STATE PROPERTY SYSTEMS AND CLOSURE SPACES: EXTRACTING THE CLASSICAL EN NONCLASSICAL PARTS

In ¹ an equivalence of the categories SP and Cls was proven. The category SP consists of the state property systems ² and their morphisms, which are the mathematical structures that describe a physical entity by means of its states and properties ^3,4,5,6,7,8. The category Cls consists of the closure spaces and the continuous maps. In earlier work it has been shown, using the equivalence between Cls and SP, that some of the axioms of quantum axiomatics are equivalent with separation axioms on the corresponding closure space. More particularly it was proven that the axiom of atomicity is equivalent to the T₁ separation axiom ⁹. In the present article we analyze the intimate relation that exists between classical and nonclassical in the state property systems and disconnected and connected in the corresponding closure space, elaborating results that appeared in ^{10, 11}. We introduce classical properties using the concept of super selection rule, i.e. two properties are separated by a superselection rule iff there do not exist ‘superposition states’ related to these two properties. Then we show that the classical properties of a state property system correspond exactly to the clopen subsets of the corresponding closure space. Thus connected closure spaces correspond precisely to state property systems for which the elements 0 and I are the only classical properties, the so called pure nonclassical state property systems. The main result is a decomposition theorem, which allows us to split a state property system into a number of ‘pure nonclassical state property systems’ and a ‘totally classical state property system’. This decomposition theorem for a state property system is the translation of a decomposition theorem for the corresponding closure space into its connected components.