Partial quantum logics revisited

Partial Boolean algebras (PBA’s) were introduced by Kochen and Specker as an algebraic model reflecting the mutual relationships among quantumphysical yes-no tests. The fact that not all pairs of tests are compatible was taken into special account. In this paper, we review PBA’s from two sides. First, we generalize the concept, taking into account also those yes-no tests which are based on unsharp measurements. Namely, we introduce partial MV-algebras, and we define a corresponding logic. Second, we turn to the representation theory of PBA’s. In analogy to the case of orthomodular lattices, we give conditions for a PBA to be isomorphic to the PBA of closed subspaces of a complex Hilbert space. Hereby, we do not restrict to purely algebraic statements; we rather give preference to conditions involving automorphisms of a PBA. We conclude outlining a critical view on the logico-algebraic approach to the foundational problem of quantum physics.