The charge related constants become directly related to α

 Let's examine these values in the c=h=k=e=Na=1 natural units and see how they connect to our previous discussion, particularly regarding α.

Key Relationships in Natural Units (c=1, h=1, e=1):

1. Fine-Structure Constant (α): α = e² / (4πε₀ħc). With e=1, ħ=h/(2π)=1/(2π), c=1, this becomes α = 1² / (4πε₀_natural * (1/2π) * 1) = 1 / (2ε₀_natural).

   • Therefore, ε₀_natural = 1 / (2α).

2. Permeability (μ₀): We know ε₀μ₀ = 1/c². With c=1, this means ε₀_natural * μ₀_natural = 1.

   • Substituting ε₀_natural = 1/(2α), we get (1/(2α)) * μ₀_natural = 1.

   • Therefore, μ₀_natural = 2α.

3. Coulomb Constant (kₑ): kₑ = 1 / (4πε₀).

   • In natural units: kₑ_natural = 1 / (4π * ε₀_natural).

   • Substituting ε₀_natural = 1/(2α), we get kₑ_natural = 1 / (4π * (1/(2α))) = 2α / (4π) = α / (2π).

Checking Against Your Table:

• α ≈ 1 / 137.035999 ≈ 0.00729735

• μ₀ (Permeability):

  • Theory predicts: μ₀_natural = 2α ≈ 2 * 0.00729735 ≈ 0.0145947

  • Your table shows: 1.4594705139e-02

  • Match! The table correctly shows μ₀ = 2α in these natural units.

• ε₀ (Permittivity):

  • Theory predicts: ε₀_natural = 1 / (2α) ≈ 1 / 0.0145947 ≈ 68.51799

  • Your table shows: 6.8517999542e+01 (which is 68.51799...)

  • Match! The table correctly shows ε₀ = 1/(2α) and ε₀μ₀ = 1.

• kₑ (Coulomb Constant):

  • Theory predicts: kₑ_natural = α / (2π) ≈ 0.00729735 / (2π) ≈ 0.0011614

  • Your table shows: 1.1614097322e-03

  • Match! The table correctly shows kₑ = α/(2π).

Interpretation:

This perfectly illustrates the point. In the SI system, μ₀ had a defined, exact value (4π × 10⁻⁷) linked to the force definition of the Ampere, and ε₀ was derived via c. In the c=h=e=1 natural units:

• The elementary charge e is the fundamental unit of charge (set to 1).

• The values of μ₀, ε₀, and kₑ are no longer arbitrary or based on historical force definitions. Instead, their numerical values are directly determined by the fine-structure constant α (and factors of π).

• μ₀ becomes 2α.

• ε₀ becomes 1/(2α).

• kₑ becomes α/(2π).

This shows how the "arbitrary" scaling factors embedded in SI (like the 10⁻⁷ in μ₀) disappear in natural units, revealing the underlying physical relationships dictated by dimensionless constants like α.