A seemingly harmless, random number with no units or dimensions has cropped up in so many places in physics and seems to control one of the most fundamental interactions in the universe.
Its name is the fine-structure constant, and it's a measure of the strength of the interaction between charged particles and the electromagnetic force. The current estimate of the fine-structure constant is 0.007 297 352 5693, with an uncertainty of 11 on the last two digits. The number is easier to remember by its inverse, approximately 1/137.
If it had any other value, life as we know it would be impossible. And yet we have no idea where it comes from.
A fine discovery
Atoms have a curious property: They can emit or absorb radiation of very specific wavelengths, called spectral lines. Those wavelengths are so specific because of quantum mechanics. An electron orbiting around a nucleus in an atom can't have just any energy; it's restricted to specific energy levels.
When electrons change levels, they can emit or absorb radiation, but that radiation will have exactly the energy difference between those two levels, and nothing else — hence the specific wavelengths and the spectral lines.
But in the early 20th century, physicists began to notice that some spectral lines were split, or had a "fine structure" (and now you can see where I'm going with this). Instead of just a single line, there were sometimes two very narrowly separated lines.
The full explanation for the "fine structure" of the spectral line rests in quantum field theory, a marriage of quantum mechanics and special relativity. And one of the first people to take a crack at understanding this was physicist Arnold Sommerfeld. He found that to develop the physics to explain the splitting of spectral lines, he had to introduce a new constant into his equations — a fine-structure constant.
The introduction of a constant wasn't all that new or exciting at the time. After all, physics equations throughout history have involved random constants that express the strengths of various relationships. Isaac Newton's formula for universal gravitation had a constant, called G, that represents the fundamental strength of the gravitational interaction. The speed of light, c, tells us about the relationship between electric and magnetic fields. The spring constant, k, tells us how stiff a particular spring is. And so on.
But there was something different in Sommerfeld's little constant: It didn't have units. There are no dimensions or unit system that the value of the number depends on. The other constants in physics aren't like this. The actual value of the speed of light, for example, doesn't really matter, because that number depends on other numbers. Your choice of units (meters per second, miles per hour or leagues per fortnight?) and the definitions of those units (exactly how long is a "meter" going to be?) matter; if you change any of those, the value of the constant changes along with it.
But that's not true for the fine-structure constant. You can have whatever unit system you want and whatever method of organizing the universe as you wish, and that number will be precisely the same.
If you were to meet an alien from a distant star system, you'd have a pretty hard time communicating the value of the speed of light. Once you nailed down how we express our numbers, you would then have to define things like meters and seconds.
But the fine structure constant? You could just spit it out, and they would understand it (as long as they count numbers the same way as we do).
The limit of knowledge
Sommerfeld originally didn't put much thought into the constant, but as our understanding of the quantum world grew, the fine-structure constant started appearing in more and more places. It seemed to crop up anytime charged particles interacted with light. In time, we came to recognize it as the fundamental measure for the strength of how charged particles interact with electromagnetic radiation.
Change that number, change the universe. If the fine-structure constant had a different value, then atoms would have different sizes, chemistry would completely change and nuclear reactions would be altered. Life as we know it would be outright impossible if the fine-structure constant had even a slightly different value.
So why does it have the value it does? Remember, that value itself is important and might even have meaning, because it exists outside any unit system we have. It simply … is.
In the early 20th century, it was thought that the constant had a value of precisely 1/137. What was so important about 137? Why that number? Why not literally any other number? Some physicists even went so far as to attempt numerology to explain the constant's origins; for example, famed astronomer Sir Arthur Eddington "calculated" that the universe had 137 * 2^256 protons in it, so "of course" 1/137 was also special.
Today, we have no explanation for the origins of this constant. Indeed, we have no theoretical explanation for its existence at all. We simply measure it in experiments and then plug the measured value into our equations to make other predictions.
Someday, a theory of everything — a complete and unified theory of physics — might explain the existence of the fine-structure constant and other constants like it. Unfortunately, we don't have a theory of everything, so we're stuck shrugging our shoulders.
But at least we know what to write on our greeting cards to the aliens.
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