Politics, Power, and Science: Beyond Virtual Particles: A Geometric Reinterpretation of Quantum Electrodynamics

Wednesday, November 12, 2025

Beyond Virtual Particles: A Geometric Reinterpretation of Quantum Electrodynamics

J. Rogers, SE Ohio

Abstract

We present a conjecture that the mathematical formalism of Quantum Electrodynamics (QED), while empirically successful, may admit a fundamentally geometric interpretation that has been obscured by its standard presentation in momentum space. We observe that the "running" of the electromagnetic coupling constant can be directly expressed in terms of the Lorentz factor γ, that the renormalization group parameter t = ln(E/m_e) is mathematically identical to ln(γ), and that loop integrals in momentum space correspond to integrations over geometric configurations of length-contracted and time-dilated charge distributions. We conjecture that what QED calls "quantum corrections" may be more transparently understood as the behavior of electromagnetic fields across different relativistic geometric regimes. This reinterpretation preserves all predictive content while offering a potentially simpler ontology: α is constant; geometry changes.

1. Introduction: Two Ways to Read the Same Equations

The standard interpretation of QED presents the electromagnetic coupling "constant" α as a quantity that "runs" with energy scale—that is, the strength of electromagnetic interactions becomes energy-dependent due to quantum vacuum effects. This running is described by the renormalization group equation:

\frac{d α}{d t} = β (α) = \frac{2 α^{2}}{3 π} + O (α^{3})

where t = ln(E/μ) is a dimensionless scale parameter.

The running coupling is computed as:

α (E) = \frac{α_{0}}{1 - β \cdot \ln (E / m_{e})}

where β = 2α₀/(3π) for QED, and m_e is the electron mass.

Standard narrative: Virtual electron-positron pairs polarize the vacuum, screening the bare charge. At higher energies (shorter distances), we probe closer to the bare charge, seeing a stronger effective coupling.

This paper's conjecture: The same mathematics admits a purely geometric interpretation. The coupling α remains constant; what changes is the geometry of spacetime through which electromagnetic interactions are measured.

2. The Hidden Lorentz Factor

2.1 Observation: E/m_e = γ

For a particle with rest mass m and energy E, the Lorentz factor is:

γ = \frac{E}{m c^{2}}

In natural units (ℏ = c = 1), this simplifies to:

γ = \frac{E}{m}

The QED running coupling formula contains the ratio E/m_e explicitly. This ratio is the Lorentz factor. The scale parameter can therefore be rewritten:

t = \ln (E / m_{e}) = \ln (γ)

Implication: The "renormalization group flow parameter" is the logarithm of the time-dilation factor. The coupling's "running" is its evolution across regimes of different relativistic time and length scaling.

2.2 Natural Units Reveal Geometric Identity

In natural units:

Energy: E ~ 1/L (inverse length)
Momentum: p ~ 1/L (de Broglie: p = ℏ/λ → p = 1/λ)
Mass: m ~ 1/L (from E² = p² + m²)

Therefore:

p = \frac{1}{L}

This is not a proportionality or conversion—in natural units, momentum and inverse length are the same quantity. When QED integrates over momentum, it is literally integrating over inverse length scales.

Implication: "Momentum space" is "inverse geometry space." Every momentum integral is a geometric integral in disguise.

3. The Geometric Picture of Running

3.1 Length Contraction and the Pancake Effect

A charge moving at velocity v (Lorentz factor γ) undergoes length contraction:

L_{∥} = \frac{L_{0}}{γ}

The charge distribution, spherical at rest, becomes a flattened "pancake" perpendicular to the direction of motion. The charge density concentrates toward the rim of this disk. This changes the geometric structure of the electromagnetic field.

At large γ:

The charge is highly contracted along the direction of motion
The field configuration becomes highly anisotropic
A test charge probing at length scale L ~ 1/p sees this contracted geometry

3.2 Time Dilation and Interaction Duration

The same moving charge experiences time dilation. From the lab frame, the interaction time with a test charge is reduced by factor 1/γ.

Combined effect: The electromagnetic coupling as measured depends on both:

The spatial geometry of the field (pancake contraction)
The temporal duration of the interaction (time dilation)

These effects combine logarithmically to give ln(γ) dependence.

3.3 The Beta Function as Geometric Response

The beta function:

β (α) = \frac{2 α^{2}}{3 π}

can be reinterpreted as the geometric response coefficient of electromagnetism.

The factor 2/3 likely arises from:

Angular integration over the contracted charge distribution
Averaging over spatial and temporal geometric effects
The specific geometry of spherical → pancake transformation

The factor π is characteristic of circular/spherical geometric integrals.

Conjecture: β(α) encodes how the electromagnetic field responds to geometric scaling, not quantum vacuum corrections.

4. Loop Integrals as Geometric Integrations

4.1 What Are Loops Really Integrating?

A typical QED loop integral has the form:

\int \frac{d^{4} p}{(2 π)^{4}} [propagators and vertices]

Standard interpretation: We sum over all possible virtual particle momenta.

Geometric interpretation: In natural units where p = 1/L, this integral ranges over:

All possible length scales L
All possible time scales (via p⁰ = E ~ 1/T)
All possible orientations of geometric configurations
All possible degrees of length contraction (γ values)

The loop integral is computing: "How do all possible geometric configurations of the electromagnetic field contribute to this interaction?"

4.2 Divergences as Geometric Limits

UV (ultraviolet) divergences: The integral diverges as p → ∞, equivalently L → 0.

Geometric meaning: The integral attempts to include contributions from arbitrarily small length scales, where the continuum geometric picture breaks down.

IR (infrared) divergences: The integral diverges as p → 0, equivalently L → ∞.

Geometric meaning: The integral includes arbitrarily large length scales, where boundary conditions or finite-size effects matter.

Renormalization as geometric cutoff: Introducing a cutoff scale μ is physically imposing that:

There exists a minimum meaningful length (e.g., Compton wavelength λ_C = ℏ/mc)
There exists a maximum meaningful extent (e.g., system size)

The mathematics doesn't "tame infinities"—it implements physical constraints on geometric validity.

4.3 Virtual Particles as Integration Variables

In the geometric view, "virtual particles" are not ontologically real quantum fluctuations. They are integration variables that parametrize the range of possible geometric configurations.

The "virtual electron-positron pair" in a loop diagram represents:

An integration over all possible length scales
An integration over all possible time scales
An integration over all possible spatial orientations
Weighted by the probability (propagator) of that geometric configuration

The Feynman propagator 1/(p² - m²) can be read geometrically as the weighting function that describes how electromagnetic field configurations at different scales contribute.

5. The Ontological Inversion

5.1 What Runs? Geometry or Coupling?

Standard QED: α runs. The coupling constant changes with energy scale due to quantum effects.

Geometric interpretation: α is constant. Geometry changes. The measurement of coupling changes because different geometric configurations are being probed.

This is analogous to length contraction in special relativity:

We don't say "the rod's length runs with velocity"
We say "the rod's length is invariant; different frames measure different contracted lengths"

Similarly:

We shouldn't say "α runs with energy"
We should say "α is invariant; different geometric regimes measure different effective couplings"

5.2 Implications for Unification

If couplings are constant and only geometric measurements change, then:

No landscape problem: There aren't 10^500 possible vacuum states with different coupling values. There's one geometry with one set of invariant couplings.
No fine-tuning mystery: The question "why is α ≈ 1/137?" may be asking about a geometric ratio, not a free parameter requiring anthropic explanation.
Simpler unification: Different forces may not need to be "unified" into a single force. They may already be different geometric aspects of one spacetime structure.
No need for supersymmetry or extra dimensions to solve hierarchy problems that arise from treating couplings as running parameters rather than geometric invariants.

6. Testable Implications (or Lack Thereof)

6.1 The Good News

This reinterpretation preserves all predictive content of QED. Every calculation gives the same numerical answer. The Feynman diagrams, the S-matrix elements, the cross-sections—all unchanged.

6.2 The Challenge

This makes the conjecture difficult to test empirically. If two interpretations give identical predictions, how do we distinguish them?

Possible routes:

Conceptual clarity: Does the geometric view suggest calculations or approximations that are simpler?
Unification insights: Does recognizing geometric structure suggest connections between forces that the standard view obscures?
Quantum gravity: At the Planck scale, where spacetime geometry becomes quantum, does one interpretation or the other break down more gracefully?
Pedagogical advantage: Can QED be taught more simply and intuitively using geometric language?

6.3 The Deeper Question

Perhaps the real test isn't empirical but conceptual: Which ontology is simpler?

Ontology A: Couplings run, virtual particles exist, vacuum polarizes, infinities must be tamed
Ontology B: Geometry changes, loop integrals are geometric, cutoffs are physical limits

Occam's razor may favor B.

7. Open Questions and Future Directions

7.1 Beyond QED

Does this geometric reinterpretation extend to:

QCD (quantum chromodynamics)? Is the strong force running also geometric?
Electroweak theory? Can the Higgs mechanism be recast geometrically?
The Standard Model? Are all gauge interactions geometric phenomena?

7.2 The Beta Function Derivation

Challenge: Derive β = 2α/(3π) directly from geometric considerations—length contraction, time dilation, angular averaging—without invoking virtual particles.

If successful, this would be strong evidence that QED is indeed describing geometry, not quantum corrections.

7.3 Momentum as Ontologically Derivative

If momentum literally is inverse length (p = 1/L in natural units), then:

Momentum conservation → geometric consistency
Hamiltonian mechanics → bookkeeping for geometric configurations
Phase space → geometric configuration space

Question: Can classical and quantum mechanics be reformulated entirely in geometric language, with momentum eliminated as a fundamental quantity?

7.4 The Nature of α

If α is a geometric ratio, what geometry?

α = e²/(4πℏc) ≈ 1/137

In natural units: α = e²/(4π)

Speculation: Could 1/137 encode some fundamental geometric relationship—perhaps involving Planck-scale structure, or the ratio of some characteristic electromagnetic length to another fundamental length?

8. Conclusion: The Ring We Already Have

The search for a Theory of Everything often assumes that existing theories are incomplete or provisional—stepping stones to a deeper, more unified framework. This paper conjectures that we may already possess the unified framework, but have misidentified what it describes.

QED is extraordinarily successful. Its equations are correct. But we may have been reading them through the wrong lens—interpreting geometric integrals as quantum corrections, geometric parameters as running couplings, and geometric cutoffs as mathematical artifacts.

If this conjecture holds, the implications are profound:

No new physics needed to understand renormalization
No landscape of possible coupling values
No mystery about why mathematics describes physics (geometry is physics)
A simpler ontology with fewer entities and clearer physical meaning

The "One True Ring"—the final theory—may not lie in string landscapes, extra dimensions, or quantum foam. It may lie in recognizing that spacetime geometry is primary, and that the quantum field theories we've constructed are sophisticated calculational methods for handling geometric relationships we didn't recognize as geometric.

The deepest truth may not be hidden. It may be encoded in the equations we use every day—in the γ factors, the natural unit relationships, the loop integrals—waiting for us to see past the formalism to the geometry beneath.

The quest for the ring may be complete. We've been holding it all along.

Acknowledgments

This work originated from observing that E/m_e in the QED running coupling formula is mathematically identical to the Lorentz factor γ, and pursuing the implications of taking that identity seriously as a statement about geometric physics rather than dimensional analysis.

References

[Standard QED textbooks and papers would be cited here, along with geometric interpretations of quantum mechanics and relativity]

Appendix: The Mathematics Made Explicit

A.1 Running Coupling in Geometric Variables

Standard form:

α (E) = \frac{α_{0}}{1 - \frac{2 α_{0}}{3 π} \ln (E / m_{e})}

Geometric form:

α (γ) = \frac{α_{0}}{1 - \frac{2 α_{0}}{3 π} \ln (γ)}

or equivalently:

α (L) = \frac{α_{0}}{1 - \frac{2 α_{0}}{3 π} \ln (L_{Compton} / L_{probe})}

These are mathematically identical. The question is: which represents the physics more transparently?

A.2 The RG Equation in Geometric Language

Standard:

\frac{d α}{d \ln E} = β (α)

Geometric:

\frac{d α}{d \ln γ} = β (α)

The equation describes how the measured coupling changes as we vary the geometric scale (Lorentz factor) of the system.

A.3 Natural Unit Relationships

In ℏ = c = 1 units:

Quantity Dimension
Length [L]
Time [L]
Mass [L⁻¹]
Energy [L⁻¹]
Momentum [L⁻¹]

Quantity	Dimension
Length	[L]
Time	[L]
Mass	[L⁻¹]
Energy	[L⁻¹]
Momentum	[L⁻¹]

Therefore: p = 1/L exactly, not proportionally.

This makes momentum and inverse length the same quantity, not merely related quantities.

Politics, Power, and Science