Quote from https://physics.nist.gov/cuu/Constants/alpha.html “The fine–structure constant α is of dimension 1 (i.e., it is simply a number) and very nearly equal to 1/137. It is the “coupling constant” or measure of the strength of the electromagnetic force that governs how electrically charged elementary particles (e.g., electron, muon) and light (photons) interact.“
Quote from wikipedia: “Based on the precise measurement of the hydrogen atom spectrum by Michelson and Morley, Arnold Sommerfeld extended the Bohr model to include elliptical orbits and relativistic dependence of mass on velocity. He introduced a term for the fine-structure constant in 1916. The first physical interpretation of the fine-structure constant α was as the ratio of the velocity of the electron in the first circular orbit of the relativistic Bohr atom to the speed of light in the vacuum. Equivalently, it was the quotient between the minimum angular momentum allowed by relativity for a closed orbit, and the minimum angular momentum allowed for it by quantum mechanics. It appears naturally in Sommerfeld’s analysis, and determines the size of the splitting or fine-structure of the hydrogenic spectral lines.”
Quote from bigthink.com: “What’s special about alpha is that it is regarded as the best example of a pure number, one that doesn’t need units. It actually combines three of nature’s fundamental constants – the speed of light, the electric charge carried by one electron, and the Planck’s constant, as explains physicist and astrobiologist Paul Davies to Cosmos magazine. Appearing at the intersection of such key areas of physics as relativity, electromagnetism and quantum mechanics”
Richard Feynman (1918 – 1988) who was instrumental in the Manhattan project has this to say about it: “There is a most profound and beautiful question associated with the observed coupling constant, e ? the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won’t recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. it's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the “hand of God” wrote that number, and “we don’t know how He pushed his pencil.” We know what kind of a dance to do experimentally to measure this number very accurately, but we don’t know what kind of dance to do on the computer to make this number come out, without putting it in secretly”
Wolfgang Pauli (1900–1958) Nobel Prize-winning physicists was obsessed with it his whole life. Pauli joked and said “When I die my first question to the Devil will be: What is the meaning of the fine-structure constant?”
Laurence Eaves, Physicist & professor at the University of Nottingham said number 137 would be the one you should use to signal aliens that we a good understanding of quantum mechanics & nuclear physics.
Werner Heisenberg: “all the quandaries of quantum mechanics would shrivel up when 137 was finally explained.” Ian Stewart a professor in physics says: “Only three constants are significant for star formation: … the fine-structure constant”
British astronomer Paul Davies declared: “God is a pure mathematician! The physical Universe does seem to be organized around elegant mathematical relationships. And one number above all others has exercised an enduring fascination for physicists: 137.035999.... It is known as the fine-structure constant... Alpha (α).”