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The Fine-Structure Constant as a Scaled Quantity

Harry Sticker

Abstract

The fine-structure constant alpha approximately 1/137 is traditionally regarded as a fundamental dimensionless parameter. I argue instead that alpha is a scaled quantity that arises only where the structural scales contributed by classical electromagnetism (e), quantum mechanics (h-bar), and special relativity (c) intersect. None of these theories, taken individually, supplies the independent scales required to define alpha. The constant first appears when relativistic corrections are added to the Schrodinger-Bohr description of hydrogen (Sommerfeld), and it becomes the structural coupling in quantum electrodynamics, where quantum and relativistic effects modify the classical electromagnetic interaction. Expressing the governing laws in canonical form reveals this dependence and eliminates representational artifacts that make alpha appear fundamental. The running of alpha in QED further demonstrates its status as a scale-dependent coupling rather than a universal constant. I conclude that alpha is a domain-specific structural ratio reflecting contingent relationships among independent physical scales.

The Fine-Structure Constant as a Scaled Quantity

Abstract

The fine-structure constant alpha approximately 1/137 is traditionally regarded as a fundamental dimensionless parameter. I argue instead that alpha is a scaled quantity that arises only where the structural scales contributed by classical electromagnetism (e), quantum mechanics (h-bar), and special relativity (c) intersect. None of these theories, taken individually, supplies the independent scales required to define alpha. The constant first appears when relativistic corrections are added to the Schrodinger-Bohr description of hydrogen (Sommerfeld), and it becomes the structural coupling in quantum electrodynamics, where quantum and relativistic effects modify the classical electromagnetic interaction. Expressing the governing laws in canonical form reveals this dependence and eliminates representational artifacts that make alpha appear fundamental. The running of alpha in QED further demonstrates its status as a scale-dependent coupling rather than a universal constant. I conclude that alpha is a domain-specific structural ratio reflecting contingent relationships among independent physical scales.

Paper Structure

This paper contains 36 sections, 17 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Historical and Conceptual Background
  3. Descartes and the Birth of Algebraic Homogenization
  4. Fourier, Buckingham, and the Rise of Dimensional Analysis
  5. Electromagnetic Units and the Illusion of Fundamentality
  6. Rationalization, IUPAP 1932, and the Ideal of Clean Form
  7. Compact Notation and the Suppression of Scale
  8. Duff and the Dimensionless-Only Thesis
  9. Scaled Quantities and Their Structural Basis
  10. Reynolds Number: Dimensionless but Domain-Bound
  11. The Relativistic Parameter $\beta$: A Ratio That Requires a Theory
  12. What Scaled Quantities Are
  13. Why $\alpha$ Belongs to This Category
  14. The Emergence of $\alpha$ at Theory Junctions
  15. Classical Electromagnetism: Scale Underdetermination and Rescaling Symmetry
  16. ...and 21 more sections

Figures (1)

  • Figure 1: Schematic structural dependence of the fine-structure constant $\alpha$ on three independent frameworks. Classical electromagnetism contributes the charge scale $e$, quantum mechanics the action scale $\hbar$, and special relativity the invariant speed $c$. The arrows indicate that $\alpha$ is definable only when all three structures are jointly operative; they are not intended to represent causal or mathematical derivation.