Politics, Power, and Science: The Running Coupling as a Classical Relativistic Effect: A Conjecture

Monday, November 10, 2025

The Running Coupling as a Classical Relativistic Effect: A Conjecture

J. Rogers, SE Ohio

Abstract

We propose that the leading logarithmic running of the electromagnetic coupling, described in Quantum Electrodynamics (QED) by the beta function, is a classical effect of special relativity. We present a geometric framework where physical laws are projections of dimensionless relationships, and "fundamental constants" are artifacts of the chosen measurement scale. We demonstrate that a Lorentz boost compresses the electromagnetic field of a charge into a transverse configuration, and that a 2D Fourier analysis of this field yields a logarithmic dependence on the boost factor $γ$ . We conjecture that the precise coefficient of this logarithm, when calculated from the exact Liénard-Wiechert potentials, will be $2 / (3 π)$ , matching the QED result. This work does not provide a final proof but rather a rigorous mathematical pathway to test a profound unification of classical and quantum phenomena.

1. Introduction: A Geometric Alternative

The standard explanation for the energy-dependence ("running") of the fine-structure constant $α$ involves the quantum vacuum and virtual particle loops. We explore an alternative hypothesis: this phenomenon is a classical, kinematic effect arising from the Lorentz transformation of electromagnetic fields.

In this framework, physics is modeled as a fibration $π : E \to B$ , where the base category $B$ contains dimensionless relationships and the total category $E$ contains their coordinate-dependent expressions. The fine-structure constant $α_{0}$ is an invariant in $B$ , while the running coupling $α (E)$ is its projection into the high-energy coordinate chart of $E$ .

2. The Mechanism: Logarithmic Running from Field Compression

2.1 The Rest Frame and the Boost

A point charge $e$ at rest has a Coulomb field $E (r) \propto 1 / r^{2}$ . A Lorentz boost along the $z$ -axis with factor $γ$ compresses this field. In the ultra-relativistic limit, the field structure approximates a "pancake" confined to the transverse plane, with a longitudinal extent scaling as $\sim 1 / γ$ .

2.2 Transverse Fourier Analysis

The dominant transverse component of the boosted field can be modeled as:

$E_{⊥} (r_{⊥}, t) \approx \frac{N}{γ} \frac{1}{r_{⊥}} F (γ t)$

where $F$ is a normalized temporal envelope and $N$ a normalization constant.

The 2D Fourier transform in the transverse plane yields a spectrum:

$∣ {\tilde{E}}_{⊥} (k_{⊥}) ∣^{2} \propto \frac{1}{γ^{2}} \frac{1}{k_{⊥}^{2}}$

This $1 / k_{⊥}^{2}$ scaling is the crucial mathematical feature.

2.3 The Emergent Logarithm

The integrated spectral weight over transverse momenta is:

$I \propto \int_{k_{⊥, min}}^{k_{⊥, max}} \frac{2 π k_{⊥} d k_{⊥}}{k_{⊥}^{2}} = 2 π \ln (\frac{k_{⊥, max}}{k_{⊥, min}})$

The upper cutoff is boost-dependent, $k_{⊥, max} \sim γ / r_{min}$ , where $r_{min}$ is a microscopic length scale. This produces the logarithmic dependence:

$I \propto \ln γ$

The effective coupling therefore takes the form:

$α (γ) = α_{0} (1 + K α_{0} \ln γ + O (α_{0}^{2}))$

where $K$ is a dimensionless constant.

3. The Central Conjecture

The pancake model demonstrates the mechanism for logarithmic running but cannot determine the precise numerical coefficient $K$ . We now state our central conjecture.

Conjecture 1. The constant $K$ is a geometric property of the exact boosted field. It is given by the finite integral:

$K = \frac{1}{2 π} \int_{S (γ)} d A N (x; γ)$

where:

$S (γ)$ is the oblate spheroid defined by an equipotential surface of the exact Liénard-Wiechert field of a charge moving with Lorentz factor $γ$ .
$N (x; γ) = \lim_{k_{⊥} \to \infty} k_{⊥}^{2} ∣ {\tilde{E}}_{⊥} (k_{⊥}; x) ∣^{2}$ is the local spectral normalization, computed from the exact field.

Conjecture 2. The evaluation of the integral in Conjecture 1 yields the value:

$K = \frac{2}{3 π}$

thereby reproducing the one-loop QED beta function coefficient.

This reduces the problem of the running coupling to a well-defined, if technically complex, calculation in classical electrodynamics.

4. A Roadmap for Proof

A proof of Conjecture 2 would require the following explicit steps, which we outline as a program for future work:

Field Specification: Begin with the exact Liénard-Wiechert potential for a charge in uniform motion:
$E (r, t) = \frac{e}{4 π ϵ_{0}} \frac{(1 - β^{2}) (R - R β)}{(R - R \cdot β)^{3}}$
where $R = r - v t$ , $R = ∣ R ∣$ , and $β = v / c$ .
Surface Definition: Define the surface $S (γ)$ at $t = 0$ . A natural choice is the spheroid:
$x^{2} + y^{2} + \frac{z^{2}}{1 - β^{2}} = R_{0}^{2}$
Fourier Transform: At each point $x$ on $S (γ)$ , compute the 2D Fourier transform ${\tilde{E}}_{⊥} (k_{⊥}; x)$ of the transverse component of the exact field.
Asymptotic Analysis: Extract the local normalization $N (x; γ)$ from the $k_{⊥} \to \infty$ asymptotics of the transform.
Geometric Integration: Perform the surface integral over $S (γ)$ . This integral is finite but requires a UV regularization consistent with a microscopic scale $r_{min}$ .
Angular Averaging: The integral will involve an angular average over directions. The claim is that this averaging yields the factor $2 / 3$ , while the Fourier transform normalization yields the factor $1 / π$ .

We have not yet completed this derivation. This paper serves to present the conceptual framework and the precise mathematical challenge that must be solved.

5. Implications and Falsifiability

If Conjecture 2 is proven true, the implications are significant:

The "running" of couplings is not a uniquely quantum phenomenon but has a classical geometric origin.
The QED beta function describes how a fixed, dimensionless charge ( $α_{0}$ ) is projected into different relativistic coordinate frames.
"Fundamental constants" are revealed as connection coefficients in a fibration of measurement, not as intrinsic properties of nature.

This framework is highly falsifiable. It makes a sharp prediction: a specific classical calculation will yield a specific numerical result. If the calculation of $K$ yields any value other than $2 / (3 π)$ , the core proposition of this paper is invalidated.

6. Conclusion

We have proposed that the logarithmic running of the electromagnetic coupling is a classical relativistic effect, arising from the geometric transformation of fields under Lorentz boosts. We have demonstrated the mechanism for the logarithm and have formulated a precise conjecture for the value of the numerical coefficient.

This paper is not a proof. It is an argument that a profound unification of classical and quantum physics is possible and a map to its potential discovery. The path to validation is clear: compute the geometric integral $K$ . We invite the community to join in this calculation. The result will either reveal a deep geometric unity underlying physics or will cleanly falsify a bold, but ultimately incorrect, hypothesis.

Appendix A: Emergence of $\ln \gamma$ from Field Geometry

A.1 Transverse Compression of the Boosted Coulomb Field

Consider a point charge $e$ moving along the $z$ -axis with speed $v$ and Lorentz factor $\gamma = 1/\sqrt{1-\beta^2}$ . In the lab frame, the transverse component of the electric field is given by the Liénard-Wiechert formula:

E_{⊥} (r_{⊥}, z, t) \approx \frac{e}{4 π ϵ_{0}} \frac{1 - β^{2}}{{(r_{⊥}^{2} + γ^{2} (z - v t)^{2})}^{3 / 2}} r_{⊥​}

For ultra-relativistic motion ( $\gamma \gg 1$ ), the field is confined to a thin “pancake” of longitudinal thickness $\sim 1/\gamma$ and transverse extent $\sim r_\perp$ .

A.2 Temporal Integration and Effective Interaction

The effective interaction experienced by a test charge can be approximated by integrating over time:

\int_{- \infty}^{\infty} ∣ E_{⊥} ∣^{2} d t \sim \int_{- \infty}^{\infty} \frac{(1 - β^{2})^{2}}{{(r_{⊥}^{2} + γ^{2} (v t)^{2})}^{3}} d t

Change variables to the contracted longitudinal coordinate $u = \gamma vt$ , $dt = du / (\gamma v)$ :

\int_{-\infty}^{\infty} |\mathbf{E}_\perp|^2 \, dt \sim \frac{(1-\beta^2)^2}{\gamma v} \int_{-\infty}^{\infty} \frac{du}{\left(r_\perp^2 + u^2 \right)^3} \sim \frac{1}{\gamma} \frac{1}{r_\perp^5} \, .

This shows that the longitudinal compression reduces the effective interaction time by $1/\gamma$ , while the transverse field grows as $\gamma$ , producing a net dependence on $\gamma$ in the effective interaction.

A.3 Transverse Fourier Transform and Spectral Weight

Perform a 2D Fourier transform in the transverse plane:

{\tilde{E}}_{⊥} (k_{⊥}) = \int d^{2} r_{⊥} e^{- i k_{⊥} \cdot r_{⊥}} E_{⊥} (r_{⊥})

For the pancake field:

∣ {\tilde{E}}_{⊥} (k_{⊥}) ∣^{2} \sim \frac{1}{γ^{2}} \frac{1}{k_{⊥}^{2​}}

Integrating over transverse momentum:

I = \int_{k_{\perp,\text{min}}}^{k_{\perp,\text{max}}} 2\pi k_\perp \, dk_\perp \, |\tilde{\mathbf{E}}_\perp(k_\perp)|^2 \sim \frac{2\pi}{\gamma^2} \int_{k_{\perp,\text{min}}}^{k_{\perp,\text{max}}} \frac{dk_\perp}{k_\perp} = \frac{2\pi}{\gamma^2} \ln \frac{k_{\perp,\text{max}}}{k_{\perp,\text{min}}}

A.4 Identification of $\ln \gamma$ Scaling

The upper transverse cutoff scales as $k_{\perp,\text{max}} \sim \gamma / r_{\text{min}}$ , yielding:

I \sim \ln γ

Thus the effective coupling acquires a logarithmic dependence on the Lorentz factor:

α_{eff} (γ) \sim α_{0} (1 + K α_{0} \ln γ + O (α_{0}^{2}))

Here, $K$ is a dimensionless number capturing the angular and Fourier normalization, as discussed in the main text. The crucial point is that the logarithm arises purely from the geometric contraction of the field under a Lorentz boost, independent of quantum vacuum effects.

A.5 Conclusion of Appendix

This appendix demonstrates that $\ln \gamma$ naturally emerges from:

The transverse compression of the boosted Coulomb field,
The reduced longitudinal interaction time, and
The integration over transverse momentum in the Fourier plane.

It provides a rigorous classical foundation for the leading logarithmic running of $\alpha$ , forming the geometric backbone of the conjecture presented in the main paper.

Politics, Power, and Science

Monday, November 10, 2025

The Running Coupling as a Classical Relativistic Effect: A Conjecture

J. Rogers, SE Ohio

Abstract

1. Introduction: A Geometric Alternative

2. The Mechanism: Logarithmic Running from Field Compression

3. The Central Conjecture

4. A Roadmap for Proof

5. Implications and Falsifiability

6. Conclusion

Appendix A: Emergence of $\ln \gamma$ from Field Geometry

A.1 Transverse Compression of the Boosted Coulomb Field

A.2 Temporal Integration and Effective Interaction

A.3 Transverse Fourier Transform and Spectral Weight

A.4 Identification of $\ln \gamma$ Scaling

A.5 Conclusion of Appendix

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Monday, November 10, 2025

The Running Coupling as a Classical Relativistic Effect: A Conjecture

J. Rogers, SE Ohio

Abstract

1. Introduction: A Geometric Alternative

2. The Mechanism: Logarithmic Running from Field Compression

3. The Central Conjecture

4. A Roadmap for Proof

5. Implications and Falsifiability

6. Conclusion

Appendix A: Emergence of ln⁡γ\ln \gammalnγ from Field Geometry

A.1 Transverse Compression of the Boosted Coulomb Field

A.2 Temporal Integration and Effective Interaction

A.3 Transverse Fourier Transform and Spectral Weight

A.4 Identification of ln⁡γ\ln \gammalnγ Scaling

A.5 Conclusion of Appendix

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Appendix A: Emergence of $\ln \gamma$ from Field Geometry

A.4 Identification of $\ln \gamma$ Scaling