Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On characterizing the standard quantum logics


Author: W. John Wilbur
Journal: Trans. Amer. Math. Soc. 233 (1977), 265-282
MSC: Primary 81.06
MathSciNet review: 0468710
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 81.06

Retrieve articles in all journals with MSC: 81.06


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0468710-X
PII: S 0002-9947(1977)0468710-X
Keywords: Complemented modular lattice, projective geometry, point, division ring, center of the ring, Archimedean field, non-Archimedean field, involutive antiautomorphism, $ \theta $-bilinear symmetric form, complete projective logic, Hilbertian space, state, normalizable, probabilistic complete projective logic, infinite dimensional space, orthonormal sequence, standard quantum logics, separable Hilbert space
Article copyright: © Copyright 1977 American Mathematical Society