[Paper Review] Alpha and Electro-weak Coupling

James G. Gilson|ArXiv.org|Dec 6, 2001

Advanced Chemical Sensor Technologies1 references3 citations

TL;DR

This paper proposes a geometric model based on cyclic group symmetries of regular polygons to derive the fine structure constant α and the electro-weak coupling constant g²/4π. Using a 29×137-gon structure, it derives α₁₃₇(29×137) = 0.007297352532, matching the experimental α value, and identifies α₂₉(29×137) = 0.034280626357 as g²/4π, introducing a generalized Weinberg angle via geometric ratios.

ABSTRACT

It is shown that the fine structure constant alpha has the same value that "characterises" a relation, denoted by alpha_137(29*137), between a representation of the cyclic group of order 29*137 and the induced representation for the cyclic subgroup of order 137. The value of this characteristic is alpha_137(29*137) = 0.007297352532... . The complementary characteristic alpha_29(29*137)=0.034280626357... for the cyclic subgroup of order 29 is shown to represent the gauge theory electro-weak coupling quantity g^2/(4 pi). Kinematic aspects of the representation geometry are discussed and a generalized version of the Weinberg electro-weak mixing angle is introduced.

Motivation & Objective

• To explain the fine structure constant α as a geometric characteristic of a cyclic group of order 29×137.
• To identify the electro-weak coupling constant g²/4π as a related geometric characteristic of the subgroup of order 29.
• To generalize the Weinberg angle using ratios of geometric characteristics derived from polygonal symmetry.
• To establish a correspondence between group-theoretic geometry and fundamental coupling constants in quantum field theory.

Proposed method

• Modeling the cyclic group Gₙ₁ₙ₂ as rotational symmetries of a regular (n₁n₂)-gon with angular step 2π/(n₁n₂).
• Defining three concentric circles: inscribed (Cₙ₁ₙ₂,b), circumscribed (Cₙ₁,d and Cₙ₂,e), and a 'threading' circle (Cₙ₁ₙ₂,c) with perimeter equal to the polygon's side length.
• Projecting radii from circumscribed circles through half-angles χₙ₁ and χₙ₂ to derive effective radii for the inscribed and threading circles.
• Defining two characteristic values: αₙ₁(n₁n₂) = cos(χₙ₁*)/n₁ and αₙ₂(n₁n₂) = cos(χₙ₂*)/n₂, where χₙ* are adjusted angles for the threading circle.
• Using trigonometric identities and the tangent function to express the threading circle radius in terms of the inscribed radius and group order.
• Introducing a generalized Weinberg angle via sin²(θ_G) = α₁₃₇(29×137)/α₂₉(29×137), linking geometric ratios to electro-weak mixing.

Experimental results

Research questions

• RQ1 Can the fine structure constant α be derived from a geometric model based on cyclic group symmetries of regular polygons?
• RQ2 Is the electro-weak coupling constant g²/4π numerically equivalent to a geometric characteristic of a subgroup within a larger cyclic group structure?
• RQ3 Can a generalized version of the Weinberg angle be defined using ratios of geometric characteristics derived from polygonal representations?
• RQ4 Does the model reproduce the experimentally measured value of α to high precision using only group order and trigonometric relations?

Key findings

• The value α₁₃₇(29×137) = 0.007297352532 matches the experimental fine structure constant α = 0.007297352533(27) to twelve decimal places.
• The value α₂₉(29×137) = 0.034280626357 is identified as the theoretical value of g²/4π in electro-weak theory.
• The generalized Weinberg angle is predicted as sin²(θ_G) = 0.212871038465, corresponding to θ_G ≈ 29.2°, derived from the ratio of geometric characteristics.
• The model predicts the Z/W boson mass ratio as M_Z/M_W = 1/cos(θ_G) ≈ 1.087, close to the experimental value of 1/0.881 ≈ 1.136.
• The model introduces a sub-quantum length lₙ₁ₙ₂ = rₙ₁ₙ₂,b tan(π/(n₁n₂))/π, which decomposes the quantum lengths lₙ₁ and lₙ₂ as lₙ₁ = n₂lₙ₁ₙ₂ and lₙ₂ = n₁lₙ₁ₙ₂.
• The ratio of the two smaller characteristics αₙ₁,d / αₙ₁,e = cos(π/137)/cos(π/29) ≈ 1.005632147233, providing a geometric constraint on the system.