The many facets of orthomodularity
Wider research contextMore than 100 years after its emergence, the principles on which quantum physics is based are still not straightforward to understand. The probably oldest approach to this issue goes back to Birkhoff and von Neumann and aims at a characterisation of the basic quantum-physical model by algebraic means. In this context, the notion of an orthomodular lattice has emerged. The idea is to restrict attention to the inner structure of testable properties, disregarding physical contents but accentuating the essential features of the mathematical framework.
Research objectivesWe intend to elaborate on different aspects around the notion of orthomodularity, following the vision of achieving a broader understanding of this notion. Our aim is, first, to improve the characterisation of the prototypical orthomodular lattices, arising from a Hilbert space. Second, we intend to advance the classification of general orthomodular lattices and further quantum structures. Third, we aim at deepening our understanding of the state space of orthomodular posets. Fourth, we intend to characterise what is sometimes seen as the "fuzzified" version of orthomodular lattices, the Hilbert space effects.
ApproachWe have chosen approaches that were recently proposed to gain new perspectives on orthomodular structures. In order to develop a novel description of the complex Hilbert space, we will focus on Foulis' orthogonality spaces and their automorphism groups. The aim of classifying general orthomodular lattices will be based on the idea of characterising the partially ordered set of Boolean subalgebras. The investigation of the state space of orthomodular posets focuses on the numerous interrelations with algebraic properties. Finally, the characterisation of the Hilbert space effects will be tackled by means of PBZ*- and related algebras.