pith. machine review for the scientific record. sign in

arxiv: 2605.02925 · v1 · submitted 2026-04-25 · ⚛️ physics.gen-ph

Recognition: unknown

The quantum mechanics of Einstein photons and generalized functions

Aleksandr Beilinson
Authors on Pith no claims yet

Pith reviewed 2026-05-09 20:43 UTC · model grok-4.3

classification ⚛️ physics.gen-ph
keywords Majorana equationsEinstein photongeneralized Green functionquantum generalized measureCasimir forcesdeinterlaced spinsSchrödinger equation
0
0 comments X

The pith

Reinterpreting Majorana equations as describing Einstein photons with deinterlaced spins yields a quantum field whose generalized measure fills all space instantly.

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

The paper attempts to show that the Majorana equations can be viewed as quantum Lorentz-covariant equations for the field of an Einstein photon featuring deinterlaced spins and a diagonal Hamiltonian. It derives the corresponding generalized Green function as a functional on finite test functions and sets up the associated Schrödinger equation. This Green function is then extended to a σ-additive quantum generalized measure defined on the dual space to the compact subspace of photon paths in L₂ space. The result is that the quantum field fills the entire space instantaneously during the evolutionary process. Casimir forces are calculated within this framework.

Core claim

The paper reinterprets the Majorana equations as quantum Lorentz-covariant equations for the field of an Einstein photon with deinterlaced spins. Its generalized Green function is constructed as a functional on finite test functions together with the Schrödinger equation. The generalized process is continued to a σ-additive quantum generalized measure on the space dual to the compact subspace of photon paths in L₂(−∞,∞). This leads to the quantum field filling the whole space instantly in the evolutionary problem. The Casimir forces arising in this field are calculated.

What carries the argument

The σ-additive quantum generalized measure extending the generalized Green function of the deinterlaced Einstein photon, acting on the dual to photon path space.

If this is right

  • The quantum field instantly fills the entire space in evolutionary problems.
  • Casimir forces can be computed for this specific photon field.
  • The photon is treated with deinterlaced spins leading to a diagonal Hamiltonian.
  • The measure is defined over paths in L₂(−∞,∞) space.

Where Pith is reading between the lines

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

  • This construction provides a way to handle quantum fields using generalized functions and measures.
  • It suggests potential non-local or instant propagation aspects in relativistic quantum photon fields.
  • Further exploration could link this to other treatments of generalized processes in quantum mechanics.

Load-bearing premise

The Majorana equations can be reinterpreted as quantum equations for an Einstein photon with deinterlaced spins, and the resulting generalized Green function extends to a σ-additive measure that causes instant space-filling.

What would settle it

Checking if the generalized Green function from the reinterpreted Majorana equations indeed extends to a σ-additive quantum measure with instantaneous space-filling behavior, or comparing computed Casimir forces to experimental values.

read the original abstract

The article consider an interpretation of Majorana equations as a quantum Lorentz covariant equations for the field of Einstein photon. A photon with "deinterlaced" spins (with diagonal Hamiltonian) is considered, its generalized Green function as a functional on finite test functions and its Schr\"{o}dinger equation are constructed. The generalized process corresponding to this Green function is continued to $\sigma$-additive quantum generalized measure on the space dual to the compact subspace of a photon paths in $L_{2}(-\infty,\infty)$. In this case, the filling of whole space by such quantum field instantly occurs in the evolutionary problem. The last part of the article gives the calculation of the "Casimir forces" arising in this field.

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

3 major / 3 minor

Summary. The manuscript reinterprets the Majorana equations as Lorentz-covariant quantum equations for the field of an 'Einstein photon' with 'deinterlaced spins' (diagonal Hamiltonian). It constructs the associated generalized Green function as a functional on finite test functions together with its Schrödinger equation, extends the corresponding generalized process to a σ-additive quantum generalized measure on the dual of the compact subspace of photon paths in L₂(−∞,∞), and asserts that this produces instantaneous filling of space by the quantum field in the evolutionary problem. The final section computes the resulting 'Casimir forces'.

Significance. If the extension to a σ-additive measure and the consequent instant space-filling property can be placed on a rigorous footing, the work would supply a novel generalized-function framework for photon fields that departs from standard quantum-field propagation. The attempt to link Majorana equations to deinterlaced-spin photons and to derive Casimir forces within this setting is an interesting direction, though it currently lacks the supporting derivations and consistency checks needed to assess its relation to established physics.

major comments (3)
  1. [Abstract and continuation section] Abstract and the section describing the continuation of the generalized process: the claim that the Green function extends to a σ-additive quantum generalized measure on the dual space is asserted without an explicit extension map, without specification of the topology on the dual of the compact subspace of L₂(−∞,∞) paths, and without verification that the resulting set function is countably additive on disjoint measurable sets. This step is load-bearing for the instant space-filling assertion in the evolutionary problem.
  2. [Reinterpretation section] Section on the reinterpretation of the Majorana equations: the passage from the standard Majorana formulation to quantum Lorentz-covariant equations for an Einstein photon with deinterlaced spins is introduced by definition of the new entities and the diagonal Hamiltonian, but no explicit transformation, equivalence proof, or consistency check with known photon properties (e.g., transversality, two polarization states) is supplied. This underpins the subsequent Green function and measure construction.
  3. [Evolutionary problem and Casimir section] Section on the evolutionary problem and Casimir forces: the instant space-filling property is used to derive the Casimir forces, yet the support properties of the measure that would justify instantaneous filling are not shown to survive the extension while preserving Lorentz covariance; without the missing additivity verification, the force calculation rests on an unestablished premise.
minor comments (3)
  1. [Abstract] Abstract: 'The article consider' is grammatically incorrect and should read 'The article considers'.
  2. [Throughout] Notation: the space L_{2}(−∞,∞) should be written L²(ℝ) for standard clarity; the phrase 'compact subspace of a photon paths' is unclear and requires precise definition.
  3. [Introduction and reinterpretation section] The terms 'Einstein photon' and 'deinterlaced spins' are introduced without a dedicated definition paragraph or reference to prior literature, which hinders readability even if they are novel to the paper.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments identify key areas requiring greater explicitness in the measure-theoretic construction and the reinterpretation of the Majorana equations. We respond point by point below and will incorporate clarifications and additional derivations in a revised version.

read point-by-point responses

  1. Referee: [Abstract and continuation section] Abstract and the section describing the continuation of the generalized process: the claim that the Green function extends to a σ-additive quantum generalized measure on the dual space is asserted without an explicit extension map, without specification of the topology on the dual of the compact subspace of L₂(−∞,∞) paths, and without verification that the resulting set function is countably additive on disjoint measurable sets. This step is load-bearing for the instant space-filling assertion in the evolutionary problem.

    Authors: We agree that the extension step requires an explicit construction. In the revision we will define the extension map via the Riesz–Markov–Kakutani representation theorem applied to the dual of the compact subspace of L₂ paths, equipped with the weak* topology induced by the finite test functions. Countable additivity on disjoint measurable sets will be verified by showing that the generalized process is continuous with respect to this topology and that finite-dimensional projections inherit σ-additivity from the underlying Lebesgue measure on path space. These additions will directly support the instant space-filling claim in the evolutionary problem. revision: yes

  2. Referee: [Reinterpretation section] Section on the reinterpretation of the Majorana equations: the passage from the standard Majorana formulation to quantum Lorentz-covariant equations for an Einstein photon with deinterlaced spins is introduced by definition of the new entities and the diagonal Hamiltonian, but no explicit transformation, equivalence proof, or consistency check with known photon properties (e.g., transversality, two polarization states) is supplied. This underpins the subsequent Green function and measure construction.

    Authors: The diagonal Hamiltonian is introduced precisely to deinterlace the spin degrees of freedom while preserving the Lorentz-covariant structure of the original Majorana equations. In the revision we will add an explicit transformation that maps the standard Majorana operator to the diagonal form, demonstrate equivalence on the space of solutions, and verify that transversality (vanishing divergence) and the two physical polarization states are retained as the diagonal entries. This will place the subsequent Green-function construction on a firmer footing. revision: yes

  3. Referee: [Evolutionary problem and Casimir section] Section on the evolutionary problem and Casimir forces: the instant space-filling property is used to derive the Casimir forces, yet the support properties of the measure that would justify instantaneous filling are not shown to survive the extension while preserving Lorentz covariance; without the missing additivity verification, the force calculation rests on an unestablished premise.

    Authors: The support of the extended measure remains the full path space because the generalized Green function satisfies an evolutionary equation whose fundamental solution is supported everywhere at each time. In the revision we will prove that this support property is invariant under the weak* extension and that Lorentz covariance is inherited directly from the covariant Majorana reinterpretation. The Casimir-force expressions will be annotated to indicate where the verified countable additivity enters the derivation. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper begins by reinterpreting the Majorana equations as Lorentz-covariant equations for an Einstein photon field with deinterlaced spins, then constructs a generalized Green function on finite test functions together with its Schrödinger equation, and finally asserts that the associated process extends to a σ-additive quantum generalized measure on the dual of the compact subspace of L₂ photon paths, yielding instant space-filling. No step reduces by construction to its own inputs: the reinterpretation supplies new definitions, the Green function and Schrödinger equation are built from those definitions, and the extension to a measure is presented as a continuation whose properties (including additivity and support) are claimed rather than being tautologically identical to the starting reinterpretation. The listed circularity patterns (self-definitional loops, fitted inputs renamed as predictions, load-bearing self-citations, imported uniqueness theorems, smuggled ansätze, or renaming of known results) are not exhibited by any quoted equation or statement in the manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 3 invented entities

The central claim rests on reinterpreting standard Majorana equations for a newly named photon concept and extending Green functions to measures without external validation or independent evidence for the new entities.

axioms (1)
  • domain assumption Majorana equations can be reinterpreted as quantum Lorentz-covariant equations for the Einstein photon field
    Explicitly stated as the starting interpretation in the abstract.
invented entities (3)
  • Einstein photon no independent evidence
    purpose: To serve as the physical object described by the reinterpreted Majorana equations
    Introduced without prior definition or external reference
  • deinterlaced spins no independent evidence
    purpose: To produce a diagonal Hamiltonian for the photon field
    Defined only within the paper's framework
  • quantum generalized measure no independent evidence
    purpose: To extend the Green function to a σ-additive object on photon path spaces
    Constructed as part of the theory with no independent handle

pith-pipeline@v0.9.0 · 5409 in / 1493 out tokens · 59457 ms · 2026-05-09T20:43:15.420925+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

46 extracted references · 1 canonical work pages

  1. [1]

    M., Shilov G

    otherlanguage* english Gel`fand I. M., Shilov G. E. Generalized functions: Properties and Operations. --- N. Y. and L.: Academic Press, 1964. --- 1 Generalized functions, 423 . --- [Trans. from rus. by E. Saletan]. otherlanguage*

    1964

  2. [2]

    M., Vilenkin N

    otherlanguage* english Gel`fand I. M., Vilenkin N. Y. Application of Harmonic Analysis. --- N. Y. and L.: Academic Press, 1964. --- 4 Generalized functions, 384 . --- [Trans. from rus. by A. Feinstein]. otherlanguage*

    1964

  3. [3]

    I., Beresteckij V

    otherlanguage* english Axiezer A. I., Beresteckij V. B. Quantum electrodynamics / L. P. Rusakova. --- 4 . --- M.: Nauka, 1981. --- 428 . --- (Russian). otherlanguage*

    1981

  4. [4]

    M., Minlos R

    otherlanguage* english Gel'fand I. M., Minlos R. A., Shapiro Z. Y. The representations of rotation group and of Lorentz group, their application. --- M.: Fizmatgiz, 1972. --- 368 . --- (Russian). otherlanguage*

    1972

  5. [5]

    D., Lifshic E

    otherlanguage* english Landau L. D., Lifshic E. M. Theory of field. --- 8 . --- M.: Fizmatlit, 2003. --- 2 Theoretical physics, 536 . --- (Russian). otherlanguage*

    2003

  6. [6]

    L., Wouthuysen S

    otherlanguage* english Foldy L. L., Wouthuysen S. A. On the Dirac theory of spin 1/2 particles and its non-relativistic limit /\!/ Phys. Rev. --- 1950. --- 78, 1. otherlanguage*

    1950

  7. [7]

    otherlanguage* english Beilinson A. A. The Cauchy process in imaginary time and the path integrals in relativistic quantum mechanics of Dirac's electrons and Einstein's photons /\!/ International Scientific Journal. --- 2018. --- 4. --- 103--116. otherlanguage*

    2018

  8. [8]

    N., Fomin S

    otherlanguage* english Kolmogorov A. N., Fomin S. V. Elements of the theory of functions and functional analysis. --- Mineola; NY: Dover, 1999. otherlanguage*

    1999

  9. [9]

    otherlanguage* english Beilinson A. A. Two isomorphic forms of quantum e.-m. field existence and Lamb and Casimir effects /\!/ Transactions on Systems. --- 2025. --- 24. --- 16--19. --- ISSN 1109-2777. --- ://wseas.com/journals/articles.php?id=9561. otherlanguage*

    2025

  10. [10]

    --- 2022

    otherlanguage* english Nobel Prize in Physics. --- 2022. otherlanguage*

    2022

  11. [11]

    Axiezer, A. I. and Beresteckij, V. B. , title =. 1981 , edition =

    1981

  12. [12]

    Гельфанд, И. М. and Шилов, Г. Е. , title =

  13. [13]

    Гельфанд, И. М. and Шилов, Г. Е. , title =. 1959 , number =

    1959

  14. [14]

    Гельфанд, И. М. and Шилов, Г. Е. , title =. 1958 , number =

    1958

  15. [15]

    Гельфанд, И. М. and Виленкин, Н. Я. , title =. 1961 , number =

    1961

  16. [16]

    Гельфанд, И. М. and Граев, М. И. and Виленкин, Н. Я. , title =. 1962 , number =

    1962

  17. [17]

    Гельфанд, И. М. and Граев, М. И. and Пятецкий -Шапиро, И. И. , title =. 1966 , number =

    1966

  18. [18]

    Gel`fand, I. M. and Shilov, G. E. , title =

  19. [19]

    Gel`fand, I. M. and Shilov, G. E. , title =. 1964 , volume =

    1964

  20. [20]

    Gel`fand, I. M. and Shilov, G. E. , title =. 1968 , volume =

    1968

  21. [21]

    Gel`fand, I. M. and Shilov, G. E. , title =. 1967 , volume =

    1967

  22. [22]

    Gel`fand, I. M. and Vilenkin, N. Ya. , title =. 1964 , volume =

    1964

  23. [23]

    Gel`fand, I. M. and Graev, M. I. and Vilenkin, N. Ya. , title =. 1966 , volume =

    1966

  24. [24]

    Beilinson, A. A. , title =. International Scientific Journal , series =. 2018 , month =

    2018

  25. [25]

    Beilinson, A. A. , title =. Proof , address =. 2023 , month =. doi:10.37394/232020.2023.3.9 , url =

    work page doi:10.37394/232020.2023.3.9 2023

  26. [26]

    Beilinson, A. A. , title =. Transactions on Systems , address =. 2025 , month =

    2025

  27. [27]

    Шилов, Г. Е. , title =. 1961 , editor =

    1961

  28. [28]

    Шилов, Г. Е. , title =. 1965 , editor =

    1965

  29. [29]

    Минлос, Р. А. , title =. Труды Московского математического общества , volume =. 1959 , pages =

    1959

  30. [30]

    Люстерник, Л. А. and Соболев, В. И. , title =. 1951 , type =

    1951

  31. [31]

    Gel'fand, I. M. and Minlos, R. A. and Shapiro, Z. Ya. , title =. 1972 , numpages =

    1972

  32. [32]

    Landau, L. D. and Lifshic, E. M. , title =

  33. [33]

    Landau, L. D. and Lifshic, E. M. , title =. 2004 , volume =

    2004

  34. [34]

    Landau, L. D. and Lifshic, E. M. , title =. 2003 , volume =

    2003

  35. [35]

    Beresteckij, V. B. and Lifshic, E. M. and Pitaevskij, L. P. , title =. 2002 , volume =

    2002

  36. [36]

    Landau, L. D. and Lifshic, E. M. , title =. 2002 , volume =

    2002

  37. [37]

    Landau, L. D. and Lifshic, E. M. , title =. 2001 , volume =

    2001

  38. [38]

    Landau, L. D. and Lifshic, E. M. , title =. 2005 , volume =

    2005

  39. [39]

    Lifshic, E. M. and Pitaevskij, L. P. , title =. 2004 , volume =

    2004

  40. [40]

    Lifshic, E. M. and Pitaevskij, L. P. , title =. 2002 , volume =

    2002

  41. [41]

    Колмогоров, А. Н. and Фомин, С. В. , title =. 1976 , edition =

    1976

  42. [42]

    Kolmogorov, A. N. and Fomin, S. V. , title =. 1999 , numpage =

    1999

  43. [43]

    Foldy, L. L. and Wouthuysen, S. A. , title =. Phys. Rev. , volume =. 1950 , numpage =

    1950

  44. [44]

    Feynman, R. P. , title =. Rev. Mod. Phys. , volume =. 1948 , numpage =

    1948

  45. [45]

    Фейнман, Р. П. , title =. Вопросы причинности в квантовой механике , address =. 1955 , editor =

    1955

  46. [46]

    Nobel Prize in Physics , year =