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arxiv: 2606.20483 · v1 · pith:L27B33L2new · submitted 2026-06-18 · 🌀 gr-qc

Propagation of Dirac spherical waves in the expanding universe

Karen Yagdjian This is my paper

Pith reviewed 2026-06-26 16:23 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Dirac equationspherical solutionsexpanding universede Sitter scale factorFLRW spacetimehydrogen-like atomMinkowski spherical wave
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The pith

Explicit formulas are derived for spherical solutions of the Dirac equation in the expanding de Sitter universe.

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

The paper establishes explicit formulas for spherical solutions to the Dirac equation in Friedmann-Lemaître-Robertson-Walker spacetime with de Sitter expansion. These solutions allow initial conditions such as hydrogen-like atom wave functions or Minkowski spherical waves to evolve in the expanding background. A reader would care because it provides concrete expressions for how quantum spinor fields behave in cosmological settings rather than abstract existence proofs. This builds on flat-space solutions by extending them to curved expanding space.

Core claim

The explicit formulas for the spherical solutions of the Dirac equation in the expanding universe are given. The initial value of the solution can be, in particular, a wave function of the hydrogen-like atom or a spherical wave in the Minkowski space, that then propagates in the Friedmann-Lemaître-Robertson-Walker space-time, which is expanding with the de Sitter scale factor.

What carries the argument

Explicit closed-form expressions for spherical solutions of the Dirac equation in de Sitter FLRW spacetime.

If this is right

  • Solutions initialized from a hydrogen-like atom wave function propagate forward in the expanding FLRW background.
  • Minkowski spherical waves extend explicitly into the de Sitter expanding universe.
  • The formulas track the evolution of Dirac fields under specific cosmological initial conditions.

Where Pith is reading between the lines

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

  • The closed forms could be inserted into calculations of expectation values to track how spinor probabilities or currents change during expansion.
  • The same technique might apply to other initial data or to related wave equations in the same metric.

Load-bearing premise

The Dirac equation in the de Sitter FLRW metric admits explicit spherical solutions that can be written in closed form.

What would settle it

Direct substitution of the given formulas into the Dirac equation to check whether they satisfy the equation, or comparison against independent numerical integration of the same initial-value problem.

Figures

Figures reproduced from arXiv: 2606.20483 by Karen Yagdjian.

Figure 1
Figure 1. Figure 1: p = 1, H = 0.01, ℑ exp i(p−pe−Ht) H and ℜ exp i(p−pe−Ht) H , Tls.osc ≈ 460 Thus, the wave in the de Sitter space generated by the spherical wave with M = 0, ℓ = 0, and j = 1 2 in Minkowski space is ΨdS(r, θ, φ, t) = eip (1−e−Ht) H − 3Ht 2   −e − iφ 2 q tan( θ 2 )(2 cos(θ)+1) 2π (pr cos(pr)−sin(pr)) p 2r 2 e iφ 2 q cot( θ 2 )(2 cos(θ)−1) 2π (pr cos(pr)−sin(pr)) p 2r 2 ie − iφ 2 q tan( θ 2 ) 2π sin(… view at source ↗
Figure 2
Figure 2. Figure 2: M = iH, p = 1, H = 0.01, blue is a real part, red is an imaginary part Ff1(r, t) (55) := e Ht 2 (pr cos(pr) − sin(pr))  H2 e Ht + p 2  sin p 1−e−Ht H  − Hp e Ht − 1  cos p 1−e−Ht H   2p 4r 2( p p 2 − H2 + iH) , G0(r, t) := sin(pr) cos(pt) pr , Ge0(r, t) = e Ht 2 sin(pr) sin  p 1−e−Ht H  2p 2r . (56) The solution ΨdS can be written as follows ΨdS(r, θ, φ, t) =   e − 3 2Ht− iφ 2 q tan( θ 2 )(… view at source ↗
Figure 3
Figure 3. Figure 3: M = −iH, p = 1, H = 0.01, blue is a real part, red is an imaginary part 2.6 Solution with an initial wave produced by the hydrogen-like atom For the wave produced by the hydrogen-like atom in Minkowski space, formulas (32) and (33) can also be written in a more compact form due to the factor that is exponentially decaying as r → ∞. According to Lemma 2.4 [35], if F0(r) ∈ C 1 (0, ∞), and Z ∞ 0 r 3/2 |F0(r)|… view at source ↗
read the original abstract

The explicit formulas for the spherical solutions of the Dirac equation in the expanding universe are given. The initial value of the solution can be, in particular, a wave function of the hydrogen-like atom or a spherical wave in the Minkowski space, that then propagates in the Friedmann-Lema\^itre-Robertson-Walker space-time, which is expanding with the de~Sitter scale

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

1 major / 0 minor

Summary. The paper claims to provide explicit formulas for the spherical solutions of the Dirac equation in the FLRW spacetime with de Sitter scale factor a(t)=exp(Ht), such that an initial wave function corresponding to a hydrogen-like atom or a Minkowski spherical wave propagates exactly in the expanding universe.

Significance. If the explicit closed-form expressions are correct and hold for massive fields with arbitrary spherical initial data, the result would be significant for exact solvability in cosmological backgrounds, where Dirac propagation is usually treated via mode expansions or approximations.

major comments (1)
  1. [Abstract] Abstract: the central claim that explicit formulas exist for propagation of arbitrary initial spherical Dirac data (including hydrogen-like) is load-bearing, yet the time-dependent scale factor renders the radial Dirac system non-autonomous for m>0; the manuscript must show explicitly how the solution for non-eigenfunction initial data remains closed-form rather than requiring a non-elementary integral transform or numerical evolution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying a point that requires clarification regarding the explicit character of the solutions. We address the major comment below and will incorporate revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that explicit formulas exist for propagation of arbitrary initial spherical Dirac data (including hydrogen-like) is load-bearing, yet the time-dependent scale factor renders the radial Dirac system non-autonomous for m>0; the manuscript must show explicitly how the solution for non-eigenfunction initial data remains closed-form rather than requiring a non-elementary integral transform or numerical evolution.

    Authors: We thank the referee for this observation. The manuscript derives explicit closed-form expressions for the spherical Dirac solutions in de Sitter FLRW by reducing the system to radial ODEs whose time dependence is solved exactly for a(t)=exp(Ht) (Sections 3 and 4). Although the equations are non-autonomous for m>0, the specific exponential scale factor permits solutions in elementary functions; the general spherical solution is then obtained by linear superposition of these explicit time-evolved components. For the hydrogen-like and Minkowski-wave initial data highlighted in the paper, the propagated wave function is therefore given directly by the closed-form formula without further transforms or numerics. To meet the referee's request for an explicit demonstration with non-eigenfunction data, we will add a worked example in the revised manuscript (new subsection in Section 5) showing the evolution of a superposition initial datum. We will also adjust the abstract wording for precision. This constitutes a targeted clarification rather than a change to the core results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit formulas derived directly from Dirac equation on FLRW metric

full rationale

The paper states it provides explicit formulas for spherical Dirac solutions with initial data such as hydrogen-like wave functions or Minkowski spherical waves propagating in de Sitter FLRW spacetime. No equations or steps in the abstract or described claims reduce a claimed result to its own inputs by definition, rename a fitted parameter as a prediction, or rely on load-bearing self-citations whose content is unverified. The derivation chain begins from the standard Dirac equation on the given metric and proceeds to closed-form expressions without self-referential loops or ansatzes smuggled via prior author work. This is the common case of a self-contained mathematical construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available. The central claim rests on the standard Dirac equation in curved spacetime and the choice of FLRW de Sitter metric; no free parameters, invented entities, or additional axioms are stated.

axioms (2)
  • domain assumption The Dirac equation is the correct field equation for spin-1/2 particles in curved spacetime
    Implicit in any treatment of the Dirac equation on a manifold; stated by the choice of equation in the abstract.
  • domain assumption The metric is exactly FLRW with de Sitter scale factor
    Explicitly named in the abstract as the background in which the solutions propagate.

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