REVIEW 3 major objections 6 minor 25 references
From Classical Trajectories to Quantum Commutation Relations
T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read This paper argues that the same set of classical trajectories admits many different Lagrangian, Hamiltonian, and quantum descriptions, because the geometric structures carrying those descriptions are underdetermined by the observed motion.
desk verdict A competent geometric-mechanics review with three nice coordinate examples; the quantum section's key claim is real but under-proved and imprecise, so the paper is worth refereeing rather than rejecting. 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 mechanism is the tensorial bookkeeping of a tangent bundle structure: on a carrier manifold $M$, the pair $(S,\Delta)$ with $S$ a $(1,1)$-tensor satisfying $S^2=0$, $\mathrm{Ker}\,S=\mathrm{Im}\,S$, $N_S=0$ and $\Delta$ a partial linear structure encodes what it means for $M$ to be a tangent bundle; the field $\Gamma$ is Newton-like precisely when $S(\Gamma)=\Delta$. That identity is tensorial, so under any diffeomorphism $\varphi$ it becomes $(\varphi^*S)(\varphi^*\Gamma)=\varphi^*\Delta$, which is the same second-order condition for a new tangent structure; this is what makes the same trajectories compatible with many Lagrangian pictures. On the quantum side, the central object is the Weyl system $W:V\to U(H)$ with $W(z+z')=W(z)W(z')e^{-i\omega(z,z')/2\hbar}$, together with the standard realization of a Weyl system on $L^2$ of a Lagrangian subspace; because the Weyl map and the realization depend on the linear and symplectic structures of $V$, changing them nonlinearly changes the measure on the 'same' subspace, hence changes which operators are self-adjoint and whether the generated algebra is a $C^*$-algebra.
What would settle it
Choose $K(|q|)=1+q^2/3$, so $Q=q+q^3/3$ and $dQ=(1+q^2)\,dq$. On $L^2(\mathbb{R},(1+q^2)\,dq)$, integration by parts for $\hat\pi=-i\,d/dq$ gives $\langle \hat\pi\psi,\varphi\rangle-\langle \psi,\hat\pi\varphi\rangle=-i\int \overline{\psi}\varphi\,2q\,dq$, which is nonzero for generic compactly supported $\psi,\varphi$; therefore $\hat\pi$ is not even symmetric on that Hilbert space. If a direct computation with the paper's definitions instead showed symmetry or self-adjointness, the claimed inequivalence would be falsified.
Extended reading notes
Core claim
The paper's central claim is that the empirical input, a family of trajectories, does not determine a unique geometric description of a dynamical system. The core reduction is tensorial: on a carrier manifold, a pair $(S,\Delta)$ encodes what it means to be a tangent bundle, and a vector field $\Gamma$ is Newton-like exactly when $S(\Gamma)=\Delta$. Since this equality is tensorial, any diffeomorphism $\varphi$ produces a new pair $(\varphi^*S,\varphi^*\Delta)$ for which the transformed field is again second-order; when $\varphi$ is a symmetry fixing $\Gamma$, the same vector field is second-order for both the old and the new tangent structure. Explicit coordinate examples turn a reparametrized free particle, a harmonic oscillator with constant frequency, and a dilation field into second-order fields with respect to newly built tangent structures. The same logic moves to the Hamiltonian side, where constants of the motion generate alternative invariant symplectic forms, and then to the quantum side, where the nonlinear relabeling $\varphi(q,p)=(qK(|q|),p)$ defines an alternative linear and symplectic vector space, an alternative Weyl system, and a generated operator algebra that is a $C^*$-algebra in one realization and not in the other. The conclusion is that nonlinearly related formulations of quantum mechanics can coexist for the same classical input.
Load-bearing premise
The load-bearing premise is the unproved assertion that the momentum operator $\hat\pi=-i\,d/dq$ is a fully meaningful self-adjoint operator on $L^2(L,dq)$ but not on $L^2(L',dQ)$, the Hilbert space built with the nonlinearly re-measured coordinate; if that assertion fails, the example does not demonstrate genuinely inequivalent quantum theories.
Editorial extensions
If this is right
- A dynamics that is not second-order with respect to a given tangent bundle structure can still be a legitimate second-order (Newton-like) system after a diffeomorphism; the worked examples include a reparametrized free particle, a harmonic oscillator with constant frequency, and the field $q\,\partial_q$.
- Reparametrizing a vector field to make it complete, as needed in the paper's motivating example of an incomplete central-force flow, no longer automatically destroys the possibility of a Lagrangian description: one can search for an alternative tangent bundle structure relative to which the reparametrized field is second-order.
- In the Hamiltonian picture, every constant of the motion gives a candidate invariant 2-form via the $T$-differential; whenever that form is nondegenerate, the same dynamical field has a second Hamiltonian description, and compatible pairs of such forms imply complete integrability.
- In the quantum picture, alternative linear structures on the same set generate alternative Weyl systems; because the measures on the 'same' Lagrangian subspace are then nonlinearly related, square-integrability changes, the momentum operator can lose self-adjointness in one realization, and the generated algebra stops being a $C^*$-algebra.
- Therefore the entire chain from trajectories to commutation relations carries modeling choices, the carrier manifold, the tangent structure, the symplectic form, and the linear structure, none of which is uniquely forced by the observed curves.
Reading between the lines
- An extension the paper does not pursue is to compute expectation values and transition probabilities for the same formal operators in the two Hilbert-space realizations; any operational difference would make the choice of linear structure empirically detectable.
- The same move, searching for a diffeomorphism that makes a given vector field second-order, can be applied systematically to incomplete flows: scanning the symmetry group of $\Gamma$ for a transformation that produces a complete second-order field would give a Lagrangian description of the reparametrized dynamics.
- The nonconstant density $\sigma(q)=dQ/dq$ is what breaks symmetry of the momentum operator; this suggests a general criterion that a nonlinear relabeling preserves the $C^*$-algebraic structure exactly when the induced density is constant, which could classify admissible coordinates.
- If alternative linear structures induce alternative Weyl systems and second-quantization procedures, then the same classical field content could in principle support inequivalent quantum field theories; the paper only gestures at second quantization, but the mechanism it describes is the same.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper argues that experimental trajectories of a classical dynamical system do not uniquely determine the geometric structures used in its Lagrangian, Hamiltonian, or quantum description. It reviews the inverse problem of the calculus of variations, the tensorial characterization of tangent bundle structures and second-order vector fields, and the construction of alternative symplectic and Poisson structures. The main positive content is a set of coordinate examples in Section 3 showing that a given vector field can be made second-order with respect to alternative tangent bundle structures, and a Section 5 example involving a nonlinear diffeomorphism of the symplectic vector space R^2 intended to show that alternative linear structures give rise to alternative Weyl systems and, allegedly, to genuinely different quantum C*-algebras.
Significance. If the claims were established, the paper would provide a useful synthesis of the classical underdetermination of Lagrangian and Hamiltonian descriptions and would extend it to the quantum setting. The classical portions are grounded in standard, independently established results (Helmholtz conditions, inverse problem literature, bi-Hamiltonian geometry), and the three examples in Section 3 are correct and instructive. The paper is, however, largely a review/synthesis rather than a new technical contribution: key results are quoted from the authors' prior work, and the only concrete quantum example is both unproved and, as written, does not support the advertised conclusion. The main value of the paper lies in its clear framing of classical ambiguities; the quantum section needs substantial revision before the paper's central thesis can be accepted.
major comments (3)
- [Section 5, Eqs. (63)-(68) and following paragraph] The example does not compare the two Weyl systems it defines. The Weyl system for (V_phi, omega_phi) has generators \hat{x}' = Q and \hat{\pi}' = -i \partial/\partial Q on L^2(L', dQ). Since phi is a vector-space isomorphism from V_phi to V and a symplectomorphism, von Neumann's theorem (which the paper itself recalls on p. 16) implies that this Weyl system is unitarily equivalent to the standard one. The subsequent comparison of \hat{x} = q and \hat{\pi} = -i \partial/\partial q on L^2(L, dq) and L^2(L', dQ) concerns a different operator, not the generator -i \partial/\partial Q of the alternative Weyl system. Therefore the claimed conclusion that "nonlinearly related formulations of quantum mechanics are possible" does not follow from the example as stated.
- [Section 5, p. 18] The phrase "algebra 'generated' by the operators \hat{x}, \hat{\pi}, I together with their adjoints" is not a well-defined C*-algebra construction when \hat{x} and \hat{\pi} are unbounded operators. The assertion that one such object is a C*-algebra and the other is not must be formulated in terms of bounded functions of the generators, for instance the Weyl unitaries or the resolvents. As written, the claim is ambiguous and cannot be verified.
- [Section 5, p. 18] The statement that \hat{\pi} = -i \partial/\partial q is self-adjoint on L^2(L, dq) but not on L^2(L', dQ) is made without proof and without specifying a domain. Writing dQ = \rho(q) dq with \rho(q) = K(|q|) + q K'(|q|), a direct integration by parts gives \langle\phi, \hat{\pi}\psi\rangle_{dQ} - \langle\hat{\pi}\phi, \psi\rangle_{dQ} = i \langle\phi, (\rho'/\rho)\psi\rangle_{dQ} for compactly supported smooth functions, so the naive differential expression is not symmetric unless \rho is constant. This computation is not supplied. If a different operator is intended (for example the generator of translations in the Q-coordinate), the claim must be restated. This omitted proof is load-bearing, as it is the only concrete evidence for the quantum part of the paper's thesis.
minor comments (6)
- [Throughout] There are typographical errors that should be corrected, including "Hamitlonian" in Section 1 and "descritpion" in Section 3.
- [Section 3, Eqs. (30), (36), (41)] The tensor S' is written as dy \otimes \partial/\partial w, whereas the convention established in Eq. (11) is S = \partial/\partial v \otimes dq. For consistency, S' should be \partial/\partial w \otimes dy.
- [Section 3, Example 3, Eq. (39)] The condition df \wedge dq \neq 0 is imprecise because f depends only on v; the intended local solvability condition is f'(v) \neq 0, which should be stated explicitly.
- [Section 5, Eq. (67)] The notation "n = 0, 1, 2,..., n,..." is sloppy; it should read "n = 0, 1, 2, \ldots".
- [Section 4, Eq. (50)] The expression "\omega_f = dd_Tf" should be written as "\omega_f = d(d_T f)" for clarity, since d_T is not an ordinary differential.
- [Appendix A, Eq. (73)] The text says that a submanifold of T*TQ is obtained, but the displayed set is a subset of TT*Q; this inconsistency should be corrected.
Circularity Check
The classical Lagrangian/Hamiltonian derivation is self-contained; Section 5's quantum inequivalence is load-bearing on a self-citation to [21] and an omitted proof.
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self citation load bearing
[Section 5, final paragraph after Eq. (70).]
"it turns out that x̂ is self-adjoint on both the Hilbert spaces, while π̂ is self-adjoint only on L2(L,dq). Consequently, the algebra "generated" by the operators x̂, π̂, I together with their adjoints on L2(L,dq) is actually a C∗-algebras, while that generated by x̂, π̂, I together with their adjoints on L2(L′,dQ) is not (see [21] for more details)."
This is the step that converts the existence of two Weyl systems into the paper's advertised conclusion that "nonlinearly related formulations of quantum mechanics are possible." No proof or computation is supplied here; the claim is deferred entirely to [21] (Ercolessi–Ibort–Marmo–Morandi), which shares an author with the present paper. Moreover, the operator π̂ = −i ∂/∂q is not unambiguously defined on L2(L′,dQ): since dQ = Q′(q)dq, the same differential expression is not the generator of Q-translations, and the self-adjointness conclusion depends on which operator is meant. Thus the quantum underdetermination rests on a load-bearing self-citation and an omitted proof rather than on the displayed construction.
full rationale
The main derivation chain in Sections 2–4 is self-contained differential geometry. The tensorial characterization of tangent bundle structures is cited to [17] and [14], but the transformation rule in Eq. (15) is a direct pullback computation, and Examples 1–3 give explicit coordinate checks. The Hamiltonian alternative descriptions in Section 4 are derived from displayed formulas, Eqs. (49)–(50), together with Cartan's identity; there are no fitted parameters and no predicted quantity that equals an input by construction. Section 5's construction of alternative Weyl systems from alternative symplectic vector-space structures is a direct application of von Neumann's theorem and the nonlinear diffeomorphism φ, so it is not circular by definition. The genuinely load-bearing and non-self-contained step is the assertion that x̂ is self-adjoint on both Hilbert spaces while π̂ is self-adjoint only on L2(L,dq), and that the generated algebra is a C*-algebra in only one case. That assertion is deferred to [21], a paper with overlapping authorship, and its meaning is ambiguous because π̂ on L2(L′,dQ) is not the generator of Q-translations. The advertised quantum conclusion therefore depends on a self-citation and an omitted proof. However, the broader thesis that classical trajectories do not fix Lagrangian, Hamiltonian, or quantum structures is independently supported by standard Helmholtz conditions, Stone–von Neumann theorem, and the explicit classical examples; hence the circularity is partial and localized rather than definitional, giving a score of 4 rather than higher.
Assumptions & free parameters
assumptions (6)
- domain assumption The set of lifted trajectories tS is a smooth submanifold M of TQ, and the second lift yields a second-order vector field Gamma on M (Section 2, Eqs. (3)-(7)).
- standard math A tangent bundle structure on a 2n-manifold is equivalent to a pair (S, Delta) with S^2=0, KerS=ImS, NS=0 and L_Delta S = -S (Section 3, theorem attributed to [17]).
- standard math A second-order vector field is characterized by S(Gamma)=Delta (Eq. (14)).
- standard math Euler-Lagrange equations are equivalent to L_Gamma theta_L - dL = 0 for a nondegenerate Lagrangian (Eq. (21)).
- standard math Stone-von Neumann theorem and von Neumann's realization theorem for Weyl systems (Section 5).
- domain assumption The momentum operator -i d/dq is self-adjoint on L^2(L,dq) but not on L^2(L',dQ), so the algebra generated by x-hat, pi-hat and I is a C*-algebra only in the first case (Section 5, from [21]).
Cite this review
Pith. "Pith review of From Classical Trajectories to Quantum Commutation Relations." pith.science (2026). https://pith.science/paper/DB74Y6XV
@misc{pith2026190806790,
author = {Pith},
title = {Pith review of: From Classical Trajectories to Quantum Commutation Relations},
year = {2026},
howpublished = {\url{https://pith.science/paper/DB74Y6XV}},
note = {Machine review of arXiv:1908.06790}
}
read the original abstract
In describing a dynamical system, the greatest part of the work for a theoretician is to translate experimental data into differential equations. It is desirable for such differential equations to admit a Lagrangian and/or an Hamiltonian description because of the Noether theorem and because they are the starting point for the quantization. As a matter of fact many ambiguities arise in each step of such a reconstruction which must be solved by the ingenuity of the theoretician. In the present work we describe geometric structures emerging in Lagrangian, Hamiltonian and Quantum description of a dynamical system underlining how many of them are not really fixed only by the trajectories observed by the experimentalist.
Reference graph
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