Independence in quantum logic approach

Abstract

An important notion in physics is the concept of independence of physical systems. One typically encounters the problem of independence of systems in the situation where 5 is a physical system and S ₁, S ₂ are two subsystems of S. The problem of independence comes then either in the form of the need to decide whether S ₁ and S ₂ are independent, or in the form of the need to impose an independence condition on S ₁, S ₂, as part of creating a suitable model of the systems involved. Relativistic quantum field theory is a case in point. We have seen in the last chapter that, on the one hand, one imposes the local commutativity (microcausality) condition on the net of local algebras, which, together with the other axioms imply other (statistical) independence conditions; on the other hand, the existence of probabilistic correlations between distant (spacelike separated) projections raises the suspicion that the (spacelike separated) local algebras are not, after all “independent” in some sense in which one expects them to be. Obviously, it is then of interest to clarify the independence relations between two subsystems of a larger quantum system, and to do this one needs intuitively and physically justifiable, and mathematically operational concepts of independence.