Regularity in quantum logic
A quantum mechanical system is said to be regular if every pairwise compatible set of questions of it is compatible. In the paper, the author shows that this property is a consequence of associating questions of the system with direct product decompositions of its state space if one assumes that the true and false questions correspond to the binary direct compositions of the state space. Such an assumption is in accordance with von Neumann's approach to quantum mechanics. The author has shown recently [Trans. Am. Math. Soc. 348, No. 5, 1839--1862 (1996; Zbl 0859.03032)] that the collection of binary direct decompositions of any algebraic, relational or topological structure \(X\) form an orthomodular poset \(\operatorname {Fact}X\). Many familiar orthomodular posets (OMPs) arise in this way. On the other hand, it was shown [loc. cit.] that the technique of building an OMP from the decompositions of \(X\) can be extended to the construction of an OMP from an arbitary relation algebra. Theorem 4.9 of the paper under review says that every such a relational OMP is regular, i.e., every pairwise compatible subset of it is included in a Boolean subalgebra; this applies also to \(\operatorname {Fact}X\). The author also studies regularity for decompositions preserving the structure of \(X\) (it is shown by an example that a subalgebra of a regular OMP need not be regular). Furthermore, he establishes a correspodence between the collection of all finite Boolean subalgebras of \(\operatorname {Fact}X\) and equivalence classes of finite direct decompositions of \(X\) (considered either as a set or as an algebraic, relational or topological structure). It turns out that the generalisation of this result to infinite Boolean subalgebras requires to replace infinite direct products with sheaves over a Boolean space. However, this does not work for \(X\) a topological structure. The paper contains the needed preliminaries on OMPs, relation algebras and sheaf theory as well as a brief review of results of the cited paper.
scientific article; zbMATH DE number 1341285