B \in \mathcal{A}(V_2)$ 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 $(\mathcal{A}(V_1),\mathcal{A}(V_2),\phi)$ 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.
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