Why is finding a mathematical basis for the fine-structure constant meaningful?

Question

I was reading QED by Richard Feynman and at the end he mentions that:

There is a most profound and beautiful question associated with the observed coupling constant, $e$ – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with an uncertainty of about 2 in the last decimal place. It has 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.)

I understand that no mathematical formula exists to compute this number but why is that necessary or even a meaningful question? Could the number just be a fundamental property of nature. Asking for a mathematical basis for this number seems to me like asking why the gravitational constant $G$ is 6.6743 × 10-11 $m^3 kg^{-1} s^{-2}$ or why the average distance from the sun to the earth is one AU? Why is the question concerning the mathematical basis for the fine-structure constant different?

Answer 1

Your alternate cases have no meaning, since they have dimensions. $\alpha$ is dimensionless. Long before I knew what it was, I read a story about if we were to contact an alien civilization, we would tell them, in binary, 137.035999177(21), and they would know how tech we were.

They would also have QED, and up to obvious factors of $\pi, 2, \sqrt 2$, all of which are well known numbers, they would see to how many digits we have measured $\alpha$, which is a reflection of our technological capabilities. Their result would not depend on units, say SI, were a meter was 1/10000 the distance from the equator to the pole (they're not from Earth), and the second is not 1/24/60/60 of a solar day since they're not from earth. Now they will have water, but a g/cc won't mean anything for mass (again, not from earth). Charge ofc, we can't even decide on its dimension given SI vs Gaussian units.

But alpha is dimensionless, so none of that matters.

But that has to do with the uncertainty on $\alpha$. The actual value is just a mystery.

For a while (I think) ppl thought it was exactly 1/137, which sparked interest in its origin...why an integer? But that's not the case. Nevertheless, it has always attracted interest--maybe it's a history of science stack exchange question?

Answer 2

I guess there is a specific history of $\alpha$ here, in particular in the early days of QFT, where QED was the only fully formulated theory, such that $\alpha$ was "the" coupling constant. For quite some time, the fact that its inverse is (very close to) an integer has led physicists to speculate that there is a "deeper" reason for 137 -- maybe the 137 encodes a deep fact about the universe, and the tiny deviation is some artifact.

This would be in contrast to a bunch of other dimensionless quantities, such as the ratios of electron to proton mass or electron to muon mass or Higgs VEV to tauon mass or whatnot. There are many of those numbers, and they presumably "just have" some value, without any specific deeper reason.

It seems that nowadays, this has fallen out of fashion a bit: For one, QED is just a limit of electroweak theory (and there is QCD as well), so it's not so clear that the QED coupling constant is the fundamental number of nature. And more importantly, coupling constants are functions of energy scale (e.g. at $M_W$, $\alpha\approx\frac1{127}$), and one thinks more in terms of effective theories. So the picture rather is that the coupling is fixed at some high scale and runs down to the low-energy limit. Then, the fact that $1/\alpha$ is almost an integer at $q^2=0$ seems more like a coincidence.

Answer 3

Physicists are inveterate askers of the question why? When we encounter something that is unexplained we immediately try to find a way to explain it. The hope is that in the search for this explanation we will discover deeper explanations for our observations.

You mention $\pi$ as a number that "just is", but when we look into the reasons for its existence we find that the ratio of the circumference to the diameter of a circle is not fixed but depends on the geometry of the universe. This led eventually to the formulation of general relativity.

Returning to the fine structure constant, we note that it is not actually a constant but depends on the energy scale. This opens the possibility that it might not be fundamental but might originate from a unified field theory, or string theory, or something we haven't thought of yet. It might also be a function of time, and a lot of effort has been expended on this possibility, though at the moment observations suggest it has not changed with time.

It is of course possible that it really "just is" and that efforts to explain its value will be futile, but even if this turns out to be the case we will have learned a lot from trying to explain it.