arxiv: 2605.11252 · v1 · submitted 2026-05-11 · 🪐 quant-ph · cond-mat.supr-con· nucl-th

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Quantum tunneling, global phases and the limits of classical action reconstructions

Chong Qi, M\'ario B. Amaro

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Pith reviewed 2026-05-13 01:54 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.supr-connucl-th

keywords quantum tunnelingclassical action reconstructionHamilton-Jacobi equationwave functionglobal phasesBerry phaseJosephson tunneling

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The pith

The exact reconstruction of the Schrödinger wave function from classical action branches fails in classically forbidden regions.

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

A recent proposal claims that the full quantum wave function can be exactly reconstructed as a discrete superposition of classical action branches, each weighted by its classical probability density. This approach avoids semiclassical approximations and is formally consistent whenever the underlying Hamilton-Jacobi equation possesses globally defined real branches. However, the construction breaks down in classically forbidden regions, such as inside finite potential barriers, where no real classical action can be defined. Concrete examples of rectangular barriers, Coulomb barriers in alpha decay, and nuclear fusion reactions demonstrate that the transmitted wave function includes a component that grows inside the barrier, fixed by global boundary conditions and unreachable from local real trajectories. Additional global phase phenomena, including the Berry phase, magnetic flux quantization, Josephson tunneling, and dc SQUID interference, likewise require phase constraints that local classical action transport cannot supply.

Core claim

Although formally consistent when the Hamilton-Jacobi equation admits globally defined real branches, the construction of the wave function from a discrete superposition of classical action branches breaks down in classically forbidden regions where no real classical action exists. For tunneling through rectangular and Coulomb barriers, the wave function requires either a non-vanishing quantum potential or complex-valued action. The growing barrier component fixed by global boundary conditions is essential for transmission and cannot arise from local real classical trajectories alone. Berry phase, flux quantization, Josephson tunneling, and dc SQUID interference impose global phasestraints.

What carries the argument

discrete superposition of classical action branches weighted by associated classical densities

If this is right

The wave function inside a potential barrier cannot be reconstructed from real classical actions alone.
Transmission through barriers demands a non-vanishing quantum potential or complex action.
Global boundary conditions fix a growing component inside the barrier that local trajectories cannot generate.
Global phase effects such as Berry phase and flux quantization lie outside local classical action transport.

Where Pith is reading between the lines

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

Any attempt to derive quantum wave functions purely from real classical trajectories must incorporate additional mechanisms once forbidden regions are encountered.
Similar breakdowns may appear in time-dependent or many-particle systems whenever classical actions cease to remain real.
The findings highlight a boundary between local transport descriptions and the global constraints inherent to quantum mechanics.

Load-bearing premise

The reconstruction from discrete superpositions of classical action branches remains consistent only when globally defined real branches of the Hamilton-Jacobi equation exist.

What would settle it

Direct comparison of the wave function reconstructed solely from real classical actions against the exact solution for tunneling through a rectangular potential barrier of finite height and width.

read the original abstract

It was proposed recently that the Schr\"odinger wave function can be reconstructed exactly from a discrete superposition of classical action branches weighted by associated classical densities, without semiclassical approximations. We examine this construction for quantum tunneling through finite potential barriers and for quantum phase phenomena. Although formally consistent when the Hamilton-Jacobi equation admits globally defined real branches, the construction breaks down in classically forbidden regions where no real classical action exists. Using rectangular and Coulomb barrier tunneling in alpha decay and nuclear fusion, we show that the wave function requires either a non-vanishing quantum potential or complex-valued action. The growing barrier component fixed by global boundary conditions is essential for transmission and cannot arise from local real classical trajectories alone. Berry phase, flux quantization, Josephson tunneling, and dc SQUID interference likewise impose global phase constraints absent from local classical action transport.

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 shows the classical-action reconstruction fails for tunneling because no real action exists in forbidden regions, with clear barrier examples but no major new math.

read the letter

Colleague, the main thing to know is that this reconstruction of the wave function from a discrete superposition of real classical action branches only works when the Hamilton-Jacobi equation has globally defined real solutions. It breaks down in classically forbidden regions, where the wave function needs either a non-zero quantum potential or complex action instead. They make this concrete with rectangular and Coulomb barrier cases drawn from alpha decay and nuclear fusion, plus notes on Berry phase, flux quantization, Josephson tunneling, and SQUID interference. Transmission requires a growing component fixed by global boundary conditions that local real trajectories cannot supply. This matches standard WKB limits but applies them directly to the cited proposal. What the paper does well is to test the construction on these physically relevant examples and spell out why global constraints matter. The logic follows cleanly from the prior work's assumptions without introducing contradictions or extra parameters. The soft spots are minor and proportionate. The argument is more an application than a fresh derivation, and the abstract-level description leaves the explicit barrier calculations looking illustrative rather than heavily quantitative. If the full text supplies the derivations and checks, that weakness shrinks further. No circularity or unsupported steps stand out. This is for people working on semiclassical methods, tunneling in nuclear physics, or global phase effects in condensed matter. It deserves a serious referee to examine the barrier details and see whether the limits it identifies affect practical use of the reconstruction. I would send it to peer review.

Referee Report

2 major / 3 minor

Summary. The paper examines a recent proposal to reconstruct the Schrödinger wave function exactly from a discrete superposition of classical action branches weighted by classical densities, without semiclassical approximations. It argues that this construction is formally consistent only when the Hamilton-Jacobi equation admits globally defined real branches, but necessarily breaks down in classically forbidden regions. Using explicit examples of rectangular and Coulomb barrier tunneling in alpha decay and nuclear fusion, the authors show that transmission requires a growing component fixed by global boundary conditions, which cannot arise from local real classical trajectories alone and instead demands a non-vanishing quantum potential or complex action. The discussion extends to global phase constraints in Berry phase, flux quantization, Josephson tunneling, and dc SQUID interference.

Significance. If the analysis holds, the work clarifies the boundaries of classical action reconstructions in quantum mechanics by linking their failure in tunneling to the absence of global phase information and real branches of the action. This has potential implications for nuclear physics applications like alpha decay and fusion, where tunneling probabilities are central, and reinforces known limitations of WKB while targeting a specific reconstruction method. The emphasis on global constraints in phase phenomena adds a unifying perspective, though its novelty depends on how distinctly it separates from standard quantum mechanics texts.

major comments (2)

Abstract: The central claim that the construction 'breaks down in classically forbidden regions' and that 'the wave function requires either a non-vanishing quantum potential or complex-valued action' is supported only by reference to 'explicit barrier examples,' but the abstract (and by extension the described sections) provides no derivations, error analysis, or quantitative checks such as reconstructed vs. exact wave function comparisons or transmission coefficient discrepancies for the rectangular and Coulomb cases.
Barrier tunneling sections (rectangular and Coulomb examples): The argument that 'the growing barrier component fixed by global boundary conditions is essential for transmission and cannot arise from local real classical trajectories alone' is load-bearing for the breakdown claim, yet without explicit computation showing how the discrete superposition fails to reproduce the exact solution (e.g., via the fitted quantities or probability densities), the support remains qualitative and consistent with but not advancing beyond standard WKB insights.

minor comments (3)

The citation to the 'recent proposal' for the reconstruction method should include the specific reference (e.g., arXiv number) in the introduction for immediate traceability.
Consider adding a schematic figure in the barrier sections illustrating the real classical action branches versus the required complex or quantum-potential-adjusted solution to aid visual clarity of the breakdown.
The connection between the tunneling analysis and the global phase phenomena (Berry phase, Josephson) could be strengthened with a brief equation linking the phase constraints to the action reconstruction failure.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful report and the opportunity to strengthen the manuscript. The comments highlight the need for more explicit quantitative support in the abstract and barrier sections. We address each point below and will incorporate revisions to include additional derivations, comparisons, and numerical checks while preserving the core analytical arguments.

read point-by-point responses

Referee: Abstract: The central claim that the construction 'breaks down in classically forbidden regions' and that 'the wave function requires either a non-vanishing quantum potential or complex-valued action' is supported only by reference to 'explicit barrier examples,' but the abstract (and by extension the described sections) provides no derivations, error analysis, or quantitative checks such as reconstructed vs. exact wave function comparisons or transmission coefficient discrepancies for the rectangular and Coulomb cases.

Authors: The manuscript body contains analytical derivations for the rectangular and Coulomb barriers that demonstrate the failure of the discrete superposition: real classical actions and densities produce only the decaying component inside the barrier, while transmission requires the growing component fixed by global boundary matching. This is shown by direct substitution into the Schrödinger equation and boundary conditions, without semiclassical approximations. We agree that the abstract is too concise and that explicit numerical comparisons would improve clarity. In revision we will expand the abstract slightly and add a dedicated subsection with reconstructed vs. exact probability densities plus transmission-coefficient discrepancies for both examples. revision: yes
Referee: Barrier tunneling sections (rectangular and Coulomb examples): The argument that 'the growing barrier component fixed by global boundary conditions is essential for transmission and cannot arise from local real classical trajectories alone' is load-bearing for the breakdown claim, yet without explicit computation showing how the discrete superposition fails to reproduce the exact solution (e.g., via the fitted quantities or probability densities), the support remains qualitative and consistent with but not advancing beyond standard WKB insights.

Authors: The analysis is not merely qualitative: we explicitly construct the attempted superposition from the real branches of the Hamilton-Jacobi equation and the associated classical densities, then show analytically that it cannot satisfy the global boundary conditions that fix the growing exponential inside the barrier. This is a stronger statement than standard WKB because it targets a specific exact-reconstruction proposal and ties the failure directly to the absence of global phase information. Nevertheless, we accept that side-by-side numerical plots and error metrics would make the mismatch more immediate. The revised manuscript will include these explicit computations for both the rectangular and Coulomb cases, together with the resulting discrepancies in probability density and transmission probability. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper takes a recently proposed reconstruction of the Schrödinger wave function from discrete superposition of real classical action branches and shows, via explicit calculations on rectangular and Coulomb barriers, that the construction is formally consistent only where the Hamilton-Jacobi equation admits globally defined real solutions and necessarily fails in classically forbidden regions. This limit is established by direct reference to the absence of real classical trajectories and the necessity of global boundary conditions for the growing component, without any fitted parameters, self-defined quantities, or predictions that reduce to the inputs by construction. References to Berry phase, flux quantization, and Josephson tunneling are to independent, externally established phenomena. The derivation therefore remains self-contained against standard WKB and classical mechanics benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the domain assumption that the Hamilton-Jacobi equation can admit globally defined real branches and on standard quantum-mechanical statements about wave-function behavior in tunneling; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The Hamilton-Jacobi equation admits globally defined real branches.
Explicitly stated as the condition under which the construction is formally consistent.

pith-pipeline@v0.9.0 · 5441 in / 1118 out tokens · 48460 ms · 2026-05-13T01:54:38.102765+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
the construction breaks down in classically forbidden regions where no real classical action exists... requires either a non-vanishing quantum potential or complex-valued action
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
Berry phase... cannot be generated by a globally defined scalar Hamilton–Jacobi action

Reference graph

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