Abstract projective geometry and the origin of the logico-algebraic approach to quantum
I wish to present two most remarkable 'representation theorems' in the realm of
projective geometry: 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. Throughout the talk I shall insist on the basic lattice theoretic
aspects of abstract projective geometry: in particular the categorical equivalence of projective
geometries and projective lattices, and the triple categorical equivalence of Hilbert geometries,
Hilbert lattices and propositional systems. It then follows 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. This fact,
which generalizes a theorem of G. Birkhoff and J. von Neumann (1936), was first discovered by C.
Piron (1964) and initiated the systematic study of a logico-algebraic approach to quantum theory,
with a particular stress on orthomodular lattices.
Isar Stubbe (CMUC, U. Coimbra)
March 07, 2006