On the connection between the categorical and the modal logic approaches to Quantum Mechanics
Giovanni CinĂ
Abstract:
This thesis aims at connecting the two research programs known as
Categorical Quantum Mechanics and Dynamic Quantum Logic. This is
achieved in three steps. First we define a procedure to extract a
Modal Logic frame from a small category and a functor into the
category of sets and relations. Second, we extend such methodology to
locally small categories. Third, we apply it to the category of
finite-dimensional Hilbert spaces to recover the semantics of Dynamic
Quantum Logic.
This process prompts new lines of research. At a general level, we
study some logics arising from wide classes of small categories. In
the case of Hilbert spaces, we investigate how to obtain richer
semantics, containing probabilistic information. We design a logic for
this semantics and prove that, via translation, it preserves the
validities of Dynamic Quantum Logic.