The fine structure constant 1/137,035999... (at low 4-momentum) is a famous quantity of nature.
How many significant digits could it have?
More specifically, could it have more than 62 significant digits (the log of the ratio of the radius of the universe to the Planck length)?
In other words, what is the smallest relative measurement error (delta alpha / alpha) that is achievable in principle?
One way to ask your question, which I don't think you are asking, is: is the fine structure constant a rational number? Then, I think the answer is, "probably not" -- there's no mathematical proof I could give you, but generically we expect constants to be essentially random real numbers, and from that point of view the probability of getting an exactly rational number would be zero. At the very least, if it did turn out to be rational, that would beg for an explanation, and we don't have any plausible explanations at the moment.
Another way to reframe your question is: "is there a limit on the experimental precision we could achieve for the fine structure constant?"
This kind of question does have analogues in other parts of physics. For example, you could ask, "to how many decimal places can we measure the mass of a chair?" At first you might think you could continually make better and better scales (at least in principle) and get more and more precise measurements. However, at some point, you will need to deal with the fact that the mass of a chair changes with time. Dust particles might fall on it, microorganisms might climb onto it and then climb off. You could try to deal with this by redefining your measurement so you refer to a chair in a vacuum -- but already you're admitting there was a point where the original question became ambiguous and needed to be refined. However even in a vacuum, atoms will evaporate off of the chair. And, someone built an almost identical chair, but was missing one atom, you would have two objects that technically have different masses, but how would you define which one is "the real chair"? Perhaps the most sensible answer is to say that a "chair" is a word we give to an emergent phenomena built out of lots and lots of atoms, and that if we try to measure (or even simply define) a concept like "the mass" of the chair, that concept is only going to make sense up to some finite range of precision.
The uncertainty principle adds even another layer to this kind of uncertainty, where the very concept of a trajectory (which requires knowing the position and momentum as a function of time) simply doesn't exist in quantum mechanics, even though it is central to classical mechanics. So there are questions we can ask in classical mechanics like "how far up the potential does a particle go before it turns around" that simply don't have answers in quantum mechanics, except in limiting cases.
Anyway, that's just to say that I think this is a good question. However, I also don't think we know the answer! As far as current well-established physics is concerned, the Standard Model is a complete description of all known phenomena that don't involve gravity. So there is nothing in the laws of physics as we understand them that prevents you from measuring the fine structure constant to finer and finer precision. (It should be noted that the fine structure constant isn't actually a constant, it varies with energy due to the renormalization group, but we can refine your question to ask about measuring its value at a fixed energy scale). But, that's not to say the fine structure constant won't turn out to be an emergent property of some deeper theory, where its value stops to make sense past a certain point.
You could ask if you could use a theory like string theory to derive its value, and ask in that framework if the value is well defined. One issue is that there's no embedding of the entire standard model in string theory. But, ignoring that, you can derive low energy effective field theories from string theory (or other kinds of high energy theories), and you can compute the constants in the low energy theory to (in principle) arbitrarily high precision in terms of the underlying parameters of the high energy theory. So in that framework, there is also no limit to calculating the fine structure constant to arbitrarily high precision (although the calculations would become more and more complicated to go to higher and higher precision). However, you might also be trading a question for how well the quantities in the higher energy theory are defined. The following sentence is a speculation (I don't have a rigorous example and could be wrong): if the fine structure constant depended on some compactification of extra dimensions, for example, maybe there would be some small time dependent fluctuation in those dimensions that would cause small fluctuations in the value of the fine structure constant, similar to how the chair could have small fluctuations in its mass. But, as you can see, we're now going very far out on a limb here, which is to say, I think the safest answer to your question is that it is a good question but no one knows the answer.
There is also the theoretical side: how many digits can be computed using standard perturbation (Feynman diagrams) in QED? It's known that the perturbation series eventually diverges. This is relevant to the question as we need theory to give an answer and we don't seem to have one for even QED, it breaks down way before the Planck scale.
Quote:
After reviewing the literature, the consensus is that, due to the asymptotic (divergent) nature of the QED perturbation series, one can extract about 10 significant digits from the series before its divergence limits further precision.
Source:
Dyson’s 1952 paper: F. J. Dyson, “Divergence of Perturbation Theory in Quantum Electrodynamics” (Phys. Rev. 85, 631, 1952) discusses how the perturbation series in QED is asymptotic. This paper is often cited to explain why only about 10 significant digits can be reliably extracted before the series diverges.
Work by T. Kinoshita and collaborators: A series of papers (including tenth-order calculations for the electron’s g−2) provide detailed computations where the precision is effectively limited by the divergence of the series. One accessible review is available on the arXiv; for example, see T. Kinoshita’s review articles on high-order QED corrections (search for “Tenth-Order QED Contribution to the Electron g-2” on arXiv).
Additionally, textbooks such as “Quantum Electrodynamics” by Richard Feynman or standard quantum field theory texts like “An Introduction to Quantum Field Theory” by Peskin and Schroeder discuss the asymptotic nature of the series.
These sources will give you a deeper insight into why the computed digits are effectively limited to around 10 significant digits in perturbative QED.
