[Paper Review] The fine structure constant and numerical alchemy

G. Dattoli|arXiv (Cornell University)|Sep 9, 2010

Various Chemistry Research Topics4 citations

TL;DR

This paper proposes a numerically exact formula for the fine structure constant α using only π and the integer 137, reproducing the experimental value to the last significant digit. The derivation is purely numerical and lacks physical motivation, framed as 'numerical alchemy'—a playful exploration of mathematical coincidences rather than a physically grounded theory.

ABSTRACT

We comment on past and more recent efforts to derive a formula yielding the fine structure constant in terms of integers and transcendent numbers. We analyse these "exoteric" attitudes and describe the myths regarding α, which seems to have very ancient roots, tracing back to Cabbala and to medieval alchemic conceptions. We discuss the obsession for this constant developed by Pauli and the cultural "environment" in which such an "obsession" grew. We also derive a simple formula for α in terms of two numbers π and 137 only. The formula we propose reproduces the experimental values up to the last significant digit, it has not any physical motivation and is the result of an alchemic combination of numbers. We make a comparison with other existing formulae, discuss the relevant limits of validity by comparison with the experimental values and discuss a criterion to recover a physical meaning, if existing, from their mathematical properties.

Motivation & Objective

• To investigate historical and cultural roots of the obsession with the fine structure constant α, particularly in Cabbala and medieval alchemy.
• To analyze the intellectual environment that shaped Wolfgang Pauli’s fascination with α, especially its connection to the number 137.
• To propose a purely numerical formula for α based on π and 137 that matches experimental values with high precision.
• To assess the validity and limitations of such formulae in comparison with experimental data.
• To explore whether any physical meaning can be extracted from the mathematical properties of these numerical coincidences.

Proposed method

• Derives a formula for α using only π and the integer 137, based on numerical experimentation rather than physical principles.
• Employs algebraic manipulation of transcendental and integer constants to achieve agreement with the measured value of α.
• Compares the proposed formula with existing numerical formulae for α, evaluating their accuracy and mathematical structure.
• Uses the experimental value of α ≈ 1/137.035999084 as a benchmark for validation.
• Applies a criterion to assess whether such numerical formulae could, in principle, be linked to deeper physical theories.
• Analyzes the formula's structure to identify patterns or symmetries that might hint at physical significance, despite its lack of theoretical foundation.

Experimental results

Research questions

• RQ1 Can the fine structure constant be expressed with perfect numerical accuracy using only π and the integer 137?
• RQ2 What historical and cultural factors contributed to the mystique surrounding the number 137 and its connection to α?
• RQ3 Why did Wolfgang Pauli exhibit such a deep personal fascination with the fine structure constant?
• RQ4 To what extent can purely numerical formulae for α be considered physically meaningful, even if they match experimental data?
• RQ5 Are there mathematical properties in such formulae that could, in principle, be linked to physical laws or symmetries?

Key findings

• The proposed formula reproduces the experimental value of the fine structure constant, α ≈ 1/137.035999084, to the last significant digit.
• The formula is derived through numerical manipulation without any physical justification, classifying it as 'numerical alchemy'.
• The paper identifies a long-standing cultural and historical fascination with the number 137, tracing back to Cabbala and medieval alchemy.
• Pauli’s obsession with α and the number 137 is contextualized within a broader intellectual tradition of seeking mystical or fundamental significance in mathematical constants.
• Despite its accuracy, the formula lacks physical motivation and cannot be derived from known physical theories.
• The paper suggests that while such formulae may not be physically meaningful, their mathematical structure could still be analyzed for potential deeper insights.