Why do we need Hilbert spaces?

TL;DR

This paper reviews the foundational role of Hilbert spaces in quantum mechanics, deriving the formalism from propositional calculus and operational principles, emphasizing non-commutativity, and proposing a toy model to motivate these features.

Contribution

It provides a detailed derivation of quantum formalism from logical and operational perspectives, highlighting the necessity of Hilbert spaces and non-commutativity.

Findings

Hilbert spaces are essential for quantum formalism

Non-commutativity is crucial in quantum logic and operational reconstructions

A toy model is proposed to motivate non-commutativity

Abstract

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.