Local primitive causality and the common cause principle in quantum field theory

Redei, Miklos and Summers, Stephen J. (2001) Local primitive causality and the common cause principle in quantum field theory.

Abstract

If $\{{\cal A}(V)\}$ 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 $({\cal A}(V_1),{\cal A}(V_2),\phi)$ is said to satisfy the Weak
Reichenbach's Common Cause Principle iff for every pair of
projections $A\in{\cal A}(V_1)$, $B\in{\cal A}(V_2)$ correlated in the normal
state $\phi$ 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
$({\cal A}(V_1),{\cal A}(V_2),\phi)$ with a locally normal and locally faithful state
$\phi$ and open bounded $V_1$ and $V_2$ satisfies the Weak Reichenbach's
Common Cause Principle.