Interpreting Bohm quantum potentials in Computing quantum waves exactly from classical action

Jean-Jacques Slotine; Winfried Lohmiller

arxiv: 2605.20443 · v1 · pith:OBOB73XDnew · submitted 2026-05-19 · 🪐 quant-ph

Interpreting Bohm quantum potentials in Computing quantum waves exactly from classical action

Winfried Lohmiller , Jean-Jacques Slotine This is my paper

Pith reviewed 2026-05-21 06:56 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Bohm quantum potentialMadelung equationsHamilton-Jacobi equationcontinuity equationFeynman kernelinitialization dependencequantum wave construction

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The pith

The Bohm quantum potential can be set to zero without loss of generality when constructing quantum waves from classical action using a Feynman-kernel initialization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that the continuity and Hamilton-Jacobi equations with the Bohm quantum potential can be solved in a way that makes the potential term disappear. The key is choosing how to start the calculation at time zero, using an initialization inspired by the Feynman kernel rather than the usual Madelung approach. This choice changes the paths taken by the action and density during the computation but leaves the final wave function unchanged. A reader cares because it explains why some derivations of quantum mechanics from classical action seem to lose the quantum potential while others keep it, yet both reach the same physical result.

Core claim

The continuity p.d.e. and the Hamilton-Jacobi p.d.e., extended by the Bohm potential, are undisputed. However, the actual action and density solutions depend on their initialization at t = 0. In this work, this initialization is motivated by the Feynman kernel, which is fundamentally different from the standard initialization of the Madelung solution. This in turn leads to different action and density solutions, and explains why in one case the Bohm quantum potential disappears and in the other does not. The resulting overall wave, however, is independent of this computational initialization.

What carries the argument

The initialization at t=0 of the action and density, motivated by the Feynman kernel, which determines whether the Bohm quantum potential remains in the solution or vanishes while keeping the same final wave.

If this is right

The proof of Lemma 3.1 extends directly to cases that include the Bohm potential explicitly.
The Bohm quantum potential term can be assumed to be zero in the wave construction without loss of generality.
The overall wave function remains the same regardless of whether the initialization makes the Bohm potential appear or disappear.
Different choices of starting conditions for the partial differential equations lead to different intermediate solutions for action and density.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This approach suggests that computational paths in solving for quantum waves can vary based on starting assumptions without affecting the physical outcome.
Similar initialization dependencies might appear in other derivations that connect classical mechanics to quantum mechanics.
Testing this with specific solvable systems like the harmonic oscillator could verify if the final wave matches across initializations.

Load-bearing premise

The Feynman-kernel initialization at t=0 produces a valid quantum wave whose action and density can be directly compared to the standard Madelung initialization while preserving the same final wave function.

What would settle it

Compute the action and density explicitly for a simple quantum system using both the Feynman-kernel initialization and the standard Madelung initialization, then check whether the final wave functions are identical and whether the Bohm potential is zero in the first case.

read the original abstract

The recent arXiv posting [11], commenting on the paper [7], argues that the proof of Lemma 3.1 in [7] is missing the Bohm quantum potential [1, 2] of the Madelung p.d.e. [9]. This short technical note extends the proof of Lemma 3.1 to introduce a Bohm quantum potential explicitly, and then shows why this term can be assumed to be zero in the wave construction, without loss of generality. The continuity p.d.e. and the Hamilton-Jacobi p.d.e., extended by the Bohm potential, are undisputed. However, the actual action and density solutions depend on their initialization at t = 0. In [7], this initialization is motivated by the Feynman kernel [4], which is fundamentally different from the standard initialization of the Madelung solution [9]. This in turn leads to different action and density solutions, and explains why in one case the Bohm quantum potential disappears and in the other does not. The resulting overall wave, however, is independent of this computational initialization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This note extends their prior lemma to fold in the Bohm term then argues it can be dropped via a Feynman-kernel initialization, but the equivalence of the final wave is asserted more than demonstrated.

read the letter

This short technical note is essentially a direct reply to a comment on the authors' earlier paper. It extends the proof of Lemma 3.1 to incorporate the Bohm quantum potential into the Hamilton-Jacobi equation and then explains why that term can still be taken as zero without affecting the final result. What stands out is how they separate the undisputed PDEs from the role of initial conditions. By using an initialization drawn from the Feynman kernel rather than the usual Madelung setup, they get different paths for the action and density. This difference accounts for the potential disappearing in their construction while the overall wave function stays independent of that choice. The paper does well in acknowledging the comment and bringing the Bohm term into the picture explicitly. It avoids claiming anything new about the equations themselves and focuses on the computational initialization as the key distinction. A softer spot is the lack of a concrete demonstration that the two solution sets produce exactly the same wave function later on. The note states that the resulting wave is independent of initialization, but it does not include the steps showing how one trajectory with a nonzero Bohm term and another without it lead to identical ψ = √ρ exp(iS/ℏ). Since the argument rests on properties from their previous work, it may not fully close the loop for someone reading this note in isolation. This is really for readers who are already engaged with the authors' approach to building quantum waves from classical action and the associated computational methods. Someone outside that thread might not see the full context for why the Feynman kernel initialization is chosen or how it differs in practice. The work shows honest engagement with the literature and the comment, so it is worth considering for peer review to verify the details of the extended proof and the independence claim.

Referee Report

1 major / 0 minor

Summary. This short technical note extends the proof of Lemma 3.1 from the authors' prior work [7] to explicitly include the Bohm quantum potential in the Hamilton-Jacobi equation. It argues that the term can nevertheless be set to zero without loss of generality because the t=0 initialization (motivated by the Feynman kernel) produces different action S and density ρ trajectories than the standard Madelung initialization, while the reconstructed wave function remains independent of that choice.

Significance. If the asserted independence of the final wave function is rigorously established, the note would clarify why the Bohm potential vanishes in the authors' computational construction while remaining consistent with the undisputed continuity and extended Hamilton-Jacobi PDEs, thereby addressing the comment on [7] and strengthening the claim that exact quantum waves can be obtained from classical action under the chosen initialization.

major comments (1)

[section on initialization dependence] In the section on initialization dependence: the central claim that the resulting overall wave is independent of the computational initialization (thereby allowing the Bohm term to be assumed zero without loss of generality) is asserted but not demonstrated. No explicit steps, equations, or verification are supplied showing that the two distinct (S, ρ) solution pairs reconstruct identical ψ = √ρ exp(iS/ℏ) at t > 0, one trajectory carrying a nonzero Bohm term and the other not.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for a more explicit demonstration in our technical note. The comment is well taken: while the independence of the final wave function is asserted, the current manuscript does not supply the step-by-step verification requested. We will revise the note to include this demonstration, thereby strengthening the argument that the Bohm term can be set to zero without loss of generality under the Feynman-kernel initialization.

read point-by-point responses

Referee: In the section on initialization dependence: the central claim that the resulting overall wave is independent of the computational initialization (thereby allowing the Bohm term to be assumed zero without loss of generality) is asserted but not demonstrated. No explicit steps, equations, or verification are supplied showing that the two distinct (S, ρ) solution pairs reconstruct identical ψ = √ρ exp(iS/ℏ) at t > 0, one trajectory carrying a nonzero Bohm term and the other not.

Authors: We agree that the independence claim requires explicit verification rather than assertion alone. In the revised manuscript we will insert a short derivation in the initialization-dependence section. Let (S_M, ρ_M) denote the solution pair obtained from standard Madelung initial data, which carries a nonzero Bohm potential Q in the extended Hamilton–Jacobi equation. Let (S_F, ρ_F) denote the pair obtained from the Feynman-kernel initial data, for which Q vanishes identically. Both pairs satisfy the same continuity equation and the same extended Hamilton–Jacobi equation, but with different initial conditions at t = 0. Crucially, the initial wave functions constructed from each pair coincide: ψ(0) = √ρ(0) exp(i S(0)/ℏ) is identical and equals the initial condition implied by the Feynman kernel. Because the time-dependent Schrödinger equation is linear and possesses unique solutions for given initial data (under standard regularity assumptions), the reconstructed waves ψ_M(t) and ψ_F(t) must be identical for all t > 0. We will write the difference δψ = ψ_M − ψ_F, note that it satisfies the homogeneous Schrödinger equation with δψ(0) = 0, and conclude δψ ≡ 0. This explicit uniqueness argument will be added together with the relevant initial-condition equations. revision: yes

Circularity Check

1 steps flagged

Bohm term set to zero wlog via initialization motivated in authors' prior work [7]

specific steps

self citation load bearing [Abstract / initialization paragraph]
"In [7], this initialization is motivated by the Feynman kernel [4], which is fundamentally different from the standard initialization of the Madelung solution [9]. This in turn leads to different action and density solutions, and explains why in one case the Bohm quantum potential disappears and in the other does not. The resulting overall wave, however, is independent of this computational initialization."

The claim that the Bohm term may be assumed zero without loss of generality rests on the specific t=0 initialization introduced in the authors' own prior paper [7]. The asserted independence of the final wave ψ = √ρ exp(iS/ℏ) under that initialization is not independently verified against the extended PDEs here; it is taken as given by the earlier construction, making the 'wlog' step circular with respect to [7].

full rationale

The note extends Lemma 3.1 to include the Bohm term in the Hamilton-Jacobi PDE but then asserts this term can be dropped without loss of generality. The only justification offered is that the action and density solutions depend on t=0 initialization, and that the initialization chosen in [7] (motivated by the Feynman kernel) produces trajectories where the Bohm term vanishes while the final wave remains the same. This independence claim is not re-derived from the extended PDEs; it is inherited from the construction already defined in the authors' previous paper. The step therefore reduces the present argument to properties of that self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The note relies on the standard Madelung continuity and Hamilton-Jacobi PDEs plus the Feynman kernel from prior work; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The continuity PDE and Hamilton-Jacobi PDE extended by the Bohm potential are the correct governing equations.
Stated as undisputed in the abstract.
domain assumption Initialization at t=0 from the Feynman kernel is a valid starting point for the action and density.
Central to the argument that the Bohm term can be set to zero.

pith-pipeline@v0.9.0 · 5719 in / 1509 out tokens · 28111 ms · 2026-05-21T06:56:34.214958+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 1 internal anchor

[1]

D. Bohm. A suggested interpretation of the quantum theory in terms of “hidden” variables, I.Physical Review, 85(2):166–179, jul 1952

work page 1952
[2]

D. Bohm. A suggested interpretation of the quantum theory in terms of “hidden” variables, II.Physical Review, 85(2):180–192, jul 1952. 5

work page 1952
[3]

Duru and H

I. Duru and H. Kleinert. Quantum Mechanics of H-atoms from path integrals. Fortschritte Phys., 1982

work page 1982
[4]

Feynman and A

R. Feynman and A. Hibbs.Quantum Mechanics and Path Integrals. McGraw-Hill, 1965

work page 1965
[5]

Kleinert.Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets

H. Kleinert.Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets. World Scientific, 2009

work page 2009
[6]

Lohmiller and J.-J

W. Lohmiller and J.-J. Slotine. Contraction analysis of nonlinear distributed systems. International Journal of Control, 78(9):678–688, 2005

work page 2005
[7]

Lohmiller and J.-J

W. Lohmiller and J.-J. E. Slotine. On computing quantum waves exactly from classi- cal action.Proceedings of the Royal Society A, 482(2336):20250413, 2026

work page 2026
[8]

Lovelock and H

D. Lovelock and H. Rund.Tensors, Differential Forms, and Variational Principles. Dover, 1989

work page 1989
[9]

Madelung

E. Madelung. Quantentheorie in hydrodynamischer Form.Zeitschrift f ¨ur Physik, 40(3-4), 1927

work page 1927
[10]

C. Pehle. Classic quantum branch sums.https://cpehle.github.io/ blog/classic_quantum_branch_sums.html

work page
[11]

G. Vattay. Comment on ‘on computing quantum waves exactly from classical action’. arXiv:2605.02621 [quant-ph], submitted 4 May 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[12]

Madelung equations.https://en.wikipedia.org/ wiki/Madelung_equations

Wikipedia contributors. Madelung equations.https://en.wikipedia.org/ wiki/Madelung_equations. Accessed: 2026-05-15. 6

work page 2026

[1] [1]

D. Bohm. A suggested interpretation of the quantum theory in terms of “hidden” variables, I.Physical Review, 85(2):166–179, jul 1952

work page 1952

[2] [2]

D. Bohm. A suggested interpretation of the quantum theory in terms of “hidden” variables, II.Physical Review, 85(2):180–192, jul 1952. 5

work page 1952

[3] [3]

Duru and H

I. Duru and H. Kleinert. Quantum Mechanics of H-atoms from path integrals. Fortschritte Phys., 1982

work page 1982

[4] [4]

Feynman and A

R. Feynman and A. Hibbs.Quantum Mechanics and Path Integrals. McGraw-Hill, 1965

work page 1965

[5] [5]

Kleinert.Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets

H. Kleinert.Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets. World Scientific, 2009

work page 2009

[6] [6]

Lohmiller and J.-J

W. Lohmiller and J.-J. Slotine. Contraction analysis of nonlinear distributed systems. International Journal of Control, 78(9):678–688, 2005

work page 2005

[7] [7]

Lohmiller and J.-J

W. Lohmiller and J.-J. E. Slotine. On computing quantum waves exactly from classi- cal action.Proceedings of the Royal Society A, 482(2336):20250413, 2026

work page 2026

[8] [8]

Lovelock and H

D. Lovelock and H. Rund.Tensors, Differential Forms, and Variational Principles. Dover, 1989

work page 1989

[9] [9]

Madelung

E. Madelung. Quantentheorie in hydrodynamischer Form.Zeitschrift f ¨ur Physik, 40(3-4), 1927

work page 1927

[10] [10]

C. Pehle. Classic quantum branch sums.https://cpehle.github.io/ blog/classic_quantum_branch_sums.html

work page

[11] [11]

G. Vattay. Comment on ‘on computing quantum waves exactly from classical action’. arXiv:2605.02621 [quant-ph], submitted 4 May 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[12] [12]

Madelung equations.https://en.wikipedia.org/ wiki/Madelung_equations

Wikipedia contributors. Madelung equations.https://en.wikipedia.org/ wiki/Madelung_equations. Accessed: 2026-05-15. 6

work page 2026