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arxiv: 2606.05226 · v1 · pith:5TAYWFK5new · submitted 2026-06-02 · ⚛️ physics.gen-ph

Klein--Gordon and Dirac Oscillators with an Apparent Mass Induced by the Momentum-Space Dual of the Fock--Lorentz Transformations

Abdelmalek Boumali , N. Jafari , Manizheh Botshekananfard This is my paper

Pith reviewed 2026-06-28 07:13 UTC · model grok-4.3

classification ⚛️ physics.gen-ph
keywords apparent massFock-Lorentz transformationsKlein-Gordon oscillatorDirac oscillatordeformed dispersionadiabatic approximationmomentum-space dual
0
0 comments X

The pith

A momentum-space dual of Fock-Lorentz transformations produces a time-dependent apparent mass that enters the Klein-Gordon and Dirac equations.

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

The paper constructs a controlled momentum-space dual of the Fock-Lorentz transformations and restricts the conformal factor to the world line vx=0. This yields an invariant mass-shell relation identical to the standard relativistic form except that the rest mass is replaced by a slowly varying apparent mass m_app(t) = m0/(1 + ct/R). Canonical quantization then produces Klein-Gordon and Dirac operators that contain this explicit time dependence. The paper verifies that the Dirac operator squares to the Klein-Gordon operator up to corrections linear in the mass derivative and suppressed by the ratio of the Compton wavelength to the FL scale R. The construction is applied to one-dimensional oscillators, where the adiabatic regime allows closed-form instantaneous spectra whose levels drift with the apparent mass and ultimately collapse toward zero as t tends to infinity.

Core claim

Restricting the FL conformal factor to the cosmological-frame world line vx=0 produces the invariant relation (E² - p²c²)(1 + ct/R)² = m0²c⁴, which is equivalent to ordinary special-relativistic dispersion with the replacement m0 → m_app(t) = m0/(1 + ct/R). Canonical quantization of this deformed shell gives Klein-Gordon and Dirac equations containing the time-dependent mass scale. Squaring the Dirac equation recovers the Klein-Gordon operator modulo first-order terms proportional to dm_app/dt that are suppressed by the Compton wavelength over the FL scale. In the adiabatic limit set by ε = c/(Rω) ≪ 1 the one-dimensional KG and Dirac oscillators admit closed-form instantaneous spectra; the D

What carries the argument

The momentum-space dual of the Fock-Lorentz transformations, which deforms the mass-shell invariant so that the rest mass is replaced by the apparent mass m_app(t) = m0/(1 + ct/R).

If this is right

  • Squaring the Dirac equation recovers the Klein-Gordon operator modulo first-order corrections proportional to dm_app/dt suppressed by the Compton wavelength over the FL scale.
  • In the adiabatic regime ε = c/(Rω) ≪ 1 the KG and Dirac oscillators possess closed-form instantaneous energy spectra.
  • For cosmological values of R the non-adiabatic corrections remain negligible.
  • In the formal limit t → ∞ the apparent mass tends to zero and the instantaneous levels for fixed quantum number collapse toward E = 0.

Where Pith is reading between the lines

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

  • The world-line ansatz isolates spectral consequences of the conformal factor and could be tested by comparing the predicted drift of oscillator levels against numerical solutions of the time-dependent wave equations.
  • The same apparent-mass replacement might be inserted into other relativistic bound-state problems whose spectra are known in the constant-mass case.

Load-bearing premise

The restriction of the FL conformal factor to the single world line vx=0 must be valid; without it the invariant relation with apparent mass does not hold and the subsequent quantization steps do not follow.

What would settle it

An explicit check, outside the vx=0 ansatz, showing that the deformed dispersion cannot be written as standard special-relativistic kinematics with any time-dependent mass function.

Figures

Figures reproduced from arXiv: 2606.05226 by Abdelmalek Boumali, Manizheh Botshekananfard, N. Jafari.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic outline of the construction developed in this work. The Fock–Lorentz trans [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Apparent mass [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Adiabatic parameter obtained from Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Instantaneous Klein–Gordon oscillator spectrum [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Instantaneous Dirac-oscillator component ladders as functions of [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
read the original abstract

We propose a controlled momentum-space dual of the Fock--Lorentz (FL) transformations and use it to derive a deformed relativistic mass shell. Restricting the FL conformal factor to the cosmological-frame world line $\vx=0$, the invariant relation takes the form $(E^{2}-\vp^{2}c^{2})(1+ct/R)^{2}=m_{0}^{2}c^{4}$, which is equivalent to the standard special-relativistic dispersion law with a time-dependent apparent mass $\mapp(t)=m_{0}/(1+ct/R)$. Canonical quantization then yields Klein--Gordon (KG) and Dirac equations containing a slowly varying mass scale. We show explicitly that squaring the Dirac equation reproduces the KG operator, modulo first-order corrections proportional to $\dot\mapp$ that are suppressed by the ratio of the Compton wavelength to the FL scale. The construction is not presented as a unique covariant phase-space theory; rather, it is a world-line ansatz designed to isolate the spectral consequences of the FL conformal factor. As applications, we study the one-dimensional KG and Dirac oscillators. In the adiabatic regime, governed by the small parameter $\epsilon=c/(R\omega)\ll1$, closed-form instantaneous spectra are obtained. The Dirac-oscillator calculation is carried out in component form and then reduced to the physical spinor spectrum, thereby avoiding the double counting of the upper and lower component ladders. Dimensionless plots illustrate the apparent-mass drift, the induced spectral evolution, and the domain of adiabatic validity. For cosmological values of $R$, non-adiabatic corrections are entirely negligible; in the formal limit $t\to\infty$ the apparent mass tends to zero and, for fixed quantum number, the instantaneous levels collapse toward $E=0$.

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 / 0 minor

Summary. The paper introduces a momentum-space dual of the Fock-Lorentz transformations and restricts the conformal factor to the cosmological world line vx=0, yielding the deformed dispersion (E² - p²c²)(1 + ct/R)² = m₀²c⁴ equivalent to a time-dependent apparent mass m_app(t) = m₀/(1 + ct/R). Canonical quantization produces KG and Dirac equations with this mass; the Dirac equation is shown to square to the KG operator up to first-order corrections in ṁ_app suppressed by the Compton-to-FL scale ratio. The construction is applied to one-dimensional KG and Dirac oscillators in the adiabatic regime (ε = c/(Rω) ≪ 1), producing closed-form instantaneous spectra, component-wise Dirac calculations to avoid double-counting, and plots of mass drift and spectral evolution; for cosmological R the non-adiabatic effects are negligible and levels collapse to E=0 as t → ∞.

Significance. If the world-line ansatz is accepted, the work supplies a concrete, parameter-controlled (R) mechanism for embedding FL-induced time dependence into relativistic quantum oscillators, with explicit adiabatic spectra and a verified squaring relation that isolates small corrections. This could serve as a toy model for cosmological modulation of quantum spectra, though the restriction to vx=0 limits covariance and the results remain tied to the specific ansatz rather than a full phase-space theory.

major comments (2)
  1. [Abstract / derivation of deformed mass shell] Abstract, paragraph beginning 'Restricting the FL conformal factor...': the restriction to vx=0 is the load-bearing step that produces the invariant relation with m_app(t); the manuscript presents it as a world-line ansatz but provides no demonstration that this form extends consistently to the phase-space operators used in the subsequent canonical quantization of the oscillators, which is required for the deformed KG and Dirac equations to be well-defined.
  2. [Abstract / quantization and squaring check] Abstract, sentence 'We show explicitly that squaring the Dirac equation reproduces the KG operator...': the claim of an explicit demonstration is central to establishing consistency of the quantized theory, yet the manuscript supplies neither the operator algebra for the time-dependent mass terms nor error estimates on the ˙m_app corrections, leaving the squaring relation unverified in the provided text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript. We respond point by point to the major comments, preserving the explicit framing of the work as a world-line ansatz.

read point-by-point responses

  1. Referee: [Abstract / derivation of deformed mass shell] Abstract, paragraph beginning 'Restricting the FL conformal factor...': the restriction to vx=0 is the load-bearing step that produces the invariant relation with m_app(t); the manuscript presents it as a world-line ansatz but provides no demonstration that this form extends consistently to the phase-space operators used in the subsequent canonical quantization of the oscillators, which is required for the deformed KG and Dirac equations to be well-defined.

    Authors: The manuscript explicitly states that the construction is a world-line ansatz and 'is not presented as a unique covariant phase-space theory.' The deformed dispersion is derived on the vx=0 world line and the resulting m_app(t) is inserted into the standard KG and Dirac operators; quantization of the one-dimensional oscillators then follows the conventional promotion of classical variables with this time-dependent mass. Because the paper does not claim a full phase-space formulation, a general demonstration of consistency beyond the world-line ansatz lies outside its stated scope. To address the concern about clarity, we will add a short paragraph in the revised manuscript that spells out how the world-line mass is used in the oscillator operators. revision: partial

  2. Referee: [Abstract / quantization and squaring check] Abstract, sentence 'We show explicitly that squaring the Dirac equation reproduces the KG operator...': the claim of an explicit demonstration is central to establishing consistency of the quantized theory, yet the manuscript supplies neither the operator algebra for the time-dependent mass terms nor error estimates on the ˙m_app corrections, leaving the squaring relation unverified in the provided text.

    Authors: The abstract reports that squaring reproduces the KG operator 'modulo first-order corrections proportional to ˙m_app that are suppressed by the ratio of the Compton wavelength to the FL scale.' The corresponding operator algebra and the indicated suppression factor are given in the main text. If the presentation of these steps is judged insufficiently detailed, we will expand the relevant derivation with additional intermediate steps in the revised manuscript. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation proceeds from explicit world-line ansatz to quantized operators with direct consistency verification.

full rationale

The paper states it adopts a world-line ansatz by restricting the FL conformal factor to vx=0, directly yielding the deformed dispersion (E²−p²c²)(1+ct/R)²=m₀²c⁴ and m_app(t)=m₀/(1+ct/R). Canonical quantization then inserts this time-dependent mass into the KG and Dirac operators, followed by an explicit algebraic check that Dirac² reproduces the KG operator (modulo ˙m_app corrections). No parameter is fitted to spectra, no result is defined in terms of itself, no self-citation chain is invoked as load-bearing, and the ansatz is openly labeled rather than smuggled. The construction is therefore self-contained against its stated inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on the background Fock-Lorentz transformations plus an explicit world-line restriction introduced to isolate the spectral effect; the apparent mass is an entity generated inside the ansatz.

free parameters (1)
  • R (FL scale)
    Large length scale appearing in the conformal factor; treated as an external cosmological parameter rather than fitted inside the paper.
axioms (2)
  • domain assumption Fock-Lorentz transformations and their conformal factor are valid background structures
    Invoked at the start of the construction without re-derivation.
  • ad hoc to paper Restriction of the conformal factor to the world line vx=0 isolates the desired spectral consequences
    Stated explicitly as the modeling choice that produces the apparent-mass dispersion.
invented entities (1)
  • apparent mass m_app(t) no independent evidence
    purpose: Encodes the deformed mass-shell relation as a time-dependent effective mass for quantization
    Generated by the world-line restriction; no independent falsifiable signature outside the model is supplied.

pith-pipeline@v0.9.1-grok · 5881 in / 1588 out tokens · 35026 ms · 2026-06-28T07:13:48.703338+00:00 · methodology

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Reference graph

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