Overcoming a challenge for Bohmian mechanics

Hrvoje Nikolic

REVIEW 3 major objections 4 minor 2 cited by

Overcoming a challenge for Bohmian mechanics

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper claims that Bohmian velocity is defined by the continuity equation rather than by the phase gradient, and constructs an explicit density-dependent velocity for the coupled-waveguide system that overcomes the recent challenge.

desk verdict A valid general point undercut by a flawed specific construction: the paper's wave functions don't satisfy its own equations. read the letter →

arxiv 2507.08049 v1 pith:X3CMRTSR submitted 2025-07-10 quant-ph hep-th

Hrvoje Nikolic This is my paper

classification quant-phhep-th

keywords Bohmianmechanicsquantumtrajectoriescontinuityequationphasegradientcoupledwaveguidesvelocityfieldprobabilitycurrenthiddenvariables

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Recent work on coupled optical waveguides reported particle motion that the standard Bohmian velocity $v=(\hbar/m)\nabla\phi$ cannot explain, and framed it as a challenge to Bohmian mechanics. This paper argues that the challenge rests on a too-narrow reading of Bohmian velocity: in general the velocity is whatever vector field makes the continuity equation $\partial_t\rho+\nabla\cdot(\rho v)=0$ hold, and the phase-gradient formula is only a special case. For the waveguide solutions $\psi_m\propto\cos(k_1x)e^{ik_2x}$ and $\psi_a\propto\sin(k_1x)e^{ik_2x}$, that more general definition yields the explicit velocity $v_x(x)=(1/L\rho(x))\,\hbar k_2/m$ in each waveguide. If this construction is valid, the reported experiment does not refute Bohmian mechanics; it only shows that the velocity must be computed from the continuity equation rather than from the phase alone.

What carries the argument

The governing mechanism is the continuity-equation definition of Bohmian velocity: instead of taking $v=(\hbar/m)\nabla\phi$ as the definition, the paper takes the continuity equation $\partial_t\rho+\nabla\cdot(\rho v)=0$ as primary and solves it for $v$. For the stationary, one-dimensional wave functions (5), this reduces to $\partial_x(\rho v_x)=0$, whose general solution is $v_x=c/\rho$; the constant $c$ is fixed by an average-velocity matching condition, yielding $v_x=1/(L\rho(x))\,\hbar k_2/m$. This object carries the argument because it supplies a velocity with the right density dependence even though the phase is a pure linear ramp $k_2x$ and the phase-gradient formula would give a constant velocity.

What would settle it

Insert the proposed solutions (5) into the coupled equations (4) and verify that the coefficients of $\cos(k_1x)$ and $\sin(k_1x)$ match in both equations for the same real coupling constant $J_0$; if they do not, the density used for (10) is not a solution of the model.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the challenge case is not a counterexample to Bohmian mechanics but an instance of a more general velocity rule. For a wave function obeying the standard Schrödinger equation, the Bohmian velocity is the phase-gradient expression, but the coupled-waveguide system obeys the two-component equations (4), which are not of that form. Requiring that the velocity satisfies the continuity equation forces the one-dimensional form $v_x(x)=c/\rho(x)$; imposing the condition that its $\rho$-weighted average equals that of the phase-gradient velocity fixes $c=\hbar k_2/(mL)$, giving equation (10). The paper also stresses that the speed measured in the experiment is not a velocity and need not agree with $v_x$, since Bohmian velocities are not directly measurable. The result is a modified, density-dependent velocity that reproduces the densities and restores the deterministic-trajectory picture.

Load-bearing premise

The argument stands on the assumption that each waveguide's probability density obeys $\partial_x(\rho v_x)=0$ by itself, even though the coupled equations (4) allow probability to flow between the two waveguides.

Editorial extensions

If this is right

The standard phase-gradient velocity is not the definition of Bohmian velocity; it is a special case that follows only when the wave function obeys the standard single-component Schrödinger equation.
In the coupled-waveguide system, the continuity-compatible velocity is $v_x(x)=1/(L\rho(x))\,\hbar k_2/m$, which varies with position even though the phase varies linearly.
The measured speed in the experiment need not equal the Bohmian velocity, because Bohmian velocities are not directly measurable; the velocity explains the densities rather than reproducing the measured speed.
The same continuity-based method can be applied to other systems whose wave equation differs from the standard Schrödinger form, whenever an average-velocity condition fixes the integration constant.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves open whether the same velocity rule emerges when the two waveguides are treated as one two-component system with the coupling terms included as sources in the continuity balance; a full two-component derivation would either confirm (10) or produce a different guidance rule.
The average-velocity condition used to fix the constant $c$ is a choice; different normalization conditions would give different density-dependent velocities, so the continuity equation alone does not uniquely determine the Bohmian trajectory in this setup.
A natural next step is to propagate a localized wave packet through the coupled equations and compare its centroid motion with trajectories from (10); the paper does not report such a comparison.
If the continuity-based construction is generally valid, it can be applied to any Schrödinger-like or effective wave equation, making the guidance rule a solvable problem in one dimension rather than a special fix for this experiment.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper has a useful reminder: Bohmian velocity is defined by a continuity equation, not necessarily by the phase gradient. The author is right that the coupled-waveguide system in the Nature paper is not a standard single-component Schrödinger equation, so the usual formula need not apply. That part is worth saying, and the reference to his earlier work [3] is appropriate.

The problem is that the specific construction doesn't hold together. The two waveguides are coupled by the ℏJ0 terms in Eq. (4), so probability can move between them. Applying the continuity equation separately to each component, as in Eq. (6), drops that source term. The correct balance for each component includes a coupling term; for the proposed solutions (5), that term is nonzero. Worse, substituting (5) into (4) gives incompatible conditions for a real J0: the first equation requires J0 = -i ℏ k1 k2 / m and the second requires the opposite sign. So the wave functions used in the paper are not solutions of the model they're supposed to describe.

The constant c in Eq. (7) is also fixed by an average-matching condition that imports the very phase-gradient velocity the paper wants to avoid. That's a red flag, though it's secondary to the inconsistency above. The final velocity (10) therefore doesn't follow from the physics of the coupled system.

If the author were to work with the full two-component wave function and the total density, or derive a proper continuity equation with sources, the idea might be repairable. But the paper as written doesn't support the claim to overcome the challenge. The broad conceptual point is sound, but the specific response is not.

For a short Matters Arising, this is borderline: the general observation is worth airing, but the technical error is central. I'd send it to a referee to confirm the algebra, because a quick check would settle it. If the referee agrees with the substitution check, the paper should be rejected or returned for a major rewrite.

Referee Report

3 major / 4 minor

Summary. This Matters Arising manuscript responds to the recent Nature challenge to Bohmian mechanics by arguing that the Bohmian velocity should be derived from the continuity equation rather than from the phase gradient of the wave function. For the coupled-waveguide model of [2], the author proposes a velocity of the form v_x(x) = c/rho(x) for each waveguide component, fixes the constant c by requiring equality of the average of this velocity with the average of the standard phase-gradient velocity (1), and obtains the explicit result v_x(x) = (1/(L rho(x))) (hbar k_2/m). The paper then claims that this 'modified Bohmian velocity' overcomes the challenge and explains the densities (8) without contradicting the measured speed in [2].

Significance. If the construction were correct, it would provide a simple and important reply to a high-profile challenge, showing that the failure of the phase-gradient formula in the coupled-waveguide system does not refute Bohmian mechanics. The paper also usefully emphasizes that the velocity in Bohmian mechanics is defined through the continuity equation, which in general does not force the phase-gradient formula. However, the specific derivation is not valid: the proposed wave functions do not solve the stated coupled equations, and the per-component source-free continuity equation on which the construction rests is not implied by the dynamics. The general conceptual point survives, but the claimed resolution of the challenge is unsupported.

major comments (3)

[§2, Eqs. (4)–(5)] The functions (5), psi_m ∝ cos(k1 x) e^{i k2 x} and psi_a ∝ sin(k1 x) e^{i k2 x}, are not solutions of the coupled equations (4) for real J0. Substituting psi_m into the first equation gives a term ℏ J0 psi_a on the right-hand side that cannot be absorbed into the left-hand side E psi_m unless J0 = 0; the second equation similarly forces J0 = 0. Thus the explicit velocity (10), which is derived from (5), is not grounded in the model (4) that the paper claims to address.
[§2, Eq. (6)] The source-free continuity equation ∂x(ρ v_x) = 0 is applied separately to ψ_m and ψ_a, but these components are not individually closed: the coupling terms ℏ J0(ψ_a − ψ_m) and ℏ J0(ψ_m − ψ_a) in (4) describe probability exchange between the waveguides. The correct per-component balance is ∂t ρ_i + ∂x J_i = S_i with a generally nonzero source S_i; for the densities (8) this source does not vanish, so the c/ρ form (7) is not a consequence of the dynamics.
[§2, Eq. (9)] The constant c in (7) is fixed by imposing the equality of the average of the modified velocity with the average of the standard phase-gradient velocity (1). This is an additional normative requirement, not a derivation from the continuity equation, which fixes the velocity only up to an arbitrary constant. Since (1) is precisely the quantity whose failure motivated the challenge, imposing (9) is a self-imposed condition that the paper does not physically justify; without it, the final velocity (10) is underdetermined.

minor comments (4)

[§2, Eq. (10)] The velocity v_x = 1/(L ρ(x)) ℏ k2/m diverges at the zeros of ρ(x), which occur periodically for the densities (8); the paper does not discuss whether particle trajectories are well-defined at these nodes.
[§1, Introduction] The notation in Eq. (1) uses ∇ϕ while the subsequent treatment is one-dimensional; this is not an error, but the reduction to ∂xϕ(x) could be stated explicitly for clarity.
[§2, Eq. (9)] The paper relies on Ref. [3] for the condition that the average velocity should match that of the phase-gradient formula, but the method of [3] is not summarized; a brief explanation of why this condition is natural would strengthen the presentation.
[§2, Discussion] The statement that the modified velocity is a 'generalization' of (1) is vague; it would help to specify the sense in which (10) reduces to (1) only in the constant-density limit.

Circularity Check

2 steps flagged · score 4.0 of 10

Explicit construction is partly self-definitional: the velocity is chosen to reproduce the density it then claims to explain, and the one free constant is fixed by the author's own prior averaging criterion.

self definitional [Eqs. (6)-(7) and final paragraph]
"The continuity equation (6) is satisfied if and only if the velocity is of the form vx(x) = c / ρ(x) , (7), where c is a constant. ... Finally note that ... the Bohmian velocity has an explanatory power, in the sense that it can explain the densities (8) in terms of motions associated with 'hidden' variables."

The velocity is defined from the density via v = c/ρ and therefore trivially satisfies ∂x(ρvx)=0 for that same density. Saying that it 'can explain the densities (8)' is just restating the construction, not giving an independent dynamical consequence of the coupled Schrödinger equations (4). The explanatory claim is circular: the target quantity, the density, is built into the definition of the explanatory quantity.
self citation load bearing [Eq. (9) and surrounding text]
"A very general method for finding the appropriate Bohmian velocity in general quantum systems has been developed in [3] ... The constant c can be fixed by requiring [3] that the average velocity is the same as the average of the velocity (1) ... This requirement leads to the final result vx(x) = 1/(Lρ(x)) ℏk2/m , (10)."

The continuity equation alone leaves c undetermined. The specific value in Eq. (10) is selected only by applying an averaging condition taken from the author's own prior work [3]. Thus the uniqueness and normalization of the proposed velocity are imported from a self-citation rather than derived from the coupled dynamics (4). The broader existence claim is independent content, but the particular 'final result' is not forced by the model; it is forced by the author's own prior criterion.

full rationale

The paper is an explicit construction, so much of its mathematical core is not circular: for any stationary density ρ, the family v = c/ρ satisfies the reduced continuity equation, and the existence of a non-phase-gradient velocity is established by that construction alone. The central phrase 'by constructing the appropriate velocity explicitly' is honest about this constructive character. However, two load-bearing components do reduce to their own inputs. First, the claim that the velocity 'can explain the densities (8)' is tautological, because v was chosen to satisfy ∂x(ρv)=0 for exactly those densities; any density could be 'explained' in the same way, so no empirical or dynamical content is added. Second, the constant c in Eq. (10) is not fixed by the coupled equations (4) but by an averaging requirement cited from the author's own earlier paper [3]. The formula (10) therefore is not uniquely determined by the physics stated in the paper; it is one member of a one-parameter family selected by a self-cited normalization condition. These features make the presentation partially circular, but not fully so: the existence of a continuity-compatible velocity and its local difference from the phase-gradient form are genuine, independent mathematical facts. The additional objection that (5) may fail to solve (4) for real J0 is a correctness concern rather than a circularity and is not scored here.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central burden is the per-component source-free continuity assumption and the imposed average-velocity normalization; no new entities are introduced. The general continuity-based definition is plausible but not sufficient to fix the velocity.

free parameters (1)

constant c in v_x = c/ρ(x) = ℏ k2 / (m L)
The continuity equation (6) determines v_x only up to an arbitrary constant c. The paper fixes c via Eq. (9), requiring the average modified velocity to equal the average standard phase-gradient velocity. This condition is not derived from the coupled Schrödinger equations; any c would keep the density stationary.

assumptions (3)

domain assumption In general quantum systems, Bohmian velocity is defined by the continuity equation rather than by the phase-gradient formula (1).
This is the paper's opening premise, supported by a reference to the author's own earlier work [3]. It is a legitimate broader definition in some Bohmian formulations, but it is an extra assumption relative to textbook Bohmian mechanics.
ad hoc to paper Each waveguide component separately satisfies the source-free continuity equation ∂x(ρ v_x)=0 (Eq. 6).
The model Eqs. (4) include the coupling terms ℏJ0(ψ_a−ψ_m) and ℏJ0(ψ_m−ψ_a), so probability can flow between the two waveguides. For stationary solutions, the per-component continuity balance has a source term; Eq. (6) omits it. This assumption is essential for obtaining v_x ∝ 1/ρ.
ad hoc to paper The average of the modified Bohmian velocity must equal the average of the standard phase-gradient velocity (Eq. 9).
This condition fixes the otherwise arbitrary constant c and is introduced without derivation. It is not implied by the coupled equations or by the continuity equation, and it is the step that selects the specific value ℏk2/(mL).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Overcoming a challenge for Bohmian mechanics." pith.science (2026). https://pith.science/paper/X3CMRTSR

@misc{pith2026250708049,
  author       = {Pith},
  title        = {Pith review of: Overcoming a challenge for Bohmian mechanics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X3CMRTSR}},
  note         = {Machine review of arXiv:2507.08049}
}

read the original abstract

Recently, Bohmian mechanics has been challenged [Nature 643, 67 (2025)] by studying a system in which the motion of particles cannot be associated only with the gradient of phase of the wave function. We point out that, in general, Bohmian velocity is defined by the continuity equation, which does not always lead to velocity depending only on the phase gradient. By constructing the appropriate velocity explicitly, we overcome the challenge.

Discussion (0). Sign in to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Actual and weak actual values in Bohmian mechanics
quant-ph 2025-12 conditional novelty 4.0 of 10

In Bohmian mechanics, Holland's local expectation values equal the real part of position-postselected weak values, and the disputed waveguide experiment's 'speed' parameter is identified as the amplitude decay rate in...
Critical Reflections on Overcoming a Challenge for Bohmian Mechanics by H. Nikolic and the Experimental Findings of Sharoglazova et al
quant-ph 2025-07 unverdicted novelty 2.0 of 10

A critical commentary claiming Nikolic's reply to Sharoglazova et al. fails because it relies on the very formula it rejects and ignores entanglement.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages · cited by 2 Pith papers

[3]

Bohmian particle trajectories in relativistic fermionic quantum field theory

Nikoli´ c, H. Bohmian particle trajectories in relativistic fermionic quantum field theory. Found. Phys. Lett.18, 123-138 (2005). 3

work page 2005
[2]

& Klaers, J

Sharoglazova, V., Puplauskis, M., Mattschas, C., Toebes, C. & Klaers, J. Energy- speed relationship of quantum particles challenges Bohmian mechanics. Nature 643, 67 (2025)

work page 2025
[1]

The Quantum Theory of Motion (Cambridge University Press, Cambridge, 1993)

Holland, P.R. The Quantum Theory of Motion (Cambridge University Press, Cambridge, 1993)

work page 1993

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [3]

Bohmian particle trajectories in relativistic fermionic quantum field theory

Nikoli´ c, H. Bohmian particle trajectories in relativistic fermionic quantum field theory. Found. Phys. Lett.18, 123-138 (2005). 3

work page 2005

[2] [2]

& Klaers, J

Sharoglazova, V., Puplauskis, M., Mattschas, C., Toebes, C. & Klaers, J. Energy- speed relationship of quantum particles challenges Bohmian mechanics. Nature 643, 67 (2025)

work page 2025

[3] [1]

The Quantum Theory of Motion (Cambridge University Press, Cambridge, 1993)

Holland, P.R. The Quantum Theory of Motion (Cambridge University Press, Cambridge, 1993)

work page 1993