Phenomenological quantum mechanics: deducing the formalism from experimental observations

Abstract: We propose an exercise in which one attempts to deduce the formalism of quantum mechanics from phenomenological observations only. Thus, the only assumed inputs are the multi-time probability distributions estimated from the results of sequential measurements of quantum observables; no presuppositions about the underlying mathematical structures are allowed. We show that it is indeed possible to derive in such a way a complete and fully functional formalism based on the structures of Hilbert spaces. However, the obtained formal description -- the bi-trajectory formalism -- turns out to be quite different from the standard state-focused formalism.