Orthogonality Spaces Arising from Infinite-Dimensional Complex Hilbert Spaces

Thomas Vetterlein

doi:10.1007/s10773-019-04230-4

Abstract

The collection of one-dimensional subspaces of an anisotropic Hermitian space is naturally endowed with an orthogonality relation and represents the typical example of what is called an orthogonality space: a set endowed with a symmetric, irreflexive binary relation. We investigate in this paper symmetry properties of orthogonality spaces. We show that two conditions concerning the existence of automorphisms of orthogonality spaces are essentially sufficient to characterise the basic model of quantum physics, the countably infinite dimensional complex Hilbert space.

Highlights

To characterise the basic quantum-physical model by algebraic means has been a major topic in the discussions around the foundations of quantum mechanics, in particular in the framework of the approach that goes back to the seminal work of Birkhoff and von Neumann [2]
In the centre of interest, we find the complex Hilbert space and the collection of those entities that correspond to the outcomes of quantum physical measurements
We focus on symmetries; we postulate the existence of certain automorphisms of orthogonality spaces

Summary

Introduction

To characterise the basic quantum-physical model by algebraic means has been a major topic in the discussions around the foundations of quantum mechanics, in particular in the framework of the approach that goes back to the seminal work of Birkhoff and von Neumann [2]. Our aim is to characterise those among them that arise from complex Hilbert spaces To tackle this problem on the basis of principles that are typical for lattice-theoretic approaches seems to be hardly possible though. We note that we have followed a similar approach in [15], where we characterised complex Hilbert spaces as partial Boolean algebras. The latter structures are based on a concept that resembles in some respects orthogonality spaces. The automorphisms are requested to leave the “uninvolved” elements fixed, like those that are orthogonal to the elements under consideration Both conditions might be seen as an expression of the flexibility inherent to the model. We review the situation once again and point out open issues in the concluding Section 4

Orthogonality Spaces

Hermitian Spaces

Conclusion

Methods