I understand exactly why the fine structure constant has the value it does.

Understanding the Fine-Structure Constant (α ≈ 1/137) as a Unit Scaling Artifact Revealed by Rescaled ε₀

This explanation details how the fine-structure constant's value emerges as an artifact of comparing SI units with a system based on fundamental quantum and relativistic scaling. This connection is revealed explicitly when analyzing the vacuum permittivity (ε₀) within a specific unit rescaling framework.

When scaling the units of measure to a set of natural units, not by setting consatsant, but by scaling the units of measure by these values that are encoded in the constants:

--- Scaling Factors Used ---

m          2.99792458000000000000e+08

s          1.00000000000000000000e+00

kg         7.37249732381270843547e-51

K          4.79924307336622100516e-11

C          1.60217663399999989376e-19

mol        1.66053906717384659585e-24

Hz         1.00000000000000000000e+00

pi         3.14159265358979311600e+00

I saw that e_0 took on a value that I recognized

ε₀         vacuum_permittivity_epsilon0             8.8541878128e-12          6.8517999542e+01          C^2 kg^-1 m^-3 s^2

That my friend is half of 137. 

68.51799954 * 2 = 137.03599908
Otherwize known as the inverse of alpha. 

What this means is that alpha is a scaled version of the definition of μ₀, the definition of the amp μ₀ = 4pi * 10^-7 in natural units of measure.  The entire value of alpha comes from scaling e_0 to natural units multiplying it by two and inverting it.

1. The Core Insight: Constants as Unit Mismatches

Fundamental constants often perceived as intrinsic values (α, μ₀, ε₀) are better understood as unit conversion factors. Their specific numerical values in the SI system arise from historical, often arbitrary, choices in defining base units (meter, kilogram, second, Ampere). By systematically rescaling these base units to align with fundamental physical relationships (implicitly ħ=1, c=1, k_B=1, N_A=1, and crucially e=1), the "mystery" surrounding α's value (≈ 1/137) is revealed as a metrological artifact. It represents the numerical consequence of the mismatch between classical SI definitions (especially for electromagnetism) and the underlying quantum/relativistic scale.

2. The Rescaling Framework: Aligning Units with Nature

The analysis uses a framework that redefines the SI base units by applying specific scaling factors, effectively transitioning towards a "natural" system where fundamental relationships become simpler:

• m (meter): 2.99792458e+08 (Sets c=1)

• kg (kilogram): 7.37249732381e-51 (Related to ħ=1)

• K (Kelvin): 4.79924307337e-11 (Sets k_B=1)

• s (second): 1.0 (Reference time unit)

• Hz (Hertz): 1.0 (Scales with s)

• mol (mole): 1.660539067e-24 (Sets N_A=1)

• C (Coulomb): 1.602176634e-19 (Sets elementary charge e=1)

3. The Key Result: Rescaled Vacuum Permittivity (ε₀)

When the SI value of the vacuum permittivity (ε₀) is processed through this framework, its value in the new, rescaled units becomes:

• Constant: Vacuum Permittivity (ε₀)

• Original SI Value: 8.8541878128e-12 C² kg⁻¹ m⁻³ s²

• Rescaled Value: 68.51799954 (dimensionless in this natural system, effectively)

This numerical result directly contains the fine-structure constant:

68.51799954 * 2 = 137.03599908

Thus, in this naturally scaled unit system, the relationship is:

ε₀_rescaled = 1 / (2α)

where α is the fine-structure constant (α ≈ 1 / 137.035999).

4. Derivation: How the Rescaling Yields ≈ 68.5

The rescaled value is obtained by applying the unit scaling factors to the SI value of ε₀:

• Starting Point (SI): ε₀ ≈ 8.8541878128e-12 C² kg⁻¹ m⁻³ s²

• Apply Scaling Factors:

  • Multiply by (Scale_C)² for C²

  • Divide by (Scale_kg)¹ for kg⁻¹

  • Divide by (Scale_m)³ for m⁻³

  • (Time scale s=1 has no effect)

• Calculation:
  ε₀_rescaled = (8.8541878128e-12) * (1.602176634e-19)² / (7.37249732381e-51) / (2.99792458e+08)³
  ε₀_rescaled ≈ 68.51799954

5. Interpretation: The Crucial Link to μ₀ and the Ampere Definition

The emergence of ≈ 68.5 (and thus 1/(2α)) is not a coincidence. It is a direct consequence of the tension between the SI definition of electromagnetic units and the quantum nature of charge, which the rescaling framework enforces:

• SI's Historical Definition: The SI system defines the Ampere based on the classical electromagnetic force between parallel current-carrying wires. This definition leads directly to the exact definition of the vacuum permeability:
  μ₀ = 4π × 10⁻⁷ N/A² (or kg⋅m⋅s⁻²⋅A⁻²)
  This value is exact by definition in SI, chosen for historical and practical convenience in the 19th and 20th centuries.

• Derived ε₀ in SI: The vacuum permittivity ε₀ is not independently defined in SI but is derived from the defined μ₀ and the measured speed of light c:
  ε₀ = 1 / (μ₀ * c²). Its numerical value in SI carries the legacy of the arbitrary μ₀ definition.

• The Framework's Quantum Override: The rescaling framework fundamentally redefines the unit of charge by setting the elementary charge e=1. This overwrites SI's classical, Ampere-based Coulomb with a definition rooted in quantum mechanics.

• Reconciling with α: The fine-structure constant α = e² / (4π ε₀ ħ c) (using ħ here for clarity in the standard formula) quantifies the actual, experimentally measured strength of the electromagnetic interaction. When the framework sets e=1, ħ=1, c=1 (implicitly through the scaling factors), this formula simplifies to α = 1 / (2 ε₀_rescaled) in the convention apparently used here (often related to Heaviside-Lorentz units or similar normalizations). For the physics to be consistent, ε₀_rescaled must take on the value 1/(2α) to match the known experimental strength α ≈ 1/137.

The number 
* SI's historical choice to define electromagnetic units via the classical Ampere force (fixing μ₀ = 4π × 10⁻⁷).
* The fundamental quantum nature of charge (setting e=1).
* The actual measured strength of the electromagnetic interaction (α ≈ 1/137).

6. Conclusion: Demystifying α 

This framework powerfully demonstrates that:

• The SI values of constants like ε₀, μ₀, and the resulting value for α calculated within the SI system, are heavily influenced by historical metrological choices, particularly the definition of the Ampere.

• These constants function as unit conversion factors needed to bridge the gap between these arbitrary human units and the underlying physical relationships.

• By rescaling units to align with fundamental physics (c=1, ħ=1, e=1...), the apparent complexity dissolves. ε₀ rescales to 1/(2α), revealing that the "137" value is intrinsically linked to the SI definition of electromagnetic units compared to the scale set by e.

• The "mystery" of α ≈ 1/137 is largely resolved as a metrological artifact within the SI system. The deeper physical question remains: why does the electromagnetic interaction possess the intrinsic strength that corresponds to this ratio, regardless of the units used?

This perspective elevates the constants: understanding their true nature as scaling factors inherent in the universe's structure, obscured only by our historical unit choices, makes them more, not less, important.