Is the existence of unbounded operators a problem for quantum mechanics? In response to Carcassi, Calderon, and Aidala

TL;DR

This paper defends the use of Hilbert spaces in quantum mechanics against claims that unbounded operators and infinite expectation values are problematic, arguing that such issues do not undermine the theory's validity.

Contribution

It refutes the claim that unbounded operators and infinite expectation values pose problems, and argues against replacing Hilbert spaces with Schwartz spaces in quantum mechanics.

Findings

Infinite expectation values do not cause issues in quantum mechanics.

Replacing Hilbert spaces with Schwartz spaces would exclude meaningful Hamiltonian evolutions.

The concept of 'physicality' in fundamental physics is inherently vague.

Abstract

In this paper I argue against Carcassi, Calderon, and Aidala's recent claim that the Hilbert spaces are unphysical and should be replaced with the Schwartz spaces in quantum mechanics, since Hilbert spaces include states with infinite expectation values for certain observables. I also review and discuss issues regarding unbounded operators in quantum mechanics raised by Streater and Wightman, Heathcote, and Lemos. I argue that the existence of infinite expectation values does not cause problems in quantum mechanics. On the other hand, replacing the Hilbert spaces with the Schwartz spaces would cause more issues, as it would exclude a class of meaningful Hamiltonian evolutions. I also discuss the question in literature whether reformulating quantum mechanics with essentially self-adjoint operators instead of self-adjoint operators may cause problems. I further analyse the hierarchies of the notions of "physicality" and possibility in fundamental physics, and suggest that "physicality" is a vague concept. Finally, I connect the problem raised by Carcassi, Calderon, and Aidala with the problem of the Hadamard condition in quantum field theory.