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

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If {A(V)} is a net of local von Neumann algebras satisfying standard axioms of algebraic relativistic quantum field theory and V₁ and V₂ are spacelike separated spacetime regions, then the system (A(V₁),A(V₂),ϕ) is said to satisfy the Weak Reichenbach's Common Cause Principle iff for every pair of projections A ∈ A(V₁), B ∈ A(V₂) correlated in the normal state ϕthere 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 V₁ and V₂ and disjoint from both V₁ and V₂, 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(V₁),A(V₂),ϕ) with a locally normal and locally faithful state ϕand open bounded V₁ and V₂ satisfies the Weak Reichenbach's Common Cause Principle.