Astrophysical tests of the stability—or not—of fundamental couplings (e.g., can the numerical value $\u223c1/137$ of the fine-structure constant *α* = *e*^{2}/*ℏc* vary with astronomical time?) are a very active area of observational research. Using a specific *α*-free non-relativistic and nonlinear isotropic quantum model compatible with its quantum electrodynamics (QED) counterpart yields the 99% accurate solution *α* = 7.364 × 10^{−3} vs its experimental value 7.297 × 10^{−3}. The $\u223c1%$ error is due to the deliberate use of mean-field Hartree approximation involving lowest-order QED in the calculations. The present theory has been checked by changing the geometry of the model. Moreover, it fits the mathematical solution of the original nonlinear integro-differential Hartree system by use of a rapidly convergent series of nonlinear eigenstates [G. Reinisch, Phys. Lett. A **498**, 129347 (2024)]. These results strongly suggest the mathematical transcendental nature—e.g., like for *π* or *e*—of *α*’s numerical value of $\u223c1/137$ and, hence, its astrophysical as well as its cosmological stability.

Since Dirac^{1,2} and Jordan^{3,4} first suggested it as a possibility, the time variation of the fundamental constants has remained a subject of fascination that motivated numerous theoretical and experimental studies. Landau^{5} envisaged the possibility that the fine-structure constant *α* ∼ 1/137 could vary with time due to the renormalization of the electric charge. Data from the natural fission reactors that operated about two billion years ago at Oklo (Gabon) had the potential of providing an extremely tight bound on the variability of *α*.^{6} This bound was revisited and the relative variation of *α* over this interval of time was found to be −0.9 × 10^{−7} < Δ*α*/*α* < 1.2 × 10^{−7} while the averaged relative growth of *α* was estimated between −6.7 × 10^{−17} and 5.0 × 10^{−17} yr^{−1}.^{7} A comprehensive review emphasized the main experimental and observational constraints that have been obtained from atomic clocks, the Oklo phenomenon, Solar System observations, meteorites dating, quasar absorption spectra, stellar physics, pulsar timing, the cosmic microwave background, and Big Bang nucleosynthesis.^{8} The deep conceptual importance of carrying out such astrophysical tests of the stability of fundamental couplings has been complemented by some evidence for such a variation,^{9} coming from high-resolution optical/UV spectroscopic measurements of the fine-structure constant *α* in absorption systems along the line of sight of bright quasars.^{10} However, this opinion is somewhat controversial: the apparent ubiquity, size, and general characteristics of the distortions are capable of significantly weakening the evidence for variations in *α* from quasar absorption lines.^{11} More recently, combining four direct measurements related to the value of *α* 13 × 10^{9} years ago with existing data, a spatial variation is preferred over a no-variation model at the 3.9*σ* level.^{12} On the other hand, Feynman^{13} was fascinated by the very numerical value $\u223c1/137$ of *α*, which he called “*one of the greatest damn mysteries of physics.*” Therefore, any indication that the numerical value of *α* could unexpectedly appear in the solution of a non-relativistic (i.e., *α*-free) mathematical description should importantly fuel this controversial issue.

The very first question is the following: can the fine-structure constant *α* = *e*^{2}/*ℏc* appear in the description of a non-relativistic physical process where the velocity of light *c* is, by definition, absent? Rather surprisingly, the answer is yes. In Refs. 14 and 15, an empirical non-relativistic connection between the electronic polarizability of atoms and *α* is proposed. In Refs. 16 and 17, the 97.7% visual transparency of graphene—a two-dimensional material with carbon atoms in a honeycomb lattice—is determined solely by *πα*. This significant 2.3% absorption of incident white light despite graphene being only one atom layer thick is a consequence of graphene’s unique electronic structure.^{18}

*α*are provided by quantum electrodynamics (QED) through its basic hypothesis of virtual photons that mediate in electromagnetic interactions.

^{19,20}The celebrated example is the description of classical Coulomb electrostatics (hence, non-relativistic) by QED’s exchange of a single virtual long-wavelength photon of probability amplitude $\u221d\alpha $.

^{21,22}Specifically, (i) the electromagnetic operator

*H*

_{em}=

*∫*

**jA**

*d*

^{3}

*x*defined by the four-vector particle current density

**j**and related potential operator

**A**is proportional to $e2/\alpha $, where

*e*is the electron charge. (ii) Therefore, the lowest-order probability amplitude to create a virtual photon |

*k*⟩ of given long-wavelength wavenumber

*k*out of the vacuum |0⟩ is

*ω*

_{k}=

*ck*. (iii) Consequently, the total number of virtual photons with all possible wave numbers that take part in the interaction is

*α*(or, equivalently, of

*c*). The use of Feynman-diagram “linked cluster theorem” constitutes a standard procedure to recover (3) as the lowest-order term of the Rayleigh–Schrödinger perturbation series and, hence, to evaluate its error. We have, indeed,

^{23}

^{,}

^{24}Thus, we obtain from series (4) the next-order correction to (3),

However, there is a snag in the above and overcoming this latter is the object of the present work. Contrary to its energy (3), **photon number (2) cannot be properly defined**. Indeed, it has a *k*^{−1} logarithmic singularity at *k* → 0 in the corresponding integral, which unfortunately occurs in the long-wavelength domain of (1).^{19} Therefore, *the above lowest-order QED sketch is not* *self-consistent*.

^{25–27}and specifically by its couple of lowest-energy electrostatic bound state eigensolutions of the non-relativistic and nonlinear Schrödinger–Poisson (SP) differential system.

^{28,29}In a

*D*= 3 three-dimensional (3d) radial harmonic potential defined by its angular frequency

*ω*, the discrete real-valued steady-state electron–electron scattering solutions

*u*

_{i}(

*X*) corresponding to the two lowest-energy

*S*= 0 eigenstates (“s” states)—ground state |

*a*) and first excited state |

*b*)—are defined by the following SP system in appropriate dimensionless units (we use parenthesis instead of brackets in |

*a*,

*b*) to emphasize eigenstate nonlinearity):

*μ*

_{i}in units of

*ℏω*. It is defined by the nonlinear eigenstate

*u*

_{i}, together with its corresponding Coulomb potential Φ

_{i}(

*X*). The initial conditions are

*ω*,

*X*=

*r*/

*L*. The linear regime $limN\u21920ui\u223c0$ where the nonlinear effects become negligible in agreement with Eqs. (6) and (7) since they become decoupled corresponds to the strongly confined case

*ω*→ ∞. Eigenstate non-orthogonality—or overlap—is defined by the inner product in agreement with normalization (12),

*ℏω*, the nonlinear eigenvalues

*μ*

_{i}can be calculated by use of either the initial conditions

*X*= 0 in (8) or the boundary conditions

*X*→ ∞ in Poisson equation (7),

*G*(

*X*′,

*X*) = 1/|

*X*′ −

*X*| of Eq. (7) at

*X*′ = 0. Equations (15) and (16) provide an excellent test for the accuracy of the numerical code: we obtained a 10

^{−8}precision by use of MATLAB’s ode45 integration code.

^{30}

*u*

_{a,b}of system (6)–(12) together with their respective Coulomb potentials Φ

_{a,b}(

*X*) given by Eqs. (8) and (15). Apply to it the lowest-order QED scheme (1)–(3) with its $\u223c1%$ error. By use of Poisson equation and with our dimensionless units, potential (17) yields the charge density,

^{28}

^{,}

**j**is the four-vector particle current density,

**A**is the related potential operator, and $ak\u2032+$ and

**a**

_{k′}are, respectively, the creation and the annihilation operators of scalar photons with wave vector

*k*′ and frequency

*ω*

_{k′}=

*ck*′.

^{24}We use the stationary configuration where

**j**reduces to its fourth charge density component (18), while

*ρ*

_{k′}is the radial Fourier component of

*ρ*. Photon amplitude (1) together with the properties of operators

**a**

^{+}and

**a**yields

*n*=

*ρ*/

*e*is the particle density defined by (18). Its three-dimensional radial Fourier component (as a function of the reduced wave number

*κ*=

*Lk*; hence,

*κX*=

*kr*) is $n\kappa =g(\kappa )/(N8\pi 3)$ with

*eW*by use of the charge density (18) yields the photon number,

*it has no singularity in the long-wavelength limit*

*κ*→ 0 since

*g*(0) = 0 as the result of normalization (12) with D = 3. Therefore, it can be used to establish a link to some appropriate specific physical property of the non-relativistic—hence,

*α*-free—nonlinear SP differential system (6)–(12).

^{31,32}

^{33}). Specifically, matrix element $\Phi aba/(\mu a\u2212\mu b)$ describes an absorption-like transition, namely, the probability amplitude for the system being in nonlinear ground eigenstate |

*a*) to populate nonlinear excited eigenstate |

*b*) as a result of interaction potential Φ

_{a}defined by probability density $|ua|2=ua2$ through (7), (8), and (15). Similarly, the second term $\Phi bab/(\mu b\u2212\mu a)$ would define the reverse process induced by interaction potential Φ

_{b}defined by $ub2$, namely, the probability amplitude for the system being in excited eigenstate |

*b*) to populate ground state |

*a*) through an emission-like transition. However, since the system is conservative, there are no external photons to induce these two processes. They are only due to the intrinsic nonlinearity of system (6)–(12) as shown by theorem (23). Therefore, we might equivalently wish to consider them within the QED framework as follows:

^{32}in the buildup of the inner product (

*a*|

*b*) as shown by theorem (23). At any extremum of (

*a*|

*b*)—e.g., at the minimum of (

*a*|

*b*), i.e., at the maximum $(a|b)max2$ of overlap probability (

*a*|

*b*)

^{2}reached at $N=Nmax$—the two nonlinearly induced emission/absorption photon amplitudes are in phase and resonantly add up. Such a resonance is defined in QED description (4) by all ⟨0|

*H*

_{w}|

*k*⟩⟨

*k*|

*H*

_{w}|0⟩ terms in the series that yield the photon gas energy. As a consequence,

**we expect photon number (22) to equal**—within the $\u223c1%$ tolerance defined by (5)—

**maximum overlap probability**$(a|b)max2$

**at**$N=Nmax$. Since the former is proportional to

*α*while the latter is independent of

*α*, we recover the value $(a|b)max2\u223c\alpha $ that has been repeatedly found—although sometimes with questionable arguments—in quantum-dot helium.

^{28,32,34–36}Quantitatively, (22) yields at $N=Nmax$

^{28}

**the numerical value**$\u223c1/137$

**of**

*α***is mathematically defined by the sole nonlinear spectral properties of dimensionless eigenvalue differential system (6)–(12)**: Thus, it is a transcendental number of mathematical origin that, like

*π*or

*e*, should probably be defined by a plethora of other mathematical models and remain invariant over cosmological times.

^{37}and second, the following

*D*= 2 two-dimensional electron gas geometry, which strongly modifies both Schrödinger and Poisson equations. Indeed, they, respectively, become in their dimensionless form (

*i*=

*a*,

*b*)

*D*= 2. The

*X*

^{−1}factor in the source term of Poisson equation(26) is due to Gauss’ theorem. When

*X*→ ∞, the electrostatic interaction energy

*e*Φ

_{i}defined by Eqs. (8), (15), (16), and (26) yields in units of $\u210f\omega limX\u2192\u221ee\Phi i(X)=X\u22121\u222b0X\u2192\u221eui2(X\u2032)X\u2032dX\u2032=N/X$ through multiplication of (26) by

*X*

^{2}and integration by parts. This result fits normalization (12) with

*D*= 2; see also (15). In contrast, dropping the

*X*

^{−1}factor in the RHS of (26) would yield the three-dimensional electron density integral $limX\u2192\u221ee\Phi i(X)=X\u22121\u222b0\u221eui2(X\u2032)X\u20322dX\u2032$ instead of the correct 2d one.

*n*=

*ρ*/

*e*is $n\kappa =h(\kappa )/(4\pi N)$, where

*J*

_{0}is the Bessel function of the first kind. Therefore, the QED probability defined by (20)–(22) becomes in the present two-dimensional parabolic electron confinement

*J*

_{0}(0) = 1 and, hence,

*h*(0) = 0 by use of normalization (12) with D = 2, we recover the same regularity of $P2d$ at

*κ*∼ 0 as in the 3d electron case (22).

Figure 1 displays the final numerical results. The intersections of photon number $P3d,2d$ with their respective square eigenstate overlap (*a*|*b*)^{2} occur as expected about the maxima of these latter (circles). In accordance with (22), we have in the 3d case $P3d=0.4714\alpha $ at maximum eigenstate overlap $(a|b)max2=3.47\xd710\u22123$ (=0.4755*α*) reached at $Nmax=6.3542$. Hence, the expected $\u223c1%$ relative error about this intersection is due to the difference between 0.4755 and 0.4714. According to (24), the 3d electron configuration yields *α*_{3d} = 3.47 × 10^{−3}/0.4714 = 7.361 × 10^{−3}. Similarly, $(a|b)max2=4.648\xd710\u22123$ (=0.6370*α*) at $Nmax=3.7477$ in the 2d case, while (29) yields $P2d=0.6309\alpha $. Thus, *α*_{2d} = 4.648 × 10^{−3}/0.6309 = 7.367 × 10^{−3} in the 2d case. It differs from *α*_{3d} = 7.361 × 10^{−3} by 0.08%. Such an excellent agreement between these two *α* values validate the present theory and exclude any fortuitous numerical coincidence. Moreover, they both approach the exact value *α* = 7.297 × 10^{−3} within less than 1% error, in perfect agreement with the QED estimation obtained by (5).

Let me conclude:

Astrophysical tests of the stability of the fine-structure constant

*α*∼ 1/137 are very active in observational research.^{6–12}I obtain

*α*= 7.364 × 10^{−3}(vs experimental*α*= 7.297 × 10^{−3}) from the mathematical solution of a dimensionless eigenvalue differential problem derived from the dual description of (i) a stationary nonlinear mean-field interacting quantum system and (ii) its lowest-order QED counterpart.This result strongly suggests the mathematical transcendental nature of

*α*and, therefore, its astrophysical as well as its cosmological stability.The corner stone of the solution lies in the physical similarity of nonlinear eigenstate overlap with induced eigenstate transitions by QED virtual photons.

Despite its 1% relative error due to deliberate simplifications, I believe that the present theory constitutes a pivotal progress toward the solution of what Feynman stressed as “

*one of the greatest damn mysteries in physics*.”^{13}

The author gratefully acknowledges the technical support from the University of Iceland, Reykjavik, as well as from UMR *Lagrange* (Observatoire de la Côte d’Azur, université de la Côte d’Azur, Nice, France).

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### Author Contributions

**Gilbert Reinisch**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Writing – original draft (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*QED: The Strange Theory of Light and Matter*

*Quantum Electrodynamics*

*A Unified Grand Tour of Theoretical Physics*

*Processus d’interaction entre photons et atomes*

*A Guide to Feynman Diagrams in the Many-Body Problem*

*Photons et atomes: Introduction a l’electrodynamique quantique*