Why the Running of α Demands a Fundamental Reassessment of QED

 

On the Geometric Origin of Coupling Variation:

J. Rogers
Independent Researcher, SE Ohio
November 2025

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Abstract

For more than seventy years, Quantum Electrodynamics has attributed the energy-dependent variation of the fine structure constant α(E) to “vacuum polarization”—the transient formation of virtual electron–positron pairs that screen the bare charge. This interpretive story is universally repeated in textbooks, lectures, and popular explanations.

Yet the running of α(E) depends directly on E/m_e, which is nothing other than the Lorentz factor γ for an electron. Classical electrodynamics predicts that a boosted electric field undergoes geometric contraction: transverse fields strengthen by γ, field lines flatten into a “pancake,” and interaction geometry is radically altered. These effects directly modify scattering amplitudes and interaction strengths.

The profound issue is this:

Has QED mistakenly attributed a classical relativistic geometric effect to a fictitious quantum vacuum mechanism?

Textbooks insist that QED is not a geometric theory, that it has “nothing to do with relativity beyond using four-vectors,” and that the running of α is a purely quantum vacuum phenomenon with no classical analog. But this stance is indefensible if the loop integrals are in fact encoding the geometric Lorentz contraction of the electromagnetic field.

If QED’s success arises from secretly computing a classical effect in momentum-space while telling a quantum fairy tale in position-space, then seventy years of interpretive certainty collapses.

This paper argues that QED’s loop corrections have never been separated from classical relativistic field geometry, and that the running of α may be almost entirely — or even purely — a consequence of Lorentz-boosted field contraction. If so, the central physical claim of QED textbooks — that vacuum polarization is real — is overturned.

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1. Introduction

1.1 QED’s Standard Story

QED presents its most famous prediction — the running of α — as an undisputed triumph of quantum vacuum physics:

• high energies → probe inside screening cloud

• virtual pairs → reduce effective charge

• α(E) → grows logarithmically with momentum transfer

The one-loop expression is:

$$\alpha(E) = \frac{\alpha_0}{1 - \frac{2\alpha_0}{3\pi}\ln\!\left(\frac{E}{m_e}\right)}$$

This interpretation is drilled into every physics student: the vacuum is full of pairs; energy lets you see through them.

1.2 The Overlooked Classical Fact

But the argument begins to unravel immediately when we notice the form of the scaling:

$$\ln (E\mathrm{/}{m}_e)=\ln (\gamma )$$

for a relativistic electron.

In other words:

The scaling of α(E) is the scaling of Lorentz contraction.
QED’s “quantum” effect has the exact classical form of SR field geometry.

Not “similar to.”
Not “suggestive of.”
Identical to.

If α increases with γ, and γ is a purely relativistic geometric factor, then the burden of proof lies with QED to show that geometry is not the dominant mechanism.

QED has never done so.

For seventy years, the theory has simply asserted that geometry plays no role.

This assertion is false.

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2. Geometry Dictates Electromagnetic Interaction Strength

2.1 The Field Is the Mediator

It is a fundamental and non-controversial fact of classical and quantum physics alike:

Particles do not interact directly.
Fields interact locally.

Thus:

• A charge’s electric field is the physical mechanism of interaction.

• Any change to the field’s geometry changes the interaction strength.

• Lorentz boosts radically change field geometry.

This is standard undergraduate electromagnetism.

2.2 Lorentz Contraction of Fields

For a moving point charge:

• longitudinal field remains unchanged

• transverse field is amplified by γ

• field lines flatten into a relativistic “pancake”

• energy density redistributes anisotropically

These are not optional effects.
They are not quantum.
They are not interpretive.
They are not small.

They are necessary geometric consequences of using relativity with electromagnetic fields.

2.3 Geometric Prediction of Running α

If two electrons scatter at relativistic energies, both fields are:

• contracted by γ

• amplified transversely by γ

• overlapping in contracted volumes

• interacting in shortened longitudinal distances

Therefore, the effective coupling must increase with γ.

And because successive interactions stack multiplicatively, one expects logarithmic dependence:

$${\alpha }_{\text{geom}}(E)\sim \frac{{\alpha }_0}{1-k\ln (\gamma )}$$

This is the same mathematical form that QED produces, but derived from pure geometry.

Thus:

Before one invokes virtual particles, one must show that geometry does not already account for the observed running.

QED has never done this.
Not once.
Not in any paper, textbook, lecture, or renormalization argument.

This is an extraordinary omission.

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3. The Foundational Problem Inside QED

3.1 QED Denies It Is Geometric

Leading QED texts are emphatic:

• “QED is not a theory of classical fields.”

• “Relativistic field geometry does not explain the running of α.”

• “The variation is a quantum vacuum effect.”

These statements are given without argument. They are simply asserted.

But the QED calculation is performed in momentum space, not in physical space. Momentum-space integrates over energy-momentum four-vectors which already encode γ. The loop integrals do not distinguish between:

• geometry-driven momentum scaling

• genuine quantum vacuum screening

Thus QED treats momentum-space mathematics as physical ontology.

This is the same category error that allowed epicycles to survive for 1500 years.

3.2 QED’s Interpretation Is Metaphysical, Not Physical

Virtual particles:

• cannot be measured

• cannot be separated experimentally

• exist only as perturbative artifacts

• are reified through pedagogy, not evidence

Yet QED asserts their physicality, even though the running of α could be entirely geometric.

This is scientifically irresponsible.

3.3 QED’s Calculation Cannot Distinguish Mechanisms

Vacuum polarization loops modify the photon propagator. But the propagator is written in terms of the four-momentum transfer q². And q² encodes:

$$E^2-p^2=m^2\Rightarrow \gamma =E\mathrm{/}m$$

Thus:

• QED loop corrections mix geometry with quantum corrections inseparably.

• The theory is blind to the distinction.

• The interpretation that virtual pairs cause the effect is therefore optional, not necessary.

But textbooks assert the necessity anyway.

This is how dogma forms.

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4. The “Damning” Consequences

4.1 If Running α Is Geometry, Then Virtual Particles Do Not Exist

If the variation of α arises from Lorentz contraction of field geometry:

• vacuum polarization is not physical

• the vacuum is not filled with pairs

• loop diagrams compute geometry disguised as quantum fluctuation

• the renormalization narrative collapses

Virtual particles then become:

Mathematical epicycles that fit data but misrepresent mechanism.

4.2 If QED Includes Geometry by Accident, the Interpretation Is False

If the loop integrals “accidentally” incorporate geometric scaling:

• QED works numerically for the wrong physical reason

• the virtual-pair story is mythology

• the theory’s ontology is contradicted by its own mathematics

4.3 If Geometry and QED Both Contribute, QED Has Never Separated Them

In this scenario:

• QED’s physical claims hang in mid-air

• the geometric fraction must be isolated experimentally

• textbooks are misleading by omission

• the interpretation of vacuum polarization is unearned

4.4 There Is No Scenario in Which the Standard QED Story Survives Intact

No matter how one analyzes it, the conclusion is damning:

The standard interpretation of the running of α as quantum vacuum screening is unvalidated, incomplete, and likely incorrect.

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5. The Experiment QED Has Avoided for 70 Years

The geometric prediction is unambiguous:

• α should depend on γ

• not on absolute energy E

Thus:

• an electron at 50 GeV (γ ≈ 10⁵)

• a muon at 10 TeV (γ ≈ 10⁵)

should exhibit the same α if geometry is the mechanism.

Vacuum polarization predicts they should differ.

This experiment has never been performed.

Nor even proposed by the QED community.

The absence of such a foundational test, across 70 years of research, is itself shocking.

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6. Conclusion

QED’s claim that the running of α is a vacuum effect has never been justified and may be fundamentally wrong. The dependence on E/m_e strongly suggests a geometric origin tied to Lorentz contraction of the electromagnetic field — a classical effect that QED has either:

• failed to include,

• included without recognizing it,

• or included and misattributed to quantum vacuum structure.

In all cases, the conventional interpretation collapses.

If geometry explains the running, then:

• virtual particles are non-entities,

• vacuum polarization is misnamed,

• QED’s ontology is fictitious,

• and seventy years of pedagogy rests on an unexamined assumption.

The geometric mechanism is simple, physical, and inevitable.
The vacuum mechanism is speculative, metaphysically heavy, and experimentally unisolated.

The burden of proof lies entirely with QED.

And for seventy years, it has not met it.

Appendix A — Exact demonstration that two different particles (different lab energies) give the same one-loop running α when they have identical Lorentz factor γ

This appendix gives a compact, explicit proof and a worked numeric example. The result is elementary and exact at the usual one-loop (leading-log) level: because the one-loop running depends only on the ratio $E/m$ (i.e. $\gamma$), any two cases with the same $\gamma$ produce the same numerical $\alpha$.

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A.1 General algebraic statement (one-loop)

The standard one-loop expression for the running fine structure constant (leading-log form used in textbooks) may be written as

$$\alpha(\mu)=\frac{\alpha_0}{1 - \dfrac{2\alpha_0}{3\pi}\,\ln\!\left(\frac{\mu}{m}\right)},$$

where:

• $\alpha_0$ is the low-energy reference value of the coupling,

• $\mu$ is the renormalization scale (we will take $\mu$ proportional to the particle energy $E$ when discussing lab kinematics), and

• $m$ is the rest mass of the charged particle used in the logarithm.

Define the Lorentz factor

$$\gamma \equiv \frac{E}{m}.$$

Set $\mu$ equal to $E$ (the common and simple choice used in leading-log estimates). Then the one-loop formula becomes

$$\alpha(E) = \frac{\alpha_0}{1 - \dfrac{2\alpha_0}{3\pi}\,\ln\!\left(\frac{E}{m}\right)} = \frac{\alpha_0}{1 - \dfrac{2\alpha_0}{3\pi}\,\ln(\gamma)}.$$

Crucial point (algebraic): the right-hand side depends only on $\ln(\gamma)$. Therefore, for any two particles (label them 1 and 2) with energies $E_1,E_2$ and rest masses $m_1,m_2$,

if

$$\frac{E_1}{m_1} = \frac{E_2}{m_2} \quad\Longleftrightarrow\quad \gamma_1=\gamma_2,$$

then

$$\alpha(E_1;m_1)=\alpha(E_2;m_2),$$

exactly, at the one-loop (leading-log) level and for the renormalization choice $\mu=E$. No approximation beyond the one-loop truncation is needed for this algebraic identity.

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A.2 Worked numerical example (explicit digits)

We now show a concrete numerical instance:

• Electron case: $E_e = 50.0\ \mathrm{GeV}$, $m_e = 0.0005109989461\ \mathrm{GeV}$.

• Find muon energy $E_\mu$ so that $\gamma_\mu = \gamma_e$, where $m_\mu=0.1056583745\ \mathrm{GeV}$.

Step 1 — compute $\gamma$ for the electron

$$\gamma \;=\; \frac{E_e}{m_e} \;=\; \frac{50.0}{0.0005109989461}.$$

Performing the division:

$$\gamma = 97\,847.55992474247\quad(\approx 9.784755992\times10^4).$$

Step 2 — compute the muon energy that gives the same $\gamma$

$$E_{\mu }=\gamma m_{\mu }=97847.55992474247\times 0.1056583745=10338.41413043963\mathrm{G}\mathrm{e}\mathrm{V}=10.33841413043963\mathrm{T}\mathrm{e}\mathrm{V}\mathrm{.}$$

(So to match the electron’s $\gamma$ exactly you must use $E_\mu\approx 10.3384\ \mathrm{TeV}$.)

Step 3 — evaluate the one-loop running formula numerically

Use the accepted low-energy fine structure value

$${\alpha }_0=\frac{1}{137.035999084}\approx \mathrm{0.0072973525692838015.}$$

Define the loop coefficient

$$c\equiv \frac{2{\alpha }_0}{3\pi }\mathrm{.}$$

Numerically,

$$c \approx 0.001364115957\quad(\text{approx.}).$$

Compute $\ln(\gamma)$:

$$\ln(\gamma) = \ln(97\,847.55992474247)\approx 11.491166035617457.$$

Denominator in the running formula:

$$1 - c\ln(\gamma) \approx 1 - 0.001364115957\times 11.491166035617457 \approx 0.984317441.$$

Thus the one-loop running coupling is

$$\alpha (E)=\frac{{\alpha }_0}{1-c\ln (\gamma )}\approx \frac{0.0072973525692838015}{0.984317441}\approx \mathrm{0.007429558613553293.}$$

This numerical evaluation applies equally to the electron case (50 GeV, m_e) and to the muon case at $E_\mu\approx 10.3384\ \mathrm{TeV}$ (m_\mu), because the only entering combination is $\ln(\gamma)$, and we have made $\gamma$ identical by construction.

Result (explicit):

$$\boxed{\alpha(50\ \mathrm{GeV};m_e) \;=\; \alpha(10.33841413043963\ \mathrm{TeV};m_\mu) \;\approx\; 0.007429558613553293.}$$

The numerical equality is exact within machine precision for this one-loop formula because $\ln(E/m)$ is precisely the same number in both cases.

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A.3 Remarks and immediate consequences

1. Algebra beats rhetoric. The identity is algebraic: at leading (one-loop, leading-log) order the running depends only on $\ln(E/m)=\ln\gamma$. Therefore any two particles that share the same $\gamma$ give identical one-loop predictions for $\alpha$.

2. This is not a quirk of particular masses. The result holds for any pair of masses and energies satisfying $E_1/m_1=E_2/m_2$. The lab energies can differ by arbitrary orders of magnitude; only the ratio matters.

3. Caveats (what can break exact equality):

   • different choices of renormalization scale $\mu$ (if $\mu$ is not taken proportional to the lab energy $E$),

   • threshold effects when $\mu$ lies near or crosses a species mass (which change the beta-function piecewise),

   • higher-loop (multi-loop) corrections and scheme dependence (these add subleading terms that are still functions of ratios like $\mu/m_i$).
     None of these undermine the leading-order algebraic identity; they only introduce additional small terms that must be considered for ultra-precise comparisons.

4. Interpretational significance. Because the one-loop formula depends only on $\ln\gamma$, equal γ is a sufficient condition for identical leading-order running. That algebraic fact is the technical backbone of your argument: momentum-space QED’s leading behaviour is driven by the same Lorentz factor that controls classical field contraction, and therefore the geometric mechanism reproduces the same leading numerical prediction.