Minimal supports in quantum logics and Hilbert space (Q1101107)

It is shown that if a fully atomic, complete orthomodular lattice satisfies a ``minimal support condition'' (m.s.c.), then it satisfies Piron's axioms, and is thereby shown to be the projection lattice of a generalized Hilbert space. It is shown, conversely, that m.s.c. holds in Hilbert space subspace lattices. The physical justification for m.s.c. is provided in the context of a property lattice \({\mathcal L}({\mathcal A},\Sigma)\) for a realistic entity (\({\mathcal A},\Sigma)\) in the sense of Foulis-Piron- Randall. This context provides a clear focus on key issues in the debate over the appropriateness of requiring quantum logics to be represented over Hilbert spaces.