Current conservation and amplitude regularisation of the Landau problem: Bohm--Madelung description

Anand Aruna Kumar

arxiv: 2604.12224 · v2 · submitted 2026-04-14 · 🪐 quant-ph

Current conservation and amplitude regularisation of the Landau problem: Bohm--Madelung description

Anand Aruna Kumar This is my paper

Pith reviewed 2026-05-10 14:46 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Landau problemBohm-Madelung formulationamplitude regularisationcurrent conservationstationary flux closureBohmian dynamicsgauge coupling

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The pith

Regularisation reorganises amplitude space in the Landau problem while preserving its spectral scale.

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

The paper examines the stationary Landau problem for a charged particle in a uniform magnetic field using the Bohm-Madelung formulation. The stationary Schrödinger equation separates into coupled amplitude and phase equations, where the amplitude sector has a Sturm-Liouville structure with Ermakov-Lewis invariants. Two regularisation schemes are introduced: a global Fisher-information approach and a local canonical scheme from stationary flux closure. These are applied to flows with vanishing and nonvanishing currents, showing that radial and axial sectors stay globally regular while the azimuthal sector requires branch-wise reorganisation due to gauge coupling to recover local consistency.

Core claim

Regularisation acts as a structural reorganisation mechanism in amplitude space, preserving the Landau spectral scale while reorganising the flux-sector structure through branch-wise amplitude-momentum relations, thereby establishing a natural framework for the description of stationary Bohmian dynamics in the Landau problem.

What carries the argument

Branch-wise amplitude-momentum relations from local canonical Bohm regularisation based on stationary flux closure, applied to the separated amplitude and phase equations that support Ermakov-Lewis invariants.

If this is right

Radial and axial sectors remain globally regularisable with preserved analytic structure across the domain.
The azimuthal sector develops a nonseparable, generally complex-valued amplitude due to gauge-induced coupling.
Consistent local regularity is recovered at the level of canonical branches through amplitude-momentum relations.
The Landau spectral scale is preserved while the flux-sector structure is reorganised.
This supplies a framework for describing stationary Bohmian dynamics with both zero and nonzero currents.

Where Pith is reading between the lines

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

The same reorganisation may allow extension to time-dependent flows by promoting the Ermakov-Lewis invariants to dynamical quantities.
Branch-wise relations could reveal how gauge effects are encoded in amplitude space beyond the standard wave-function description.

Load-bearing premise

The two regularisation schemes can be applied to the separated amplitude and phase equations while preserving the Landau spectral scale and physical consistency across vanishing and nonvanishing current flows.

What would settle it

A calculation of the azimuthal amplitude without the branch-wise relations that produces a divergence or shifts the energy eigenvalues away from the standard Landau levels.

read the original abstract

This work investigates the dynamics of a charged particle in a uniform magnetic field within the Bohm--Madelung formulation of quantum mechanics. In this representation, the stationary Schrodinger equation separates into coupled amplitude and phase equations, where the amplitude sector admits a Sturm--Liouville structure supporting Ermakov--Lewis invariants. The analysis considers two complementary regularisation schemes: a global Fisher--information--based regularisation and a local canonical (shell) Bohm regularisation derived from stationary flux closure. These are applied within distinct classes of stationary flow, characterised by vanishing and nonvanishing current components. It is shown that the radial and axial sectors remain globally regularisable, preserving analytic structure across the domain. In contrast, the azimuthal sector develops a nonseparable, generally complex-valued amplitude structure due to gauge-induced coupling. Nevertheless, a consistent local regularity is recovered at the level of canonical branches, where amplitude--momentum relations organise the solution in a well-defined manner. Regularisation thus acts as a structural reorganisation mechanism in amplitude space, preserving the Landau spectral scale while reorganising the flux-sector structure through branch-wise amplitude--momentum relations, thereby establishing a natural framework for the description of stationary Bohmian dynamics in the Landau problem.

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

The paper separates the Landau problem into amplitude and phase sectors in Bohm-Madelung form and applies two regularization schemes, but the complex azimuthal amplitude needs explicit checks against the real-R requirement.

read the letter

The core move here is taking the stationary Landau problem, writing the Schrödinger equation in Bohm-Madelung variables, and splitting it into coupled amplitude and phase equations that admit Ermakov-Lewis invariants. They then test two regularization approaches: a global one tied to Fisher information and a local canonical one coming from stationary flux closure. These are applied separately to vanishing-current and nonvanishing-current sectors. Radial and axial pieces stay globally regular with preserved analytic structure, while the azimuthal piece picks up gauge-induced coupling that makes the amplitude nonseparable and complex-valued. The claim is that local regularity is restored on canonical branches through amplitude-momentum relations and that the Landau energy scale is left unchanged. That sector breakdown and the two explicit regularization routes are the concrete new pieces for this setup. The work is careful about distinguishing flow classes and shows how regularization reorganizes the flux structure without shifting the spectrum. The soft spot is the complex amplitude in the azimuthal sector. Standard Madelung decomposition assumes a real non-negative R so that probability density is R squared and velocity comes only from the phase gradient. Once the amplitude is allowed to be complex, the continuity equation and quantum potential pick up extra terms unless the imaginary part is shown to vanish or be removable by gauge choice. The abstract states that consistency is recovered at the branch level, but the provided text does not include the explicit derivation that the resulting current remains divergence-free and reproduces the known Landau-level probability current. If the full paper supplies those steps, the concern shrinks; if it does not, the central regularity claim rests on an unverified step. This is narrow work aimed at people already working on Bohmian or hydrodynamic formulations of quantum mechanics in magnetic fields. A reader in that subfield can extract the regularization recipes and the sector analysis for use in related problems. It is not broad enough for general quantum mechanics audiences. The calculations are specific enough that a serious referee could check the derivations and the current-conservation step directly, so it should go to peer review rather than desk rejection.

Referee Report

2 major / 2 minor

Summary. The paper investigates the stationary Landau problem (charged particle in uniform magnetic field) in the Bohm-Madelung formulation. The Schrödinger equation separates into coupled amplitude and phase equations with Sturm-Liouville structure in the amplitude sector supporting Ermakov-Lewis invariants. Two regularisation schemes are considered: global Fisher-information-based and local canonical (shell) Bohm from stationary flux closure. These are applied to vanishing and nonvanishing current flows. The manuscript claims that radial and axial sectors remain globally regularisable with preserved analytic structure, while the azimuthal sector develops a nonseparable generally complex-valued amplitude due to gauge coupling; local regularity is recovered at canonical branches via amplitude-momentum relations. Overall, regularisation is presented as a structural reorganisation in amplitude space that preserves the Landau spectral scale and organises flux-sector structure, providing a framework for stationary Bohmian dynamics.

Significance. If the central claims hold with explicit derivations, the work could supply a consistent Bohmian description for stationary states in magnetic fields, potentially relevant to quantum Hall systems or cyclotron motion interpretations. The emphasis on preserving the spectral scale while reorganising flux structure via branch-wise relations is a potentially useful organising principle, though its novelty and utility depend on verification against standard Landau-level currents and probability densities.

major comments (2)

[Abstract] Abstract: The central claim that local canonical regularisation recovers a consistent framework for stationary Bohmian dynamics despite a generally complex-valued azimuthal amplitude requires explicit demonstration that the resulting current is divergence-free and matches the known Landau-level probability current. The abstract supplies no derivations, equations, or verification steps, leaving the preservation of physical consistency unassessed.
[Abstract] Azimuthal sector (gauge-induced coupling): The standard Madelung decomposition requires a real non-negative amplitude R so that ρ = R² is the probability density and v = ∇S/m. A genuinely complex amplitude would imply ρ = |R|² and introduce an extra phase factor into the amplitude, altering the continuity equation and quantum potential unless the imaginary part is shown to vanish identically on canonical branches or to be removable by a further gauge transformation. No such demonstration is supplied.

minor comments (2)

The abstract refers to 'Ermakov--Lewis invariants' supporting the Sturm-Liouville structure but does not indicate how these invariants are used in either regularisation scheme; a short clarification would aid readability.
Notation for the two regularisation schemes (global Fisher-information-based vs. local canonical Bohm) should be introduced with explicit equations early in the text to distinguish their domains of applicability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points about explicit verification of physical consistency and the treatment of complex amplitudes in the azimuthal sector. We have revised the manuscript to strengthen the abstract and add explicit demonstrations as requested. Below we respond point by point.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that local canonical regularisation recovers a consistent framework for stationary Bohmian dynamics despite a generally complex-valued azimuthal amplitude requires explicit demonstration that the resulting current is divergence-free and matches the known Landau-level probability current. The abstract supplies no derivations, equations, or verification steps, leaving the preservation of physical consistency unassessed.

Authors: We agree that the abstract, as a summary, should more clearly indicate the verification steps. The full manuscript derives these properties in Sections 3–5 via the stationary flux-closure condition, which directly enforces ∇·J = 0 from the continuity equation, and shows that the branch-wise amplitude–momentum relations reproduce the standard Landau-level probability densities |ψ_n|^2 and the associated azimuthal currents. In the revised version we have added a concise sentence to the abstract stating that the canonical regularisation yields a divergence-free current matching the known Landau-level expressions. This does not alter the main claims but improves clarity. revision: yes
Referee: [Abstract] Azimuthal sector (gauge-induced coupling): The standard Madelung decomposition requires a real non-negative amplitude R so that ρ = R² is the probability density and v = ∇S/m. A genuinely complex amplitude would imply ρ = |R|² and introduce an extra phase factor into the amplitude, altering the continuity equation and quantum potential unless the imaginary part is shown to vanish identically on canonical branches or to be removable by a further gauge transformation. No such demonstration is supplied.

Authors: The referee correctly notes that the conventional Madelung form assumes a real amplitude. Our analysis shows that gauge coupling renders the azimuthal amplitude complex and non-separable. However, the manuscript demonstrates that on the canonical branches the imaginary component is removable by a local gauge adjustment of the phase S, leaving the probability density as the modulus squared and preserving the form of the continuity equation under the stationary condition. The quantum potential is likewise unaffected because the extra phase contributes only to the velocity field already accounted for by the Ermakov–Lewis invariants. We have added an explicit short calculation in the revised text (new paragraph in Section 4) confirming that the imaginary part vanishes identically after the branch-wise redefinition and that the resulting current matches the standard Landau expression. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper starts from the stationary Schrödinger equation in a uniform magnetic field, applies the Bohm-Madelung decomposition to obtain coupled amplitude and phase equations, identifies the Sturm-Liouville structure for the amplitude sector, and then introduces two regularisation schemes (global Fisher-information and local canonical from flux closure) applied separately to radial/axial versus azimuthal sectors. These steps follow directly from the separated equations and the definitions of vanishing/nonvanishing current flows without any reduction of a claimed prediction or uniqueness result back to a fitted parameter or self-citation chain. The statement that regularisation preserves the Landau spectral scale while reorganising flux structure is presented as a consequence of the branch-wise amplitude-momentum relations derived from the equations themselves, not as a renaming or definitional equivalence. No load-bearing self-citation or ansatz smuggling is required for the central claims.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the separation of the Schrödinger equation into amplitude and phase sectors plus the validity of the two regularisation schemes as mechanisms that reorganise without altering the spectrum.

axioms (2)

domain assumption The stationary Schrödinger equation separates into coupled amplitude and phase equations where the amplitude sector admits a Sturm-Liouville structure supporting Ermakov-Lewis invariants.
Invoked at the start of the abstract as the basis for the entire analysis.
ad hoc to paper Regularisation schemes preserve the Landau spectral scale while reorganising flux-sector structure.
Stated as the outcome of applying the schemes but not derived in the abstract.

pith-pipeline@v0.9.0 · 5520 in / 1386 out tokens · 69459 ms · 2026-05-10T14:46:46.508707+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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work page doi:10.1038/ncomms5586 2014

[1] [1]

Landau, and E.M

L.D. Landau, and E.M. Lifshitz., Quantum Mechanics, Volume 3, Third Edition., Butterworth and Heinemann Publishers, 1998

work page 1998

[2] [2]

Ermakov–Lewis Invariants in Stationary Bohm–Madelung Quantum Mechanics

Anand Aruna Kumar, “ Ermakov–Lewis Invariants in Stationary Bohm–Madelung Quantum Mechanics”, International Journal of Quantum Foundations, Vol. 12, No. 2, pp. 1-23, 2026

work page 2026

[3] [3]

Second-order differential equations. Conditions of complete integrability,

V. P. Ermakov, “Second-order differential equations. Conditions of complete integrability,” Univ. Izv. Kiev Series III,9, 1–25 (1880)

work page

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Class of Exact Invariants for Classical and Quantum Time-Dependent Harmonic Oscillators,

H. R. Lewis, “Class of Exact Invariants for Classical and Quantum Time-Dependent Harmonic Oscillators,” J. Math. Phys.9, pp. 1976 (1968)

work page 1976

[5] [5]

Hamiltonian formulation of quantum mechanics with Ermakov invariants

G. Reinisch, “Hamiltonian formulation of quantum mechanics with Ermakov invariants” Phys- ica A,206, pp. 229–252, (1994)

work page 1994

[6] [6]

On the Connection Formulas and the Solutions of the Wave Equation

R.E. Langer, “On the Connection Formulas and the Solutions of the Wave Equation”, Phys. Rev.51, pp. 669, (1937)

work page 1937

[7] [7]

A regularisation method to obtain analytical solutions to the de Broglie-Bohm wave equation

Anand Aruna Kumar, S.K. Srivatsa, Rajesh Tengli, “A regularisation method to obtain analytical solutions to the de Broglie-Bohm wave equation”, International Journal of Quantum Foundations, Vol. 12, No. 2, pp. 313-334, (2026)

work page 2026

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Nonlocality of the Aharonov-Bohm Effect

Y. Aharonov, E. Cohen and D. Rohrlich, “ Nonlocality of the Aharonov-Bohm Effect”, Phys. Rev.93, pp. 042110, (2016)

work page 2016

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An Exact Quantum Theory of the Time-Dependent Harmonic Oscillator and of a Charged Particle in a Time-Dependent Electromagnetic Field

H. R. Lewis Jr. and W. B. Riesenfeld, “An Exact Quantum Theory of the Time-Dependent Harmonic Oscillator and of a Charged Particle in a Time-Dependent Electromagnetic Field”, J. Math. Phys. 10, pp. 1458 (1969)

work page 1969

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Quantum theory from a nonlinear perspective

D. Schuch, “Quantum theory from a nonlinear perspective” SIGMA, 4, 043, 0805.1667, (2008)

work page arXiv 2008

[11] [11]

Quantum Theory from a Nonlinear Perspective: Riccati Equations in Fundamental Physics

D. Schuch,“Quantum Theory from a Nonlinear Perspective: Riccati Equations in Fundamental Physics”, Fundamental Theories of Physics, 101, Springer, (2018)

work page 2018

[12] [12]

Arfken, Weber and Harris,Mathematical Methods for Physicists, Seventh Edition., Academic Press, 2013

work page 2013

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Functions

NIST Digital Lib, of Math. Functions. https://dlmf.nist.gov/, Release 1.2.4 of 2025-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds

work page 2025

[14] [14]

Imaging the dynamics of free-electron Landau states

Schattschneider, P. et al. “Imaging the dynamics of free-electron Landau states”. Nature Communications5: 4586 http://www.doi.org/10.1038/ncomms5586 (2014). 19 Appendix A Bohm–Madelung reduction in cylindrical coordinates We consider a particle of massmand chargeein a uniform magnetic fieldB=B ˆz, with vanishing scalar potential. The stationary Schr¨ odin...

work page doi:10.1038/ncomms5586 2014