A characterisation of orthomodular spaces by Sasaki maps

TL;DR

This paper characterizes orthomodular spaces using Sasaki maps on orthosets, providing conditions under which these structures are isomorphic to spaces of one-dimensional subspaces of Hilbert or orthomodular spaces.

Contribution

It introduces Sasaki maps on orthosets and establishes conditions for their isomorphism to orthomodular spaces, extending the understanding of the structure of quantum logic.

Findings

Orthosets with enough Sasaki maps are isomorphic to orthomodular space point sets.

Conditions are provided for the orthomodular space to be a Hilbert space over real, complex, or quaternionic fields.

The work links Sasaki projections with the structure of orthomodular lattices.

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 $\mathbb H$.