Pith. sign in

REVIEW 5 major objections 4 minor 60 references

Multipole decomposition of the thermal one-loop self-energy correction for a bound atomic electron

T0 review · 5 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Expanding the thermal one-loop self-energy of a bound electron in $\alpha Z$ reproduces the quantum-mechanical blackbody Stark and Zeeman shifts and yields new relativistic, quadrupole, and diamagnetic thermal corrections.

desk verdict Solid TQED derivation of BBRZ plus new small thermal corrections, with two delegated steps a referee should ask the authors to put in the paper. read the letter →

arxiv 2507.15422 v1 pith:3YLVWR55 submitted 2025-07-21 physics.atom-ph

J. J. Lopez-Rodriguez , A. Bobylev , P. Kvasov , T. Zalialiutdinov , D. Solovyev This is my paper · ORCID
classification physics.atom-ph
keywords finite-temperatureQEDthermalStarkshiftblackbodyZeeman(BBRZ)one-loopself-energymultipoleexpansiondiamagnetichydrogenatomradiation
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

The paper sets out to show that finite-temperature quantum electrodynamics, applied to a single bound electron, gives the same blackbody-radiation shifts of atomic levels as ordinary quantum-mechanical perturbation theory, and that it also predicts shifts quantum mechanics misses. Starting from the one-loop self-energy correction, in which the electron emits and reabsorbs a photon whose propagator carries the Planck factor, the authors expand in the relativistic parameter $\alpha Z$ (fine-structure constant times nuclear charge). In the dipole limit the real part reproduces the known thermal Stark shift, and the next order yields the thermal blackbody Zeeman shift; the paper states that its Eq. (23) completely coincides with the quantum-mechanics result. The same expansion produces relativistic corrections to the Stark shift, a thermal quadrupole interaction, and a thermal diamagnetic shift, the last one reaching about 19 Hz for the $n=100$, $l=0$ state of hydrogen at 300 K. If the analysis is right, one QED expression is the common source of all these shifts, which matters for precision spectroscopy in atomic clocks.

What carries the argument

The central object is the thermal photon propagator, whose second term carries the Planck distribution $n_\beta(\omega)$, inserted into the one-loop self-energy of the bound electron. The argument is carried by the multipole decomposition of $\sin(\omega r_{12})/(\omega r_{12})$ into spherical Bessel functions and vector spherical harmonics, followed by a Taylor expansion in $\omega r \sim \alpha Z$; successive orders isolate the dipole Stark shift, the magnetic-dipole Zeeman shift, and the quadrupole interaction. A second essential move is splitting the Dirac spectrum into positive- and negative-energy states: the negative continuum, with denominators of order $2mc^2$, is what produces the diamagnetic thermal shift after a state-independent constant is cancelled. Nonrelativistic limits are taken by reducing the Dirac magnetic-moment operator $\boldsymbol\mu = e[\mathbf r\times\boldsymbol\alpha]/2$ to $\mu_B(\mathbf l+2\mathbf s)$ and by using completeness of Schr\"odinger states.

What would settle it

A direct calculation of the full sum over negative-energy Dirac states in Eq. (B4), without discarding the state-independent term, would settle the issue: any residual constant would invalidate Eq. (33). Alternatively, measuring the blackbody-radiation frequency shift of a hydrogen Rydberg state with $n=100$, $l=0$ at 300 K and comparing it with the predicted 19 Hz diamagnetic contribution would test the result experimentally.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the thermal one-loop self-energy of a bound electron contains the entire multipole response of an atom to blackbody radiation. In the nonrelativistic dipole limit its real part is exactly the dynamical Stark shift obtained by second-order perturbation theory, and the next order in $\alpha Z$ is the blackbody Zeeman shift; the paper says Eq. (23) completely coincides with the known quantum-mechanics result. The same expansion also yields a relativistic correction to the Stark shift, a correction from relativistic wave-function components, a thermal quadrupole interaction, and a diamagnetic term that emerges from the negative-energy part of the Dirac spectrum. The paper concludes that finite-temperature QED and quantum-mechanical perturbation theory are consistent, and that the one-loop expression is the more complete starting point because it generates effects the quantum-mechanical approach does not contain.

Load-bearing premise

The derivation assumes that a state-independent constant coming from the negative-energy part of the Dirac spectrum exactly cancels the matching constant in the thermal Stark shift; if the cancellation is incomplete, the predicted diamagnetic thermal shift does not follow.

Editorial extensions

If this is right

  • The thermal Stark shift for hydrogen-like atoms is no longer an input from perturbation theory but a consequence of the one-loop QED expression; the paper's numerical tables agree with the established values.
  • The thermal Zeeman shift for hyperfine and clock transitions follows from the same diagram as the Stark shift, and the formula gives fractional shifts near $-1.3\times 10^{-17}$ at 300 K for hydrogen, caesium, rubidium, and strontium clock transitions.
  • Relativistic corrections to the Stark shift and the wave function enter at the $10^{-6}$ relative level, comparable to dynamical corrections, so precision frequency measurements at the hertz level must include them.
  • The thermal diamagnetic shift scales strongly with principal quantum number, reaching about 19 Hz for $n=100$, $l=0$ at 300 K, making Rydberg states a possible place to observe it.
  • Regularizing the resonant denominators with finite level widths changes BBR-induced line widths substantially for low-lying states at room temperature, and for magnetic-dipole transitions the induced rate can dominate the spontaneous one when the transition lies near the Planck peak.

Reading between the lines

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

  • Beyond the paper, one could apply the same multipole expansion to higher multipoles, such as electric octupole or magnetic quadrupole terms, for transitions where the leading electric-dipole term is forbidden; the paper mentions such terms but does not evaluate them numerically.
  • The load-bearing cancellation in Appendix B is the step a skeptical reader should test: a direct numerical summation over negative-energy states, without discarding the state-independent constant, would verify whether Eq. (33) survives.
  • For optical clock error budgets, the hydrogen results suggest that analogous many-electron calculations are needed, since relativistic and diamagnetic corrections are at the level of the dynamical corrections already included in clock shift models.
  • A Rydberg-state measurement of the BBR shift at $n\sim 100$ would be a clean test: the predicted 19 Hz diamagnetic contribution is large enough to separate from the Stark shift by its distinct scaling with state and temperature.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 4 minor

Summary. The manuscript derives the finite-temperature one-loop self-energy correction for a bound electron, expanding the thermal photon field in multipoles and in powers of αZ. It recovers the nonrelativistic thermal Stark shift and the thermal Zeeman shift, and obtains new relativistic corrections (Eqs. (26) and (31)), a thermal quadrupole interaction (Eq. (25)), and a thermal diamagnetic shift (Eq. (33)). Numerical estimates are given for hydrogen states and for transitions relevant to atomic clocks, with comparisons to existing results where available.

Significance. The paper's strength is the unified TQED derivation: starting from a single self-energy expression, it reproduces known QM results without fitting parameters and yields falsifiable predictions for small relativistic corrections and the diamagnetic shift (e.g., 19.1 Hz for the n=100 Rydberg state at room temperature). The numerical tables for the thermal Stark shift agree well with Farley and Wing, and the BBRZ estimates for clock transitions match previous literature. However, the derivation relies on delegated results from the authors' previous papers, and the new diamagnetic result has an apparent sign inconsistency. If these issues are resolved, the paper would be a useful reference for thermal shift calculations in precision spectroscopy.

major comments (5)
  1. [Appendix B, Eq. (B4); Section II.F] The cancellation of the state-independent negative-continuum contribution is load-bearing but is not derived in this manuscript. The text states that this constant "exactly cancels the analogous state-independent part of the Stark thermal shift" and cites Eq. (141) of [21], adding that "it is sufficient to merely discard all such contributions." This step is essential both for the reduction to Eq. (16) and for isolating the diamagnetic shift Eq. (33). Please provide the explicit derivation of this cancellation or reproduce the relevant result from [21], because an incomplete cancellation would add T^2 and T^4 terms to Eq. (33).
  2. [Section II.F, Eqs. (32)-(33)] There is a sign inconsistency between Eq. (32) and Eq. (33). Eq. (32) gives ΔE(−)_a = −e^2/(3π) ∫ dω nβ(ω) ω^3 ⟨a|r^2|a⟩, and since ∫_0^∞ dω nβ(ω) ω^3 = π^4/(15 β^4) > 0, the resulting shift in Eq. (33) should carry a negative sign unless an additional sign is explained. The text asserts a positive diamagnetic shift, which is the physically expected sign in QM, but the derivation as written yields the opposite. This internal inconsistency must be resolved; as written, the sign of the new diamagnetic result is ambiguous.
  3. [Section II.B, Eq. (16)] The reduction from the relativistic expression (11) to the nonrelativistic Stark shift (16) is the core of the claimed consistency between TQED and QM, but the paper only says "Leaving aside the first three terms... see [14] for details." Please include the derivation or a detailed outline in an appendix so the reader can verify the treatment of the negative continuum and the conversion to length form. Without this, the central consistency claim is asserted rather than demonstrated in this manuscript.
  4. [Section II.C, Eq. (23)] The statement that Eq. (23) "completely coincides with the result known from the quantum mechanics approach" overstates its content: Eq. (23) is the leading contribution from the nearest state a′ within the full sum of Eq. (22), which is the actual QM result. The remaining sum over intermediate states is only discussed later (Table VIII). Please rephrase to avoid implying that Eq. (23) alone represents the full QM result.
  5. [Section II.B, Eq. (11)] The derivation begins with Eq. (11), which is quoted from [14,21] with "omitting intermediate calculations." While it is acceptable to build on prior work, the paper should state the assumptions and domain of validity of Eq. (11), such as the Furry picture, the one-electron nature of the bound state, and the neglect of thermal corrections to the fermion propagator. This is necessary because the subsequent multipole decomposition and the claimed consistency with QM are entirely based on this starting point.
minor comments (4)
  1. [Section IV heading] The section heading "IV . CONSCLUSION" contains a typo; it should read "CONCLUSION."
  2. [Eq. (20)] Eq. (20) contains the term "(α1 α2)(r1 r1)", which appears to be a typo for "(α1 α2)(r1 r2)" based on comparison with Eq. (A4) and the surrounding derivation.
  3. [Table V] Table V uses "0.0" both for exactly zero and for values that are numerically insignificant; please define this notation in the caption or in the text.
  4. [Section II.D] When introducing Γ = Γa + Γa′ for two excited states, the text could clarify whether interference or cross terms are neglected in the regularization procedure.

Circularity Check

1 steps flagged · score 4.0 of 10

The diamagnetic thermal shift is obtained only after canceling a state-independent negative-continuum term by citing Eq. (141) of the authors' earlier paper [21].

  1. self citation load bearing [Appendix B, after Eq. (B4); also referenced near Eq. (16) in Section II B]
    "Thus, the term withαmatrices gives a state-independent contribution. This constant, however, exactly cancels the analogous state-independent part of the Stark thermal shift, when forming a cubic dependence onω(see Eq. (141) in [21]). In the simplest case, it is sufficient to merely discard all such contributions."

    The derivation of the diamagnetic shift, Eq. (33) and Eq. (B5), reduces to keeping only the r^2 term in Eq. (B4) after discarding the ⟨a|α^2|a⟩ term. The paper does not derive this cancellation; it delegates the proof to Eq. (141) of Ref. [21], a prior paper by the same author (D. Solovyev). This cancellation is load-bearing: if it is incomplete, the state-independent T^4 contribution would survive and Eq. (33) would not be the physical thermal shift. Thus the new diamagnetic result is not self-contained within this manuscript, even though the leading-order Stark and Zeeman results are independently benchmarked.

full rationale

The central Stark and Zeeman derivations are not circular: the paper starts from the relativistic thermal one-loop self-energy expression (11), expands in αZ, and reduces to the nonrelativistic QM forms (16) and (22)–(23), then compares numerically against independent QM results from Farley and Wing [2] and standard BBRZ values [8, 12]. No fitted parameters or private empirical inputs are used. The one genuine self-citation issue is in Appendix B: the state-independent negative-continuum term ⟨a|α^2|a⟩ is canceled by invoking Eq. (141) of the authors' own earlier paper [21], with the paper stating that it is 'sufficient to merely discard all such contributions.' That cancellation is exactly what isolates the diamagnetic shift Eq. (33) and also underpins the disappearance of the negative-continuum constant in the leading-order Stark reduction. Because this step is load-bearing for a claimed new correction and is justified only by self-citation rather than by a proof in the present text, the derivation is partially circular. The leading-order consistency claim remains independently supported, so the paper falls at the lower end of the circularity scale.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on several domain assumptions from finite-temperature QED and on two specific self-cited results from [14] and [21]. No free parameters are fitted; all numerical inputs are physical constants or literature values. No invented entities are introduced.

assumptions (5)
  • domain assumption The finite-temperature photon propagator is the sum of the vacuum propagator and a thermal term proportional to the Planck distribution nβ(ω), Eq. (10).
    Standard TQED; the paper uses this propagator without derivation.
  • domain assumption The one-loop thermal self-energy shift is given by Eq. (11) with Iβ defined by Eq. (12), taken from [14,21].
    The paper states 'omitting intermediate calculations for brevity' and relies on these cited derivations.
  • ad hoc to paper The state-independent contribution from the negative Dirac continuum exactly cancels the analogous state-independent part of the thermal Stark shift, citing Eq. (141) of [21].
    This cancellation is asserted, not proved; it is load-bearing for the diamagnetic shift Eq. (33).
  • domain assumption The multipole expansion of sin(ωr12) is truncated at the order shown, with terms of order ω5 and higher discarded; ωr << 1 for room-temperature BBR and hydrogenic states.
    Used throughout the αZ expansion; the paper provides parametric estimates for the discarded terms.
  • domain assumption Including an infinite series of zero-temperature self-energy insertions into the thermal loop resums to a Lorentzian denominator with width Γ = Γa + Γa', Eq. (27).
    Adopted from [14,29,30,33] without re-derivation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Multipole decomposition of the thermal one-loop self-energy correction for a bound atomic electron." pith.science (2026). https://pith.science/paper/3YLVWR55

@misc{pith2026250715422,
  author       = {Pith},
  title        = {Pith review of: Multipole decomposition of the thermal one-loop self-energy correction for a bound atomic electron},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3YLVWR55}},
  note         = {Machine review of arXiv:2507.15422}
}
abstract

In this paper, we present a comprehensive analysis of the one-loop self-energy correction at finite temperature for the bound electron. In this approach, we study the influence of thermal radiation on atomic systems. Along the way, we found well-known effects, including thermal Stark and Zeeman shifts, as well as thermal quadrupole interactions and relativistic corrections to the multipole expansion of photon field operators. We show that the corresponding contributions arise from the decomposition of the fully relativistic expression in terms of the $\alpha Z$ parameter. The presented analysis unambiguously determines the consistency of the quantum electrodynamics theory at finite temperature (TQED) with the perturbation theory of quantum mechanics (QM). Although our analysis mainly focuses on the hydrogen atom model, their potential implications for precision spectroscopic experiments are discussed.

Figures

Figures reproduced from arXiv: 2507.15422 by the authors.

Figure 1
Figure 1. FIG. 1: Thermal one-loop self-energy correction for the energy of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Insertions of ordinary (zero-temperature) self-energy loops [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

60 extracted references · 54 canonical work pages

  1. [21]

    Solovyev, Annals of Physics415, 168128 (2020)

    D. Solovyev, Annals of Physics415, 168128 (2020)

    2020

  2. [14]

    Solovyev, L

    D. Solovyev, L. Labzowsky, and G. Plunien, Phys. Rev. A92, 022508 (2015)

    2015

  3. [1]

    T. F. Gallagher and W. E. Cooke, Phys. Rev. Lett.42, 835 (1979)

    1979

  4. [2]

    J. W. Farley and W. H. Wing, Phys. Rev. A23, 2397 (1981)

    1981

  5. [3]

    T. L. Nicholson, S. L. Campbell, R. B. Hutson,et al., Nature Communications6(2015), 10.1038/ncomms7896

    work page doi:10.1038/ncomms7896 2015

  6. [4]

    Pizzocaro, P

    M. Pizzocaro, P. Thoumany, B. Rauf,et al., Metrologia54, 102 (2017)

    work page 2017

  7. [5]

    Hellwig, R

    H. Hellwig, R. Vessot, M. Levine,et al., IEEE Transactions on Instrumentation and Measurement19, 200 (1970)

    work page 1970

  8. [6]

    van Rooij, J

    R. van Rooij, J. S. Borbely, J. Simonet,et al., Science333, 196–198 (2011)

    work page 2011

Show all 60 references
  1. [7]

    Matveev, C

    A. Matveev, C. Parthey, K. Predehl,et al., Physical Review Let- ters110(2013), 10.1103/PhysRevLett.110.230801

    2013 doi

  2. [8]

    W. M. Itano, L. L. Lewis, and D. J. Wineland, Phys. Rev. A25, 1233 (1982)

    1982

  3. [9]

    Simon, P

    E. Simon, P. Laurent, and A. Clairon, Phys. Rev. A57, 436 (1998)

    1998

  4. [10]

    Beloy, N

    K. Beloy, N. Hinkley, N. Phillips,et al., Phys. Rev. Lett.113, 260801 (2014)

    2014

  5. [11]

    U. I. Safronova, Phys. Rev. A82, 022504 (2010)

    2010

  6. [12]

    J. Han, Y . Zuo, J. Zhang,et al., The European Physical Journal D73, 9 (2019)

    2019

  7. [13]

    Riehle,Frequency Standards: Basics and Applications(Wi- ley, 2006)

    F. Riehle,Frequency Standards: Basics and Applications(Wi- ley, 2006)

    2006

  8. [15]

    Dolan and R

    L. Dolan and R. Jackiw, Phys. Rev. D9, 3320 (1974)

    1974

  9. [16]

    J. F. Donoghue and B. R. Holstein, Phys. Rev. D28, 340 (1983)

    1983

  10. [17]

    J. F. Donoghue, B. R. Holstein, and R. W. Robinett, Annals of Physics164, 233 (1985)

    1985

  11. [18]

    Labzowsky, G

    L. Labzowsky, G. Klimchitskaya, and Y . Dmitriev,Relativistic Effects in the Spectra of Atomic Systems(Institute of Physics Publishing, 1993)

    1993

  12. [19]

    I. I. Sobel’man,Introduction to the Theory of Atomic Spectra (Pergamon, 1972)

    1972

  13. [20]

    A. I. Akhiezer and V . B. Berestetskii,Quantum Electrodynam- ics(Wiley-Interscience, New York, 1965)

    1965

  14. [22]

    I. S. Gradshteyn, I. M. Ryzhik, and A. Jeffrey,Table of inte- grals, series, and products, 5th ed. (Academic Press, San Diego, 1994)

    1994

  15. [23]

    D. A. Varshalovich, A. M. Moskalev, and V . K. Khersonskii, Quantum Theory of Angular Momentum(World Scientific, Sin- gapore, 1989)

    1989

  16. [24]

    G. Łach, M. DeKieviet, and U. D. Jentschura, Physical Review A81(2010), 10.1103/physreva.81.052507

    2010 doi

  17. [25]

    Lindgren,Relativistic Many-Body Theory(Springer New York, 2011)

    I. Lindgren,Relativistic Many-Body Theory(Springer New York, 2011)

    2011

  18. [26]

    S. G. Porsev and A. Derevianko, Phys. Rev. A74, 020502 (2006)

    2006

  19. [27]

    U. D. Jentschura and M. Haas, Phys. Rev. A78, 042504 (2008)

    2008

  20. [28]

    Persson, S

    H. Persson, S. Salomonson, P. Sunnergren,et al., Phys. Rev. A 56, R2499 (1997)

    1997

  21. [29]

    Low, Phys

    F. Low, Phys. Rev.88, 53 (1952)

    1952

  22. [30]

    Andreev, L

    O. Andreev, L. Labzowsky, G. Plunien,et al., Phys. Rep.455, 135 (2008)

    2008

  23. [31]

    G. Łach, M. DeKieviet, and U. D. Jentschura, Phys. Rev. Lett. 108, 043005 (2012)

    2012

  24. [32]

    Tang, Y .-F

    Z.-M. Tang, Y .-F. Wei, B. K. Sahoo,et al., Phys. Rev. A110, 043108 (2024)

    2024

  25. [33]

    Zalialiutdinov, D

    T. Zalialiutdinov, D. Solovyev, L. Labzowsky,et al., Phys. Rep. 737, 1 (2018)

    2018

  26. [34]

    U. D. Jentschura and G. S. Adkins,Quantum Electrodynam- ics: Atoms, Lasers and Gravity(WORLD SCIENTIFIC, 2022) https://www.worldscientific.com/doi/pdf/10.1142/12722

    2022 doi

  27. [35]

    L. D. Landau and E. Lifshitz,Quantum Mechanics: Non- Relativistic Theory, V ol. V olume 3 (Physics and Matematics lit- erature, 2001)

    2001

  28. [36]

    Zalialiutdinov, A

    T. Zalialiutdinov, A. Anikin, and D. Solovyev, Journal of Ex- perimental and Theoretical Physics135, 605 (2022)

    2022

  29. [37]

    Czarnecki, U

    A. Czarnecki, U. D. Jentschura, and K. Pachucki, Phys. Rev. Lett.95, 180404 (2005)

    2005

  30. [38]

    I. P. Grant, Journal of Physics B: Atomic, Molecular and Opti- cal Physics42, 055002 (2009)

    2009

  31. [39]

    Horbatsch and E

    M. Horbatsch and E. A. Hessels, Phys. Rev. A93, 022513 (2016)

    2016

  32. [40]

    Sucher, Reports on Progress in Physics41, 1781 (1978)

    J. Sucher, Reports on Progress in Physics41, 1781 (1978)

    1978

  33. [41]

    Seager, D

    S. Seager, D. Sasselov, and D. Scott, The Astrophysical Journal 523, L1 (1999)

    1999

  34. [42]

    Seager, D

    S. Seager, D. Sasselov, and D. Scott, The Astrophysical Journal Supplement Series128, 407 (2000)

    2000

  35. [43]

    Jentschura, Phys

    U. Jentschura, Phys. Rev. A69, 052118 (2004)

    2004

  36. [44]

    Kolachevsky, M

    N. Kolachevsky, M. Haas, U. Jentschura,et al., Phys. Rev. A 74, 052504 (2006)

    2006

  37. [45]

    Wiese and J

    W. Wiese and J. Fuhr, Journal of Physical and Chemical Refer- ence Data38, 565 (2009). 14

    2009

  38. [46]

    Popov R. V . and Maiorova A. V ., Optics and Spectroscopy122, 366 (2017)

    2017

  39. [47]

    K. W. Martin, B. Stuhl, J. Eugenio,et al., Phys. Rev. A100, 023417 (2019)

    2019

  40. [48]

    Kramida, Yu

    A. Kramida, Yu. Ralchenko, J. Reader, and and NIST ASD Team, NIST Atomic Spectra Database (ver. 5.12), [Online]. Available:https://physics.nist.gov/asd[2025, January 31]. National Institute of Standards and Technology, Gaithersburg, MD. (2024)

    2024

  41. [49]

    M. S. Safronova and U. I. Safronova, Phys. Rev. A83, 012503 (2011)

    2011

  42. [50]

    Zalialiutdinov and D

    T. Zalialiutdinov and D. Solovyev, Phys. Rev. A109, 012816 (2024)

    2024

  43. [51]

    Solovyev, V

    D. Solovyev, V . Sharipov, L. Labzowsky,et al., Journal of Physics B: Atomic, Molecular and Optical Physics43, 074005 (2010)

    2010

  44. [52]

    S. G. Porsev and A. Derevianko, Phys. Rev. A74, 020502(R) (2006)

    2006

  45. [53]

    Beigman and U

    I. Beigman and U. Safronova, Soviet Physics JETP60, 2045 (1971), English translation: JETP33(6), 1102 (1971)

    1971

  46. [54]

    G. W. F. Drake, Phys. Rev. A3, 908 (1971)

    1971

  47. [55]

    A. M. Puchkov and L. N. Labzovskii, Optics and Spectroscopy 106, 153 (2009). Appendix A: Derivation of thermal Zeeman and quadrupole shifts

    2009

  48. [56]

    Derivation of thermal Zeeman shift To obtain the thermal Zeeman shift of the atomic level in the framework of QED theory, one should consider a correction of the next order with respect to the dipole approximation in the one-loop self-energy operator described by the Feynman g...

  49. [57]

    To derive the Zeeman thermal shift, we use only half of the last contribution in the right-hand side of Eq

    and references therein. To derive the Zeeman thermal shift, we use only half of the last contribution in the right-hand side of Eq. (A1). The result can be written as ⟨an|(α1α2)r2 12|na⟩=(A4) −⟨an|([r1 ×α 1]·[r 2 ×α 2])|na⟩+ +⟨an|(α1α2)(r2 1 +r 2 2)|na⟩ −⟨an|(r1α2)(r2α1)−(α 1α...

  50. [58]

    (A4), i.e.⟨an|(r 1α2)(r2α1) + (α1α2)(r1r2)|na⟩

    Derivation of thermal quadripole interaction To construct the quadrupole moment, one should consider the remaining terms arising from the last two summands in the right-hand side of Eq. (A4), i.e.⟨an|(r 1α2)(r2α1) + (α1α2)(r1r2)|na⟩. Let us first transform this matrix element ...

  51. [59]

    The integrals are implied in the sense of the principal value according to Sokhotski-Plemelj theorem

    + ω2 20 (r4 1 +r 4 2 −4(r 1r2)(r2 1 +r 2 2))|na⟩ −(A13) e2 6π X n ∞ˆ 0 dω ω3nβ(ω) ω3 an ω2an −ω 2 (a|r1ir1j|n)(n|r2ir2j|a) + e2 6π X n ∞ˆ 0 dω ω3nβ(ω) 2εan ε2an −ω 2 ⟨an| ω2 5 (r1r2)2 + ω2 10 r2 1r2 2|na⟩ with corresponding notations for relativistic (nonrelativistic) energies...

  52. [60]

    The remaining terms in Eq. (A16) The derivation of the thermal magneto-dipole and electric quadrupole interactions from the one-loop correction to the self-energy of the bound electron left out the terms propor- tional toω 3. The importance of these contributions to the freque...

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.