Transitivity and homogeneity of orthosets and the real Hilbert spaces

TL;DR

This paper explores the structure of orthosets, which are sets with an orthogonality relation, and shows how under certain symmetry conditions they correspond to real Hilbert spaces, revealing new insights into their geometric and algebraic properties.

Contribution

It characterizes orthosets with transitivity and homogeneity conditions as arising from projective Hermitian spaces, linking orthogonality structures to real Hilbert spaces and their automorphism groups.

Findings

Orthosets under transitivity and homogeneity are from projective Hermitian spaces.

Automorphisms preserving orthogonal complements relate to quadratic spaces over ordered fields.

Automorphism actions imply the scalar field is a subfield of the real numbers.

Abstract

An orthoset (also called an orthogonality space) is a set $X$ equipped with a symmetric and irreflexive binary relation $\perp$, called the orthogonality relation. In quantum physics, orthosets play a central role. In fact, a Hilbert space gives rise to an orthoset in a canonical way and can be reconstructed from it. A complex Hilbert space can be seen as a real Hilbert space endowed with a complex structure. This fact motivates us to explore characteristic features of real Hilbert spaces by means of the abelian groups of rotations of a plane. Accordingly, we consider orthosets together with the groups of automorphisms that keep the orthogonal complement of a given pair of distinct elements fixed. We establish that, under a transitivity and a homogeneity assumption, an orthoset arises from a projective (anisotropic) Hermitian space. To find conditions under which the latter's scalar division ring is $\mathbb R$ is difficult in the present framework. However, restricting considerations to divisible automorphisms, we can narrow down the possibilities to positive definite quadratic spaces over an ordered field. The further requirement that the action of these automorphisms is quasiprimitive implies that the scalar field is a subfield of $\mathbb R$.