Fractional Angular Momenta in Electron Beams and Hydrogen-Like Atoms

Irismar G. da Paz; Robert Ducharme

arxiv: 2508.17395 · v3 · submitted 2025-08-24 · 🪐 quant-ph

Fractional Angular Momenta in Electron Beams and Hydrogen-Like Atoms

Robert Ducharme , Irismar G. da Paz This is my paper

Pith reviewed 2026-05-18 21:24 UTC · model grok-4.3

classification 🪐 quant-ph

keywords fractional angular momentumDirac equationelectron beamshydrogen-like atomsGouy phasewave-particle dualityKlein-Gordon equation

0 comments

The pith

Factorizing the Klein-Gordon equation with Dirac matrices mixes angular momentum states to produce fractional angular momenta in both electron beams and hydrogen-like atoms.

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

The paper extends an earlier result on relativistic Gaussian electron beams to solutions of the Dirac equation for hydrogen-like atoms. It shows that the factorization method not only adds spin but also mixes angular momentum states such that eigenstates decompose into opposite-spin terms, each carrying fractional contributions from both spin angular momentum and orbital angular momentum operators. This leads to observable effects including a fractional Gouy phase in beams and fractional wave-particle duality in atoms and beams. A sympathetic reader would care because the mixing appears as a built-in feature of the Dirac approach rather than an extra assumption, suggesting a more unified picture of relativistic angular momentum in both free particles and bound systems.

Core claim

Factorization of the Klein-Gordon equation using Dirac matrices does more than introduce spin; it also produces a specific mixing of the angular momentum states leading to the presence of fractional angular momenta. Eigenstates of the Dirac equation can be decomposed into two terms of opposite spin, each being eigenstates of both the SAM and OAM operators. The same method used for beams is applied to hydrogen-like atoms, yielding fractional angular momenta together with related effects such as fractional Gouy phase in beams and fractional wave-particle duality in both atoms and beams.

What carries the argument

Decomposition of Dirac equation eigenstates into two terms of opposite spin, each an eigenstate of both the spin angular momentum and orbital angular momentum operators. This decomposition extracts the fractional contributions to total angular momentum.

If this is right

Fractional angular momenta appear in the Dirac solutions for hydrogen-like atoms as well as for electron beams.
A fractional Gouy phase emerges in diverging relativistic electron beams.
Fractional wave-particle duality is present in the dynamics of both atoms and beams.
The angular momentum mixing is a general consequence of the Dirac factorization of the Klein-Gordon equation.

Where Pith is reading between the lines

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

The same decomposition technique might be tested in other bound-state systems such as positronium or muonic atoms to check for consistent fractional splitting.
Interference experiments with relativistic electron beams could search for the predicted fractional phase shift as a function of beam divergence.
If confirmed, the result would link the introduction of spin to a broader restructuring of angular momentum operators in relativistic quantum mechanics.

Load-bearing premise

Eigenstates of the Dirac equation can be decomposed into two terms of opposite spin, each being eigenstates of both the spin angular momentum and orbital angular momentum operators.

What would settle it

A direct calculation of the angular momentum expectation values in the Dirac solutions for the hydrogen atom that yields strictly integer values for both spin and orbital parts, with no fractional splitting, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2508.17395 by Irismar G. da Paz, Robert Ducharme.

**Figure 1.** Figure 1: FIG. 1: Twisted wavefront for one bi-spinor component in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

read the original abstract

In an earlier letter [Ducharme \textit{et al.} Phys. Rev. Lett. \textbf{126}, 134803 (2021)], a solution to the Dirac equation for a relativistic Gaussian electron beam showed that for a diverging beam the spin of each electron is the sum of fractional contributions from both the spin angular momentum (SAM) and orbital angular momentum (OAM) operators. Fractional angular momenta emerge when eigenstates of the Dirac equation can be decomposed into two terms of opposite spin. Each of these terms being eigenstates of both the SAM and OAM operators. Building on this understanding, the same method used to calculate fractional angular momenta in beams is applied here to solutions of the Dirac equation for hydrogen-like atoms. The results strengthen the idea that factorization of the Klein-Gordon equation using Dirac matrices equation does more than introduce spin, it also produces a specific mixing of the angular momentum states leading to the presence of fractional angular momenta and related effects such as fractional Gouy phase in the case of beams and fractional wave-particle duality in both atoms and beams.

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

This extends the 2021 beam decomposition to hydrogen-like atoms but the key step of splitting Dirac solutions into opposite-spin terms each eigen to both SAM and OAM looks questionable for bound states.

read the letter

The main takeaway is that the authors apply their earlier factorization of the Klein-Gordon equation via Dirac matrices to the atomic case, claiming this produces fractional angular momenta plus a fractional wave-particle duality effect in hydrogen-like atoms, just as they found for beams in the 2021 PRL. The new element is the explicit extension to bound states and the duality claim for atoms rather than beams alone. They keep the same two-term decomposition of the wave function into opposite-spin pieces that are each eigenstates of both the spin and orbital operators. If that decomposition is valid, the mixing follows and the fractional contributions appear naturally. That part is internally consistent with their prior work and gives a unified picture across beams and atoms. The approach is straightforward once the 2021 method is accepted, and it does organize some angular-momentum phenomena under one roof. The soft spot is whether the decomposition actually works for the atomic solutions. Standard Dirac hydrogen wave functions have radial parts fixed by the Coulomb potential and the boundary conditions at zero and infinity. Nothing in the usual construction guarantees that the upper and lower components can be split into two terms that each keep definite Sz and Lz eigenvalues while still solving the Dirac equation. If the split has to be imposed by hand, the fractional pieces become an artifact of the splitting rather than a physical consequence of the factorization. The abstract gives no new derivations or numerical checks for the atomic case, so the claim rests almost entirely on carrying over the beam result. This paper is mainly for theorists already working on relativistic angular momentum in beams or atoms who want to see the same method tried on bound states. A reader who knows the 2021 letter will understand the extension quickly, but the work does not stand on its own without that background. I would send it to peer review so referees can test whether the decomposition holds under the atomic boundary conditions or whether extra assumptions are required.

Referee Report

2 major / 2 minor

Summary. The manuscript extends the authors' 2021 PRL results on fractional angular momenta in relativistic electron beams to hydrogen-like atoms by applying a similar decomposition of Dirac equation solutions into two opposite-spin terms, each claimed to be an eigenstate of both SAM and OAM operators. This is said to arise from the factorization of the Klein-Gordon equation with Dirac matrices, leading to fractional angular momenta and fractional wave-particle duality in atoms as well as beams.

Significance. If the decomposition is valid and not an artifact, the findings could significantly impact interpretations of angular momentum and duality in relativistic quantum mechanics for both free and bound electron states. The work attempts to unify beam and atomic phenomena under this framework, but its significance depends on providing rigorous justification for the atomic extension beyond the prior beam analysis.

major comments (2)

[Application to hydrogen-like atoms] Application to hydrogen-like atoms: The manuscript does not demonstrate that the standard Dirac hydrogen wavefunctions allow the two-term decomposition where each term is simultaneously an eigenstate of S_z and L_z. The radial functions are determined by the Coulomb potential and boundary conditions, which may not preserve the required eigenvalues under such a splitting, unlike potentially in the free beam case. This is load-bearing for the fractional angular momentum claim in atoms.
[Abstract] Abstract: The central conclusion that the factorization produces mixing leading to fractional effects in atoms is not backed by independent derivations or verifications shown in the manuscript; it appears to follow directly from the 2021 work without addressing bound-state specifics.

minor comments (2)

The abstract repeats the beam results extensively; focus more on the new atomic findings and any differences.
[References] Add references to classic works on Dirac equation solutions for hydrogen to better situate the proposed decomposition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments provided. We address each major comment point by point below, indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

Referee: [Application to hydrogen-like atoms] Application to hydrogen-like atoms: The manuscript does not demonstrate that the standard Dirac hydrogen wavefunctions allow the two-term decomposition where each term is simultaneously an eigenstate of S_z and L_z. The radial functions are determined by the Coulomb potential and boundary conditions, which may not preserve the required eigenvalues under such a splitting, unlike potentially in the free beam case. This is load-bearing for the fractional angular momentum claim in atoms.

Authors: We agree that an explicit demonstration for the hydrogen-like atom case strengthens the manuscript. The standard Dirac solutions for the Coulomb problem are known analytically and can be decomposed into two opposite-spin terms following the same procedure as in the free-beam case. Because S_z and L_z act only on the angular and spin degrees of freedom while the radial functions satisfy the same central-potential boundary conditions for the full wave function, the eigenvalues are preserved under the splitting. In the revised manuscript we will add a dedicated subsection that writes out the decomposition explicitly for the standard (n, κ, m_j) solutions, verifies the action of S_z and L_z on each term, and confirms consistency with the radial boundary conditions. This addresses the load-bearing concern directly. revision: yes
Referee: [Abstract] Abstract: The central conclusion that the factorization produces mixing leading to fractional effects in atoms is not backed by independent derivations or verifications shown in the manuscript; it appears to follow directly from the 2021 work without addressing bound-state specifics.

Authors: The manuscript applies the identical decomposition method to the closed-form Dirac hydrogen wave functions, which already incorporates the bound-state specifics (Coulomb potential, normalizability at the origin and at infinity). Nevertheless, we accept that the abstract and main text could make this independence more transparent. We have revised the abstract to emphasize the bound-state verification and will add a short paragraph in the main text outlining how the radial equation and boundary conditions remain satisfied after the two-term splitting. These changes clarify that the atomic results are not merely carried over from the 2021 beam analysis. revision: partial

Circularity Check

1 steps flagged

Atomic fractional angular momenta rest on the same two-term opposite-spin decomposition introduced in authors' 2021 PRL

specific steps

self citation load bearing [Abstract]
"Fractional angular momenta emerge when eigenstates of the Dirac equation can be decomposed into two terms of opposite spin. Each of these terms being eigenstates of both the SAM and OAM operators. Building on this understanding, the same method used to calculate fractional angular momenta in beams is applied here to solutions of the Dirac equation for hydrogen-like atoms."

The quoted decomposition (opposite-spin terms each simultaneously eigen to SAM and OAM) is the load-bearing premise taken from the authors' 2021 PRL; the atomic results are obtained simply by reusing that premise and method, so the claimed fractional contributions and fractional wave-particle duality in atoms are not independently derived from the standard Dirac hydrogen solutions.

full rationale

The paper states that fractional angular momenta emerge from a specific decomposition of Dirac eigenstates into opposite-spin terms that are each eigenstates of both SAM and OAM, then explicitly applies the identical method to hydrogen-like atoms. This decomposition is taken directly from the cited prior work by the same authors; no independent derivation from the Dirac operator plus Coulomb potential is supplied that would guarantee preservation of the S_z and L_z eigenvalues for the bound radial functions. The central claim therefore reduces to an extension of the self-cited result rather than a fresh derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard Dirac equation and the decomposition assumption carried from prior literature; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

domain assumption Eigenstates of the Dirac equation for the systems considered can be decomposed into two terms of opposite spin that are each eigenstates of both the SAM and OAM operators.
Invoked in the abstract as the mechanism producing fractional angular momenta in both beams and atoms.

pith-pipeline@v0.9.0 · 5716 in / 1411 out tokens · 47532 ms · 2026-05-18T21:24:37.171539+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/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

factorization of the Klein-Gordon equation using Dirac matrices ... produces a specific mixing of the angular momentum states leading to ... fractional angular momenta
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

fractional contributions from both the spin angular momentum (SAM) and orbital angular momentum (OAM) operators

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

27 extracted references · 27 canonical work pages · 1 internal anchor

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that provide explicit formula for fractional OAM and SAM and a clear understanding in terms of Berry phase [6]. As theoretical interest in the interplay of SAM and OAM continues, there is growing number of experimental approaches to measuring and exploiting it [7]. From the quantum ﬁeld theorist perspective, Fukushima and Pu

work page
[2]

Fractional Angular Momenta in Electron Beams and Hydrogen-Like Atoms

have expressed surprise at results from electron beam studies [9] and acknowledged the need for more research into the issues raised. The starting point for this work is an observation re- lated to a solution of the Dirac equation [10] that applies to a tightly focused Gaussian electron beam. Speciﬁcally, that the spin of the electrons emerges as the sum ...

work page internal anchor Pith review Pith/arXiv arXiv 2025
[3]

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R. Ducharme, I. G. da Paz, and A. G. Hayrapetyan, Phys. Rev. Lett. 126, 134803 (2021)

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[1] [1]

As theoretical interest in the interplay of SAM and OAM continues, there is growing number of experimental approaches to measuring and exploiting it [7]

that provide explicit formula for fractional OAM and SAM and a clear understanding in terms of Berry phase [6]. As theoretical interest in the interplay of SAM and OAM continues, there is growing number of experimental approaches to measuring and exploiting it [7]. From the quantum ﬁeld theorist perspective, Fukushima and Pu

work page

[2] [2]

Fractional Angular Momenta in Electron Beams and Hydrogen-Like Atoms

have expressed surprise at results from electron beam studies [9] and acknowledged the need for more research into the issues raised. The starting point for this work is an observation re- lated to a solution of the Dirac equation [10] that applies to a tightly focused Gaussian electron beam. Speciﬁcally, that the spin of the electrons emerges as the sum ...

work page internal anchor Pith review Pith/arXiv arXiv 2025

[3] [3]

Allen, M

L. Allen, M. W. Beijersbergen, R.J.C. Spreeuw, and J.P. Woerdman, Phys. Rev. A 45, 8185 (1992); L. Allen, M. Padgett, and M. Babiker, Prog. Opt. 39 , 291 (1999); G. Molina-Terriza, J.P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007); S. Franke-Arnold, L. Allen, and M.J. Pad- gett, Laser & Photon. Rev. 2, 299 (2008); K. Y. Bliokh and A. Aiello, J. Opt. 15...

work page 1992

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K. Y. Bliokh, Y. P. Bliokh, S. Savel’ev, and F. Nori, Phys. Rev. Lett. 99, 190404 (2007)

work page 2007

[5] [5]

Uchida and A

M. Uchida and A. Tonomouro, Nature (London) 464, 737 (2010); J. Verbeeck, H. Tian, and P. Schattschnei- der, Nature (London) 467, 301 (2010); B. J. McMor- ran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, Science 331, 192 (2011); J. Verbeeck, P. Schattschneider, S. Lazar, M. Stoger-P ¨ollach, S. L ¨oﬄer, A. Ste...

work page 2010

[6] [6]

K. Y. Bliokh, M, Alonso, E. Ostrovskaya, and A. Aiello, Phys. Rev. A 82, 63825 (2010)

work page 2010

[7] [7]

K. Y. Bliokh, M. R. Dennis, and F. Nori, Phys. Rev. Lett. 107, 174802 (2011)

work page 2011

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M. V. Berry, J. Opt. A 6, 259 (2004)

work page 2004

[9] [9]

J. B. G ¨otte, K. O. Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. Barnett, and M. Padgett, Opt. Ex- press 16, 993 (2008); D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, Opt. Express, 18, 16480 (2010). V. V. Kotl- yar, A. A. Kovalev, and A. P. Porﬁrev, Opt. Express 27, 11236-11251 (2019)

work page 2008

[10] [10]

Fukushima, and S

K. Fukushima, and S. Pu, Strongly Interacting Mat- ter under Rotation (Springer International Publishing, Cham, 381-396, 2021)

work page 2021

[11] [11]

S. M. Barnett, Phys. Rev. Lett. 118, 114802 (2017)

work page 2017

[12] [12]

Ducharme, I

R. Ducharme, I. G. da Paz, and A. G. Hayrapetyan, Phys. Rev. Lett. 126, 134803 (2021)

work page 2021

[13] [13]

D. M. Greenberger and A. Yasin, Phys. Let. A 128, 391 (1988)

work page 1988

[14] [14]

Bohr, Nature 121, 580 (1928)

N. Bohr, Nature 121, 580 (1928)

work page 1928

[15] [15]

W. K. Wootters and W. H. Zurek, Phys. Rev. D 19, 473 (1979)

work page 1979

[16] [16]

Jaeger, A

G. Jaeger, A. Shimony, L. Vaidman, Phys Rev A., 51, 1 (1995)

work page 1995

[17] [17]

Khatiwada, and X.-F

P. Khatiwada, and X.-F. Qian, Phys. Rev. Res. 7, 3 (2025)

work page 2025

[18] [18]

Komar, Phys

A. Komar, Phys. Rev. D 18, 1887 (1978)

work page 1978

[19] [19]

H. W. Crater and P. Van Alstine, Phys. Rev. D 36,3007 (1987)

work page 1987

[20] [20]

P. A. M. Dirac, Proc. R. Soc. A, 246 326 (1958)

work page 1958

[21] [21]

D. G. Curie, T. F. Jordan, and E. C. G. Sudarshan, Re- views of Modern Physics 35, 2 (1963)

work page 1963

[22] [22]

H. W. Crater and J. Schiermeyer, Phys. Rev. D 82, 9 (2010)

work page 2010

[23] [23]

P. A. M. Dirac, Reviews of Modern Physics 21 3 (1949)

work page 1949

[24] [24]

P. A. M. Dirac, The Principles of Quantum Mechanics (Clarendon Press, Oxford, 1930)

work page 1930

[25] [25]

Barut and A.J

A.O. Barut and A.J. Bracken, Phys. Rev. D, 23 10, (1981)

work page 1981

[26] [26]

Penrose, The Road to Reality, (Alfred A

R. Penrose, The Road to Reality, (Alfred A. Knopf, 2004)

work page 2004

[27] [27]

Feshbach and F

H. Feshbach and F. Villars, Rev. Mod. Phys., 30, 24 (1958)

work page 1958