The Fine Structure Constant, identified by the Greek letter α is a number used in quantum physics calculations. The current estimate is that it's around 0.0072973525628 - or roughly 1/137.
It has been called the “coupling constant” - or the measure of the strength of the electromagnetic force that governs how electrically charged elementary particles (e.g., electron, muon) and light (photons) interact. It was introduced in 1916 by physicist Arnold Sommerfeld in order to explain the observed splitting (or fine structure) of the energy levels of the hydrogen atom.
It is a crucial factor in many of the fundamental calculations in quantum physics.
There is no explanation for the constant's value, other than calculations which show that if it deviated by more than ± 4% or so, the structure of the universe would by so radically different that life could not have evloved, and so no-one would be around to observe and quantify it.
A further mystery is that it appears not to be constant. When discrepancies were originally identified, it was assumed that they could be explained by experimental error. The predominat curent view is that it is changing by a mesureable ammount each year. (Technical details here )
Further info : Introduction to the constants for nonexperts US NIST Physical Measurement Laboratory
Also see :
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