Are Hilbert Spaces Unphysical? Hardly, My Dear!

TL;DR

This paper defends the physical legitimacy of Hilbert spaces in quantum mechanics by correcting misconceptions and analyzing objections, ultimately arguing that Hilbert spaces remain a valid and useful mathematical framework.

Contribution

It refutes recent claims that Hilbert spaces are unphysical, clarifies misconceptions, and discusses the relevance of infinite-expectation-value states in quantum theory.

Findings

Incorrectness of the finite-to-infinite expectation value mapping

Rebuttal of the claim that Hilbert spaces turn potential infinities into actual infinities

Infinite-expectation-value states are physically innocuous

Abstract

It is widely accepted that the states of any quantum system are vectors in a Hilbert space. Not everyone agrees, however. The recent paper ``The unphysicality of Hilbert spaces'' by Carcassi, Calder\'on and Aidala is a thoughtful dissection of the mathematical structure of quantum mechanics that seeks to pinpoint supposedly unsurmountable difficulties inherent in postulating that the physical states are elements of a Hilbert space. Its pivotal charge against Hilbert spaces is that by a change of variables, which is a change-of-basis unitary transformation, one ``can map states with finite expectation values to those with infinite ones''. In the present work it is shown that this statement is incorrect and the source of the error is spotted. In consequence, the purported example of a time evolution that makes ``the expectation value oscillate from finite to infinite in finite time" is also faulty, and the assertion that Hilbert spaces ``turn a potential infinity into an actual infinity'' is unsubstantiated. Two other objections to Hilbert spaces on physical grounds, both technically correct, are the isomorphism of separable Hilbert spaces and the unavoidable existence of infinite-expectation-value states. The former turns out to be quite irrrelevant but the latter remains an issue without a fully satisfactory solution, although the evidence so far is that it is physically innocuous. All in all, while the authors' thesis that Hilbert spaces must be given up deserves some attention, it is a long way from being persuasive as it is founded chiefly on a misconception and, subsidiarily, on immaterial or flimsy arguments.