The source of the orthomodular law

Abstract

A central theme in quantum logic is to consider generalizations of the lattice of closed subspaces of a Hilbert space as models for the propositions of a quantum mechanical system. It is observed that ortholattices satisfies the identity known as the orthomodular law. The orthomodular lattices (OMLs) and orthomodular posets (OMPs) as abstract models for the propositions of a quantum mechanical system are well established. It is instructive to visualize how the validity of the orthomodular law in ortholattices follows in a transparent way from basic properties of Hilbert spaces. The orthomodular law has several equivalent formulations that essentially highlight different aspects of its nature. OMLs are exactly those ortholattices L where the partial ordering of L is determined by the partial orderings of its Boolean subalgebras. Thus, OMLs are exactly the locally Boolean ortholattices. Orthomodular law has a major role in the theoretical foundations of quantum mechanics and the subsequent development of dimension theory for certain OMLs.