Quantum logic and a theorem on commensurability

M. K. Mukherjee

Thomas A. Brody

Alex Patterson

Using Functor Substitutability for the basis of creating an intuitionist interpretation of the results of that subscribe. Laying the Grounds for Challenging Goedel's Incompleteness Theorems with the Aspects of the Poincare Group. Proceeding with: Propositional Quantum Logic, Transitioning “Non-Commensurable” Smoothly to Functor-Auxiliary Substitutability and Detachment.

Thomas A. Brody

Dion M. F. Hartmann

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. \n\nThe 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.

C. S. Sharma, M K Mukherjee

Two theorems are proved. In the first properties of an important mapping from an orthocomplemented lattice to itself are studied. In the second the authors extend the characterisation theorem of Zierler (1961) to get a very useful theorem characterising orthomodular lattices Since quantum logics are merely sigma -complete orthomodular lattices, the principal result is, for application in quantum physics, a characterisation theorem for quantum logics.