Physics – Quantum Physics
Scientific paper
2007-10-10
Chapter in: Handbook of Quantum Logic and Quantum Structures: Quantum Structures (Eds. K. Engesser, D. M. Gabbay and D. Lehman
Physics
Quantum Physics
42 pages
Scientific paper
With this chapter we provide a compact yet complete survey of two most remarkable "representation theorems": every arguesian projective geometry is represented by an essentially unique vector space, and every arguesian Hilbert geometry is represented by an essentially unique generalized Hilbert space. C. Piron's original representation theorem for propositional systems is then a corollary: it says that every irreducible, complete, atomistic, orthomodular lattice satisfying the covering law and of rank at least 4 is isomorphic to the lattice of closed subspaces of an essentially unique generalized Hilbert space. Piron's theorem combines abstract projective geometry with lattice theory. In fact, throughout this chapter we present the basic lattice theoretic aspects of abstract projective geometry: we prove the categorical equivalence of projective geometries and projective lattices, and the triple categorical equivalence of Hilbert geometries, Hilbert lattices and propositional systems.
Steirteghem Bart Van
Stubbe Isar
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