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 V1 and V2 are spacelike separated spacetime regions, then the system (A(V1),A(V2), f) 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 f 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 (A(V1),A(V2), f) with a locally normal and locally faithful state f and suitable bounded V1 and V2 satisfies the Weak Reichenbach’s Common Cause Principle.