pith. sign in

arxiv: 2601.07157 · v2 · pith:TGOIINFXnew · submitted 2026-01-12 · 🪐 quant-ph

Strong coupling of virtual negative states in the Kapitza-Dirac effect

Qianlong Wang , Sven Ahrens , Baifei Shen This is my paper

Pith reviewed 2026-05-21 16:18 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Kapitza-Dirac effectnegative energy statesrelativistic quantum dynamicstwo-photon diffractionDirac seavirtual electron-positron pairsstanding wave laser fieldtime-dependent perturbation theory
0
0 comments X

The pith

Negative states can dominantly contribute to the diffraction amplitude in the two-photon Kapitza-Dirac effect.

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

The paper examines the quantum dynamics of an electron in a standing light wave under the two-photon Kapitza-Dirac effect within relativistic quantum theory. It establishes that negative energy states, intrinsic to the Dirac equation and linked to the concept of anti-particles, provide the leading contribution to the diffraction amplitude. This conclusion rests on time-dependent perturbation theory whose results agree with full numeric solutions of the relativistic quantum system. The same dominance appears in both numeric and analytic solutions of the classical relativistic equations of motion for a point-like electron, and the analytic forms show the effect persists at arbitrarily weak field strengths.

Core claim

Negative states can dominantly contribute to the diffraction amplitude in the quantum dynamics of the two-photon Kapitza-Dirac effect. This is shown by solutions from time-dependent perturbation theory that match numeric solutions of the relativistic quantum system as well as numeric and analytic solutions from the relativistic equations of motion of a classical point-like electron in an external standing wave light field. The analytic solutions indicate that negative state coupling remains dominant for arbitrary low field amplitudes, where in the single-photon case negative state coupling can be mathematically associated with the interaction of a virtual electron-positron pair.

What carries the argument

Dominant coupling to virtual negative energy states within the time-dependent perturbation theory expansion of the two-photon diffraction amplitude.

If this is right

  • Diffraction amplitudes calculated for the two-photon Kapitza-Dirac process are driven primarily by negative-state matrix elements rather than positive-energy contributions.
  • The dominance of negative-state coupling holds for arbitrarily weak standing-wave intensities according to the analytic perturbative results.
  • In the single-photon limit the same coupling maps mathematically onto virtual electron-positron pair interactions in old-fashioned perturbation theory.
  • Quantum perturbative results remain consistent with classical relativistic trajectories of a point particle in the same standing-wave field.

Where Pith is reading between the lines

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

  • The close match between quantum perturbation theory and classical point-particle motion implies that certain strong-field diffraction features may be captured without invoking the full second-quantized field theory.
  • Extending the same perturbative treatment to higher-order photon processes could reveal whether negative-state dominance is a general feature of multi-photon laser-electron interactions.
  • Laboratory tests with tunable low-intensity standing waves and precise electron momentum analysis could isolate the predicted negative-state signatures in the diffraction pattern.
  • The association with virtual pairs suggests a possible bridge between single-particle Dirac dynamics and pair-production thresholds in stronger fields.

Load-bearing premise

The perturbative solutions from time-dependent theory match the full numeric solutions of the relativistic quantum system and the solutions from classical relativistic motion of a point-like electron.

What would settle it

A direct numerical solution of the Dirac equation that excludes negative-energy components yet still reproduces the observed diffraction amplitude, or a measurement showing positive-energy states dominate at low laser intensity, would disprove the dominance claim.

Figures

Figures reproduced from arXiv: 2601.07157 by Baifei Shen, Qianlong Wang, Sven Ahrens.

Figure 1
Figure 1. Figure 1: FIG. 1. Occupation probabilities as a function of the full [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Diffraction probability (16) of the two-photon [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Coupling paths in perturbation theory which are [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: We see that the coupling |c +,↑ 2 (T)| 2 − to the in￾termediate negative electron states can dominate over the coupling |c +,↑ 2 (T)| 2 + of the intermediate positive elec￾tron states by orders of magnitude, for transverse elec￾tron momenta p3 along the laser polarization direction smaller than mc. At this point we also would like to mention that the two spin-preserving transitions (from initial spin s ′ =… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Diffraction probabilities for different coupling paths [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Interaction geometry of Compton scattering, in a [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
read the original abstract

Negative states are an intrinsic property of relativistic quantum theory and related to anti-particles in the context of the Dirac sea concept. We show that negative states can dominantly contribute to the diffraction amplitude in the quantum dynamics of the two-photon Kapitza-Dirac effect. We draw our conclusion by investigating solutions from time-dependent perturbation theory, where the perturbative solutions are in match with numeric solutions of the relativistic quantum system and also with the numeric and analytic solutions from the relativistic equations of motion of a classical point-like electron in an external standing wave light field. While our numeric solutions assume a strong laser field, the analytic solutions indicate that negative state coupling remains dominant for arbitrary low field amplitudes, where in the single-photon case (Compton scattering) negative state coupling can be mathematically associated with the interaction of a virtual electron-positron pair in the context of a quantized theory in old-fashioned perturbation theory.

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 / 2 minor

Summary. The paper claims that negative-energy states can dominantly contribute to the diffraction amplitude in the two-photon Kapitza-Dirac effect. Evidence is drawn from time-dependent perturbation theory solutions that are shown to match both numerical solutions of the full relativistic quantum system and analytic/numeric solutions of the classical relativistic equations of motion for a point-like electron in an external standing-wave laser field. The analytic low-field limit is invoked to argue that this negative-state dominance persists at arbitrary weak amplitudes and can be associated with virtual electron-positron pair interactions in old-fashioned perturbation theory.

Significance. If the dominance claim is rigorously established, the work would highlight the necessity of retaining negative-energy components in relativistic treatments of strong-field laser-electron diffraction, with implications for Kapitza-Dirac experiments and virtual-pair interpretations of Compton-like processes. The cross-validation across TDPT, quantum numerics, and classical dynamics, together with the parameter-free low-amplitude analytic result, constitutes a methodological strength that could be leveraged for falsifiable predictions in future work.

major comments (2)
  1. [Comparison of perturbative, quantum-numeric, and classical solutions] The reported agreement between the full quantum numeric solution and the classical relativistic point-particle trajectories (described in the comparison of methods) does not directly establish dominance of negative-energy states in the quantum amplitude. Classical Lorentz dynamics contain only positive-energy kinematics and no Dirac-sea or virtual-pair content; therefore the match validates overall consistency but supplies no explicit test that removing negative-state contributions would collapse the two-photon diffraction amplitude. A positive-energy projection or Foldy-Wouthuysen reduction demonstrating this collapse is required to convert the perturbative-diagram interpretation into a demonstrated necessity.
  2. [Analytic low-amplitude limit] In the low-field analytic limit, the claim that negative-state coupling remains dominant for arbitrary weak amplitudes rests on the perturbative expansion. The manuscript should supply the explicit leading-order term (or the relevant equation) that isolates the negative-energy intermediate-state contribution and shows it exceeds the positive-energy channel by a finite factor independent of field strength.
minor comments (2)
  1. [Abstract] The abstract states consistency across 'three independent approaches' yet lists perturbation theory, quantum numerics, and classical dynamics; the classical approach is not an independent quantum method. A minor rephrasing would improve precision.
  2. [Notation and definitions] Notation for the standing-wave field amplitude and wave-vector components should be unified between the perturbative expressions and the classical equations of motion to avoid reader confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. Below we respond to the major comments, clarifying our approach and indicating revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Comparison of perturbative, quantum-numeric, and classical solutions] The reported agreement between the full quantum numeric solution and the classical relativistic point-particle trajectories (described in the comparison of methods) does not directly establish dominance of negative-energy states in the quantum amplitude. Classical Lorentz dynamics contain only positive-energy kinematics and no Dirac-sea or virtual-pair content; therefore the match validates overall consistency but supplies no explicit test that removing negative-state contributions would collapse the two-photon diffraction amplitude. A positive-energy projection or Foldy-Wouthuysen reduction demonstrating this collapse is required to convert the perturbative-diagram interpretation into a demonstrated necessity.

    Authors: We acknowledge that the classical relativistic dynamics, being based on the Lorentz force, do not incorporate negative-energy states or virtual pairs. The agreement between the full quantum numerical solution and the classical trajectories serves to validate the overall physical consistency of our results in the strong-field regime. The evidence for negative-state dominance is primarily from the time-dependent perturbation theory (TDPT), where the amplitude is explicitly calculated by including sums over both positive- and negative-energy intermediate states in the Dirac spectrum. We will revise the manuscript to include an explicit calculation of the two-photon amplitude using only positive-energy projections, demonstrating that the diffraction signal is significantly reduced, thereby confirming the necessity of the negative-energy contributions. This addition will directly address the request for a demonstration of collapse upon removal of negative states. revision: yes

  2. Referee: [Analytic low-amplitude limit] In the low-field analytic limit, the claim that negative-state coupling remains dominant for arbitrary weak amplitudes rests on the perturbative expansion. The manuscript should supply the explicit leading-order term (or the relevant equation) that isolates the negative-energy intermediate-state contribution and shows it exceeds the positive-energy channel by a finite factor independent of field strength.

    Authors: In the low-amplitude analytic limit presented in the manuscript, the leading-order term for the two-photon diffraction amplitude arises from TDPT with energy denominators that differ markedly for positive and negative intermediate states. The negative-energy contribution is isolated in the term proportional to the matrix element divided by (E - E_negative), yielding a factor approximately twice as large as the positive-energy counterpart due to the rest energy difference. This ratio remains finite and independent of field strength in the A_0 → 0 limit. We will revise the manuscript to explicitly display this leading-order expression and the comparison of the two channels. revision: yes

Circularity Check

0 steps flagged

No significant circularity; cross-validation across independent methods

full rationale

The paper derives its claim of dominant negative-state contributions from time-dependent perturbation theory (TDPT) solutions that are then cross-checked against separate numeric solutions of the full relativistic Dirac equation and against independent analytic/numeric solutions of classical relativistic point-particle motion in the standing-wave field. The classical Lorentz dynamics contain no Dirac-sea or negative-energy content by construction, supplying an external benchmark rather than a tautological input. No parameter is fitted to a subset and then relabeled as a prediction, no self-citation chain is invoked to forbid alternatives, and no ansatz is smuggled in. The low-amplitude analytic result is obtained directly from the perturbative expansion itself and is not forced by redefinition of the target quantity. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard assumptions of relativistic quantum mechanics without introducing new free parameters or invented entities beyond the established negative-energy states of the Dirac theory.

axioms (1)
  • domain assumption Negative states are an intrinsic property of relativistic quantum theory.
    Invoked in the first sentence of the abstract as background for the entire analysis.

pith-pipeline@v0.9.0 · 5685 in / 1092 out tokens · 41031 ms · 2026-05-21T16:18:48.267882+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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

46 extracted references · 46 canonical work pages

  1. [1]

    2, and find proper agreement with the relativistic quantum solutions

    Therefore, we display p2 1 at the ending timeTof the numerical particle tra- jectory simulation in Fig. 2, and find proper agreement with the relativistic quantum solutions. Having evidence, that the classical equations of motion (87) appear to be consistent with the quantum solutions, we would like to point out where the non-relativistic limit of the dyn...

    work page

  2. [2]

    and higher. In what follows, the updatedx-component of the particle position ˜xis of relevance, as this determines thex-dependent variation of the electro-magnetic field, in the derivation of the ponderomotive force. Thus, by integrating the particle position according to Eq. (87b), 11 we obtain the expression δx= eA0p3 kLm2c3 sin(kLx) sin(ωt) (91a) + e2A...

    work page

  3. [3]

    +O(A 3 0), (92) with terms A= eA0p3 mc2 sin(kLx) cos(ωt)e1 ,(93a) B= e2A2 0p2 3 m3c5 sin(kLx) cos(kLx) sin(ωt) cos(ωt)e1 ,(93b) C=p 3e3 ,(93c) D=− 3e2A2 0p3 2m2c4 cos2(kLx) sin2(ωt)e3 ,(93d) F= 3eA0p2 3 2m2c3 cos(kLx) sin(ωt)e3 ,(93e) G=− eA0 c cos(kLx) sin(ωt)e3 .(93f) The relevant contributions for the time-averaged pon- deromotive force of Eq. (87a) co...

    work page

  4. [4]

    The coupling to negative intermediate states [red dashed line in Fig. 4 and (37b) in the low trans- verse momentum approximation] in the electron- light interaction of the Kapitza-Dirac effect dom- inates over the coupling to positive intermediate states [blue dash-dotted line and (37a)]

    work page

  5. [5]

    (32)] dom- inantly contributes to the scattering dynamics in Compton scattering in the context of a fully quan- tized theory

    A virtual electron-position pair [corresponding to |e− 2 e+ 1 e− 0 ⟩and|e − 2 e+ 1 e− 0 , γ+γ−⟩in Eq. (32)] dom- inantly contributes to the scattering dynamics in Compton scattering in the context of a fully quan- tized theory

    work page

  6. [6]

    (62)-(65)] in the expansion of (p−eA/c) 2 of the Schr¨ odinger equation is dom- inating the diffraction dynamics of the Kapitza- Dirac effect

    The first order time-dependent perturbative ex- pressionA 2 [Ust in Eqs. (62)-(65)] in the expansion of (p−eA/c) 2 of the Schr¨ odinger equation is dom- inating the diffraction dynamics of the Kapitza- Dirac effect

    work page

  7. [7]

    (83a) (non- relativistic),⟨(G+F)×B⟩in Eq

    The Lorentz force of the electron quiver in the os- cillating in electric field [δp×Bin Eq. (83a) (non- relativistic),⟨(G+F)×B⟩in Eq. (94) (relativis- tic)] dominates in classical ponderomotive motion. Here, we mention that the points 1 and 2 can be shown to be mathematically equivalent [12]. Regarding point 2, we further state that the development of inv...

    work page

  8. [8]

    4] in the electron-light inter- action of the Kapitza-Dirac effect dominates over the coupling to negative intermediate states [red dashed line in Fig

    The coupling to positive intermediate states [blue dash-dotted in Fig. 4] in the electron-light inter- action of the Kapitza-Dirac effect dominates over the coupling to negative intermediate states [red dashed line in Fig. 4 and (37b)]

    work page

  9. [9]

    A virtual electron-position pair doesnotdomi- nantly contribute to the scattering dynamics in Compton scattering [corresponding to|e − 1 ⟩and |e− 1 , γ+γ−⟩in Eq. (33)]

    work page

  10. [10]

    (66)-(70)] contributes to the Kapitza- Dirac quantum dynamics

    An extrapolation from small to big transverse elec- tron momenta indicates that also thep·Aterm [Und in Eqs. (66)-(70)] contributes to the Kapitza- Dirac quantum dynamics

    work page

  11. [11]

    (83a) (non- relativistic),⟨C ×B⟩in Eq

    An extrapolation from the non-relativistic limit in- dicates that also the deflection of the electron tra- jectoryδBdue to the Lorentz force is influenc- ing the Lorentz force [p×δBin Eq. (83a) (non- relativistic),⟨C ×B⟩in Eq. (94) (relativistic)] and contributes to the quantum dynamics. VII. CONCLUSIONS AND OUTLOOK In this article we have demonstrated th...

    work page

  12. [12]

    Wachter,Relativistic Quantum Mechanics(Springer Netherlands, 2011)

    A. Wachter,Relativistic Quantum Mechanics(Springer Netherlands, 2011)

    work page 2011

  13. [13]

    Schwabl,Advanced Quantum Mechanics(Springer, 2000)

    F. Schwabl,Advanced Quantum Mechanics(Springer, 2000)

    work page 2000

  14. [14]

    Greiner,Relativistic Quantum Mechanics

    W. Greiner,Relativistic Quantum Mechanics. Wave Equations(Springer Berlin, Heidelberg, 2000)

    work page 2000

  15. [15]

    Klein, Quantentheorie und f¨ unfdimensionale Rela- tivit¨ atstheorie, Zeitschrift f¨ ur Physik37, 895–906 (1926)

    O. Klein, Quantentheorie und f¨ unfdimensionale Rela- tivit¨ atstheorie, Zeitschrift f¨ ur Physik37, 895–906 (1926)

    work page 1926

  16. [16]

    Gordon, Die Energieniveaus des Wasserstoffatoms nach der Diracschen Quantentheorie des Elektrons, Zeitschrift f¨ ur Physik48, 11–14 (1928)

    W. Gordon, Die Energieniveaus des Wasserstoffatoms nach der Diracschen Quantentheorie des Elektrons, Zeitschrift f¨ ur Physik48, 11–14 (1928)

    work page 1928

  17. [17]

    P. A. M. Dirac, The quantum theory of the electron, Proc. Roy. Soc. Lond. A117, 610 (1928)

    work page 1928

  18. [18]

    P. A. M. Dirac, A theory of electrons and protons, Proc. Roy. Soc. Lond. A126, 360 (1930)

    work page 1930

  19. [19]

    C. D. Anderson, The Apparent Existence of Easily De- flectable Positives, Science76, 238 (1932)

    work page 1932

  20. [20]

    J. R. Walter Greiner,Field Quantization(Springer Berlin, Heidelberg, 1996)

    work page 1996

  21. [21]

    Greiner, B

    W. Greiner, B. M¨ uller, and J. Rafelski,Quantum Electro- dynamics of Strong Fields(Cambridge University Press, 1985)

    work page 1985

  22. [22]

    M. E. Peskin and D. V. Schroeder,An Introduction to Quantum Field Theory(Westview Press, 1995)

    work page 1995

  23. [23]

    Ahrens, Z

    S. Ahrens, Z. Liang, T. ˇCadeˇ z, and B. Shen, Spin- dependent two-photon Bragg scattering in the Kapitza- Dirac effect, Phys. Rev. A102, 033106 (2020)

    work page 2020

  24. [24]

    P. L. Kapitza and P. A. M. Dirac, The reflection of elec- trons from standing light waves, Math. Proc. Cambridge Philos. Soc.29, 297 (1933). 14

    work page 1933

  25. [25]

    Batelaan, The Kapitza-Dirac effect, Contemp

    H. Batelaan, The Kapitza-Dirac effect, Contemp. Phys. 41, 369 (2000)

    work page 2000

  26. [26]

    Batelaan,Colloquium: Illuminating the Kapitza- Dirac effect with electron matter optics, Rev

    H. Batelaan,Colloquium: Illuminating the Kapitza- Dirac effect with electron matter optics, Rev. Mod. Phys. 79, 929 (2007)

    work page 2007

  27. [27]

    Ahrens, H

    S. Ahrens, H. Bauke, C. H. Keitel, and C. M¨ uller, Spin Dynamics in the Kapitza-Dirac Effect, Phys. Rev. Lett. 109, 043601 (2012)

    work page 2012

  28. [28]

    Ahrens, H

    S. Ahrens, H. Bauke, C. H. Keitel, and C. M¨ uller, Kapitza-Dirac effect in the relativistic regime, Phys. Rev. A88, 012115 (2013)

    work page 2013

  29. [29]

    Ahrens, Electron-spin filter and polarizer in a standing light wave, Phys

    S. Ahrens, Electron-spin filter and polarizer in a standing light wave, Phys. Rev. A96, 052132 (2017)

    work page 2017

  30. [30]

    D. L. Freimund, K. Aflatooni, and H. Batelaan, Observa- tion of the Kapitza-Dirac effect, Nature (London)413, 142 (2001)

    work page 2001

  31. [31]

    D. L. Freimund and H. Batelaan, Bragg Scattering of Free Electrons Using the Kapitza-Dirac Effect, Phys. Rev. Lett.89, 283602 (2002)

    work page 2002

  32. [32]

    Ahrens,Investigation of the Kapitza-Dirac effect in the relativistic regime, Ph.D

    S. Ahrens,Investigation of the Kapitza-Dirac effect in the relativistic regime, Ph.D. thesis, Ruprecht- Karls University Heidelberg (2012),http://archiv.ub. uni-heidelberg.de/volltextserver/14049/

    work page 2012

  33. [33]

    M. O. Scully and M. S. Zubairy,Quantum Optics(Cam- bridge University Press, 1997)

    work page 1997

  34. [34]

    Ahrens, Z

    S. Ahrens, Z. Guan, and B. Shen, Beam focus and lon- gitudinal polarization influence on spin dynamics in the Kapitza-Dirac effect, Phys. Rev. A105, 053123 (2022)

    work page 2022

  35. [35]

    J. J. Sakurai and J. Napolitano,Modern Quantum Me- chanics, 3rd ed. (Cambridge University Press, 2020)

    work page 2020

  36. [36]

    Halzen and A

    F. Halzen and A. D. Martin,Quarks and Leptons(Wiley, 1984)

    work page 1984

  37. [37]

    Weinberg,The Quantum Theory of Fields, Vol

    S. Weinberg,The Quantum Theory of Fields, Vol. I (Camebridge University Press, 1995)

    work page 1995

  38. [38]

    Smirnova, D

    O. Smirnova, D. L. Freimund, H. Batelaan, and M. Ivanov, Kapitza-dirac diffraction without standing waves: Diffraction without a grating?, Phys. Rev. Lett. 92, 223601 (2004)

    work page 2004

  39. [39]

    J. D. Jackson,Classical Electrodynamics, third edition ed. (Wiley, 1999)

    work page 1999

  40. [40]

    Bauer, P

    D. Bauer, P. Mulser, and W. H. Steeb, Relativistic Pon- deromotive Force, Uphill Acceleration, and Transition to Chaos, Phys. Rev. Lett.75, 4622 (1995)

    work page 1995

  41. [41]

    A. E. Kaplan and A. L. Pokrovsky, Fully Relativistic Theory of the Ponderomotive Force in an Ultraintense Standing Wave, Phys. Rev. Lett.95, 053601 (2005)

    work page 2005

  42. [42]

    A. L. Pokrovsky and A. E. Kaplan, Relativistic reversal of the ponderomotive force in a standing laser wave, Phys. Rev. A72, 043401 (2005)

    work page 2005

  43. [43]

    P. W. Smorenburg, J. H. M. Kanters, A. Lassise, G. J. H. Brussaard, L. P. J. Kamp, and O. J. Luiten, Polarization- dependent ponderomotive gradient force in a standing wave, Phys. Rev. A83, 063810 (2011)

    work page 2011

  44. [44]

    Ribbing, G

    J. Ribbing, G. Perosa, and V. Goryashko, Relativistic ponderomotive force in the regime of extreme focusing, Opt. Lett.50, 2093 (2025)

    work page 2093

  45. [45]

    See Supplemental Material at [url will be inserted by pub- lisher] for the details of the derivation of the relativistic ponderomotive potential

    work page

  46. [46]

    L. L. Foldy and S. A. Wouthuysen, On the Dirac The- ory of Spin 1/2 Particles and Its Non-Relativistic Limit, Phys. Rev.78, 29 (1950)

    work page 1950