A characterisation of orthomodular spaces by Sasaki maps
Bert Lindenhovius, Thomas Vetterlein
TL;DR
The paper investigates when an orthoset $(X,\perp)$ can be realized as the projective space of a Hermitian/orthomodular space by introducing Sasaki maps that mimic projection-like behavior. It shows that irreducible, point-closed Sasaki spaces of rank at least 4 are isomorphic to $(P(H),\perp)$ for some orthomodular space $H$, and that their associated subspace lattices $\mathcal{C}(X,\perp)$ become orthomodular lattices with a full Sasaki set of projections. This connection links the orthoset framework to lattice-theoretic projections and, via Solér's theorem and Wigner-type results, characterizes infinite-rank cases as classical Hilbert spaces over $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$. The work provides a direct, physically meaningful pathway from orthogonality structures to the standard Hilbert-space formalism in quantum theory.
Abstract
Given a Hilbert space $H$, the set $P(H)$ of one-dimensional subspaces of $H$ becomes an orthoset when equipped with the orthogonality relation $\perp$ induced by the inner product on $H$. Here, an \emph{orthoset} is a pair $(X,\perp)$ of a set $X$ and a symmetric, irreflexive binary relation $\perp$ on $X$. In this contribution, we investigate what conditions on an orthoset $(X,\perp)$ are sufficient to conclude that the orthoset is isomorphic to $(P(H),\perp)$ for some orthomodular space $H$, where \emph{orthomodular spaces} are linear spaces that generalize Hilbert spaces. In order to achieve this goal, we introduce \emph{Sasaki maps} on orthosets, which are strongly related to Sasaki projections on orthomodular lattices. We show that any orthoset $(X,\perp)$ with sufficiently many Sasaki maps is isomorphic to $(P(H),\perp)$ for some orthomodular space, and we give more conditions on $(X,\perp)$ to assure that $H$ is actually a Hilbert space over $\mathbb R$, $\mathbb C$ or $\mathbb H$.