The fine-structure constant, written α, is the number that sets how strongly light and electrons interact, and it is about 1/137 because that is what every measurement finds. Nobody knows why. The constant has no units, so its value is not an accident of metres or seconds, and no theory has ever derived it from anything deeper. The official 2022 value of 1/α is 137.035999177: twelve digits, measured rather than explained. A pure number that sits inside every atom, with no reason anyone can give for it, is why 137 has haunted physics for more than a century.
Richard Feynman turned the puzzle into a duty. In his 1985 book QED: The Strange Theory of Light and Matter he wrote that the number had 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.”
It is an odd thing to worry about. Most numbers in physics carry units: a speed in metres per second, a mass in kilograms. This one carries nothing. An alien civilisation, with its own rulers and its own clocks, would measure exactly the same value. And yet it decides the size of atoms, and so the colours of everything. Change it, and the world would look different.
A split in hydrogen’s red line
The case begins with a line of light. Glowing hydrogen does not give off a smear of colour but a set of sharp lines, and one of them is red. In 1887 Albert Michelson and Edward Morley looked at that red line very closely and concluded that “the red hydrogen line must be a double line”. Two colours, a hair apart. Nobody could say why.
Twenty-six years later, in 1913, Niels Bohr offered a picture of the atom. The electron travels on fixed orbits around the nucleus; when it jumps from one orbit to another it gives off light of one exact colour. Each jump, one line. Bohr’s model explained hydrogen’s lines, but not the tiny split. In Bohr’s atom the red line should have been single.
In 1916 Arnold Sommerfeld, in Munich, took up the problem with two changes. First, he let the orbits be ellipses, not only circles. Second, he added Einstein’s special relativity. An electron on an ellipse swings close to the nucleus and moves fast there, and relativity says a fast-moving electron gains a little mass. Its energy shifts, very slightly, and one line becomes two.
The size of the split depended on a new number. Sommerfeld added to Bohr’s equations what he called die charakteristische Konstante unserer Feinstrukturen, the characteristic constant of our fine structures, and wrote it with the Greek letter α. Its value was about 1/137. It even has a picture of its own: in Bohr’s smallest orbit, the electron moves at α times the speed of light, about 2,200 kilometres per second. Fast enough for relativity to leave a mark, but only a small one. The tiny gap in a red line had given physics its strangest number.
Why the fine-structure constant has no units
Written out in full, the constant is
α = e² / (4π ε₀ ħ c)
where e is the charge of the electron, ε₀ is the electric constant of empty space, ħ is Planck’s constant and c is the speed of light. Four constants of nature, each measured in its own units. Put them together and the units cancel: first the coulombs, then the metres, the seconds and the kilograms. The video animates them leaving one by one. Nothing is left but a pure number.
This is why the missing units matter. The speed of light depends on what we call a metre and what we call a second. α does not. Measure it in feet, in hours, in units invented on another planet, and the answer is the same. Any civilisation, using any units, gets this value.
What it measures is the strength of the grip between electrons and light. Feynman described it through a single idea, an amplitude: “the amplitude for a real electron to emit or absorb a real photon”. Square that amplitude and you get about 1/137. In 1985 he noted that physicists liked to remember it as “the inverse of its square: about 137.03597”, a figure since sharpened to the twelve digits above. It is a small number, which means the grip is loose.
And α reaches all the way into the atom. Atomic energies scale with α². Make α a little larger or smaller and those energies change; the size of atoms would change, the colours of the light they give off would change, and so would all of chemistry. Every flame, every leaf, every drop of water depends on this one fraction.
The men who tried to explain 137
Arthur Eddington, one of the most famous astronomers of his day, believed α did not need to be measured at all. It could be derived, he thought, by pure reasoning. Around 1929 he argued that its inverse must be exactly 136. Not about. Exactly. Then the experiments came out closer to 137, and Eddington argued again: now it must be exactly 137. A man who reasoned his way to one number, then reasoned his way to the next. The story goes that a satirical magazine gave him a nickname for it: Sir Arthur Adding-One.
Wolfgang Pauli, one of the founders of quantum mechanics, was fascinated by 137 for the rest of his life. He even worked with the psychoanalyst Carl Jung on what its significance might be, and a remark often attributed to him runs: “When I die my first question to the Devil will be: What is the meaning of the fine structure constant?” He died in Zurich in December 1958. The story goes that he died in room 137 of a hospital there. Whether or not it is true, it is told and retold.
Feynman had no answer either. In the same passage of QED, more than half a century after Sommerfeld, he gave the case its most famous summary.
“It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man.” — Richard Feynman, QED: The Strange Theory of Light and Matter, 1985
Look at what all these attempts had in common. Each tried to build 137 out of pure numbers, and each one, set against the measurement, failed. Today nobody derives α. It is simply measured, and the line Feynman wanted on every physicist’s wall has its own story in why Feynman told physicists to hang 137 on the wall.
Is the fine-structure constant really constant?
If we cannot explain α, we can at least measure it, and that turned out to be a mystery of its own. In 2018 a team in Berkeley, working with caesium atoms, found 1/α = 137.035999046. In 2020 a team in Paris, working with rubidium atoms, found 137.035999206. Put the two side by side and they agree to about one part in a billion. Yet they differ by more than their stated uncertainties: about five standard deviations. So one of them is missing something, or our theory is. Nobody knows which yet. For now, physicists use an official recommended value; in 2022 CODATA gave 137.035999177.
That is the present. Astronomers have also looked into the past. Light from a distant quasar, travelling for billions of years, crossed clouds of gas on its way to us, and the gas left its mark in the light, and with it α’s value from then. Between about 2000 and 2011, some studies claimed tiny changes: a constant of nature, drifting over time. The most precise recent measurement, published in 2022, used the ESPRESSO spectrograph on the light of the quasar HE 0515−4414 and found no change at the level of about one part in a million. As far as we can tell, it holds still.
So here is where the case stands. We know the number to twelve digits. We still do not know why it is that number. Sommerfeld found it in a split red line; Eddington and Pauli chased it; Feynman told us to worry.
And that’s how we found out what the number is. Why it is that number is still on the wall.
