Fundamental Nature of the Fine-Structure Constant

Arnold Sommerfeld introduced the fine-structure constant that determines the strength of the electromagnetic interaction. Following Sommerfeld, Wolfgang Pauli left several clues to calculating the fine-structure constant with his research on Johannes Kepler’s view of nature and Pythagorean geometry. The Laplace limit of Kepler’s equation in classical mechanics, the Bohr-Sommerfeld model of the hydrogen atom and Julian Schwinger’s research enable a calculation of the electron magnetic moment anomaly. Considerations of fundamental lengths such as the charge radius of the proton and mass ratios suggest some further foundational interpretations of quantum electrodynamics.


Introduction
In addition to introducing the fine-structure constant [1]- [5], Arnold Sommerfeld added elliptic orbits to Bohr's atomic model deriving the Bohr-Sommerfeld model [6,7]. Then Wolfgang Pauli was influenced by Sommerfeld's search for the Platonic connections that were implied by the mystery of the fine-structure constant [2,8,9]. As the fine-structure constant determines the electromagnetic strength its theoretical origin was for Pauli a key unsolved physical problem, also considered especially significant by Max Born, Richard Feynman and many other physicists [10].
The fine-structure constant, alpha, has a variety of physical interpretations from which various determinations have been made; atom interferometry and Bloch oscillations, the neutron Compton wavelength measurement, AC Josephson effect, quantum Hall effect in condensed matter physics [11], hydrogen and muonium hyperfine structure, precision measurements of helium fine-structure, absorption of light in graphene [12], also the topological phenomena in condensed matter physics [13], the relative optical transparency of a plasmonic system [14], elementary particle lifetimes [15] and the anomalous magnetic moment of the electron in quantum electrodynamics [16]. Slightly different values of the fine-structure constant are found from different experimental measurements [17] and finally there is also the question of its variation from the subatomic to the cosmological triangles [24] together with the prime constant [28], described as a binary expansion corresponding to an indicator function for the set of prime numbers. Calculation of the inverse fine-structure constant as an approximate derivation from prime number theory: with an approximate value of α −1 ≃ 137.035 999 168, with three prime numbers and the prime constant. The square of the diagonal of a "prime constant rectangle" is The polygon circumscribing constant κ is the reciprocal of the Kepler-Bouwkamp constant [29] related to "Pauli's triangle" with sides approximately proportional to 1, ϕ, √ ϕ √ 5 with the golden ratio ϕ = (1 + √ 5)/2 [30]. 180 − 23 = 157 and 360 − 23 = 337. 23 + 37 = 60 and 60/ϕ ≃ 37 [24]. The triangles 85, 132, 157 and 175, 288, 337 are primitive Pythagorean triples. Defining ρ for p(k) as the k-th prime: with the prime constant ρ ≃ 0.414 682 509 851 111 and κ again as a reciprocal of the Kepler-Bouwkamp constant [29]. Also, ρ ≃ with the approximate value of α −1 ≃ 137.035 999 168, same value as determined in Eq. (2) from above. Plato's favorite symbolic number 5040 = 7!. The polygon circumscribing constant, the reciprocal of the Kepler-Bouwkamp constant [29], is also formulated as a converging series involving the Riemann zeta function ζ(s) found in the perturbative determination of the electron magnetic moment anomaly from quantum electrodynamics.
From optical physics, α/2π ≃ exp(−πϕn) where n is the index of refraction for water. Snell's Law: n 1 sin θ 1 = n 2 sin θ 2 , where θ 1 is the angle between the ray and the surface normal in the first medium, θ 2 is the angle between the ray and the surface normal in the second medium and n 1 and n 2 are indices of refraction (n 1 ≃ 1 in a near vacuum and n 2 > 1 in a transparent substance). With angle of incidence 45 • the angle of refraction is 32 • (Pauli's World Clock [24]) for water and n 2 ≃ 1.
relating the golden ratio and inverse fine-structure constant. Other approximations, sin The eccentricity of a golden ellipse [36] (14). The constant µ is the real fixed point of the hyperbolic cotangent and λ is the Laplace limit of Kepler's equation [37]. Kepler's equation for calculating orbits: where M is the mean anomaly, E is the eccentric anomaly and ε is the eccentricity [38]- [42]. With suggestive elliptic connections from the study of polarized light, Tse and MacDonald [43], in their theory of "magneto-optical Faraday and Kerr effects of thin topological insulator films," find a "Faraday angle equal to the fine structure constant" and an approximate π/2 Kerr rotation [44,45]. If the eccentricity ε = λ, which is the Laplace limit of Kepler's equation, and E = tan −1 (α −1 ) ≃ π/2 in the parametric form of Kepler's equation with radians; then the mean anomaly M is: ) and the sec(2π/7) ≃ ϕ [24]. From Kepler's triangle (1, √ ϕ, ϕ) and the heptagon, tan(2π/7) ≃ √ π/2 and tan −1 ( √ ϕ) is an approximate heptagon angle 2π/7, with √ ϕ ≃ 4/π. The diameter of the circumscribing sphere of the regular dodecahedron with side equal to one is ρκ ≃ 2πγ. Also, the outer radius The square root of phi is described by the dodecahedron proportions. The silver constant from the heptagon: The regular heptagon is related to the origin of calculus, the cycloid curve, the least action principle and the squaring of the circle; see the historical references in [24]. Silver The real fixed point of the hyperbolic cotangent is µ and γ ≃ µ/λπ [46], with the Laplace limit of Kepler's equation again, see Eq. (14). The cosh ρ ≃ sec(π/7) = 2/ √ S. In another reference relating to relevant geometry Li, Ji and Cao discovered that Fibonacci spirals found on conical patterns in nature can be effectively modeled as a least energy configuration [47].
also gives the value for α −1 ≃ 137.035 999 168 with Eq. (19) and the value for τ ≃ 137.038 431 610. The factor of 7 also appears as 2 × 7 2 = 98 and 7 3 = 343. Eq. (20) represents a particular quartic plane curve, different combinations of the coefficients of the general curve give rise to the lemniscate of Bernoulli. Gauss's and Euler's study of the arc length of Bernoulli's lemniscate, a polar curve having the general form of a toric section, led to later work on elliptic functions. Some forms for Gauss's constant [56]: Lemniscate constant L = πG ≃ 2.622 057 554 292 119 ≃ cosh ϕ and G ≃ 0.8346 is Gauss's constant, the reciprocal of the arithmetic-geometric mean of 1 and √ 2, the basis for his exploration of the lemniscate function [54]. Gauss's constant is also linked with the beta function, the gamma function at argument 1/4 and Jacobi theta functions.
The fundamental length g M is from the work of Mendel Sachs [66] on the spinorquaternion formulation for the wave function of the electron in hydrogen and prediction for the Lamb shift of the hydrogen spectrum. Sachs presents another viewpoint on the anomalous magnetic moment of the electron, developing a continuous field concept without assuming point charges and their problems with infinities. The explanation has some correlations with the golden ratio geometry of Wolfgang Pauli's World Clock [24].
W (x), the Lambert W -function, is an analog of the golden ratio for exponentials as exp[−W (1)] = W (1). The sin α −1 ≈ 6Ω/πϕ. The cosh Ω ≃ λ/Ω and the csch ν ≃ µ, where µ again is equal to the coth µ and ν is the real fixed point of the hyperbolic secant: Also, m w /m H • ≃ ν/µ ≃ √ ϕ/2 ≃ 2/π, an approximation with the golden ratio geometry. From the Foundation Stone of classical harmonic theory, the alpha harmonic is equal to the sum of the golden ratio harmonic and the omega constant harmonic, relating the Greek Pythagorean form of the fine-structure constant to the golden ratio geometry [24].

Conclusion
From Arnold Sommerfeld's introduction of the fine-structure constant and Wolfgang Pauli's search for an explanation, approximate values for the fine-structure constant have been determined. With an extension of the Keplerian intuition regarding the fundamental geometry of basic polygons, conic sections and Platonic polyhedra included in Wolfgang Pauli's World Clock geometry; the mathematical and physical model for the calculations is an alternative to accounting for individual contributions of the interactions between field quanta and begins to address some of the questions raised by Richard Feynman, Freeman Dyson, Paul Dirac and others about quantum electrodynamics [70].
The polygon circumscribing constant, reciprocal of the Kepler-Bouwkamp constant, has interesting geometric connections with the torus topology relating the relativity of Einstein with the geometry of classical and quantum mechanics. The nature of this topology was pursued by Wolfgang Pauli and is suggested in the Keplerian paradigm of Pythagorean harmonic proportions from ancient geometry [24]. Quantum mechanics is thus found to be an approximation theory based on the classical mathematical problem of squaring the circle, with all of its complex analogies and philosophical implications.