Rowan Brad Quni-GudzinasThe fine-structure constant alpha ~ 1/137.036 is one of the most precisely measured numbers in physics, yet its origin remains unexplained. A new preprint reframes alpha not as an abstract coupling constant, but as a projective cross-ratio: the classical electron radius divided by the reduced Compton wavelength.
Full preprint: https://doi.org/10.5281/zenodo.20100109
#Physics #QuantumMechanics #Mathematics #ProjectiveGeometry #Science #Research #Academic #ParticlePhysics
Fine-Structure Constant as a Cross-Ratio: A Geometric Reframing of α
The fine-structure constant $\alpha \approx 1/137.036$ is one of the most precisely measured numbers in physics, yet its origin remains unexplained. The standard presentation—$\alpha = e^2/(4\pi\varepsilon_0\hbar c)$ with dimensionful constants—frames it as an abstract coupling strength, obscuring its geometric character. This document proposes a reframing: $\alpha$ is naturally understood as the cross-ratio of two measurable length scales characterizing the electron—the classical electron radius $r_e$ and the reduced Compton wavelength $\bar{\lambda}_C$. The reframing makes three contributions: (a) it reveals $\alpha$’s projective invariance structure, explaining why $\alpha$ is independent of unit choices and coordinate rescalings in geometric rather than algebraic terms; (b) it connects to the integer-ratio structure of precision experiments, where $\alpha$ is determined from rational observables (quantum Hall filling factors $\nu = p/q$, Penning trap frequency ratios $N_s/N_c)$; and (c) it integrates five complementary mathematical formalisms—adelic, projective, topological, syntactic, and hierarchical—that illuminate different aspects of $\alpha$’s cross-ratio nature. The document includes Python-verified quantitative results, historical context (Eddington, Wyler), experimental grounding, and an honest acknowledgment of limitations—most notably that $\alpha = r_e/\bar{\lambda}_C$ is an algebraic identity, not a first-principles derivation, and the contribution lies in conceptual reframing rather than numerical prediction.
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