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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?

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    $\begingroup$ Your 3 cases are very different from each other. The gravitational constant is an artefact of an arbitrary choice of units, and so it has absolutely no meaning even within basic physics. The average distance between Sun and Earth is the definition of one AU. You are correct that it could just be that the fine structure constant is some arbitrary physical constant. The point he is trying to raise, however, is that IF the fine structure constant is some mathematically meaningful constant, then this meaning would also carry over to physics, making it interesting. ie meaning is only in hindsight $\endgroup$
    – naturallyInconsistent
    Commented Jun 11 at 4:56
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    $\begingroup$ well, if dimensionless numbers are different then why are we not trying to find a mathematical formula for the ratio of an electron's rest mass to a proton's rest mass, that clearly just seems like the way nature is so how is the fine structure constant different? $\endgroup$
    – Gunnar
    Commented Jun 11 at 5:44
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    $\begingroup$ Oh, you can definitely bet that someone is trying to find a mathematical formula for the ratio of an electron's rest mass to a proton's rest mass. It would be rather interesting and meaningful (again in hindsight if it is found) if such a formula exists at all. It is just that this is not as fundamentally interesting as fine structure constant. $\endgroup$
    – naturallyInconsistent
    Commented Jun 11 at 6:00
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    $\begingroup$ @Prahar but asking "Why?" does not seem to be a sensible question in physics. Physicists rather try to find how nature works. Feynman states this in the very same book earlier, "while I am describing to you how Nature works, you won't understand why Nature works that way . But you see, nobody understands that. I can't explain why Nature behaves in this peculiar way." and also in this clip: youtube.com/watch?v=MO0r930Sn_8. $\endgroup$
    – Gunnar
    Commented Jun 11 at 16:36
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    $\begingroup$ Why do the up and down quarks have charges $2/3e$ and $-1/3e$, respectively? Why those ratios specifically? Answer: It is a consequence of gauge anomaly cancellation. Without this specific ratio of charges, the standard model would not be a well-defined theory. $\endgroup$
    – Prahar
    Commented Jun 11 at 16:52

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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?

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    $\begingroup$ It was first "obviously" 1/136, and then it was updated to be "obviously" 1/137, and now we cannot pretend that it is integer. $\endgroup$
    – naturallyInconsistent
    Commented Jun 11 at 5:42
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    $\begingroup$ One of the shorter Physical Review papers ever (DOI10.1103/PhysRev.82.554.2) is: "The most exact value at present for the ratio of proton to electron mass is 1836.12+/-0.05. It may be of interest to note that this number coincides with $6\pi^{5}$ = 1836.12." along with one footnote. $\endgroup$
    – Jon Custer
    Commented Jun 11 at 12:41
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    $\begingroup$ @naturallyInconsistent and JEB: You said integer, but I think you mean rational number? Otherwise I'm very confused. $\endgroup$
    – Mr. Bultitude
    Commented Jun 12 at 0:04
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    $\begingroup$ I think you can safely assume that the pretence was that it was the multiplicative inverse of an integer, i.e. $1$ divided by an integer. $\endgroup$
    – Lee Mosher
    Commented Jun 12 at 2:42
  • $\begingroup$ The story was actually wrong. The value may be well-known, binary system may be recognizable but the binary representation of the value is a matter of agreement and cannot be deducted. $\endgroup$
    – user13964273
    Commented Jun 14 at 12:27
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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.

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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.

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