The unphysicality of Hilbert spaces

TL;DR

The paper critiques the use of Hilbert spaces in quantum mechanics, arguing they introduce unphysical infinities and suggesting alternative mathematical frameworks like Schwartz spaces for representing quantum states.

Contribution

It highlights the unphysical implications of Hilbert space completeness in quantum theory and advocates for considering alternative spaces such as Schwartz spaces.

Findings

Hilbert spaces require properties that lead to infinite expectations.

Completeness in infinite dimensions can produce physically unrealistic states.

Schwartz spaces ensure finite expectations and are mathematically well-behaved.

Abstract

We argue that Hilbert spaces are not suitable to represent quantum states mathematically, in the sense that they require properties that are untenable by physical entities. We first demonstrate that the requirements posited by complex inner product spaces are physically justified. We then show that completeness in the infinite-dimensional case requires the inclusion of states with infinite expectations, coordinate transformations that take finite expectations to infinite ones and vice versa, and time evolutions that transform finite expectations to infinite ones in finite time. This means we should be wary of using Hilbert spaces to represent quantum states as they turn a potential infinity into an actual infinity. The main point of the paper, then, is to raise awareness that results that rely on the completeness of Hilbert spaces may not be physically significant. While we do not claim to know what a physically more appropriate closure should be, we note that Schwartz spaces, among other things, guarantee that the expectations of all polynomials of position and momentum are finite, their elements are uniquely identified by these expectations, and they are closed under Fourier transform.