You ask some tough and interesting questions.
Question: How do alpha and pi differ, fundamentally and philosophically?
Alpha, the fine structure constant, tells us the strength of the electromagnetic field. My physicist colleagues don’t know any reason why the values of the four fundamental forces have the values that they do. Of course, my colleagues and I would like to know why.
We usually say that pi is the ratio of the circumference of a circle to its diameter, but this ratio has the value 3.14159… , in Euclidean plane geometry. Consider Euclidean spherical geometry. The equator is a circle, and straight lines are great circles. Follow a great circle beginning at the equator on the prime meridian, 0 degrees longitude, to the North Pole. Then follow continue the great circle along the 180 degree longitude line. Those two segments of a great circle form a diameter of the equator. But the ratio of the circumference to the diameter in this case is 2. Indeed the ratio of the circumference to the diameter of a circle in spherical geometry varies according to the diameter. Small circles are almost plane, and for them the ratio of close to 3.14159…. For large circles, the ratio is smaller, and the example I gave has a value of two.
Question: Why should pi be the value it is? We can easily see from simple geometry that pi has to lie between 2 and 4, but why exactly the value it is? And why should pi figure into Dirac’s constant? And how on earth (or anywhere else) could e raised to the power i*pi equal to -1? No one knows, but the mathematics all work out, and the things we build using these constants and their relationships reliably work in the real world. So, is alpha just another constant that will eventually be shown to relate mathematically to other better-known constants via some simple yet impossibly profound equation?
I agree that your questions about the mathematical constants and their relations are deep. I suggest that we ask our pal Herb. He’s a mathematician after all.
As for alpha, I confess that it is not a constant! As the energy with which two charged particles meet up with each other increases, alpha begins to increase. Thus, alpha as a function of energy is not a constant.
Here’s a little Feynman quotation for you:
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 about 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.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the “hand of God” wrote that number, and “we don’t know how He pushed his pencil.” We know what kind of a dance to do experimentally to measure this number very accurately, but we don’t know what kind of dance to do on the computer to make this number come out, without putting it in secretly!
Question: It seems to me that, if we can truly see billions of years into the past by looking at just-arrived light from far away and long ago, then something has been constant, or essentially constant, for a very long time. And if enormous spaces and masses and energies have behaved or operated consistently for such long times, then some small number of constants (or perhaps an uncountably large number of constants) must exist at or near the heart of the matter, so to speak. Which, to me, says that there’s nothing comparatively special or mysterious or differentiating among alpha and pi and the other constants. They just are what they are, and what’s to discuss?
Physicists do what you suggest; look for evidence that the constants of nature (but not of mathematics) might change over time or from one place to another. Looking back in time, as you suggest, and examining the properties of atomic spectra on Earth with those we see from the most distant stars and galaxies show that alpha has not changed by more than a part in a billion, or so, since the earliest times we can see. Physicists have also examined the decay products from the amazing natural reactor in a uranium mine in Gabon, and this shows very small, if any, change in alpha over billions of years. But it is a good question.
The startlingly large ratio between the mass of a proton and the mass of an electron is I believe not so startling: the ratio of 1,835 or so is an extraordinarily tiny number all things considered. Maybe that it’s so SMALL should be the startling aspect!
These days, physicists would say that the proton is not a fundamental particle. Three quarks, two ups and a down, are bound together by a bunch of gluons. Most of what we see as the mass of the proton has to do with converting the binding energy of the quarks into mass by E = mc2. So we’d say that the question is what is the source of the mass difference between the electron and the two quarks? For the moment, I’ll add that some of the electron’s mass comes from the now famous Higgs boson recently discovered at CERN, or really the Higgs mechanism, a kind of cosmic quantum viscosity.
Question: To how many digits do we know alpha so far? Do you expect that, like pi and e, it will turn out to be a transcendental?
OK. Here you go. First is the best value of alpha from our best knowledge of the constants in the formula, the electron’s charge, Planck’s constant, the speed of light, even pi, and 4! The epsilon is the permittivity of free space.
Physicists usually remember alpha as 1/137, so here it is as the reciprocal of alpha.
Here’s the best experimental value. Basically this is within 0.25 parts per billion from the best theoretical value. Even the experimental value required calculating more than 12,000 Feynman diagrams through 10th order. A truly heroic feat.
As for your question as to whether alpha might be a transcendental number. I have to say no, never.
All physical measurements yield rational numbers. When we measure something, we always, by necessity, always record only a certain number of digits. If the number has units, then it is obviously a rational number (the ratio of two whole numbers). Theorists work with real numbers, when they wish to compare things to Nature, but they always truncate their result to a rational number. Thus, I’d say, that no number that any physicist ever measures will ever be a transcendental number.
I might add, that computers also compute in rational numbers. Indeed, any particular computer has a finite possible number of numbers it might ever use.
Question: Does what I say and ask make any sense at all to you?
As always, your ideas are sensible, and you ask good questions. As you can see from the Feynman quotation, the best physicists ask them too.