Propositional systems, Hilbert lattices and generalized hilbert spaces

Abstract

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.

Classical Limit and Quantum Logic

“…As it is well known, those propositions can be endowed with an orthomodular lattice structure (Kalmbach 1983). Additionally, a solid axiomatic foundation for quantum mechanics can be used to explain in an operational way many important features of the Hilbert space formalism (Varadarajan 1968;Stubbe and Van Steirteghem 2007; see also Holik et al 2013Holik et al , 2014Holik et al , 2015 for more recent developments, and for the relationship between the quantum-logical approach and quantum probability theory). But the feature relevant to our discussion is that the logic associated to all varieties of quantum theories is not Boolean, due to the fact that it is not distributive.…”

Orthogonality and Dimensionality

“…Having studied the close relationship between orthomatroids and propositional systems, we will now adapt Piron's Representation Theorem [5][6][7] to orthomatroids. This theorem can be stated as:…”

“…As for the second requirement, a central result, Piron's celebrated Representation Theorem, provides conditions for an orthomodular lattice to be isomorphic to the lattice of all closed subspaces of a Hilbert space (or, more precisely, to a generalized Hilbert space, that is where it is not required that the used field is a "classical" one) [5][6][7]. However, one of those conditions, namely the covering law, "presents a [...] delicate problem.…”

Logic, Geometry And Probability Theory

“…On the other hand, lattice theory is deeply connected to geometry: projective geometry can be described in terms of lattices and related also to vector spaces [22]. As an example, any vector space has associated a projective geometry and a lattice of subspaces.…”