Response to "Are Hilbert Spaces Unphysical? Hardly, My Dear!''

TL;DR

This paper defends the physical relevance of Hilbert spaces by clarifying the distinction between coordinate transformations and basis changes in quantum mechanics, countering recent criticisms.

Contribution

It clarifies the difference between coordinate transformations and basis changes, rebutting a recent critique of the unphysicality of Hilbert spaces.

Findings

Coordinate transformations can change expectation values.

Basis changes do not alter expectation values.

The critique conflates two distinct concepts.

Abstract

A recent criticism of our paper ``The unphysicality of Hilbert spaces'' by Nivaldo Lemos refutes our central argument that a state with finite expectation value can be mapped to a state with infinite expectation value by a coordinate transformation. By conflating coordinate transformation with change of basis in quantum mechanics, Lemos argues that expectation values are invariant under change of variables. In the present work, we clarify the distinction between coordinate transformation and change of basis, and rebut Lemos' main argument.