Fine-structure constant

Sommerfeld introduced α in 1916 to explain the fine splitting of hydrogen's spectral lines, where it appears as the ratio of an electron's orbital speed in the first Bohr orbit to c. Written as α = e²/(4πε₀ħc), every unit cancels: it is a bare number, the same for any observer using any unit system, and therefore something a fundamental theory ought to predict. None does. Pauli was famously preoccupied by it, and Feynman called it "one of the greatest damn mysteries of physics".

Its smallness is what makes quantum electrodynamics calculable — perturbation series in α converge fast, which is why QED predicts the electron's magnetic moment to twelve digits. Beware the "1/137" shorthand: it is 1/137.036, and the exact 1/137 is numerology. α also runs with energy, rising to about 1/128 at the mass of the Z boson, because vacuum polarisation screens charge less at short distances.