Jauch-Piron States

Abstract

In the first part of the book the extensions of classical principles of measure and probability theory to noncommutative projection structures were invesgtiated. It was seen that basic tools of classical analysis can be established for the quantum measure spaces given by ordered structures of projections. One of the most essential achievements along this line is the Gleason Theorem that guarantees the existence of quantum integral and underlines the physical meaning of the basic quantum calculus. In the present chapter we focus on a completely new type of results that hold only in the presence of noncommutativity and thus demonstrate a considerable difference between classical and quantum measure theory. In general, it will be shown that topological properties of states can be derived from their algebraic lattice-theoretic properties, which is not the case for measures on boolean structures.