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“The fine-structure constant is ubiquitous throughout physics. I’ve already noted its connection to the electromagnetic interaction. In atomic physics, the binding energy, fine-structure splitting, and Lamb shift are all proportional to powers of α. In condensed matter physics, α characterizes Josephson junction oscillations and quantum Hall resistance steps. In addition, α is an important component of our system of fundamental constants. [Physics Today]”
Gerald Gabrielse“Following the path of earlier unificationists, one of Eddington's aims was to reduce the contingencies in the description of nature, for example, by explaining the fundamental constants of physics rather than accepting them as merely experimental data. One of these constants was the fine-structure constant ..., which entered prominently in Dirac's theory and was known to be about 1/137.”
Helge Kragh, Quantum Generations: A History of Physics in the Twentieth Century“Physicists love this number not just because it is dimensionless, but also because it is a combination of three fundamental constants of nature. Why do these constants come together to make the particular number 1/137.036 and not some other number?”
John Archibald Wheeler, Geons, Black Holes, and Quantum Foam: A Life in Physics“But some numbers, called dimensionless numbers, have the same numerical value no matter what units of measurement are chosen. Probably the most famous of these is the "fine-structure constant," .... Physicists love this number not just because it is dimensionless, but also because it is a combination of three fundamental constants of nature.”
John Archibald Wheeler, Geons, Black Holes, and Quantum Foam: A Life in Physics