The paper On complemented lattices was the third paper in the new theory of orthomodular lattices which started in 1936 with Birkhoff and von Neumann’s idea of developing a new many-valued logic for quantum mechanics by using the lattice of closed subspaces C( H ) of a Hilbert space H as the valuation lattice.

Birkhäuser Boston eBooksJan 1, 19906 citations

International Journal of Theoretical Physics

Group-Valued Measures on the Lattice of Closed Subspaces of a Hilbert Space

John Harding, Ekaterina Jager, D. A. Smith

Published May 1, 2005• 7 citations

Publisher

DOI: 10.1007/s10773-005-3981-x

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{\mathcal H}

correspond to direct product decompositions H ≃ A × A ⊥

\mathcal{H}\simeq A\times A^\perp

of the Hilbert space, a result that lies at the heart of the superposition principle. In [10] it was shown that the direct product decompositions of any set, group, vector space, and topological space form an orthomodular poset. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. It is the purpose of this note to prove a version of Wigner’s theorem: for an infinite set X, the automorphism group of the orthomodular poset Fact(X) of direct product decompositions of X is isomorphic to the permutation group of X. The structure Fact(X) plays the role for direct product decompositions of a set that the lattice of equivalence relations plays for surjective images of a set. So determining its automorphism group is of interest independent of its application to quantum mechanics. Other properties of Fact(X) are determined in proving our version of Wigner’s theorem, namely that Fact(X) is atomistic in a very strong way.