Fine Structure Constant.pdf

The fine-structure constant α is a dimensionless number in physics with a value known to 12 significant figures, but its mathematical significance remains unclear. Richard Feynman described α as a "magic number" with no understanding of its origin. Mathematician I.J. Good argued that a numerological explanation for α would only be acceptable if it came from a more fundamental theory providing a Platonic explanation of its value.

In contrast, the value of π was fully understood by Archimedes over 2,000 years ago, and Euler discovered a deep relation between π and complex numbers. However, physicists have struggled to find a similar mathematical explanation for α.

The paper proposes a mathematical model for α, denoted by the Cyrillic letter Ж, which arises from a fundamental Platonic theory called renormalization. This theory involves a flow of change of scale, which physicists think of as energy, and numbers get renormalized under this flow. The direction of the flow depends on whether numbers increase or decrease, and the standard convention is that energy increases.

The model connects Ж to π and e, answering Feynman's question about the origin of α. The paper also introduces another Cyrillic letter Ч, related to Ж by the formula Ч/γ = Ж/π, where γ is Euler's constant. This formula helps with the computation of Ж, and the paper provides a fast algorithm to produce a decimal expansion of Ж to 12 decimal places.

The main strategy of the paper involves extending the Archimedes-Euler approach to π from the commutative world of complex numbers to the non-commutative world of quaternions. The key idea is based on von Neumann's hyperfinite factor of type II, which is constructed by an infinite sequence of iterated exponentials. The paper also uses the Hirzebruch formalism, which introduces a formal algebraic process of multiplicative sequences, and the Todd exponential, whose generating function is the Bernoulli function.

The paper aims to provide a rigorous and elegant mathematical model for α, which can be interpreted as giving a rigorous foundation for the results derived from Feynman diagrams without the heavy computational work. The model is based on solid mathematical foundations and provides a Platonic explanation for the value of α, answering Feynman's question about the origin of this mysterious number.