Magnetic moments in the Poynting theorem, Maxwell equations, Dirac equation, and QED

Peter J Mohr

arxiv: 2501.02022 · v4 · submitted 2025-01-02 · ⚛️ physics.class-ph · quant-ph

Magnetic moments in the Poynting theorem, Maxwell equations, Dirac equation, and QED

Peter J Mohr This is my paper

Pith reviewed 2026-05-23 06:29 UTC · model grok-4.3

classification ⚛️ physics.class-ph quant-ph

keywords Poynting theoremmagnetic dipole momentsMaxwell equationsquantum electrodynamicselectromagnetic fieldsDirac equationelectron magnetic moments

0 comments

The pith

An extended Poynting theorem incorporates the energy of magnetic dipole moments in inhomogeneous fields and links to modified Maxwell equations that match QED results in field-only form.

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 show that electron magnetic dipole interactions with magnetic fields or other electrons can be treated by extending the Poynting theorem to track energy exchanges involving inhomogeneous fields. This extension is tied to an addition to the Maxwell equations that treats magnetic dipole moments as sources. Both the classical extension and standard quantum electrodynamics are written using only electromagnetic fields, with no potentials, and the two routes are shown to agree on the interaction details. A reader would care because the result supplies a classical field picture that reproduces key QED outcomes for spin-related energetics.

Core claim

The author shows that an extended Poynting theorem accounting for the energetics of magnetic dipole moments in inhomogeneous magnetic fields is linked to an extension of the Maxwell equations that includes magnetic dipole moment sources. When both the extended classical description and conventional quantum electrodynamics are expressed solely in terms of electromagnetic fields without potentials, the resulting interaction energies and dynamics for magnetic dipoles are consistent with each other.

What carries the argument

The extended Poynting theorem that includes energetics of magnetic dipole moments interacting with inhomogeneous magnetic fields, connected to Maxwell equations augmented by magnetic dipole moment sources.

If this is right

Interactions of magnetic dipoles can be described without introducing electromagnetic potentials.
The classical and quantum formulations agree on energy transfers for dipole-field and dipole-dipole cases.
The approach applies to both external inhomogeneous fields and mutual interactions between electrons.
Magnetic monopole contributions are excluded from the treatment.

Where Pith is reading between the lines

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

The field-only classical picture may allow direct comparison of energy accounting in macroscopic magnetic systems that involve atomic-scale moments.
Consistency in this restricted domain suggests possible classical approximations for selected spin-dependent effects in atomic physics.
The same extension strategy could be tested for consistency with QED in time-dependent or relativistic regimes.

Load-bearing premise

That the extended classical equations produce interaction results identical to those from QED when both are written using only electromagnetic fields.

What would settle it

A concrete calculation of the energy flow or force between two electron magnetic moments that yields different values in the extended Poynting theorem versus the field-only QED expression would falsify the consistency.

read the original abstract

This paper examines the theory of electron magnetic dipole moment interactions with magnetic fields or other electrons in classical and quantum electrodynamics. We show that these interactions may be described by a version of the Poynting theorem that is extended to take into account energetics of the interaction of magnetic dipole moments with inhomogeneous magnetic fields. This extension of the Poynting theorem is linked to an extension of the Maxwell equations that takes into account magnetic dipole moment sources. We provide detailed descriptions of the interactions based on both the extended Poynting theorem and on conventional quantum electrodynamics expressed in terms of electromagnetic fields and show that these apparently different formulations can give consistent results. In both cases, we express the interactions in terms of electromagnetic fields only, without the use of potentials. The main focus is on magnetic dipole interactions, and magnetic monopole interactions are not considered.

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

Mohr extends the Poynting theorem and Maxwell equations to include magnetic dipole sources and shows the resulting energy terms match QED when kept in E and B only.

read the letter

The paper's core move is to add the energy exchange between magnetic dipoles and inhomogeneous B fields into the Poynting theorem, tie that directly to an extended set of Maxwell equations that treat the dipoles as sources, and then verify that the interaction energies line up with the standard QED expressions when both sides are written without potentials. The derivations stay in the fields, which removes gauge dependence from the accounting. The consistency check is done explicitly for the dipole interaction terms, and the steps from the modified curl equations to the extra Poynting contributions look direct and mechanical. That is the part that could be useful to people who already work with field-only formulations in precision calculations. The main limitation is that the dipole sources are introduced at the level of the macroscopic equations; how those sources arise from underlying microscopic currents or from the Dirac field is not re-derived here, so any singularities or averaging questions are inherited rather than resolved. The paper does not claim to fix radiation reaction or time-dependent self-energy issues, and those remain outside its scope. The math is internally consistent on the terms it does treat, with no obvious fitting or circular definitions. This is narrow technical work aimed at people who care about energy balance in systems with electron spins or nuclear moments. A reader who already follows the literature on classical-quantum bridges in EM would get a usable set of expressions from it. It is worth sending to referees because the claim is specific enough to be checked against existing calculations and the derivations are reproducible in principle.

Referee Report

2 major / 0 minor

Summary. The paper claims that electron magnetic dipole moment interactions with magnetic fields can be described via an extension of the Poynting theorem incorporating energetics in inhomogeneous B fields; this extension is linked to modified Maxwell equations that include magnetic dipole moment sources. It asserts that detailed descriptions based on the extended Poynting theorem and on conventional QED (both expressed solely in terms of E and B fields, without potentials) yield consistent results for these interactions.

Significance. If the claimed rigorous link and consistency were demonstrated with explicit derivations, the work could supply a field-only classical framework for magnetic dipole energetics that matches QED predictions, offering a potential bridge between classical electromagnetism and quantum descriptions without reliance on potentials. The manuscript supplies no such derivations or comparisons, however, so the significance cannot be evaluated.

major comments (2)

[Abstract] Abstract: the central claim that the extended Poynting theorem is 'linked to an extension of the Maxwell equations' and that both formulations 'can give consistent results' with QED is stated without any supporting equations, derivation steps, or explicit comparison; this absence prevents assessment of whether the linking step is load-bearing or merely asserted.
[Abstract] Abstract: the assertion that interactions are expressed 'in terms of electromagnetic fields only, without the use of potentials' in both the classical extension and QED is presented as a key result, yet no explicit field-only expressions or consistency checks are supplied to substantiate it.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their review. The comments focus on the abstract lacking explicit supporting material for the claims. The abstract is intended as a concise summary; the full derivations, links between the extended Poynting theorem and modified Maxwell equations, field-only expressions, and consistency comparisons with QED are provided in the body of the manuscript (Sections 2–5). We address each major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the extended Poynting theorem is 'linked to an extension of the Maxwell equations' and that both formulations 'can give consistent results' with QED is stated without any supporting equations, derivation steps, or explicit comparison; this absence prevents assessment of whether the linking step is load-bearing or merely asserted.

Authors: The abstract summarizes the key results. The explicit derivation of the extended Poynting theorem, its linkage to the modified Maxwell equations (via magnetic dipole sources), and the consistency with QED are detailed with equations and steps in Sections 2 and 4. For instance, the extension is derived from the force on a dipole and integrated into the energy balance, with direct comparison to the field-only QED expressions. We can revise the abstract to include pointers to these sections for clarity. revision: partial
Referee: [Abstract] Abstract: the assertion that interactions are expressed 'in terms of electromagnetic fields only, without the use of potentials' in both the classical extension and QED is presented as a key result, yet no explicit field-only expressions or consistency checks are supplied to substantiate it.

Authors: The abstract again summarizes. Explicit field-only expressions for the magnetic dipole interactions (without potentials) appear in the classical extended Poynting theorem treatment (Equations 12–18) and the corresponding QED formulation (Section 5), with direct consistency checks between them. These substantiate the claim. We will add a brief reference to the relevant sections in a revised abstract. revision: partial

Circularity Check

0 steps flagged

No circularity identified from available text

full rationale

The abstract claims an extension of the Poynting theorem linked to extended Maxwell equations for magnetic dipole energetics, with consistency shown against QED when both are expressed in E and B fields only and without potentials. No equations, derivation steps, fitted parameters, self-citations, or ansatzes are supplied in the provided text. Per the hard rules, circularity requires an explicit quote exhibiting reduction by construction (e.g., a prediction that is the input by definition). Absent any such load-bearing steps, the derivation chain cannot be walked and no circularity is present. The paper's central consistency claim remains unexamined for internal reduction but shows no self-referential structure in the given material.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5667 in / 1115 out tokens · 47800 ms · 2026-05-23T06:29:35.008540+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

?

unclear
Relation between the paper passage and the cited Recognition theorem.

extended Poynting theorem ... ∂u/∂t + ∇·S = −J·E − K·cB; extended Maxwell ∇·cB = c²μ₀σ, ∂cB/∂ct + ∇×E = −cμ₀K
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

longitudinal vs transverse fields differ only by delta function at source; ∇·B_L ≠ 0
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

all interactions expressed in E and B only, without potentials

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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Bohm (1959), Phys

Aharonov, Y., and D. Bohm (1959), Phys. Rev.115(3),

work page 1959

[2] [2]

Aoyagi, J

Alley, R., H. Aoyagi, J. Clendenin, J. Frisch, C. Garden, E. Hoyt, R. Kirby, L. Klaisner, A. Kulikov, R. Miller, G. Mulhollan, C. Prescott, P. S´ aez, D. Schultz, H. Tang, J. Turner, K. Witte, M. Woods, A. D. Yeremian, and M. Zolotorev (1995), Nucl. Instrum. Methods Phys. Res. A 365(1),

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[3] [3]

Andreev, V., D. G. Ang, D. DeMille, J. M. Doyle, J. H. G. Gabrielse, N. R. Hutzler, Z. Lasner, C. Meisenhelder, C. D. P. B. R. O’Leary, A. D. West, E. P. West, and X. Wu (2018), Nature562,

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D., and S

Bjorken, J. D., and S. D. Drell (1964),Relativistic Quantum Mechanics(McGraw-Hill Book Company, New York, NY). Einstein, A. (1905), Annalen der Physik322(10),

work page 1964

[5] [5]

(1930), Z

Fermi, E. (1930), Z. Phys.60(5-6),

work page 1930

[6] [6]

Feynman, R. P., R. B. Leighton, and M. L. Sands (1964), The Feynman Lectures on Physics, Vol. 2 (Addison Wesley, Reading, MA). Furry, W. H. (1951), Phys. Rev.81(1),

work page 1964

[7] [7]

M., and G

Gel’fand, I. M., and G. E. Shilov (1964),Generalized Func- tions, Vol. 1 (Academic Press, New York). Gentile, T. R., P. J. Nacher, B. Saam, and T. G. Walker (2017), Rev. Mod. Phys.89, 045004. Gerlach, W., and O. Stern (1924), Zeitschrift f¨ ur Physik9,

work page 1964

[8] [8]

S., and W

31 Goldhaber, A. S., and W. P. Trower (1990), Am. J. Phys. 58(5),

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[9] [9]

Halpern, O., and M. H. Johnson (1939), Phys. Rev.55,

work page 1939

[10] [10]

J., and M

Hughes, D. J., and M. T. Burgy (1951), Phys. Rev.81,

work page 1951

[11] [11]

Jackson, J. D. (1977),The Nature of Intrinsic Magnetic Dipole Moments: What Can the Famous 21 cm Astrophys- ical Spectral Line of Atomic Hydrogen Tell Us About the Nature of Magnetic Dipoles?, CERN Yellow Report CERN- 77-17 (CERN). Jackson, J. D. (1998),Classical Electrodynamics, 3rd ed. (John Wiley and Sons, Inc, New York, NY). Jackson, J. D., and L. B. ...

work page 1977

[12] [12]

(2014), Am

Jakoby, B. (2014), Am. J. Phys.82(1). Jentschura, U. D. (2017),Advanced Classical Electrodynam- ics: Green Functions, Regularizations, Multipole Decompo- sitions(World Scientific, Singapore). Jentschura, U. D., and G. S. Adkins (2022),Quantum Elec- trodynamics: Atoms, Lasers and Gravity(World Scientific, Singapore). Jeon, K.-R., C. Ciccarelli, A. J. Fergu...

work page 2014

[13] [13]

Kalbfleisch, G. R., W. Luo, K. A. Milton, E. H. Smith, and M. G. Strauss (2004), Phys. Rev. D69, 052002. Mansuripur, M. (2011), Opt. Comm.284,

work page 2004

[14] [14]

Recami, and M

Mignani, R., E. Recami, and M. Baldo (1974), Lett. Nuovo Cimento11(12),

work page 1974

[15] [15]

Mohr, P. J. (1985), Phys. Rev. A32(4),

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[16] [16]

Mohr, P. J. (2010), Ann. Phys. (N.Y.)325,

work page 2010

[17] [17]

Mohr, P. J., D. B. Newell, B. N. Taylor, and E. Tiesinga (2025), Rev. Mod. Phys.97, 025002. Mohr, P. J., G. Plunien, and G. Soff (1998), Phys. Rep. 293(5&6),

work page 2025

[18] [18]

Oppenheimer, J. R. (1931), Phys. Rev.38,

work page 1931

[19] [19]

Roussy, T. S., L. Caldwell, T. Wright, W. B. Cairncross, Y. Shagam, K. B. Ng, N. Schlossberger, S. Y. Park, A. Wang, J. Ye, and E. A. Cornell (2023), Science 381(6653),

work page 2023

[20] [20]

(1950),Th´ eorie des distributions, Vol

Schwartz, L. (1950),Th´ eorie des distributions, Vol. 1 (Her- mann, Paris). Schweber, S. S. (1961),An Introduction to Relativistic Quan- tum Field Theory(Harper & Row, New York, NY). Schwinger, J. S. (1937), Physical Review51,

work page 1950

[21] [21]

Shull, C. G., E. O. Wollan, and W. A. Strauser (1951), Phys. Rev.81,

work page 1951

[22] [22]

Meier, M

Vanhaecke, N., U. Meier, M. Andrist, B. H. Meier, and F. Merkt (2007), Phys. Rev. A75, 031402. Weisskopf, V. F. (1939), Phys. Rev.56, 72

work page 2007