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Direct, analytic solution for the electromagnetic vector potential in any gauge

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

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Pith's one-line read The vector potential in any electromagnetic gauge is one analytic formula: the retarded vector potential plus a gradient correction built from the difference between the retarded and chosen scalar potentials.

desk verdict A correct and clearly written derivation, but the central formula is the standard gauge transformation from the Lorenz gauge; the 'no gauge condition' framing overstates what is actually proven. read the letter →

arxiv 2507.02104 v1 pith:QD5DLMTW submitted 2025-07-02 physics.class-ph

Kuo-Ho Yang , Robert D. Nevels This is my paper · ORCID
classification physics.class-ph MSC 78A2535Q61 PACS 03.50.De41.20.-q
keywords classicalelectrodynamicsMaxwell'sequationsvectorpotentialgaugeinvarianceretardedpotentialsLorenzCoulombvelocity
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

This paper claims that, for any time-dependent charge-current distribution, the electromagnetic vector potential in every gauge can be written with a single analytic formula: $\mathbf{A}(r,t) = \mathbf{A}_c(r,t) + c\,\nabla \int [\Phi_c(r,t) - \Phi(r,t)]\,dt$, where $\mathbf{A}_c$ and $\Phi_c$ are the retarded potentials and $\Phi$ is the scalar potential of the gauge one wants. The derivation starts directly from Maxwell's equations for the potentials and never imposes a gauge condition, so the formula is presented as valid for the Lorenz, Coulomb, velocity, generalized Kirchhoff, and multipolar gauges alike. If correct, choosing a gauge reduces to choosing a scalar potential: once $\Phi$ is known, the corresponding vector potential is fully determined by quadrature, and the electric and magnetic fields are the retarded fields, propagating at speed $c$ from the physical sources.

What carries the argument

The load-bearing object is eq. (10), the identity $\mathbf{A}(r,t) = \mathbf{A}_c(r,t) + c\,\nabla\!\int[\Phi_c(r,t)-\Phi(r,t)]\,dt$, together with the split $\mathbf{A}=\mathbf{A}_1+\mathbf{A}_2$ that produces it. $\mathbf{A}_1$ is the standard retarded vector potential from the current density; the correction $\mathbf{A}_2$ is found by differentiating its source term, using the scalar-potential Maxwell equation to replace $\partial(\nabla\cdot\mathbf{A})/\partial t$ with $4\pi$ times the charge density, and solving the resulting wave equation with the retarded Green function. This machinery converts the gauge ambiguity of the potentials into a single scalar difference, $\Phi_c-\Phi$, making the vector potential a direct functional of the scalar potential.

What would settle it

Take a uniformly moving point charge, compute the Coulomb-gauge scalar potential, insert it into eq. (20) to obtain $\mathbf{A}$, and then form $\mathbf{E}$ and $\mathbf{B}$; if these fields differ from the retarded fields of the same charge at any point outside the source, the central identity is false. Up to the allowed $\nabla\Omega(r)$ term, agreement is what the paper predicts.

Watch

Extended reading notes

Core claim

The central claim is the identity $\mathbf{A} = \mathbf{A}_c + c\,\nabla\int(\Phi_c - \Phi)\,dt$, obtained from the sourced wave equations for the potentials with no gauge condition. Writing $\mathbf{A} = \mathbf{A}_1 + \mathbf{A}_2$, the paper identifies $\mathbf{A}_1$ as the standard c-retarded vector potential of the current density and shows that $\mathbf{A}_2$ obeys a wave equation whose source is the gradient of $\Phi_c - \Phi$; the scalar-potential Maxwell equation is used to eliminate the time derivative of $\nabla\cdot\mathbf{A}$ and close the system. The resulting vector potential has a gauge-invariant part, $\mathbf{A}_c$, common to every gauge, and a gauge-dependent part fixed entirely by the chosen scalar potential. The paper then verifies that any such potentials give the same electric and magnetic fields as the retarded potentials, so field propagation at speed $c$ is preserved regardless of gauge.

Load-bearing premise

The load-bearing premise is that the scalar potential in the desired gauge is already known; eq. (10) determines $\mathbf{A}$ only after $\Phi$ has been found, and the derivation also assumes no boundary surfaces and drops an additive time-independent gradient $\nabla\Omega(r)$.

Editorial extensions

If this is right

  • In any gauge, the vector potential follows by quadrature once the scalar potential in that gauge is known; no gauge-specific vector-potential wave equation has to be solved.
  • The Lorenz, Coulomb, velocity, generalized Kirchhoff, and multipolar vector potentials all result from substituting the corresponding scalar potential into the same formula.
  • In every gauge, the electric and magnetic fields equal those built from the retarded potentials, so all gauges describe fields that propagate at speed $c$ from the physical charge-current distribution.
  • Two gauges are always related by the gauge function $\chi = c\int(\Phi-\Phi')\,dt$, which is exactly the gradient term that carries potentials from the Lorenz gauge to any other gauge.

Reading between the lines

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

  • Editorial inference: numerically, one could solve only the scalar-potential equation in the desired gauge and then construct $\mathbf{A}$ by a quadrature, turning gauge choice into a post-processing step rather than a separate partial-differential-equation solve.
  • Editorial inference: because the derivation assumes no boundary surfaces, applying eq. (10) in bounded domains or with artificial numerical boundaries would require a treatment of surface terms that the paper does not provide.
  • Editorial inference: a direct test would be to evaluate eq. (10) for a moving point charge in the Coulomb gauge and compare against an independent Coulomb-gauge calculation; agreement up to the allowed time-independent gradient $\nabla\Omega(r)$ would support the identity.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper claims to derive a direct, analytic solution for the electromagnetic vector potential A in any gauge, starting from Maxwell's equations for potentials and without imposing a gauge condition. The derivation splits A into a retarded part Ac and a part A2, and obtains A2 = c grad of the time integral of (Phi_c - Phi), where Phi_c is the retarded scalar potential and Phi is the scalar potential of the chosen gauge. The final formula is A = Ac + c grad integral(Phi_c - Phi)dt. The paper then applies this formula to the velocity gauge, the Lorenz gauge, the Coulomb gauge, the generalized Kirchhoff gauge, and the Poincaré gauge, and argues that the electric and magnetic fields derived from the formula are gauge invariant and propagate at speed c from the physical sources.

Significance. The identity at the center of the paper, eq. (10), is correct and provides a compact universal relation between the vector potential in an arbitrary gauge and the scalar potential of that gauge. The paper has useful features: eq. (11) explicitly verifies the divergence equation, Appendix A gives a Fourier-transform derivation, and the applications to several gauges show the breadth of the formula. However, the advertised result is not a direct solution for A without a gauge condition: eq. (10) expresses A only in terms of the scalar potential Phi, which must still be found by solving a gauge-conditioned scalar equation. The formula is in fact the standard gauge transformation from the Lorenz-gauge potentials, as the authors themselves note in Section 4. The paper also leaves undiscussed the homogeneous solutions of the wave equation in the step from eq. (7) to eq. (8), and the indefinite time integrals are not defined and may diverge for static or DC components. These issues are local in the sense that the algebraic identity is sound, but they affect the central claim and the stated scope of the paper.

major comments (3)
  1. [Sec. 2, Eq. (10), and Sec. 6] The central claim that eq. (10) is a direct analytic solution for A in any gauge without using a gauge condition is not supported. Eq. (10) determines A only in terms of the scalar potential Phi of the desired gauge; to obtain a concrete vector potential one must first solve a gauge-conditioned scalar equation, e.g., eq. (13) for the velocity gauge or the Poisson equation for the Coulomb gauge. Section 6 concedes this: 'as soon as the scalar potential is solved, the corresponding vector potential is completely determined.' Moreover, eqs. (33)-(34) show that eq. (10) is exactly the gauge transformation from the Lorenz-gauge potentials (Phi_c, Ac). The derivation is therefore a restatement of a gauge relation rather than an independent construction of A from the sources.
  2. [Sec. 2, Eqs. (7)-(8)] The step from eq. (7) to eq. (8) cancels the wave operator on dA2/dt without discussing the homogeneous kernel. Eq. (7) determines dA2/dt only up to an arbitrary solution H(r,t) of the homogeneous wave equation. The paper states that A2 is determined only up to an additive time-independent gradient of the form grad Omega(r), but this is incomplete: the kernel of the wave operator also contains time-dependent solutions. Eq. (8) is a particular solution, and the claim that eq. (10) is the universal solution for any gauge requires additional boundary or initial conditions (for example, no incoming radiation) that are not stated.
  3. [Sec. 2, Eqs. (8), (10); Sec. 3, Eq. (15); Appendix A] The indefinite time integrals in eqs. (8), (10), (15), (20), (28), (34), and (39) have no specified lower limit, and they generally do not converge for static or DC components. For a time-independent charge distribution, Phi_c and Phi are time-independent, so the integral of Phi_c - Phi grows linearly in t; for a periodic source, the integral is periodic only if the time-average of Phi_c - Phi vanishes. Thus the statement that the solution is valid for an 'arbitrary time-dependent charge-current distribution' is too broad. The same limitation appears in the Fourier derivation: eq. (A.5) divides by omega, so the omega = 0 component is not treated.
minor comments (4)
  1. [Sec. 1, Introduction] The phrase 'potentialswithout' in the first paragraph is a typo and should read 'potentials without'.
  2. [Sec. 4, Eqs. (27)-(34)] The discussion of Jackson's gauge-transformation method is helpful, but the text should explicitly state that this equivalence means eq. (10) is an identity among potentials rather than a new solution method; as written, the presentation risks confusing the two.
  3. [Sec. 5, Eq. (45)] The derivation of grad Phi(P) from the line-integral expression in eq. (44) is not shown; an intermediate step would improve readability, especially because the Poincaré gauge example is otherwise concise.
  4. [Appendix A, Eq. (A.5)] The Fourier derivation divides by omega without discussing the omega = 0 case; a remark that static and time-independent parts are excluded or handled by a limiting procedure would make the appendix self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central identity is derived from Maxwell's equations, not assumed; the required scalar potential is an acknowledged input, not a renamed prediction.

full rationale

The paper's central formula, Eq. (10), is obtained by an explicit algebraic derivation from Maxwell's equations for potentials, not by fitting or by importing a result. A is split into the retarded source part Ac and a correction A2; Eq. (5) is differentiated and Eq. (2) is used to eliminate the divergence term, yielding Eq. (7), whose solution is Eq. (8). Substitution back into Eq. (10) is then verified in Eq. (11). No target quantity is used as an input, no parameter is fitted, and no 'prediction' is constructed from data. The paper explicitly concedes in Section 6 that the scalar potential must first be solved before the vector potential is fully determined; this is a scope limitation on the word 'solution', but it is not circularity because Eq. (10) is an identity connecting A and Phi, not a definition of either. The agreement with Jackson's gauge-transformation method, acknowledged in Eqs. (33)-(34), is an external consistency check rather than a load-bearing self-citation. Self-citations to Yang [3,8] and to Yang and McDonald [13] are used only for provenance of the velocity-gauge result and for earlier Coulomb-gauge work; the derivation in this paper is self-contained. Thus, while the title's promise of a 'direct solution' is somewhat stronger than what is proven, the derivation itself is not circular.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central formula rests on standard wave-equation Green's functions and on the retarded-potential assumption, plus a set of auxiliary choices: no boundaries, a specified turn-on time, a zero additive gradient, and the convergence of indefinite time integrals. None of these introduces new physics, but they define the conditions under which eq. (10) is a genuine solution.

assumptions (4)
  • domain assumption Sources are localized, turned on at t0, and there are no boundary surfaces.
    States in Section 2: 'We consider localized charge and current densities... turned on at t0' and 'We assume that there are no boundary surfaces anywhere.' These conditions justify the use of retarded Green's functions and the choice of causal solutions.
  • ad hoc to paper The additive time-independent gradient grad Omega(r) in the solution for A2 is set to zero.
    Section 2, after eq. (10): 'A2 in eq. (8) and consequently the full potential A in eq. (10) are only determined to within an additive time-independent gradient function grad Omega(r), which is set to zero in this paper for brevity.' This choice is arbitrary and not derived from the equations.
  • domain assumption The wave equation for dA2/dt has only the retarded particular solution, with no homogeneous waves.
    Implicit in the step from eq. (7) to eq. (8), where the wave operator is cancelled without discussing the homogeneous solution. The assumption relies on radiation conditions and the source being switched on at t0.
  • ad hoc to paper The indefinite time integral integral (Phi_c - Phi) dt is well defined with some lower limit not specified.
    The formula (10) uses an indefinite integral over time; the paper does not specify the lower limit or impose convergence conditions on Phi_c - Phi as t approaches plus or minus infinity, which can fail for sources with static components.

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Cite this review

Pith. "Pith review of Direct, analytic solution for the electromagnetic vector potential in any gauge." pith.science (2026). https://pith.science/paper/QD5DLMTW

@misc{pith2026250702104,
  author       = {Pith},
  title        = {Pith review of: Direct, analytic solution for the electromagnetic vector potential in any gauge},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QD5DLMTW}},
  note         = {Machine review of arXiv:2507.02104}
}
read the original abstract

We derive an analytic solution for the electromagnetic vector potential in any gauge directly from Maxwell's equations for potentials for an arbitrary time-dependent charge-current distribution. No gauge condition is used in the derivation. Our solution for the vector potential has a gauge-invariant part and a gauge-dependent part. The gauge-dependent part is related to the scalar potential.

Discussion (0). Continue with ORCID 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

  1. Comment on work of Yang and Nevels "Direct, analytic solution for the electromagnetic vector potential in any gauge" (arXiv:2507.02104)

    physics.class-ph 2025-07 reject novelty 4.0 of 10

    A comment asserting that Eq. (10) of Yang and Nevels is wrong, supported by a Coulomb-gauge calculation for a charge that suddenly starts moving; the calculation appears to contain errors.

  2. Reply to "Comments on work of Yang and Nevels 'Direct, analytic solution for the electromagnetic vector potential in any gauge' (arXiv:2507.02104)" by Onoochin (arXiv:2507.08042)

    physics.class-ph 2025-09 conditional novelty 2.0 of 10

    The velocity-gauge vector potential A_2 = c∫(Phi_c - Phi^(v))dt is shown to satisfy the wave equation by four independent analytic proofs.

Reference graph

Works this paper leans on

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