The Fine-Structure Constant: A Unit Ratio Disguised as a Fundamental Constant

J. Rogers, SE Ohio, 07 Apr 2025, 1750

Abstract: The fine-structure constant α, traditionally viewed as a fundamental dimensionless constant of nature, is shown here to be a direct consequence of the relationships between our definitions of electric charge, current, mass, and length. This paper demonstrates that α can be expressed as 2π times a ratio of SI unit definitions, implying that its numerical value reflects the metrological structure of our unit system rather than a deep mystery of physical reality. This understanding helps us improve our understanding of how unit systems work. This understanding is more important than the mystery.

---

1. Introduction

The fine-structure constant, α ≈ 1/137.035999, appears ubiquitously in quantum electrodynamics and has been historically regarded as one of the most enigmatic constants in physics. However, a closer examination reveals that its value is not mysterious but rather reflects the interdependent structure of the SI unit definitions.

---

2. Formula for α

If we write out the traditional form of the fine-structure constant:

$$\alpha =\frac{e^2}{4\pi {\epsilon }_0\mathrm{\hslash }c}$$
By going clear back to the definition of the amp through many levels of the definitions we arrived numerically at 

2pi * e**2  * 10^-7 /( Hz_kg * c) = 1/ 137.03599 

• 2pi is the actual real geometric physics. 
• e is the definition of charge it was originally defined by the amp
•  Hz_kg is the ratio between frequency and kg h/c^2 This defines kg versus second through Hz.
• c is the definition of the meter against the second 
• 10^-7 is the definition of the force for amp.
  
  

We express the fine-structure constant as:

$$\mathrm{<span\; style="background-color:\; white;"><span\; style="font-family:\; times;">\alpha </span></span>}<span\; style="background-color:\; white;"><span\; style="font-family:\; times;">=</span></span>\mathrm{<span\; style="background-color:\; white;"><span\; style="font-family:\; times;">2</span></span>}\mathrm{<span\; style="background-color:\; white;"><span\; style="font-family:\; times;">\pi </span></span>}<span\; style="background-color:\; white;"><span\; style="font-family:\; times;">\cdot </span></span><span\; style="background-color:\; white;"><span\; style="font-family:\; times;">(</span></span>\frac{{\text{<span style="background-color: white;"><span style="font-family: times;">(charge\_definition</span></span>}}^{\mathrm{<span\; style="background-color:\; white;"><span\; style="font-family:\; times;">2</span></span>}}<span\; style="background-color:\; white;"><span\; style="font-family:\; times;">\cdot </span></span>\text{<span style="background-color: white;"><span style="font-family: times;">amp\_definition)</span></span>}}{\text{<span style="background-color: white;"><span style="font-family: times;">(kg\_definition</span></span>}<span\; style="background-color:\; white;"><span\; style="font-family:\; times;">\cdot </span></span>\text{<span style="background-color: white;"><span style="font-family: times;">meter\_definition)</span></span>}}<span\; style="background-color:\; white;"><span\; style="font-family:\; times;">)</span></span>$$

This representation highlights that the only non-arbitrary geometric component is 2π, while the rest is a scaling ratio between the definitions of SI base units.  ' Unit_definition' refers to the numerical definition of the specified unit of measure charge in the SI unit system.

What this shows is is the ratio that scales the 2π is built into the relationship between length, mass, time and the way we define ampere and charge. This ratio does tie back into how our math and units of measures model space and time for motion and force and charge.  

---

3. Unit Dependency

The SI base units are not independent when physical laws are expected to remain invariant. For instance: Force depends on  on the definitions of the kilogram and meter, and they all depend on the definition of the second. Ampere is used to define the elemental charge through a chain of other constants.

Interdependence of units

• 
  

  3.1 Why Force Depends on kg and Meter Definitions

  Force (F = ma) explicitly ties mechanical units to electromagnetism:

  1. Mass (kg) and length (meter) and time (second) define the newton (N = kg·m/s²).

  2. Electromagnetic forces (e.g., between currents in Ampère’s law) must match mechanical forces to preserve consistency in physical laws.
     Example: If we redefine the kilogram (e.g., scale it by 10), the newton scales by 10. To keep Ampère’s force law consistent, the ampere must rescale by 10, propagating changes to charge because of the relationships between these units in our unit system of measurement, keeping α the same.

  3. 
     

  3.2 Compensatory Rescaling

  • 
    

  • If we fix e (elementary charge) as a constant (e.g., e = 1 in natural units), the ampere’s definition changes. This requires scalings in the definitions of the ampere, kilogram, or meter to preserve the internal coherence of the system and these definitional changes result in the same numerical value of α.

  • 
    

  • This forces compensatory adjustments to kg and meter definitions to preserve F = ma and F_EM = (μ₀ I₁ I₂ L) / (2π d)  (Ampère's law).

    
    Simply because of how our unit system and the definitions of how the units of force, length, charge, and mass are defined, that defines this ratio.  
    
    This is essentially a system with N variables but only N-1 degrees of freedom.  If you set any three, you fix the last unit of measure scaling.

• 
  

---

4. Interlocking Definitions

The ratio:

$$\frac{{\text{<span style="background-color: white;"><span style="font-family: times;">(charge\_definition</span></span>}}^{\mathrm{<span\; style="background-color:\; white;"><span\; style="font-family:\; times;">2</span></span>}}<span\; style="background-color:\; white;"><span\; style="font-family:\; times;">\cdot </span></span>\text{<span style="background-color: white;"><span style="font-family: times;">amp\_definition)</span></span>}}{\text{<span style="background-color: white;"><span style="font-family: times;">(kg\_definition</span></span>}<span\; style="background-color:\; white;"><span\; style="font-family:\; times;">\cdot </span></span>\text{<span style="background-color: white;"><span style="font-family: times;">meter\_definition)</span></span>}}$$

is simply a reflection of how force (F = ma) and electromagnetic definitions relate in the SI system. Changes to one definition propagate through the system via interdependent constraints. Thus, α is not revealing a fundamental physical secret, but a consistent balancing of units in our unit system.  This is the definition of how mass and length define force, how the amp units defines the charge units. 

4.1 Explicit Example

1. Original SI (pre-2019):
   Ampère defined via force (arbitrary human value) between parallel wires.
   Kilogram defined via a physical artifact (Le Grand K).
   Meter defined via wavelength of light.
   α emerged from balancing these definitions.

    

   Revised SI (post-2019): This fixed the values by decree.
   Ampère defined by fixing e = 1.602176634 \times 10^{-19} , \text{C}.
   Kilogram defined via Planck’s constant (h). 

    

   Result: α remains unchanged because the ratio of unit definitions is preserved.  

---

5. Conclusion

When reinterpreted through the lens of unit definition, α ceases to be a mystery. It is 2π multiplied by a dimensionless ratio of unit definitions, tied together by physical laws like F = ma and electromagnetic force expressions. Its constancy across measurements simply reflects the coherence of our metrological framework.

This reframing demystifies α and underscores the importance of examining how unit choices shape our perception of constants and to see how the choices we make in our models of the universe relate to each other.  

Appendix, program to allow setting of constants and seeing what amp force becomes.

For e = 1.0              Amp is 10^-44.59057921229325

For e = 1.0              Amp is 10^-2.935014538797864 for c=h=e=1

For e = 1.602176634e-19  Amp is 10^-6.999999999763218

For e = 2.56697e-38      Amp is 10^30.59057921276681

cat alphasolver01.py 

import math

def solve_for_x(alpha, Hz_kg, c, e):

    """

    Solves 2π * e^2 * 10^x / (Hz_kg * c) = alpha for x, given e as a parameter.

    

    Parameters:

        alpha (float): the value of the fine-structure constant (e.g., 1/137.03599).

        Hz_kg (float): the scaling ratio between frequency and kilogram (unit-specific constant).

        c (float): speed of light in m/s.

        e (float): the electric charge unit value (e.g., elementary charge).

    

    Returns:

        x (float): the exponent such that 10^x satisfies the equation.

    """

    

    # Our equation is:

    # 2π * e^2 * 10^x / (Hz_kg * c) = alpha

    # Rearranging:

    # 10^x = (alpha * Hz_kg * c) / (2π * e^2)

    numerator = alpha * Hz_kg * c

    denominator = 2 * math.pi * e**2  # Include e^2 here

    ten_power_x = numerator / denominator

    # To solve for x, we take the logarithm (base 10)

    x = math.log10(ten_power_x)

    return x

# Example parameter values:

# alpha: fine-structure constant

alpha = 1/137.035999084

# c: speed of light in m/s

c = 299792458  # exact SI value

h = 6.62607015e-34  # Planck's constant in Joule-seconds

# Calculate Hz_kg = h / c^2

Hz_kg = h / (c ** 2)

# e: elementary charge. Let's test with e = 1 for simplicity.

e_value = 1.0  

print(f"For e = {e_value:<16} Amp is 10^{solve_for_x(alpha, Hz_kg, c, e_value)}")

print(f"For e = {e_value:<16} Amp is 10^{solve_for_x(alpha, 1, 1, e_value)} for c=h=e=1")

e_value = 1.602176634e-19 

print(f"For e = {e_value:<16} Amp is 10^{solve_for_x(alpha, Hz_kg, c, e_value)}")

e_value = 1.602176634e-19**2 

print(f"For e = {e_value:<16.8} Amp is 10^{solve_for_x(alpha, Hz_kg, c, e_value)}")