The fine structure constant

Mathematical constants such as $pi$ and $e$ have values that are determined by their definitions. For example, you can define $pi$ as the ratio of the circumference of a circle to its diameter, and then use this definition to calculate $pi$ to as many decimal places as you want. Constants like this cannot have any other value than the value they have.

By contrast, we know of no reason why the fine structure constant has the value it has. And we can imagine a universe in which it has a different value. So this leads physicists to wonder why it is approximately $1/137$ instead of, say, $1/140$, or something much smaller or much larger.

In short, dimensionless physical constants are not mathematical constants. Or at least they aren’t in our current physical theories. Instead they are simply parameters of our models.

Just because something doesn't have units, doesn't make it inherently interesting enough to "sit with." The ratio of my height to the length of a football doesn't have units, but it also isn't a very fundamental or interesting quantity.

The parameter that controls the strength of a force is something we can derive from measurements, and it has many implications for how things around us behave, but at the end of the day if it was a slightly different number, QED wouldn't be a very different theory.

Also once you take into account the running of couplings, the fine structure constant looks even less fundamental. If we measure the strength of the electromagnetic interaction at a different every scale, we get a different number. The fine structure constant can only be defined as the strength of the EM interaction at a specific energy scale.

First, I would argue that $pi$, in fact, does have a meaning, and an explanation as to its value. The ratio of a circle's radius to its diameter is dependent on the curvature of the space that the circle is occupying. The $pi$ that you're used to is the value for space with zero curvature; if the space is strongly postively or negatively curved, the value of that ratio will be a different number. Try drawing a circle on a globe - the circumference of that circle will be smaller than $pi$ times the distance from one side of the circle to the other.

This means that measuring the ratio of a circle's circumference to its diameter tells you something about the curvature of the space you're living in. So the value of $pi$ does have a specific meaning and explanation - the measured value is explained by the fact that the space we live in is locally well-approximated by a "flat" space, without any strong curvature.

Euler's number also has a meaning and explanation: we first discovered its value when we started playing around with compound interest. It was eventually discovered that if you started compounding the interest faster and faster, in smaller and smaller increments, the amount of money you had was described by an exponential function with $e$ as the base. Since then, $e$ has been associated with continuous processes, and often when converting a discrete process into a continuous one, you'll encounter $e$ somewhere. Its specific value depends on our definition of the derivative, but the fact that we see it so often in our measurements means that natural processes are often well-approximated by continuous processes.

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Getting back to the main point: for a long time in physics, there's been the expectation that the numbers that we come across in our measurements are not arbitrary. When we measured the emission line spectrum of hydrogen, we did not stop there and say, "Well, we have these arbitrary frequencies that hydrogen emits, things must just be like that." We looked for a model to explain where those frequencies came from. Eventually, we came up with a simple formula that predicted all of the spectral lines using only a single constant, the Rydberg constant.

We could have also stopped there, saying, "Well, the Rydberg constant has this arbitrary value, and that's the end of discussion," but we didn't. We expected, again, that the value of that constant had some deeper explanation, and wasn't arbitrary. It turned out that we were correct again: the Rydberg constant has the value it does because its value is set by a quantum-mechanical model of the hydrogen atom, which only depends on such fundamental things like $hbar$ (the conversion factor between energy and angular frequency), the fundamental quantum of free charge (i.e. the charge on the electron), the mass of the electron, the permittivity of free space, and the speed of light.

Even then, we continued to expect that those constants had some deeper explanation. Some of them are fairly shallow - for example, the value of $hbar$ is simply the result of the fact that our "human" units of energy and time are not really compatible with each other. In more "natural" units, the value of $hbar$ can be set to one. Similarly, the value of the speed of light, and the permittivity of free space, are artefacts of our units for distance and time not really being compatible, and their values can also be set to one with a more "natural" choice of units, where mass, energy, time, and distance are all the same unit and intercompatible with each other.

But the other constants in that formula do have a deeper explanation. Quantum electrodynamics dictates that the electron charge that we measure is actually the "dressed" electron charge; the "natural" or "bare" electron charge is roughly 1, but interactions between the electron and the quantum vacuum induce a "vacuum polarization" that reduces the apparent charge that we see. So the value of the measured electron charge is determined by the strength of the electromagnetic interaction, which is parametrized by the fine structure constant $alpha$. And, according to the Higgs mechanism, the mass of the electron is determined by the coupling strength of the Yukawa coupling between the electron and the Higgs field.

So now, in the present day, we have some constants, like the above Yukawa coupling strength and $alpha$, which might seem arbitrary. But every time we encountered a seemingly arbitrary constant in the past, there was always some deeper explanation for why it had the value it did. Is it possible that there just isn't such a meaning for these constants this time? Yes, it's definitely possible. But based on our past experience in science, there is no reason to expect that to be true.

Distinct from $pi$ and $e$, the fine structure constant does not have a meaning defined simply in terms of mathematical formulae, which are valid without reference to physics or indeed to anything outside of mathematics. In quantum electrodynamics the likelihood of an interaction between a photon and an electron, as represented by the Feynman diagram

is determined by the coupling constant, usually designated by $e$ (clearly not to be confused with Euler's number). The fine structure constant is given (in natural units) by

$$alpha = frac{e^2}{4pi},$$

but, distinct from $e$, the fine structure constant is defined to be dimensionless. Thus if $e$ is measured in units of charge, the fine structure constant is defined as

$$alpha = frac{e^2}{4pihbar cepsilon_0}$$

so as to be dimensionless. The fine structure constant is a measure of the probability, or frequency, of interactions between photons and electrons, contingent on the distribution of matter. It is dimensionless for the same reason that any probability is dimensionless.