I review how geometry enters probability theory, and how it enters quantum mechanics. The logic is similar in both cases, but the latter case is richer and more interesting.
These are the notes written for the talk given at the workshop Rethinking foundations of physics 2016. In section 2, a derivation of the the quantum formalism starting from propositional calculus (quantum logic) is reviewed, pointing out which are the basic requirements that lead to the use of Hilbert spaces. In section 3, a similar analysis is done following for the reconstruction of quantum theory using an operational approach. In both cases, non-commutativity plays a crucial role. Finally, in section 4a toy model which try to motivate non-commutativity is proposed. Despite this last section is interesting to read, the analysis performed there is not complete. This toy model will be re-formulated in a rigorous way (and extended) in future works.
This paper collects and extends the lectures I gave at the “XXIV International Fall Workshop on Geometry and Physics” held in Zaragoza (Spain) during September 2015. Within these lectures I review the formulation of Quantum Mechanics, and quantum theories in general, from a mathematically advanced viewpoint, essentially based on the orthomodular lattice of elementary propositions, discussing some fundamental ideas, mathematical tools and theorems also related to the representation of physical symmetries. The final step consists of an elementary introduction the so-called ([Formula: see text]-) algebraic formulation of quantum theories.