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 ), φ) 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 φ 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 ), φ) with a locally normal and locally faithful state φ and suitable bounded V 1 and V 2 satisfies the Weak Reichenbach's Common Cause Principle.