Why Is 1/137 One of the Greatest Unsolved Problems In Physics? (Script) (Patreon)

It’s the job of physicists to worry about numbers, but there’s one number that physicists have stressed about more than any other. That number is 0.00729735256 - approximately 1/137. This is the fine structure constant, and it appears everywhere in our equations of quantum physics, and we’re still trying to figure out why.

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The fine structure constant, designated as the greek letter alpha, just looks like one of the many constants of nature that power our laws of physics. Like the speed of light, the gravitational constant, or Planck’s constant. But there’s something so weird and so compelling about this number that many of the founders of quantum mechanics obsessed over it.

Paul Dirac called it “the most fundamental unsolved problem of physics.” Wolfgang Pauli said, “When I die my first question to the Devil will be: What is the meaning of the fine structure constant?” Even Richard Feyman pondered its mysteries his entire life. In 1985 he wrote that "all good theoretical physicists put this number up on their wall and worry about it."

But what is it about this one number that makes it the worthy subject of the obsession of savants? Before we get to that, let me tell you the story of its discovery.

As with much of quantum mechanics, it started with us watching the light produced as electrons flicked between energy levels in atoms. This process results in the emission of photons of specific energies that we observe as spectral lines - sharp peaks in the light observed when we break it up into a spectrum of different wavelengths.

For example this is the spectrum of Hydrogen. Hydrogen atoms only emit light with these specific energies. Other elements have other spectral lines. Explaining spectral lines was a major driver of the development of quantum mechanics, and one of its first great successes, first with the Bohr model explaining hydrogen lines, then the Schrodinger equation for heavier elements.

But there was a problem. As our measurement apparatus improved, we saw that the single spectral lines were actually a little off the calculated values, and moreover each single line was revealed to be composed of two lines at almost but not quite identical energies.

It was Arnold Sommerfeld who managed to explain the discrepancy by including the effects of Einstein’s still-new relativity, as well as the fact that the energy levels of electrons with opposite spins are separated slightly by their interaction with their own orbital magnetic field.

Sommerfeld found something peculiar: that the difference in energy between the fine lines was always a multiple of one particular number: the square of the charge of the electron, divided by four times pi, the permittivity of free space, Planck's constant and the speed of light. OK, big deal. We see combinations of these sorts of important constants throughout the laws of physics. But the weird thing with this particular combination is that it has no units.

How can that be? The charge of the electron is in Couloumbs, the speed of light in meters per second squared, vacuum permittivity and Planck's constant also have their units. But when you bring these together all units completely cancel out. We’re left with just a number - a pure number. This number just happens to be 1/137.035999, the fine structure constant.If this number only appeared in the formula for the fine structure splitting of spectral lines it would be just a fun oddity, except this started to show up everywhere.

For example, the repulsive energy between two electrons is 137 smaller than a photon with wavelength equal to the distance between the electrons.

And the orbital speed of an electron in the ground state of the Bohr model of the hydrogen atom is 137 slower than the speed of light.

And the energy of that ground state electron is smaller than the rest mass energy of the electron by a factor of 137 squared. And that’s just the tip of the iceberg for the appearances of the fine structure constant in the laws of physics.

There’s no obvious reason that these various ratios of properties should all work out to be 1/137, or that number to some power. It’s clear the number is trying to tell us something important about the universe, and now more than 100 years after Sommerfeld discovered the structure constant, I’d like to tell you what it means. Except … I can’t, because we still don’t know. But we do at least have a few ideas. To explore them, let's talk about couplings.

Whenever two particles get close to each other there's a chance they will interact, and they can interact in many different ways, which we can visualize with Feynman Diagrams.

We have an episode about those if you’re curious. These diagrams are used to add up the probabilities of particles interacting by all the different ways that interaction could happen. Those probabilities depend on many things, like the particles’ positions and momenta, spins, charges, masses, etc.

These factors multiply a sort of base probability to make the interaction more or less likely. That base probability comes from the coupling constant or coupling strength for the interaction. And that’s exactly what the fine structure constant is: it’s the coupling of the electromagnetic force.

The square of alpha is the base probability that an electron will emit or absorb a photon, or in the case of two electrons interacting in Feynman diagrams - it’s the base probability at each vertex, each interaction between electron and virtual photon, adjusted by all these other parameters I mentioned.

So the Fine Structure Constant sets the "strength" of the electromagnetic force. The more chance of interaction between the electron and electromagnetic fields, the more of an EM disturbance each electron will make. So it’s starting to make sense why the fine structure constant appears in all of these formulas that depend on the electromagnetic force. But the big questions remain: why does alpha take on the value that it does, and why does this specific combination of other fundamental constants come out to be exactly alpha?

When I say that alpha takes on a specific value, I’m not telling you everything. Sometimes it doesn’t. In fact the fine structure constant isn’t as constant as it sounds. It changes with the energy of the interaction. The higher the energy, the larger the constant. In the insane energies right after the Big Bang, the coupling constant for the EM field - which was then joined with the other forces, would have been close to 1, but it quickly dropped to lower values as the energy dropped and the forces separated. We’re now at the bottom of the energy scale, and the fine structure constant has bottomed out at 1/137.035999.

There’s no reason that we know of that it shouldn’t have dropped all the way to zero rather than stopping at this minimum strength - but we should be glad of this fact, because an alpha=0 would mean no electromagnetism, and that would mean no fridge magnets, among other inconveniences like no atoms.

And actually we’re luckier than you think. This constant sets the size of atoms - a larger value means electrons would be closer to nuclei, making them more tightly bound and less able to participate in chemical bonds. A smaller value would mean electrons were less tightly bound, making atoms and molecules less stable. It’s been estimated that if the fine structure constant were just a few percent different, carbon would never have formed inside stars, making life impossible.

We don’t know why our universe ended up with the particular value for this or many of the other fundamental constants. Many physicists believe that these constants were set more or less randomly at the beginning of the universe. It would be surprising that they landed on just the right values to allow for the formation of life - unless of course there are many, many universes with different values for the constants. Then it’s not surprising at all that we find ourselves in one of the ones capable of producing us. We’ve talked about this anthropic argument in the past.

The fact that the fine structure constant has such a convenient value isn’t the weirdest thing about it. The weirdest thing is that it’s dimensionless.

Imagine you were able to send a very short message to an alien civilization. Just a handful of bits - enough to encode one number. What number would you choose to ensure they knew that the message came from an intelligent species?

You could try the various constants of nature to demonstrate that you knew advanced physics. The problem is, most of these constants require you to choose units of measurement. Transmit, say, the value for the speed of light - 299,792,458 m/s, and you also have to explain what a meter and a second are. Try the gravitational or Planck’s constant and you also have to define the kilogram. There’s no way for the alien civilization to recognize these numbers without knowing our units for distance, time, mass, electric charge, etc.

But the fine structure constant is unitless. It’s equal to 1/137-ish for everyone in the universe. Even if you just transmit the number 137, those aliens are going to realize that something’s up.

That’s handy for interstellar communication, but it also tells us that’s something's -seriously- up with the fine structure constant. Now being unitless on its own isn’t anything special. We can come up with all sorts of unitless values - just take the ratio between two things with the same units, like the ratio of the mass of the electron and proton, or the coefficient of friction of an inclined plane. But these things don’t pop up in all these unexpected places like the fine structure constant does.

So what do we make of this number that is both unitless and ubiquitous. Let’s start by thinking about the similarly prolific constants of nature - the ones that actually have units. Those units tell us a lot about what those constants mean. They tell us that the constants of nature represent relationships. For example, the speed of light is the translation factor between the dimensions of space and time in relativity; and it’s also the relationship between mass and energy in Einstein’s famous equation.

The gravitational constant is the relationship between mass, distance, and gravitational force, Planck's Constant is the relationship between the uncertainty in measuring position and velocity. The list goes on. The relationship is defined by the units of the constant.

But without any units, it's not immediately clear what kind of relationship the Fine Structure Constant represents. So here’s an idea: perhaps this odd little number represents a relationship between relationships.

If the other constants of nature tie various physical parameters together, perhaps the fine structure constant is what ties those constants together. Think about it this way - if the constants of nature were set randomly at the big bang, and were set independently to each other - then we wouldn’t necessarily expect there to be any one way of combining them that’s particularly special. Sure, you could find a combo where the units cancel out - but that combo wouldn’t necessarily have physical significance. The fact that this canceling gives the fine structure constant, and the fine structure constant also represents the relationship between many real, physical aspects of the universe, seems to be telling us something. It hints at a connection between the other fundamental constants - perhaps pointing to an underlying common mechanism that set the values for the constants at the Big Bang. Or perhaps it hints at a deeper connection between the properties of the elementary particles, like the mass and charge of the electron.

Finally, it could be that the fine structure constant is not a physical constant, but a mathematical one, like pi, but perhaps we haven't realized this is the case because our mathematics are not advanced enough yet

This is pretty speculative, but the specialness of the fine structure constant warrants speculation. And we’ve been speculating on this problem for a century as this funny little recurring number popped up again and again in our studies of the subatomic world. Back to Feynman one last time. He called it “one of the greatest damn mysteries of physics” and poetically mused that “the hand of God wrote that number, and we don't know how He pushed his pencil.” In other words, to build a universe it may be that only one number needs to be decided in the beginning and from it all other constants naturally follow. And perhaps that number was 1/137, the fine structure constant - whose value sets the rules of this particular space time.