Astrophysical tests of the stability—or not—of fundamental couplings (e.g., can the numerical value 1/137 of the fine-structure constant α = e2/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 1% 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 1/137 and, hence, its astrophysical as well as its cosmological stability.

Since Dirac1,2 and Jordan3,4 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. Landau5 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 < Δα/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 × 109 years ago with existing data, a spatial variation is preferred over a no-variation model at the 3.9σ level.12 On the other hand, Feynman13 was fascinated by the very numerical value 1/137 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 = e2/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 

Actually possible links between non-relativistic physical systems and 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 α.21,22 Specifically, (i) the electromagnetic operator Hem = jAd3x defined by the four-vector particle current density j and related potential operator A is proportional to e2/α, 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
Ak=k|Hem|0ωkα,
(1)
since ωk = ck. (iii) Consequently, the total number of virtual photons with all possible wave numbers that take part in the interaction is
Pqed=k|Ak|2α.
(2)
(iv) On the other hand, the corresponding energy of this gas of virtual photons is
Eqed=k|Ak|2ωke2.
(3)
We recover the classical non-relativistic Coulomb energy: it is independent of 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,
Eqed=k0|Hem|kk|Hem|0ωk[0|Hem|kk|Hem|0]2(ωk)3+2[0|Hem|kk|Hem|0]3(ωk)5+.
(4)
The negative signs in those denominators in (4) of numerator terms to the odd powers are canceled by the property that the matrix elements of the creation and annihilation operators do always have opposite signs.24 Thus, we obtain from series (4) the next-order correction to (3),
Eqed=k|k|Hem|0|2ωk+o(e2/α)4(c)3=k|Ak|2ωk+o(e2α)e2[1+o(α)].
(5)
Therefore, the lowest-order QED description (1)(3) yields an error of 1%.

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.

It is possible to find a specific non-relativistic quantum-electrostatic system whose lowest-order QED description similar to (1)(3) yields a well-defined photon number Pqedα free of any long-wavelength singularity. It is provided by quantum-dot helium25–27 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 ui(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):
d2dX2+D1XddX+Ci(X)14X2ui(X)=0,
(6)
d2dX2+2XddXCi(X)=ui2(X),i=a,b,
(7)
where
Ci(X)=μieΦi(X)
(8)
is related to eigenvalue (or chemical potential) μi in units of ℏω. It is defined by the nonlinear eigenstate ui, together with its corresponding Coulomb potential Φi(X). The initial conditions are
ui(0)=ui0,duidXX=0=0,
(9)
Ci(0)=Ci0,dCidXX=0=0,
(10)
and the eigenstate’s regular boundary condition reads
limxui(X)=0.
(11)
For all eigenstates, the normalization condition is
i0ui2(X)XD1dX=N.
(12)
It is, indeed, defined by the quantum–classical order parameter that only depends on the confining parabolicity ω,
N=e2/Lω1ω.
(13)
The characteristic length L=/2mω is the “harmonic length” of the oscillator. It defines the reduced radial coordinate X = r/L. The linear regime limN0ui0 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),
(a|b)=1N0uaubXD1dX.
(14)
In units of ℏω, the nonlinear eigenvalues μi can be calculated by use of either the initial conditions X = 0 in (8) or the boundary conditions X → ∞ in Poisson equation (7),
μi=Ci(0)+eΦi(0)=limXCi(X)+NX,
(15)
where
eΦi(0)=0G(0,X)ui2X2dX=01Xui2X2dX=0ui2XdX,
(16)
by use of the 3d Green function 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 
Now, define the electrostatic potential as
W(X)=Φb(X)Φa(X),
(17)
by use of its two first nonlinear eigenstates ua,b of system (6)(12) together with their respective Coulomb potentials Φa,b(X) given by Eqs. (8) and (15). Apply to it the lowest-order QED scheme (1)(3) with its 1% error. By use of Poisson equation and with our dimensionless units, potential (17) yields the charge density,
ρ(X)=e4πNL3[ub2(X)ua2(X)].
(18)
Use the nonlinear electromagnetic interaction operator defined (in Gaussian units) by potential (17),28,
Hw=NjAd3x=2πL3/2N2πcakρk*+ak+ρkd3kk,
(19)
where j is the four-vector particle current density, A is the related potential operator, and ak+ and ak are, respectively, the creation and the annihilation operators of scalar photons with wave vector k′ and frequency ωk = ck′.24 We use the stationary configuration where j reduces to its fourth charge density component (18), while ρk is the radial Fourier component of ρ. Photon amplitude (1) together with the properties of operators a+ and a yields
Ak=k|Hw|0ck=2πL3/2N2παnkk3/2,
(20)
where 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 nκ=g(κ)/(N8π3) with
g(κ)=0[ub(X)]2[ua(X)]2sinκXκXX2dX.
(21)
The lowest-order QED description of the classical non-relativistic electrostatic potential energy eW by use of the charge density (18) yields the photon number,
P3d=k|Ak|2=L2π30|Ak|24πk2dk=απ0g2(κ)dκκ,
(22)
in agreement with (2). However, contrary to (2), 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).
We have the following theorem about nonlinear eigenstate overlap (14):31,32
(a|b)=eWabμbμa=eΦabaμaμb+eΦbabμbμa.
(23)
The respective Coulomb potentials are labeled here as superscripts when necessary in order to avoid any confusion with the real-valued matrix element Φabi=(a|Φi|b)=(b|Φi|a)=Φbai. The physical significance of theorem (23) can be illustrated from lowest-order QED (actually, first-order time-independent perturbation theory in quantum mechanics33). Specifically, matrix element Φaba/(μaμb) 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 Φa defined by probability density |ua|2=ua2 through (7), (8), and (15). Similarly, the second term Φbab/(μbμa) would define the reverse process induced by interaction potential Φb 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:
  1. Nonlinearity yields eigenstate overlap (14).

  2. Eigenstate overlap (14) creates virtual QED photons (22).

  3. Virtual photons (22) induce eigenstate transitions (23).

These photons interfere32 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|Hw|k⟩⟨k|Hw|0⟩ terms in the series that yield the photon gas energy. As a consequence, we expect photon number (22) to equal—within the 1% 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)max2α that has been repeatedly found—although sometimes with questionable arguments—in quantum-dot helium.28,32,34–36 Quantitatively, (22) yields at N=Nmax28 
α=π(a|b)max20g2(κ)κ1dκN=Nmax.
(24)
Therefore, the numerical value 1/137 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.
Two results confirm the quite specific role played in (24) by the maximum overlap probability (a|b)max2 at Nmax (see Fig. 1): first, the direct mathematical solution at Nmax of Hartree’s original integro-differential system by use of a rapidly convergent series of its nonlinear eigenstates,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)
d2dX2+1XddX+Ci14X2ui=0,
(25)
d2dX2+2XddXCi=ui2X.
(26)
The normalization condition is still given by (12) with 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 eΦi defined by Eqs. (8), (15), (16), and (26) yields in units of ωlimXeΦi(X)=X10Xui2(X)XdX=N/X through multiplication of (26) by X2 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 limXeΦi(X)=X10ui2(X)X2dX instead of the correct 2d one.
FIG. 1.

Photon number P3d,2d (broken lines) vs square eigenstate overlap (a|b)2 (continuous line) in units of α. The circles indicate the maxima of (a|b)2 corresponding to the actual minima of eigenstate overlap (a|b).

FIG. 1.

Photon number P3d,2d (broken lines) vs square eigenstate overlap (a|b)2 (continuous line) in units of α. The circles indicate the maxima of (a|b)2 corresponding to the actual minima of eigenstate overlap (a|b).

Close modal
In this 2d electron frame, the charge density (18) now becomes:
ρ=e4πNL2ub2ua2,
(27)
while the corresponding 2d Fourier component of n = ρ/e is nκ=h(κ)/(4πN), where
h(κ)=0[ub2(X)ua2(X)]J0(κX)XdX
(28)
and J0 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
P2d=α20h2(κ)dκκ.
(29)
Since J0(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=0.4714α at maximum eigenstate overlap (a|b)max2=3.47×103 (=0.4755α) reached at Nmax=6.3542. Hence, the expected 1% relative error about this intersection is due to the difference between 0.4755 and 0.4714. According to (24), the 3d electron configuration yields α3d = 3.47 × 10−3/0.4714 = 7.361 × 10−3. Similarly, (a|b)max2=4.648×103 (=0.6370α) at Nmax=3.7477 in the 2d case, while (29) yields P2d=0.6309α. Thus, α2d = 4.648 × 10−3/0.6309 = 7.367 × 10−3 in the 2d case. It differs from α3d = 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:

  1. Astrophysical tests of the stability of the fine-structure constant α ∼ 1/137 are very active in observational research.6–12 

  2. 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.

  3. This result strongly suggests the mathematical transcendental nature of α and, therefore, its astrophysical as well as its cosmological stability.

  4. The corner stone of the solution lies in the physical similarity of nonlinear eigenstate overlap with induced eigenstate transitions by QED virtual photons.

  5. 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).

The author has no conflicts to disclose.

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

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

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