Transitivity and homogeneity of orthosets and inner-product spaces over subfields of $${{\mathbb {R}}}$$

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 an elementary role. In particular, a Hilbert space gives rise to an orthoset in a canonical way and can be reconstructed from it. We investigate in this paper the question to which extent real Hilbert spaces can be characterised as orthosets possessing suitable types of symmetries. We establish that orthosets fulfilling a transitivity as well as a certain homogeneity property arise from (anisotropic) Hermitian spaces. Moreover, restricting considerations to divisible automorphisms, we narrow down the possibilities to positive definite quadratic spaces over an ordered field. We eventually show that, under the additional requirement that the action of these automorphisms is quasiprimitive, the scalar field embeds into $${{\mathbb {R}}}$$ R .