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

If (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 ((V₁), (V₂), ) is said to satisfy the Weak Reichenbach's Common Cause Principle iff for every pair of projections A(V₁), B(V₂) correlated in the normal state 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 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 ((V₁), (V₂), ) with a locally normal and locally faithful state and suitable bounded V₁ and V₂ satisfies the Weak Reichenbach's Common Cause Principle.