Axiomatic foundations of quantum physics: Critiques and misunderstandings. Piron's question-proposition system

Some of the most frequent misconceptions about axiomatic quantum physics are discussed with the aim of clarifying their true significance, taking Piron's approach as conceptual framework. In particular, we deal with the following topics: the wrong identification of Piron's questions and Mackey's questions, and some curious alleged empirical consequences; the role of propositions as suitable equivalence classes of questions, their partial order structure, and the paradoxical consequences of the erroneous assignment to questions of some lattice properties involving propositions; the logical and the empirical purport of some “negative” theorems; the standard Hilbert space model of the theory and the consequent “metaphysical disasters” related to some identifications, which are peculiar of this model. A controversy between Foulis-Piron-Randall and Hadjisavvas-Thieffine-Mugur-Schächter is analyzed on the basis of the proposed Hilbert space model (in which Piron's questions are realized by Hilbertian “effects,” i.e., linear bounded operatorsF such that which clarify the different point of views. As an example, we treat the unsharp localization operators inL2(ℝ).