Nonordered quantum logic and its YES-NO representation

Krystyna Bugajska

Thomas Vetterlein

Partial Boolean algebras (PBAs) were introduced by Kochen and Specker as an algebraic model reflecting the mutual relationships among quantum-physical yes–no tests. The fact that not all pairs of tests are compatible was taken into special account. In this paper, we review PBAs from two sides. First, we generalise 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 PBAs. 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 ourselves to purely algebraic statements; we rather give preference to conditions involving automorphisms of a PBA. We conclude by outlining a critical view on the logico-algebraic approach to the foundational problem of quantum physics.

John Bell

Gianpiero Cattaneo, María Luisa Dalla Chiara, Roberto Giuntini +1 more

Enrico G. Beltrametti, Gianni Cassinelli

Olivier Darrigol

This chapter is devoted to several attempts at showing the necessity of quantum mechanics. It first examines whether necessity arguments can be extracted from the history of quantum mechanics, both on the matrix-mechanics side and on the wave-mechanics side. It then recounts the stunning mathematical discovery (1979) that the only one-parameter deformation of the Lie algebra of classical Hamiltonian mechanics is the Moyal algebra, which is isomorphic to the algebra of infinitesimal evolutions of quantum mechanics. The chapter goes on with an account of “quantum logic” as an attempt to justify the matrix-density representation of quantum states and the unitary evolution of these states by a natural logic of Yes–No experimental questions. The last section is a discussion of Lucien Hardy’s (2001) and more recent proposals of “natural axioms” for statistical measurement correlations, leading again to the matrix-density formulation of quantum mechanics.