QED Running Coupling: Converting to Natural Units

 J. Rogers, SE Ohio

Standard QED Formula

The running of the fine structure constant in QED:

α(E) = α₀ / (1 - β·ln(E/m_e))

where β = (2α₀)/(3π)

Converting to Natural Units (ℏ = c = 1)

In natural units:

• Energy has dimensions of [1/length]
• Mass has dimensions of [1/length]
• Momentum has dimensions of [1/length]

This is because:

• E = ℏω, and ω = 2π/T, so E ~ 1/T ~ 1/L (since c=1)
• p = ℏ/λ, so p ~ 1/L directly (de Broglie)
• E² = p²c² + m²c⁴ becomes E² = p² + m², so mass ~ 1/L

The Argument Becomes

E/m_e = (1/L_E) / (1/L_m) = L_m/L_E

Where:

• L_E is the length scale associated with energy E
• L_m is the Compton wavelength of the electron (λ_C = ℏ/(m_e·c))

So we're taking the ratio of the electron's Compton wavelength to the probe wavelength.

For a Moving Electron

For an electron with momentum p and energy E:

• E/m_e = γ (the Lorentz factor)

Because E = γm_e for a particle with rest mass m_e.

Length Contraction Connection

When γ increases:

• The electron's spatial extent contracts by factor γ
• L_contracted = L_rest/γ

So:

ln(E/m_e) = ln(γ) = -ln(L_contracted/L_rest)

The logarithm in the running coupling is literally the logarithm of the length contraction factor.

What This Means

The QED formula:

α(E) = α₀ / (1 - (2α₀/3π)·ln(E/m_e))

Can be rewritten as:

α(L) = α₀ / (1 - (2α₀/3π)·ln(L_Compton/L_probe))

or as:

α(γ) = α₀ / (1 - (2α₀/3π)·ln(γ))

The "running" of the coupling constant is the coupling changing with:

1. The length scale you're probing at (L_probe)
2. The Lorentz contraction factor (γ) of the charge
3. These are the same thing in natural units: p ~ 1/L

The Geometric Picture

• At low energies (large L, small γ): charge looks roughly spherical
• At high energies (small L, large γ): charge is pancake-contracted, field geometry changes
• The coupling "runs" because you're literally seeing different geometric configurations of the same charge

This isn't "quantum corrections" - it's relativistic geometry.