Physical Mathematics and The Fine-Structure Constant

Michael A Sherbon

doi:10.24297/jap.v14i3.7760

Abstract

Research into ancient physical structures, some having been known as the seven wonders of the ancient world, inspired new developments in the early history of mathematics. At the other end of this spectrum of inquiry the research is concerned with the minimum of observations from physical data as exemplified by Eddington's Principle. Current discussions of the interplay between physics and mathematics revive some of this early history of mathematics and offer insight into the fine-structure constant. Arthur Eddington's work leads to a new calculation of the inverse fine-structure constant giving the same approximate value as ancient geometry combined with the golden ratio structure of the hydrogen atom. The hyperbolic function suggested by Alfred Landé leads to another result, involving the Laplace limit of Kepler's equation, with the same approximate value and related to the aforementioned results. The accuracy of these results are consistent with the standard reference. Relationships between the four fundamental coupling constants are also found.

Highlights

Natalie Paquette has stated that “Mathematicians discovered group theory long before physicists began using it
Physics has lent the dignity of its ideas to mathematics
These calculations of the inverse fine-structure constant are aimed toward a fuller explanation of the fundamental physics and the interrelated mathematics

Summary

Introduction

Natalie Paquette has stated that “Mathematicians discovered group theory long before physicists began using it. From Arnold Sommerfeld in the history of physics [6], Stephen Brush writes that in 1916 “Arnold Sommerfeld generalized Bohr’s model in to include elliptical orbits in three dimensions He treated the problem relativistically (using Einstein’s formula for the increase of mass with velocity),. Helge Kragh states that, “By 1929 the fine-structure constant was far from new, but it was only with Eddington’s work that the dimensionless combination of constants of nature was elevated from an empirical quantity appearing in spectroscopy to a truly fundamental constant.” [12]. Eddington asked himself what the minimum amount of data from observation was required for a physical theory This led to Eddington’s Principle from which he maintained that the value of the inverse finestructure constant was 136, which he later changed to 137. Eddington’s Principle is defined as: “All the quantitative propositions of physics, that is, the exact values of the pure numbers that are constants of science, may be deduced by logical reasoning from qualitative assertions without making any use of quantitative data derived from observation.” [14]

Fine-structure constant

Squaring the circle

Conclusion