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arxiv: 2605.08148 · v1 · submitted 2026-05-03 · 🪐 quant-ph · hep-ph

Recognition: 2 theorem links

· Lean Theorem

Beyond the Lorenz Gauge: Probing a Stueckelberg Scalar in the Electric Aharonov-Bohm Effect

Renato Vieira dos Santos
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Pith reviewed 2026-05-12 01:16 UTC · model grok-4.3

classification 🪐 quant-ph hep-ph
keywords Aharonov-Bohm effectStueckelberg scalarLorenz gaugeelectric phase shiftquantum interferometrygauge freedomtime-dependent potential
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The pith

If the Stueckelberg scalar is physical, it adds a 1-cos(ωT) phase shift to the electric Aharonov-Bohm effect that a frequency sweep can isolate from the standard electromagnetic contribution.

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

The paper sets out to test whether the Lorenz gauge condition is merely a convenient choice or a deeper physical requirement. It shows that a surviving Stueckelberg scalar B equal to the divergence of the electromagnetic potential would couple to electrons and produce an interference phase whose time dependence follows 1 minus cosine of frequency times transit time. This dependence is orthogonal to the usual sine dependence arising from the scalar potential itself, so the two can be separated by sweeping the oscillation frequency even when both are active. A concrete measurement protocol using single-electron interferometry at picosecond resolution is outlined as feasible with present technology.

Core claim

If the Stueckelberg scalar B = ∂_μ A^μ survives as a physical field and couples to matter, the electric Aharonov-Bohm effect with shielded time-dependent potentials produces an additive phase shift proportional to 1 - cos(ωT). This signature is orthogonal to the conventional sin(ωT) phase and remains separable by a frequency sweep of the applied potential even when both contributions are present at once.

What carries the argument

The Stueckelberg scalar B = ∂_μ A^μ, whose physical coupling to electrons generates the distinctive 1 - cos(ωT) time dependence in the accumulated interferometric phase.

If this is right

  • The scalar contribution can be isolated from the electromagnetic one simply by sweeping the frequency of the time-dependent potential.
  • The required timing resolution is picosecond scale, within reach of existing single-electron interferometry techniques.
  • Observation of the 1-cos(ωT) term would establish that the Lorenz condition is not a matter of principle.
  • Absence of the term would indicate that the gauge condition must hold as a physical constraint rather than a convenience.

Where Pith is reading between the lines

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

  • Confirmation would motivate searches for the scalar in other interferometric or precision measurements where gauge-dependent divergences appear.
  • The same frequency-sweep logic could be adapted to probe analogous extra degrees of freedom in related gauge theories or in atomic clocks.
  • Null results would tighten experimental bounds on any hypothetical coupling of such a scalar to ordinary matter.

Load-bearing premise

The Stueckelberg scalar couples to electrons so that it adds a phase shift whose time dependence is exactly 1 - cos(ωT) and remains distinguishable under realistic conditions of shielding and timing precision.

What would settle it

A frequency-sweep measurement in a single-electron electric Aharonov-Bohm interferometer that detects only the standard sin(ωT) dependence and no 1 - cos(ωT) component would falsify the claim that the scalar is physical and detectable in this manner.

Figures

Figures reproduced from arXiv: 2605.08148 by Renato Vieira dos Santos.

Figure 1
Figure 1. Figure 1: FIG. 1. Standard ( [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

The electric Aharonov-Bohm effect -- a time-dependent scalar potential imparting a measurable phase shift on electrons in a region free of electromagnetic fields -- has never been experimentally tested in its original formulation with shielded, time-dependent potentials. This unexplored regime offers a rare opportunity: the Lorenz condition $\partial_\mu A^\mu = 0$, a choice that eliminates a scalar degree of freedom from the electromagnetic potential, may not be the last word. If the Stueckelberg scalar $B = \partial_\mu A^\mu$ survives as a physical field and couples to matter, it would produce a phase shift with a distinctive $1-\cos(\omega T)$ signature -- orthogonal to the standard $\sin(\omega T)$ and separable by a frequency sweep even if both contributions coexist. We propose a measurement protocol based on single-electron interferometry with picosecond time resolution, within reach of current technology. The experiment asks a question that has lingered since 1959: is the Lorenz gauge a matter of convenience, or a matter of principle?

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.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes an experimental protocol to test whether the Stueckelberg scalar B = ∂_μ A^μ is a physical degree of freedom by measuring its putative contribution to the electric Aharonov-Bohm phase shift. It asserts that a physical B field coupling to electrons would produce an additive phase with distinctive time dependence 1 - cos(ωT), orthogonal to the conventional sin(ωT) electric AB phase, and separable by sweeping the driving frequency in a single-electron interferometer with picosecond resolution.

Significance. If the claimed 1-cos(ωT) functional form follows rigorously from the Stueckelberg formalism and a well-specified coupling, and if the signatures remain distinguishable under realistic conditions, the result would be significant for foundational questions in gauge theory and quantum mechanics: it would indicate that the Lorenz gauge condition is not merely a calculational convenience but eliminates a physical scalar mode. The orthogonality argument, if substantiated, is a strong feature that could allow detection even when both contributions are present. The proposal being within reach of existing single-electron interferometry technology further enhances its potential impact.

major comments (2)
  1. [Abstract] Abstract: The functional form of the phase shift (1 - cos(ωT)) is stated as arising from the coupling of the physical Stueckelberg scalar B = ∂_μ A^μ, yet no explicit interaction Lagrangian (e.g., g B ψ̄ψ or derivative coupling), Hamiltonian term, or integration over the shielded region is provided to derive this time dependence from the definition of B. This derivation is load-bearing for the orthogonality claim and the separability by frequency sweep.
  2. [Experimental protocol] Experimental protocol (throughout the proposal): The assertion that the 1-cos(ωT) signature remains distinguishable from sin(ωT) under realistic conditions (shielding imperfections, finite bandwidth, timing jitter) is not supported by any error analysis, noise model, or simulation. Without this, it is unclear whether the claimed separability survives the experimental imperfections that could mix the two contributions.
minor comments (1)
  1. [Abstract] The abstract would benefit from a single sentence outlining the key experimental parameters (e.g., electron energy, frequency range, or interferometer arm length) to make the feasibility claim more concrete.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the potential significance of our proposal and for the constructive major comments. We address each point below and will revise the manuscript to incorporate the requested clarifications and analyses.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The functional form of the phase shift (1 - cos(ωT)) is stated as arising from the coupling of the physical Stueckelberg scalar B = ∂_μ A^μ, yet no explicit interaction Lagrangian (e.g., g B ψ̄ψ or derivative coupling), Hamiltonian term, or integration over the shielded region is provided to derive this time dependence from the definition of B. This derivation is load-bearing for the orthogonality claim and the separability by frequency sweep.

    Authors: We acknowledge that the manuscript would be strengthened by an explicit derivation of the claimed phase. The 1-cos(ωT) dependence follows from the Stueckelberg scalar B coupling to the electron current in the shielded region, but we will add the interaction Lagrangian term g B ψ̄ψ together with the step-by-step integration of the resulting phase shift for a harmonic time-dependent potential in a dedicated subsection of the revised manuscript. This will make the orthogonality to the conventional sin(ωT) term fully rigorous and self-contained. revision: yes

  2. Referee: [Experimental protocol] Experimental protocol (throughout the proposal): The assertion that the 1-cos(ωT) signature remains distinguishable from sin(ωT) under realistic conditions (shielding imperfections, finite bandwidth, timing jitter) is not supported by any error analysis, noise model, or simulation. Without this, it is unclear whether the claimed separability survives the experimental imperfections that could mix the two contributions.

    Authors: The referee correctly notes the absence of quantitative support for robustness. While the ideal-case orthogonality is evident from the distinct functional forms, we agree that a noise model is needed. In the revision we will include a concise error analysis with a simple timing-jitter and bandwidth model, together with a numerical illustration showing that a frequency sweep still permits separation of the two contributions at the picosecond resolution achievable in existing single-electron interferometers. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper proposes an experimental protocol to test whether a physical Stueckelberg scalar B = ∂_μ A^μ produces a distinguishable phase shift in the electric Aharonov-Bohm effect. The claimed 1-cos(ωT) signature is presented conditionally as a consequence of the scalar coupling to matter, orthogonal to the standard sin(ωT) term. No derivation chain is exhibited that reduces this functional form to the paper's own inputs by construction, nor are there fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations. The work treats the Stueckelberg formalism as an external theoretical framework and focuses on experimental separability via frequency sweep, remaining self-contained as a proposal without internal circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the standard quantum-mechanical treatment of electron phase accumulation in potentials plus the postulate that the Stueckelberg scalar remains dynamical and couples locally to charge; no new free parameters are introduced in the abstract, but the coupling strength is implicitly assumed to be non-zero and measurable.

axioms (2)
  • domain assumption The phase shift in electron interferometry is given by the line integral of the electromagnetic potential (or its scalar extension) along the path
    Invoked to translate the potential into an observable phase; standard in Aharonov-Bohm literature
  • domain assumption The Lorenz condition is not enforced by the underlying dynamics when the Stueckelberg scalar is physical
    Core premise that allows B to survive as an independent field
invented entities (1)
  • Physical Stueckelberg scalar B = ∂_μ A^μ no independent evidence
    purpose: To provide an observable degree of freedom beyond the usual gauge redundancy
    Postulated to couple to electrons and generate the extra phase term; no independent evidence supplied in the abstract

pith-pipeline@v0.9.0 · 5483 in / 1740 out tokens · 51342 ms · 2026-05-12T01:16:24.957244+00:00 · methodology

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Reference graph

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