Orthogonality and Symmetry

Abstract

Wider research context

More than 100 years after its emergence, the principles on which quantum physics is based are still not straightforward to understand. In the spirit of the probably oldest approach, which goes back to Birkhoff and von Neumann, algebraic, relational, or categorical structures abstracted from the basic quantum-physical model have been proposed for investigation. Considerations concern, for instance, the inner structure of physically testable properties or quantum physical states, described by orthomodular lattices or orthosets.

Objectives

Our research aims at an improved understanding of the basic mathematical formalism on which quantum physics is based. We proceed on three different levels. First, we intend to improve our knowledge about algebraic structures closely linked to the quantum physical model, most notably orthomodular lattices and their state spaces. Second, we focus on the reconstruction of the Hilbert space model of quantum mechanics, at first place by means of orthosets and their symmetries. Third, we consider operator algebras, where the aim is to establish their properties in relation to derived structures like the orthoset of their states.

Approach

The notion of orthogonality is in the centre of our interest. Based solely on this structural feature, quite a number of common structures can be described, among them the Hilbert spaces. Accordingly, our main tool are orthosets, which are solely based on a symmetric, irreflexive binary relation. The mutual relation between orthosets and the underlying structure -- which can be an algebraic structure, an inner-product space, or an operator algebra -- is to be investigated. A particular focus lies on the mutual relationship between the respective symmetry groups. Moreover, research is supposed to be done on a categorical level, to which end we consider dagger categories as appropriate.