The many facets of orthomodularity

More than 100 years after its emergence, the principles on which one of the most important physical theories quantum mechanics is based are still not straightforward to understand. The probably oldest approach to this issue goes back to the two mathematicians Garrett Birkhoff and John von Neumann, whose seminal work The logic of quantum mechanics has been of considerable influence. The idea is to describe the inner structure of testable properties of a physical system by algebraic means. In this context, the notion of orthomodularity was coined. The totality of quantum physical properties of a system is, on the one hand, ordered in a natural way: one property can be stronger than another one. This structure is, on the other hand, complemented: any property possesses a negation. We are thus led to an algebraic structure called ortholattice. The present case is in a certain sense special; for any two comparable properties there is a third one that is complementary to the first one and whose disjunction with the latter leads to the second one. An ortholattice with this property is called an orthomodular lattice. The aim of the project is to elaborate on different aspects around orthomodularity, a notion that plays an important role not only in the context of the foundations of quantum mechanics, but generally in mathematics. Our group, consisting of twelve mathematicians and physicists from Austria and the Czech Republic, plans to work on the following topics. We aim, first, at a deeper understanding of the mentioned prototypical example of an orthomodular lattice, the lattice of closed subspaces of a complex Hilbert space. We use to this end the notion of an orthogonality space, which is due to David Foulis. We consider, second, orthomodular lattices in a broader context and include several further, closely related structures in the discussion. We are concerned, third, with the state space of orthomodular structures. A state assigns a real number between 0 and 1 to each element of a ortholattice and can be seen as an analogue of a state of a quantum physical system. We consider, forth, effect algebras. These structures can be considered as a generalisation of orthomodular lattices, similarly to the case of fuzzy sets which, possessing continuous membership degrees, generalise ordinary sets.

Even after more than 100 years since its invention, quantum physics seems odd at first glance. One possible approach to enable a deeper insight focuses solely on the most basic model used by this theory: the Hilbert space. The idea is to reduce the Hilbert space, which is a linear space equipped with an inner product, to a structure of a simplest possible type. This issue was a central focus of the project. A Hilbert space can be reduced to a certain algebraic structure, a so-called orthomodular lattice. This fact motivated us to place the concept of orthomodularity at the centre of our considerations. Orthomodularity is actually a core concept in mathematics, a notion that may describe all possible factorisations of some structure. The project work centred around a number of issues arising in this context, dealing with various structures, like orthomodular lattices, effect algebras, and quantales. We elaborated in particular on a much more far-reaching reduction of the Hilbert space. The basic quantum physical model can actually be built solely upon the concept of orthogonality. The (pure) states of a quantum mechanical system are modelled by one-dimensional subspaces of a Hilbert space, and for two such subspaces to be orthogonal means that the probability of the transition between the corresponding states is zero. The collection of all one-dimensional subspaces, equipped with the orthogonality relation, is a so-called orthoset. Remarkably, this orthoset alone determines the structure of the Hilbert space essentially uniquely. We were able to give an surprisingly simple answer to the question of what characteristic property an orthoset arising from a inner product space has. The so-called linear orthosets are defined by a simple combinatorial property. A further challenge was to define a category based on orthosets, that is, to define structure-preserving mappings between orthosets in a suitable way. The essential requirement here is to be able to establish an analogy to linear spaces. After lengthy and moderately successful attempts, we came to the conclusion that a very simple property, the so-called adjointability of mappings between orthosets, achieves the desired effect. Requiring that the inclusion map of subspaces of orthosets is adjointable emerged as an alternative criterion for the representability by inner product spaces. The results are remarkable in that orthosets offer only a minimal structure, which merely specifies that certain pairs of elements are in relationship with each other. Yet the notion of an orthoset has proven to be powerful enough to offer a new perspective on linear spaces, a structure that is omnipresent in mathematics.