REVIEW 5 minor 12 references
Are Hilbert Spaces Unphysical? Hardly, My Dear!
T0 review · 0 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read This paper defends the Hilbert-space framework of quantum mechanics by showing that the leading objection—that a unitary change of variables can turn finite expectation values into infinite ones—misidentifies the position operator in the…
desk verdict A sound, narrowly-scoped refutation of the central technical example in Carcassi et al.; the core math is right, and the only soft spot is how much weight the word 'misconception' carries. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the unitary operator $U$ induced by a change of variables $y=f(x)$, $f'>0$, acting on wavefunctions as $\tilde{\psi}(y)=\psi(f^{-1}(y))\sqrt{f'(f^{-1}(y))}$. The identity carrying the argument is $\tilde{X}=U X U^{\dagger}$, namely the operator of multiplication by $f^{-1}(y)$, which forces $\langle\tilde{X}\rangle=\langle X\rangle$. A second mechanism is the commutation-relation contradiction $[X_H(0),\dot{X}_H(0)]=i\hbar$, used to show that the proposed time evolution cannot be generated by a Hamiltonian of the standard form $P^2/2+V(X)$.
What would settle it
Compute both sides of $\int f^{-1}(y)|\tilde{\psi}(y)|^2\,dy=\int x|\psi(x)|^2\,dx$ numerically for a smooth, non-Gaussian $\psi\in L^2(\mathbb{R})$ and a smooth diffeomorphism $f$ with $f'>0$, such as $f(x)=\tan(\frac{\pi}{2}\operatorname{erf}(x))$; any discrepancy would refute the paper's invariance claim.
Extended reading notes
Core claim
Under a change of variables $y=f(x)$, with $f$ a diffeomorphism satisfying $f'>0$, the induced map $U$ defined by $\tilde{\psi}(y)=\psi(f^{-1}(y))\sqrt{f'(f^{-1}(y))}$ is unitary, and the transformed position operator is $\tilde{X}=U X U^{\dagger}$, which is multiplication by $f^{-1}(y)$, not by $y$. Consequently $\langle\tilde{X}\rangle=\int f^{-1}(y)|\tilde{\psi}(y)|^2\,dy=\int x|\psi(x)|^2\,dx=\langle X\rangle$, so expectation values are unchanged. What the targeted paper actually showed is that $\int f(x)|\psi(x)|^2\,dx$ can be infinite even when $\int x|\psi(x)|^2\,dx$ is finite—a statement about the unbounded observable $f(X)$, not about unitary equivalence. The time-dependent version is also ruled out, because a Heisenberg-picture position operator of the form $X\cos\omega t+f(X)\sin\omega t$ would have $[X_H(0),\dot{X}_H(0)]=i\hbar$, contradicting the proposed dynamics.
Load-bearing premise
The refutation depends on the convention that when you change coordinates you must change the position operator along with the wavefunction; if you instead treat the new coordinate itself as the physical position, the finite-to-infinite transition is real but says something about an unbounded function of position, not about unitary equivalence.
Editorial extensions
If this is right
- For any physical coordinate change, the observable must be transformed as $O'=U O U^{\dagger}$; otherwise one is computing the expectation value of a different operator, not the same observable in a new basis.
- The finite-to-infinite example survives only as a statement about the unbounded operator $f(X)$: a finite $\langle X\rangle$ does not imply a finite $\langle f(X)\rangle$.
- A time-dependent coordinate transformation is not automatically a time evolution, and the particular transformation proposed in the targeted paper cannot be generated by a Hamiltonian of the standard one-dimensional form.
- The isomorphism of separable Hilbert spaces does not make physical systems identical, because Hamiltonians and other observables need not be unitarily equivalent, as shown by the hydrogen atom versus the harmonic oscillator.
- Infinite-expectation-value states are unavoidable for unbounded self-adjoint operators, but the paper argues they are physically innocuous and that replacing $L^2(\mathbb{R}^{3n})$ by Schwartz space creates new difficulties, including symmetric operators with no self-adjoint extension.
Reading between the lines
- If the paper is right, any alleged basis-change paradox should be checked by whether the observable was transformed covariantly; this gives a quick diagnostic for similar claims in other representations.
- The argument suggests a physical criterion for admissible coordinates: a coordinate change is harmless exactly when the relevant observables remain well-defined in the new variables, and pathological functions produce infinite expectations only for $f(X)$, not for $X$.
- The Schwartz-space discussion points to a testable mathematical program: define essential self-adjointness intrinsically on Schwartz space without referring to a Hilbert-space completion; if this cannot be done, completeness remains necessary for the spectral theorem.
- A numerical check of the identity $\int f^{-1}(y)|\tilde{\psi}(y)|^2\,dy=\int x|\psi(x)|^2\,dx$ over a family of non-Gaussian square-integrable states and smooth diffeomorphisms would corroborate the paper's central calculation.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper is a critical response to Carcassi, Calderon, and Aidala's claim that Hilbert spaces are unphysical. Its central rebuttal is that the main objection, namely that a change of variables viewed as a unitary change of basis can map states with finite position expectation values into states with infinite ones, is based on transforming the wavefunction without transforming the position operator. The manuscript shows that under the standard unitary transformation law the wavefunction and the position operator transform together: with y=f(x), the transformed position operator is multiplication by f^{-1}(y), not by y, and expectation values are preserved (Section II.A, Eqs. (8)-(15)). It then argues that the time-dependent example of Carcassi et al. is not generated by a physically reasonable one-dimensional Hamiltonian of the form P^2/2+V(X) (Section II.B), that the isomorphism of separable Hilbert spaces is physically irrelevant, that infinite expectation values are unavoidable but probably innocuous, and that replacing L^2 by Schwartz space creates new technical problems. The conclusion is that the case against Hilbert spaces is not persuasive.
Significance. The central calculation is correct and easily checkable: Eq. (14) cleanly shows that a coordinate change acts on the position operator as multiplication by the inverse map, and Eq. (15) restores the equality of expectation values. The manuscript is valuable because it identifies the exact step in the recent challenge where the state vector is transformed but the observable is not, doing so without fitted parameters or additional structure. It also honestly concedes the two technically correct objections concerning the isomorphism of separable Hilbert spaces and the unavoidable existence of infinite-expectation-value states, and it gives a reasonable if not decisive argument that these objections do not invalidate Hilbert-space quantum mechanics. The paper would benefit from a more careful statement of the convention dependence of its 'error' attribution, but the mathematical core is sound.
minor comments (5)
- [Section II.A, Eq. (16)] The paper's repeated characterization of Carcassi et al.'s step as 'a misconception' and 'a fateful misinterpretation' is stronger than the mathematics strictly warrants. Equation (16) correctly shows that multiplying the transformed wavefunction by y gives ⟨y⟩ = ⟨f(X)⟩, which can be infinite; the real point is that multiplication by y is not the covariant transform of the position operator. Rephrasing the conclusion as 'their conclusion does not follow under the standard transformation law' would be more precise and would avoid overstating the result.
- [Section II.B] The contradiction with the Heisenberg equation assumes a time-independent Hamiltonian of the specific form H = P^2/2 + V(X), as stated around Eq. (19). If the authors intend to rule out all possible Hamiltonian time evolutions, a more general argument would be needed; as written, the sufficient point is that Carcassi et al. provide no Hamiltonian at all, so their time-dependent coordinate transformation is not shown to be a bona fide unitary time evolution.
- [Section V] The claim that T = X^3 P + P X^3 on Schwartz space is symmetric but has no self-adjoint extension is correct, but the discussion is quite terse. A sentence outlining why the deficiency indices are unequal would make the example more self-contained and easier to verify.
- [General] There are several typographical errors that should be corrected: 'isomomorphic' in Section III, 'appearences' in Section V, 'discusssion' in Section II.B, and 'irrrelevant' in the abstract.
- [Section II.A] The phrase 'of no consequence' immediately after Eq. (16) could be expanded: the finite-to-infinite transition is a statement about the unbounded observable f(X), not about the position operator, and this explains why it is physically harmless. Such an expansion would clarify the argument for readers who use the alternative convention that y is the physical position coordinate.
Circularity Check
No circularity: the paper's refutation is a self-contained application of standard unitary-transformation rules, with no fitted parameters or self-citations.
full rationale
The paper does not derive its target conclusion from its own assumptions in a circular way. Its central claim is that, under the coordinate change y = f(x), the unitary U defined in Eq. (8) transforms the position operator to μultiplication by f^{-1}(y), so expectation values are preserved. This is a direct computation from the standard transformation law O' = U O U†, which the paper explicitly invokes and cites to external textbooks (Messiah, Prugovecki). No parameter is fitted and no part of Carcassi et al.'s conclusion is assumed. The potential convention-dependence is honestly acknowledged in Eq. (16): if one instead defines the position observable in the new coordinate as multiplication by y, then ⟨y⟩ = ⟨f(x)⟩, which can be infinite; the paper labels this 'of no consequence' rather than hiding it. This is a substantive interpretive disagreement, not a circular derivation. All citations are to independent external sources, not to the author's own prior work, and the mathematical steps are explicitly written out and verifiable. The finite-to-infinite transition is therefore not a prediction that reduces to a fit or to a self-citation; it is a claim about how observables must transform under basis changes, argued from standard quantum-mechanical principles.
Assumptions & free parameters
assumptions (6)
- domain assumption Quantum observables are self-adjoint operators and unitary transformations act as U on states and U O U^\dagger on operators.
- domain assumption The change of variables y=f(x) is a diffeomorphism with f'(x)>0 and preserves probability as |psi(x)|^2 dx = |psi~(y)|^2 dy.
- domain assumption A physically reasonable one-dimensional Hamiltonian has the standard form H = P^2/2 + V(X_H).
- domain assumption Coordinate transformations cannot change physical predictions.
- standard math Separable Hilbert spaces are all unitarily isomorphic and unitarily equivalent operators have the same spectrum.
- standard math Kato's theorem: atomic and molecular Hamiltonians are self-adjoint.
Cite this review
Pith. "Pith review of Are Hilbert Spaces Unphysical? Hardly, My Dear!." pith.science (2026). https://pith.science/paper/GXNN3TBQ
@misc{pith2026250103294,
author = {Pith},
title = {Pith review of: Are Hilbert Spaces Unphysical? Hardly, My Dear!},
year = {2026},
howpublished = {\url{https://pith.science/paper/GXNN3TBQ}},
note = {Machine review of arXiv:2501.03294}
}
read the original 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.
Reference graph
Works this paper leans on
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Reviewed August 10, 2026 · model on record in the stance chip above.
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