On characterizing the standard quantum logics

Abstract:

Let $\mathcal {L}$ be a complete projective logic. Then $\mathcal {L}$ has a natural representation as the lattice of $\langle { \cdot , \cdot } \rangle$-closed subspaces of a left vector space V over a division ring D, where $\langle {\cdot ,\cdot } \rangle$ is a definite $\theta$-bilinear symmetric form on V, $\theta$ being some involutive antiautomorphism of D. Now a well-known theorem of Piron states that if D is isomorphic to the real field, the complex field or the sfield of quaternions, if $\theta$ is continuous, and if the dimension of $\mathcal {L}$ is properly restricted, then $\mathcal {L}$ is just one of the standard Hilbert space logics. Here we also assume $\mathcal {L}$ is a complete projective logic. Then if every $\theta$-fixed element of D is in the center of D and can be written as $\pm d\theta (d)$, some $d \in D$, and if the dimension of $\mathcal {L}$ is properly restricted, we show that $\mathcal {L}$ is just one of the standard Hilbert space logics over the reals, the complexes, or the quaternions. One consequence is the extension of Piron’s theorem to discontinuous $\theta$. Another is a purely lattice theoretic characterization of the lattice of closed subspaces of separable complex Hilbert space.