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Long-range magnetic interaction within quantum electrodynamics formalism

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Pith's one-line read This work derives the long-range magnetic interaction between atoms from one-photon exchange in quantum electrodynamics, showing that the classical spin-spin potential emerges as the leading surviving term and that a new 1/R oscillatory mag

desk verdict Plausible QED derivation of the known spin-spin term plus a genuinely new 1/R hyperfine channel, but the new channel's coefficient is internally inconsistent by a factor 3/2 and the numerics have unit errors. read the letter →

arxiv 2607.22435 v1 pith:3ZQO4OUS submitted 2026-07-24 physics.atom-ph

D. Solovyev , T. Zalialiutdinov , A. Anikin , A. Bobylev , P. Kvasov , J. J. Lopez-Rodriguez , A. Moshkin , D. Zinenko
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D. Glazov
This is my paper · ORCID
classification physics.atom-ph
keywords long-rangeinteractionmagneticdipole-dipolespin-spincouplingquantumelectrodynamicsS-matrixhyperfinestructurehydrogen-antihydrogenthermalradiation
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

Using the S-matrix formalism of quantum electrodynamics, this work derives the long-range magnetic interaction between two atoms from one-photon exchange. It shows that the classical dipole-dipole spin-spin potential arises as the leading surviving term, and that when the atoms can change hyperfine state a new 1/R oscillatory magnetic interaction appears, with no classical static analogue. The authors demonstrate that this QED picture accounts for the magnetic interaction of hydrogen atoms in s-states and explains why the hydrogen-antihydrogen interaction has opposite sign at zero and finite temperature. The finite-temperature imaginary part is interpreted as a thermally induced decay or annihilation channel. A reader would care because the derivation puts a well-known empirical potential on first-principles footing and predicts a new long-range channel that could affect precision spectroscopy and matter-antimatter experiments.

What carries the argument

The load-bearing object is the one-photon exchange S-matrix element for two one-electron atoms, combined with a Taylor expansion of the photon propagator e^{±i|k0|r12}/r12 in the electronic radius vectors. Retaining the second-order term (r_A·r_B) after angle averaging, and rewriting the Dirac α-matrix products via the identity (α_A·α_B)(r_A·r_B) = (α_A·r_B)(r_A·α_B) + ([r_A×α_A]·[r_B×α_B]), the magnetic moment operator μ = e[r×α]/2 emerges naturally. The same operator, evaluated between hyperfine sublevels with energy difference Δ_HFS, carries the new oscillatory 1/R interaction.

What would settle it

Compute the diagonal matrix elements of the 'rem' terms in Eq. (20) for the 1s state of hydrogen in the non-relativistic limit. If any is non-zero, Eq. (21) fails as stated; if they all vanish, one can also look for the predicted 1/R oscillatory hyperfine shift in a pair of hydrogen atoms at separations where the 1/R^3 term is suppressed.

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Extended reading notes

Core claim

The central claim is that the one-photon exchange S-matrix element, when the photon propagator is expanded in powers of the ratio of atomic size to internuclear separation, yields the magnetic interaction energy ΔE_AB^(m) = ⟨μA·μB⟩/R^3 − 3⟨(μA·R)(μB·R)⟩/R^5 for two atoms in unchanged states, in agreement with the classical magnetic dipole-dipole result. Beyond that, for transitions between hyperfine sublevels the same expression reduces to ΔE_HFS^(m) = −Δ_HFS^2 μ_0^2/(3R) e^{iΔ_HFS R}, a 1/R interaction with oscillatory dependence on internuclear distance that has no classical analogue. The derivation identifies the magnetic moment operator μ = e[r×α]/2 as the object coupling the atoms, and

Load-bearing premise

The derivation of the pure spin-spin potential relies on dropping the remaining quadrupole-type terms in Eq. (20) because they would change the angular momentum by two units; if any of those terms has a non-vanishing diagonal matrix element in the s-states considered, the interaction is not the pure 1/R^3 form.

Editorial extensions

If this is right

  • The classical spin-spin magnetic interaction between neutral atoms is a leading-order QED one-photon exchange effect, not an add-on to the two-photon van der Waals potential.
  • For two hydrogen atoms in 1s states, the interaction energy is μ_0^2/R^3 ≈ 87.6 MHz at R=10 a.u., consistent with known spin-spin corrections at molecular distances.
  • A new resonant magnetic channel, scaling as 1/R with cos/sin(Δ_HFS R), couples hyperfine sublevels of distant hydrogen atoms and survives where the static dipole-dipole term is absent.
  • At finite temperature the magnetic interaction falls off as 1/R^2 and is purely imaginary to leading order, giving a thermally induced decay channel.
  • For hydrogen-antihydrogen the sign of the interaction reverses relative to hydrogen-hydrogen, so the thermal imaginary part becomes a blackbody-stimulated annihilation rate that grows with temperature.

Reading between the lines

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

  • The 1/R hyperfine channel could be probed using cold-atom precision spectroscopy at separations near 0.1 μm, where the predicted shift approaches the current 1s-2s frequency uncertainty; the authors note the effect but do not propose an experiment.
  • The same multipole expansion applied to nuclear spins suggests comparable, isotope-dependent magnetic long-range forces between atoms with nonzero nuclear magnetic moments, a testable extension beyond the electronic magnetic moment considered here.
  • If the dropped quadrupole-type terms do contribute for excited or non-s states, the method should predict additional angular-momentum-dependent corrections to Eq. (21) that could be checked against relativistic calculations.
  • The blackbody-induced annihilation mechanism offers a qualitative cosmological handle on the matter-antimatter asymmetry, but the authors leave quantitative modeling to future work.
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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

4 major / 5 minor

Summary. The paper claims to derive the long-range magnetic dipole-dipole interaction between two neutral atoms from the one-photon exchange S-matrix in QED. After a multipole expansion of the photon propagator, it obtains Eq. (21), which reproduces the classical spin-spin potential. It then extends the formalism to hyperfine transitions, where the de-excitation of one atom and excitation of the other gives a resonant 1/R interaction, Eqs. (25) and (28). Numerical estimates are given for hydrogen, including a thermal contribution and a sign change for hydrogen-antihydrogen systems. The central claim is that QED provides a first-principles derivation of the classical magnetic interaction and predicts a new hyperfine-mediated 1/R channel.

Significance. If the derivation and numerics were fully correct, the paper would provide a useful QED derivation of the classical magnetic dipole-dipole potential and identify a new long-range channel whose strength depends on the hyperfine splitting. The S-matrix framework is standard and the elastic limit leading to Eq. (21) is plausible. However, the quantitative claims are currently not well-defined: the central hyperfine coefficient differs by a factor of 3/2 between Eqs. (25) and (28), and the numerical section contains unit inconsistencies. These issues must be resolved before the paper's conclusions can be accepted. The work does not include machine-checked proofs or reproducible code, so the analytic derivation and numerical tables carry the evidentiary weight.

major comments (4)
  1. [§III, Eqs. (25) and (28)] The central quantitative result is ambiguous because the same hyperfine interaction is quoted with two different coefficients. Eq. (25) gives ΔE_HFS = −(2/9)Δ_HFS^2 μ0^2 /R e^{iΔ_HFS R} (for the real and imaginary parts), while Eq. (28) gives the same expression with coefficient −1/3. Both are stated to follow from identical angular algebra for s-states. The numerical estimates in Sec. IV use the −1/3 version, but no justification is given for discarding the −2/9 version. This factor of 3/2 directly changes the predicted strength of the new 1/R channel and all derived quantities. The authors should redo the angular algebra, show the intermediate matrix elements of μ_A μ_B and (μ_A R)(μ_B R), and state which coefficient is correct.
  2. [§IV, numerical evaluation] The units are internally inconsistent. The text says 'Taking the Bohr magneton in atomic units, μ0=1/2', but the reported value ΔE_AB ≈ 1.331×10^{-5}/R^3 a.u. corresponds to μ0^2 = α^2/4, i.e., μ0 = α/2, not 1/2. The thermal coefficient −1.85×10^{-10} i/R^2 labelled 'a.u.' is, from the numbers, actually in relativistic units with μ0=√α/2 and β in r.u.; converting to atomic units changes the numerical value by α^{-2}. Consequently, the quoted 87.6 MHz and 2.43×10^4 s^{-1} do not follow from the stated equations with any single consistent unit system. The numerical section should be recalculated in one explicitly chosen unit system, with the Bohr magneton convention stated and used consistently throughout.
  3. [§II, Eq. (20)] The neglect of the 'rem' terms is justified by a quadrupole selection rule (Δl=2). For the diagonal s-state matrix elements actually relevant to Eq. (21), this justification is not the operative reason: in the non-relativistic limit, using ⟨α_i r_j⟩ ∝ δ_ij, the six operators in Eq. (20) sum exactly to zero. The conclusion (21) therefore survives, but the stated argument is incorrect. The derivation should be corrected by showing the actual cancellation (or by presenting the surviving matrix elements) rather than invoking a selection rule that does not apply to the diagonal case.
  4. [§IV, Eq. (27)] The statement that after angular algebra for s-states ΔE_AB = μ0^2/R^3 omits the angular dependence present in Eq. (21). For a spherically symmetric s-state pair, the expectation value of the dipolar interaction depends on the orientation of the electron spins relative to R; averaging over spin projections or over the direction of R generally gives zero, while the value μ0^2/R^3 corresponds to a specific geometry (parallel moments perpendicular to R). The manuscript should specify the quantization axis, spin state, and orientation assumptions used to obtain Eq. (27), and reconcile this with the general tensor form of Eq. (21). This also affects the sign discussion for hydrogen-antihydrogen.
minor comments (5)
  1. [§II, Eq. (12)] The angular average ⟨(r_AB R)^2⟩ = r_AB^2 R^2/3 is used without comment about the fact that this is a spherical average; the paper should state that this is the orientation-averaged result, and whether the final potentials are meant to be orientation-averaged or fixed-geometry.
  2. [§II, Eq. (19)-(20)] The notation 'rem' in Eq. (19) and the subsequent 'rem' terms in Eq. (20) is undefined. A sentence defining the decomposition of the matrix element would improve clarity.
  3. [§III, Eq. (25)] The real part is written as cos(Δ_HFS R) while the imaginary part uses sin(|Δ_HFS|R); the modulus convention is introduced but the real part omits the modulus. This should be made consistent.
  4. [§IV, text after Eq. (28)] The text refers to 'Im ΔE_HFS^β = 2.43×10^4 s^{-1}' immediately after discussing the thermal contribution from Eq. (27), but Eq. (27) is not the hyperfine channel. The notation is confusing and should be clarified.
  5. [§V, Conclusions] The interpretation of the imaginary part as a thermally induced annihilation rate and the speculative cosmological conclusion about the absence of antimatter should be explicitly labeled as qualitative speculation; as written, they are presented as a firm result.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the QED derivation is self-contained; the HFS strength uses the measured hyperfine splitting as a physical input, and the only notable issue (Eq. 25 vs 28 factor 3/2) is an internal consistency defect, not a fitted or self-referential prediction.

full rationale

The derivation chain is not circular. Eq. (21) follows from the one-photon S-matrix element (1) through the multipole expansion (10)-(13), the amplitude (17), and the magnetic-moment reexpression (18)-(19). No parameter in Eq. (21) is fitted to the result; the spin-spin form emerges from standard QED current-matrix elements. The HFS expressions (25),(26),(28) use the physically measured hyperfine splitting Delta_HFS and Bohr magneton mu0 as inputs; they do not fit a parameter to the predicted R-dependent interaction, so the 1/R channel is not a fitted input renamed as a prediction. Ref. [20] (same group) is cited for the coordinate-space propagator (8)-(9) and Ref. [31] (same group) for the quadrupole classification in (20); these are technical intermediate results, not the conclusion, and neither invokes a self-citation uniqueness theorem. The factors 1/R^3 and 1/R arise from the expansion, not from assuming the answer. The discrepancy between Eq. (25) (-2/9) and Eq. (28) (-1/3) is a real internal consistency problem for the quantitative claim, but it is a mathematical error/ambiguity, not a circularity: neither coefficient is constructed by definition from the other. The paper is therefore not circular in any of the enumerated senses; at most there are minor self-citations used as background.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central formulas rest on standard QED input plus two thin assumptions: the truncation/angular averaging of the r12 expansion and the neglect of Eq. (20) rem terms. No new entities are introduced. Δ_HFS is a measured input, so the HFS channel is not a parameter-free prediction.

free parameters (1)
  • Δ_HFS (1s hyperfine splitting of hydrogen) = 2.15878×10^-7 a.u.
    Measured spectroscopic input; the HFS channel coefficient scales as Δ_HFS^2, so the numerical prediction is not parameter-free.
assumptions (4)
  • domain assumption One-photon exchange (1) is the leading S-matrix contribution to the magnetic interaction
    Used throughout; assumes higher-order diagrams and nuclear magnetic moments do not affect the leading result.
  • ad hoc to paper Multipole expansion (10)-(12) truncated at second order, with angular average ⟨(rAB R)^2⟩=r_AB^2 R^2/3, captures the magnetic dipole interaction
    Load-bearing for Eq. (17); anisotropic and higher-order pieces are not explicitly bounded.
  • ad hoc to paper Neglect of 'rem' terms in Eq. (20) because they are quadrupole interactions with selection rule Δl=2
    States the terms can be neglected, but no matrix elements are shown for s states.
  • domain assumption Finite-temperature photon propagator (9) and its ε→0 limit give the factor -2iR/β
    Adopted from Refs. [28,29] and used for Eq. (22).

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Cite this review

Pith. "Pith review of Long-range magnetic interaction within quantum electrodynamics formalism." pith.science (2026). https://pith.science/paper/3ZQO4OUS

@misc{pith2026260722435,
  author       = {Pith},
  title        = {Pith review of: Long-range magnetic interaction within quantum electrodynamics formalism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3ZQO4OUS}},
  note         = {Machine review of arXiv:2607.22435}
}
read the original abstract

Within the framework of quantum electrodynamics, the interaction between two atoms at large distances is analyzed. Using the S-matrix formalism, an expression for the magnetic interaction potential is derived, which agrees with the well-known result of classical electrodynamics. However, quantum electrodynamics goes beyond this conventional result and allows one to treat a wide range of problems related to the structure of atomic energy levels. In particular, it is shown that the asymptotic behavior of the interaction potential can deviate from the classical prediction, depending on the atomic states involved. As an example, dispersion coefficients are calculated for the s-states of hydrogen atoms, where the long-range potential reduces to a spin-spin interaction. The results obtained open up the possibility of a straightforward comparative analysis of long-range interaction potentials between atoms of matter and antimatter. The applicability of this approach is demonstrated for the hydrogen-antihydrogen system.

Figures

Figures reproduced from arXiv: 2607.22435 by the authors.

Figure 1
Figure 1. FIG. 1. Feynman diagrams depicting one-photon exchange [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. A diagram of radius vectors for two one-electron [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

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