Quantum intuitionistic logic and the Gelfand representation theorem

Abstract

The subject of this thesis is on the shared horizon of theoretical physics and logic: Quantum logic. In 1936, Birkhofff and von Neumann already formulated a quantum logic. But, this logic has some undesirable properties concerning the interpretation of the logical conjunction and disjunction as ``and'' and ``or''. Therefor, Landsman has recently set out to develop a new logic for quantum mechanics, which is set up inside a topos and thus is intuitionistic.

The construction of Landsmans intuitionistic quantum logic relies on the Gelfand representation theorem. However, to apply the theorem a constructive proof is needed. Coquand and Spitters recently published such a proof, but the material that was published is too condensed for our taste. So in this thesis we attempt to expand on this constructive proof of the Gelfand representation theorem and check all details the original authors left out. As a result, more insight is gained in the constructive proof of the Gelfand representation theorem, providing a stronger foundation for intuitionistic quantum logic.