The Common Cause Principle in algebraic relativistic quantum field theory

Algebraic quantum field theory (AQFT) predicts correlations between projections A, B lying in von Neumann algebras A(V1), A(V2) associated with spacelike separated spacetime regions V1, V2 in Minkowski spacetime. These spacelike correlations predicted by AQFT lead naturally to the question of the status of Reichenbach's Common Cause Principle within AQFT. The aim of this chapter is to investigate the problem of status of Reichenbach's Common Cause Principle within AQFT.

Since the correlated projections belong to algebras associated with spacelike separated regions, a direct causal influence between them is excluded by the Special Theory of Relativity. Consequently, compliance of AQFT with Reichenbach's Common Cause Principle would mean that for every correlation between projections A and B lying in von Neumann algebras A(V1) and A(V2), respectively, associated with spacelike separated spacetime regions V1, V2, there must exist a projection C possessing the probabilistic properties that qualify it to be a Reichenbachian common cause of the correlation between A and B. However, since observables and hence also the projections in AQFT must be localized, one also has to specify the spacetime region V with which the von Neumann algebra A(V) is associated that contains the common cause C.

Intuitively, the region V should be disjoint from both V1 and V2 but should not be causally disjoint from them in order to leave room for a causal effect of C on the events A and B so that C can account for the correlation between A and B.