Second-order stationary solutions for fermions in an external Coulomb field

I.I.Safronov; V.P.Neznamov

arxiv: 1907.03579 · v1 · pith:2DNDPTNOnew · submitted 2019-07-03 · ⚛️ physics.gen-ph · hep-th

Second-order stationary solutions for fermions in an external Coulomb field

V.P.Neznamov , I.I.Safronov This is my paper

Pith reviewed 2026-05-25 09:25 UTC · model grok-4.3

classification ⚛️ physics.gen-ph hep-th

keywords second-order equationsCoulomb fieldDirac equationenergy spectrumimpermeable barrierfermion stationary statespositron confinementsupercritical nuclei

0 comments

The pith

Self-conjugate second-order equations for fermions in a Coulomb field produce the Dirac energy spectrum for attraction and an impermeable barrier for repulsion.

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

The paper develops stationary solutions to second-order self-conjugate equations for spinor wavefunctions of fermions in an external Coulomb field. These equations separate positive-energy and negative-energy states, making a probabilistic interpretation possible. In an attractive Coulomb field the energies match those of the Dirac equation exactly, although the probability densities differ by small amounts. In a repulsive field the equations yield an impermeable potential barrier whose radius depends on the classical electron radius and the fermion energy. The barrier is consistent with existing data on electron structure and scattering, and it suggests possible confinement of positrons inside very heavy nuclei.

Core claim

For the Coulomb field of attraction, the energy spectrum of the second-order equation coincides with the spectrum of the Dirac equation, while the probability densities of states are slightly different. For a Coulomb field of repulsion, there exists an impermeable potential barrier with radius depending on the classical electron radius and on the electron energy. The existence of the impermeable barrier does not contradict the results of experiment for determining the inner electron structure and does not affect (in the lowest order of perturbation theory) the Coulomb electron scattering cross section. The existence of the impermeable barrier can lead to positron confinement in supercritical

What carries the argument

Self-conjugate second-order equations with spinor wavefunctions that allow separation of positive and negative energy stationary states for probabilistic interpretation.

If this is right

The energy spectrum matches the Dirac equation for attractive potentials.
Probability densities differ slightly from those of the Dirac equation.
An impermeable barrier appears in repulsive fields with radius set by classical electron radius and energy.
The barrier leaves the Coulomb scattering cross section unchanged to lowest order in perturbation theory.
Positron confinement becomes possible in nuclei with atomic number 170 or greater.

Where Pith is reading between the lines

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

The barrier could modify predictions for pair production in strong fields beyond the lowest order.
Differences in probability densities might be detectable through precision measurements of atomic transition rates.
The method could be applied to other external fields to obtain alternative descriptions of fermion behavior.

Load-bearing premise

The second-order equations provide the correct self-conjugate description of fermions that permits a probabilistic interpretation via separated positive- and negative-energy stationary states.

What would settle it

Observation of a probability density for the 1S state in hydrogen that deviates significantly from the Dirac prediction, or the detection of a barrier-induced effect in electron scattering at energies where the predicted barrier radius exceeds the classical electron radius.

read the original abstract

We have studied self-conjugate second-order equations with spinor wavefunctions for fermions moving in an external Coulomb field. For stationary states, the equations are characterized by separated states with positive and negative energies, which render probabilistic interpretation possible. For the Coulomb field of attraction, the energy spectrum of the second-order equation coincides with the spectrum of the Dirac equation, while the probability densities of states are slightly different. For a Coulomb field of repulsion, there exists an impermeable potential barrier with radius depending on the classical electron radius and on the electron energy. The existence of the impermeable barrier does not contradict the results of experiment for determining the inner electron structure and does not affect (in the lowest order of perturbation theory) the Coulomb electron scattering cross section. The existence of the impermeable barrier can lead to positron confinement in supercritical nuclei with $Z \geq 170$ in case of realization of spontaneous emission of vacuum electron-positron pairs.

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

The paper offers a second-order wave equation for fermions in Coulomb fields that matches the Dirac spectrum for attraction and introduces an energy-dependent impermeable barrier for repulsion.

read the letter

This paper gives a second-order formulation for fermions in an external Coulomb field. For attractive potentials the energies line up with the Dirac spectrum, though the probability densities shift slightly. For repulsive potentials it adds an impermeable barrier whose radius depends on the classical electron radius and the particle energy. The authors note that this barrier leaves the lowest-order Coulomb scattering cross section unchanged and does not clash with existing data on electron structure. They also suggest it could confine positrons around supercritical nuclei with Z at or above 170 during spontaneous pair creation. The new element is that barrier and the claim that the equations support separated positive- and negative-energy stationary states, which they say allows a probabilistic interpretation. The work is consistent on the attractive side, which is a useful check against the large existing Dirac literature. The separation into positive and negative energies is the move that lets them avoid some usual second-order problems. The soft spots sit in the unshown algebra. Self-conjugacy of the operator and the clean separation property are asserted as features of the equations, but the explicit form and the demonstration are not visible in the abstract. The barrier construction also brings in the classical electron radius as an extra scale, which is not derived from the wave equation itself and may need more justification. This is for readers who follow alternatives to the Dirac equation in strong fields and the modeling of supercritical atoms. A specialist looking for concrete, checkable modifications with possible implications for pair production would find it relevant. The paper shows clear engagement with the Dirac results and makes specific, falsifiable claims, so it deserves a serious referee who can examine the operator and any numerical checks. I would send it to peer review.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces self-conjugate second-order equations with spinor wavefunctions for fermions in an external Coulomb field. Stationary states are claimed to separate into positive- and negative-energy components, permitting a probabilistic interpretation. For attractive Coulomb fields the energy spectrum is asserted to coincide with that of the Dirac equation (while probability densities differ slightly); for repulsive fields an impermeable barrier appears whose radius depends on the classical electron radius and the electron energy. The barrier is stated not to contradict electron-structure experiments or to modify the Coulomb scattering cross section at lowest order in perturbation theory, and is suggested to imply positron confinement inside supercritical nuclei with Z ≥ 170.

Significance. If the second-order formulation is rigorously self-conjugate and the barrier construction is correct, the work would supply an alternative description of fermions in strong fields and a concrete mechanism for positron trapping in supercritical nuclei. The claimed spectral coincidence with the Dirac equation for attraction is a non-trivial point of contact with established theory.

major comments (2)

[Abstract] Abstract (and the paragraph on stationary states): the central claims rest on the assertion that the second-order equations are self-conjugate and permit separation of positive- and negative-energy stationary states, yet neither the explicit operator nor the demonstration of self-conjugacy or the separation property is supplied; without these steps the probabilistic interpretation and the subsequent spectrum and barrier results cannot be verified.
[Abstract] Abstract (repulsive-field paragraph): the impermeable barrier is introduced with a radius that depends on the classical electron radius (a free parameter listed in the axiom ledger) and on electron energy, but no derivation of the barrier, its functional form, or its explicit radius expression is given; this construction is load-bearing for the claims of non-contradiction with experiment and of positron confinement for Z ≥ 170.

minor comments (1)

The abstract states that probability densities are 'slightly different' but supplies neither the quantitative difference nor any comparison table or plot.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major comments point by point below and will revise the manuscript to supply the missing explicit derivations.

read point-by-point responses

Referee: [Abstract] Abstract (and the paragraph on stationary states): the central claims rest on the assertion that the second-order equations are self-conjugate and permit separation of positive- and negative-energy stationary states, yet neither the explicit operator nor the demonstration of self-conjugacy or the separation property is supplied; without these steps the probabilistic interpretation and the subsequent spectrum and barrier results cannot be verified.

Authors: We agree that the explicit second-order operator and the demonstrations of self-conjugacy and separation into positive- and negative-energy stationary states are not supplied in sufficient detail. These elements are required to verify the probabilistic interpretation. In the revised manuscript we will insert the explicit form of the self-conjugate operator together with the step-by-step proof of self-conjugacy and the separation property for stationary solutions. revision: yes
Referee: [Abstract] Abstract (repulsive-field paragraph): the impermeable barrier is introduced with a radius that depends on the classical electron radius (a free parameter listed in the axiom ledger) and on electron energy, but no derivation of the barrier, its functional form, or its explicit radius expression is given; this construction is load-bearing for the claims of non-contradiction with experiment and of positron confinement for Z ≥ 170.

Authors: We acknowledge that the derivation of the barrier, its functional form, and the explicit radius expression (depending on the classical electron radius and energy) are not provided. The revised manuscript will include a concise derivation of the impermeable barrier from the second-order equation in the repulsive case, together with the explicit radius formula, to support the statements on experimental consistency and positron confinement. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central results derived from introduced equations without reduction to inputs.

full rationale

The paper introduces its own second-order self-conjugate equations and derives stationary solutions for the Coulomb problem. The claimed spectrum coincidence with the Dirac equation and the repulsive barrier are presented as consequences of solving those equations, not as inputs or self-citations. No load-bearing step reduces a prediction to a fitted parameter, self-definition, or prior author result by construction. The self-conjugacy and separation properties are stated as features of the chosen equations rather than derived from the target claims.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the postulated form of the self-conjugate second-order operator and the physical identification of its solutions with fermions; the classical electron radius enters the barrier radius as an input scale.

free parameters (1)

classical electron radius
Barrier radius depends on this quantity; treated as an external scale rather than derived.

axioms (1)

domain assumption The second-order equations are self-conjugate and admit a probabilistic interpretation once positive- and negative-energy states are separated.
Invoked in the stationary-states paragraph of the abstract as the justification for the whole approach.

pith-pipeline@v0.9.0 · 5687 in / 1281 out tokens · 37117 ms · 2026-05-25T09:25:40.045270+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 4 internal anchors

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work page internal anchor Pith review Pith/arXiv arXiv 2018
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V. P. Neznamov, I. I. Safronov , and V. Е. Shemarulin. J. Exp. Theor. Phys. 128, 64 (2019), arxiv:1904.05791

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[1] [1]

P. A. M.Dirac. The Principles of Quantum Mechanics (Clarendon, Oxford, 1958)

work page 1958

[2] [2]

V. P. Neznamov. Theor. Math. Phys. 197, 1823 (2018)

work page 2018

[3] [3]

V. P. Neznamov and I. I. Safronov . J. Exp. Theor. Phys. 127, 647 (2018), arxiv:1809.08940

work page internal anchor Pith review Pith/arXiv arXiv 2018

[4] [4]

V. P. Neznamov, I. I. Safronov, and V. Е. Shemarulin. J. Exp. Theor. Phys. 127, 684 (2018), arxiv:1810.01960

work page internal anchor Pith review Pith/arXiv arXiv 2018

[5] [5]

V. P. Neznamov, I. I. Safronov , and V. Е. Shemarulin. J. Exp. Theor. Phys. 128, 64 (2019), arxiv:1904.05791

work page internal anchor Pith review Pith/arXiv arXiv 2019

[6] [6]

L. L. Foldy and S. A. Wouthuysen. Phys. Rev. 78, 29 (1950)

work page 1950

[7] [7]

J. D. Bjorken and S. D. Drell , Relativistic Quantum Mechanics (McGraw-Hill College, New York, 1965)

work page 1965

[8] [8]

I. Ya. Pomeranchuk and Ya. A. Smorodinsky. J. Phys. USSR 9, 97 (1945)

work page 1945

[9] [9]

Pieper and W

W. Pieper and W. Griener, Zs. Phys. 218, 327 (1969)

work page 1969

[10] [10]

Ya. B. Zeldovich and V. S. Popov, Sov. Phys. Usp. 14, 673 (1972)

work page 1972

[11] [11]

A. S. Davydov, Quantum Mechanics (Fizmatlit, Moscow, 1973; Pergamon, Oxford, 1965)

work page 1973

[12] [12]

K. M. Case, Phys. Rev. 95, 1323 (1954)

work page 1954

[13] [13]

V. P. Neznamov and A. J. Silenko. J. Math. Phys. 50, 122302 (2009); arxiv: 0906.2069 (math-ph)

work page internal anchor Pith review Pith/arXiv arXiv 2009

[14] [14]

V. P. Neznamov, Part. Nucl. 37, 86 (2006)

work page 2006

[15] [15]

V. P. Neznamov, Part. Nucl. 43, 15 (2012). 23

work page 2012

[16] [16]

V. B. Berestetskii, Е. М. Lifshitz, and L. P. Pitaevskii, Course of Theoretical Physics, Vol. 4: Quantum Electrodynamics (Fizmatlit, Moscow , 2006, Pergamon, Oxford 1982)

work page 2006

[17] [17]

L. D. Landau and Е. М. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: Non-Relativistic Theory (Nauka, Moscow, 1963; Pergamon, New York, 1977)

work page 1963

[18] [18]

B. L. Voronov, D. M. Gitman , and I. V. Tyutin , Theor. Math. Phys. 150, 34 (2007)

work page 2007

[19] [19]

D. M. Gitman, I. V. Tyutin , and B. L. Voronov , Self-adjoint E xtensions in Quantum Mechanics (Springer Sience, New York, 2012)

work page 2012

[20] [20]

Pruefer, Math

H. Pruefer, Math. Ann. 95, 499 (1926)

work page 1926

[21] [21]

Ulehla and M

I. Ulehla and M. Havlíček, Appl. Math. 25, 358 (1980)

work page 1980

[22] [22]

Ulehla, M

I. Ulehla, M. Havlíček and J. Hořejší, Phys. Lett. A 82, 64 (1981)

work page 1981

[23] [23]

Ulehla, Preprint RL-82-095 (Rutherford Laboratory 1982)

I. Ulehla, Preprint RL-82-095 (Rutherford Laboratory 1982)

work page 1982

[24] [24]

H. A. Bethe and E. E. S alpeter. Quantum Mechanics of One- and Two-Electron Atoms (Springer, Berlin, 1957)

work page 1957

[25] [25]

Dittrich and P.Exner, J

J. Dittrich and P.Exner, J. Math. Phys.26, 2000 (1985)

work page 2000

[26] [26]

Hanneke, T

G .Gabrielse, D. Hanneke, T. Kinoshita, M. Noi , and B. Odom, Phys. Rev. Lett. 97, 030802 (2006)

work page 2006

[27] [27]

V. P. Neznamov, I. I. Safronov, and V. Е. Shemarulin, Vopr. At. Nauki Tekh., Ser.: Theor. Prikl. Fiz., No. 1, 63 (2018)

work page 2018