On the Invariant Nature of the Fundamental Electromagnetic Coupling Constant

J. Rogers, SE Ohio, 20 Oct 2025, 1130

Abstract

A persistent ambiguity in fundamental physics lies in distinguishing between constants that are artifacts of human-chosen unit systems and those that represent true, invariant features of reality. While constants like c, h, and G can have their numerical values changed by a choice of units, dimensionless constants such as the fine-structure constant, α, are believed to be more fundamental. This paper presents a formal dimensional consistency proof that the linear electromagnetic coupling constant, amp_force_natural (defined as α/2π), is a true scale-invariant constant. By postulating two foundational geometric constraints—one governing the nature of the spacetime substrate and the other governing the structure of the electron—we demonstrate that the value of amp_force_natural must remain unchanged under any arbitrary rescaling of length and mass. This result formally separates this constant from metrological artifacts and elevates it to the status of a fundamental, irreducible geometric property of the universe.

1. Introduction

The role of fundamental constants in physics is a subject of deep philosophical and physical inquiry. The development of natural unit systems, such as those proposed by Planck and Stoney, has shown that many dimensional constants (e.g., the speed of light c, the Planck constant h, the gravitational constant G) can be set to unity. This suggests that their specific numerical values in systems like the SI are artifacts of our anthropocentric choice of units for mass, length, and time.

However, dimensionless constants, most famously the fine-structure constant α, cannot be so easily dismissed. Its value of approximately 1/137 is independent of any unit system, suggesting it represents a deep, unexplained truth about the structure of reality.

This paper focuses on the closely related linear electromagnetic coupling constant, amp_force_natural = α/2π, which appears as the first-order term in the calculation of the electron's anomalous magnetic moment. We will provide a rigorous proof that amp_force_natural is not merely dimensionless by convention, but is a true scale-invariant constant. The proof rests on the mutual consistency of two fundamental geometric constraints that must hold true in any self-consistent description of our universe.

2. The Two Foundational Geometric Constraints

Our proof is built upon two axioms. The first describes the universal nature of the spacetime substrate when viewed through a natural unit system. The second provides a specific, empirical definition of amp_force_natural derived from the properties of the electron within a well-defined set of units.

2.1 The Spacetime Substrate Constraint

In any system of natural units where the speed of light c and the Planck constant h are set to unity (c = 1, h = 1), the dimensions of Mass [M], Energy [E], Frequency [f], and inverse Length [L⁻¹] become equivalent. This implies a fundamental, scale-free relationship between the dimensions of Mass and Length:

$$\mathrm{<span\; style="background-color:\; white;">M</span>}<span\; style="background-color:\; white;">\propto </span>{\mathrm{<span\; style="background-color:\; white;">L</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">1</span>}}$$

This constraint dictates the inherent geometry of the background spacetime substrate. It is a universal rule stating that mass and length are not independent physical dimensions but are inversely related once the mediating constants c and h are absorbed into the definitions. Any valid coordinate transformation must respect this underlying grammar.

2.2 The Electron Structure Constraint and the Definition of amp_force_natural

The second constraint is an empirical law concerning the nature of the electron itself, which we will use to define amp_force_natural. We analyze the electron's properties within a specific and well-defined natural unit system, which we will call the Stoney-GUT Scale, where c = h = e = 1, with e being the elementary charge. This is a common and powerful choice for analyzing the structure of grand unified theories (GUTs).

In this specific c=h=e=1 system, all quantities can be expressed as dimensionless ratios relative to a fundamental scale. The classical electron radius, r_e, is defined in SI units as e²/(4πε₀m_ec²). The fine-structure constant, α, is e²/(2ε₀hc). A direct substitution shows:

$${\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">e</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">\alpha </span>}<span\; style="background-color:\; white;">\cdot </span>\frac{\mathrm{<span\; style="background-color:\; white;">h</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">e</span>}}\mathrm{<span\; style="background-color:\; white;">c</span>}}$$

In a system where c=h=1, this becomes:

$${\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">e</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">\alpha </span>}<span\; style="background-color:\; white;">\cdot </span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">e</span>}}}$$

Rearranging this equation, we get a product of mass and length:

$${\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">e</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">e</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">\alpha </span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}}$$

We now define the constant amp_force_natural as this specific, dimensionless product, which is physically realized by the electron's properties in the c=h=e=1 framework:

$$\text{<span style="background-color: white;">amp\_force\_natural</span>}<span\; style="background-color:\; white;">\equiv </span>{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">e</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">e</span>}}<span\; style="background-color:\; white;">(</span>\text{<span style="background-color: white;">in a </span>}\mathrm{<span\; style="background-color:\; white;">c</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">h</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">e</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1</span>}\text{<span style="background-color: white;"> system</span>}<span\; style="background-color:\; white;">)</span>$$

Therefore, our second constraint is the physical law that the electron's structure necessarily fixes this product to the invariant value of α/2π. This axiom defines the electron not as a point particle with arbitrary properties, but as a specific, stable geometric structure where its mass-nature and its spatial-nature are bound by this observable, invariant product.

3. The Invariance Proof

The proof proceeds by subjecting this system of constraints to an arbitrary coordinate transformation and demonstrating that the value of amp_force_natural remains unchanged.

Let us consider an arbitrary rescaling of our fundamental unit of length by a non-zero, dimensionless factor k.

Step 1: The Rescaling Transformation
Let our initial coordinate system be described by units of length L and mass M. We introduce a new coordinate system (the "primed" system) where the unit of length L' is defined as:

$${\mathrm{<span\; style="background-color:\; white;">L</span>}}^{\mathrm{<span\; style="background-color:\; white;">\prime </span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">k</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">L</span>}$$

Step 2: Application of the Substrate Constraint
The Spacetime Substrate Constraint (M ∝ L⁻¹) is a fundamental law that must hold true in all coordinate systems. Therefore, the rescaling of length forces a corresponding inverse rescaling of mass. The new unit of mass M' must be:

$${\mathrm{<span\; style="background-color:\; white;">M</span>}}^{\mathrm{<span\; style="background-color:\; white;">\prime </span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">M</span>}}{\mathrm{<span\; style="background-color:\; white;">k</span>}}$$

Step 3: Testing the Electron Constraint in the New System
We now test if the Electron Structure Constraint remains valid in the new, primed coordinate system. We calculate the product of the electron's mass and classical radius as measured in the primed units (m'_e, r'_e).

$${\text{<span style="background-color: white;">Product</span>}}^{\mathrm{<span\; style="background-color:\; white;">\prime </span>}}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">e</span>}}^{\mathrm{<span\; style="background-color:\; white;">\prime </span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">e</span>}}^{\mathrm{<span\; style="background-color:\; white;">\prime </span>}}$$

Substituting the transformations from Steps 1 and 2:

$${\text{<span style="background-color: white;">Product</span>}}^{\mathrm{<span\; style="background-color:\; white;">\prime </span>}}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>\frac{{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">e</span>}}}{\mathrm{<span\; style="background-color:\; white;">k</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">k</span>}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">e</span>}}<span\; style="background-color:\; white;">)</span>$$

Step 4: The Invariant Result
The arbitrary scaling factor k cancels out perfectly:

$${\text{<span style="background-color: white;">Product</span>}}^{\mathrm{<span\; style="background-color:\; white;">\prime </span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">k</span>}}{\mathrm{<span\; style="background-color:\; white;">k</span>}}<span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">e</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">e</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">e</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">e</span>}}<span\; style="background-color:\; white;">)</span>$$

Since we know from our initial premise that m_e · r_e = amp_force_natural, we have proven that:

$${\text{<span style="background-color: white;">Product</span>}}^{\mathrm{<span\; style="background-color:\; white;">\prime </span>}}<span\; style="background-color:\; white;">=</span>\text{<span style="background-color: white;">amp\_force\_natural</span>}$$

The numerical value of the product is identical in both the original and the rescaled coordinate systems. This holds true for any and all possible values of the scaling factor k.

4. Discussion and Implications

The success of this dimensional consistency proof is not a trivial mathematical result. It has profound implications for our understanding of fundamental constants.

4.1. A True Geometric Invariant: We have proven that amp_force_natural is a true geometric invariant. Unlike G, whose numerical value is entirely dependent on our choice of kilograms and meters, or c and h, which can be absorbed into the dimensions themselves, amp_force_natural is a pure number whose value is independent of any and all possible choices of scale. It is a structural constant of the universe, analogous to geometric constants like π.

4.2. Separating Physics from Metrology: This proof provides a formal methodology for distinguishing genuine physical invariants from metrological artifacts (unit conversion factors). Constants that remain invariant under this type of dimensional analysis represent the true, irreducible geometric structure of reality. Constants whose values can be altered are part of the coordinate system—the map, not the territory.

5. Conclusion

We have presented a formal proof demonstrating that the linear electromagnetic coupling constant, amp_force_natural, is a scale-invariant feature of our universe. This was achieved by showing that two foundational geometric constraints—one describing the spacetime substrate and the other describing the electron's structure—are mutually consistent under any arbitrary rescaling of units.

This result confirms that the value of amp_force_natural is not an accident of our measurement system. It is a fundamental, dimensionless constant that reflects the unchangeable geometric relationship between mass and length as embodied by the electron. The goal of fundamental physics, therefore, should be to not only measure such invariants with greater precision, but to derive their values from the single, underlying geometric principle that governs all of reality.