If you’ve heard a bit about physics, you might have heard that each of the fundamental forces (electromagnetism, the weak nuclear force, the strong nuclear force, and gravity) has a coupling constant, a number, handed down from nature itself, that determines how strong of a force it is. Maybe you’ve seen them in a table, like this:
If you’ve heard a bit more about physics, though, you’ll have heard that those coupling constants aren’t actually constant! Instead, they vary with energy. Maybe you’ve seen them plotted like this:
The usual way physicists explain this is in terms of quantum effects. We talk about “virtual particles”, and explain that any time particles and forces interact, these virtual particles can pop up, adding corrections that change with the energy of the interacting particles. The coupling constant includes all of these corrections, so it can’t be constant, it has to vary with energy.
Maybe you’re happy with this explanation. But maybe you object:
“Isn’t there still a constant, though? If you ignore all the virtual particles, and drop all the corrections, isn’t there some constant number you’re correcting? Some sort of `bare coupling constant’ you could put into a nice table for me?”
There are two reasons I can’t do that. One is an epistemological reason, that comes from what we can and cannot know. The other is practical: even if I knew the bare coupling, most of the time I wouldn’t want to use it.
Let’s start with the epistemology:
The first thing to understand is that we can’t measure the bare coupling directly. When we measure the strength of forces, we’re always measuring the result of quantum corrections. We can’t “turn off” the virtual particles.
You could imagine measuring it indirectly, though. You’d measure the end result of all the corrections, then go back and calculate. That calculation would tell you how big the corrections were supposed to be, and you could subtract them off, solve the equation, and find the bare coupling.
And this would be a totally reasonable thing to do, except that when you go and try to calculate the quantum corrections, instead of something sensible, you get infinity.
We think that “infinity” is due to our ignorance: we know some of the quantum corrections, but not all of them, because we don’t have a final theory of nature. In order to calculate anything we need to hedge around that ignorance, with a trick called renormalization. I talk about that more in an older post. The key message to take away there is that in order to calculate anything we need to give up the hope of measuring certain bare constants, even “indirectly”. Once we fix a few constants that way, the rest of the theory gives reliable predictions.
So we can’t measure bare constants, and we can’t reason our way to them. We have to find the full coupling, with all the quantum corrections, and use that as our coupling constant.
Still, you might wonder, why does the coupling constant have to vary? Can’t I just pick one measurement, at one energy, and call that the constant?
This is where pragmatism comes in. You could fix your constant at some arbitrary energy, sure. But you’ll regret it.
In particle physics, we usually calculate in something called perturbation theory. Instead of calculating something exactly, we have to use approximations. We add up the approximations, order by order, expecting that each time the corrections will get smaller and smaller, so we get closer and closer to the truth.
And this works reasonably well if your coupling constant is small enough, provided it’s at the right energy.
If your coupling constant is at the wrong energy, then your quantum corrections will notice the difference. They won’t just be small numbers anymore. Instead, they end up containing logarithms of the ratio of energies. The more difference between your arbitrary energy scale and the correct one, the bigger these logarithms get.
This doesn’t make your calculation wrong, exactly. It makes your error estimate wrong. It means that your assumption that the next order is “small enough” isn’t actually true. You’d need to go to higher and higher orders to get a “good enough” answer, if you can get there at all.
Because of that, you don’t want to think about the coupling constants as actually constant. If we knew the final theory then maybe we’d know the true numbers, the ultimate bare coupling constants. But we still would want to use coupling constants that vary with energy for practical calculations. We’d still prefer the plot, and not just the table.
4 gravitons 
The presentation cited below, at pages 38-40 (see also pages 57-59) argues that the widely touted claim that the MSSM gives rise to gauge unification isn’t accurate based upon data developed since the claim was originally made. Quoting Woit: “At a [2018] summer school lecture, Ben Allanach says the prediction is off by 5 sigma, i.e. that if you try and predict the strong coupling at the Z mass this way, you get 0.129 +/- 0.002, whereas the measured value is 0.119 +/- 0.002.” The presentation is: https://indico.cern.ch/event/684125/contributions/2884178/attachments/1674904/2688628/bsm-1.lecturer.pdf
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Yeah I’m including the plot as a representative “thing people draw”, not as an argument for MSSM Grand Unification.
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Also it is worth noting that while the value of the coupling constants is a single experimentally determined parameter of the Standard Model per coupling constant (although you can choose the energy scale at which you measure it to be whatever you want), that the beta function that governs how the coupling constant runs with energy scale is determined purely from theory and not based upon experimental measurements.
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Yup. I was thinking that was an implication of the second part of my post, but it’s not a particularly clear one. I’ll think about rewriting to make the distinction clearer.
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Using only General Relativity and the mass and spin of the electron: The ratio of the radius of the resulting ‘wildly’ naked Kerr ring singularity to the tiny Schwarzschild radius for the electron mass IS the ratio of alpha to alpha_g. This entire calculation uses nothing about electromagnetism – yet generates a value for alpha (EM) from only alpha_g. Why is this not a unification hint?
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The Kerr metric by definition is for a charged object, so you are presupposing electromagnetism. The “black hole / electron coincidence” pertains to the magnetic moment.
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The Kerr metric is not for a charged object. The Kerr-Newman metric is for a charged spinning object.
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If you were talking about this, it is in fact a statement for Kerr-Newman, not just Kerr.
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Was not talking about that. The spin angular momentum of the electron is hbar/2, the mass m_e. Taking only those two constants and the Kerr solution, one obtains the fact that the ratio of the ring radius to the Schwarzschild radius of the electron – is exactly the ratio of the Coulomb force to the Gravitational force. – alpha/alpha_g in your image above.
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Ok, at this point I know this won’t be productive, but I have to ask: alpha and alpha_g at which energy?
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Energy – not relevant in this model.
You study N=4 super Yang-Mills – so you start with a simple limited theory looking for how physics works there, which is generally a good idea. I am here working in another toy space – a ‘purely GR’ world with no EM, no strong force, no QM, etc. What can we learn from this toy modal? – we can find alpha! There is no reason why alpha should show up in a world of only GR.
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In order to match alpha, you are presumably comparing to some published/experimentally found value of alpha. What value is that, and what energy does it correspond to?
(As an aside: you not realizing that was what I was asking about probably means you didn’t read my post. Not reading the post and then commenting about something at best tangentially related is spam, and future comments in that vein are likely to get deleted.)
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Just realized you may have just been intending to reply to ohwilleke’s comment about unification. This blog supports threaded replies, so in general you can make that kind of thing clear by putting the comment as a reply to his comment rather than the post.
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Dear 4 Gravitons, there is another reason, actually more fundamental to my eyes, why couplings run: it’s because of the Wilsonian approach to what a QFT is. The couplings run because the infrared wavelengths alone (those that one can actually probe) should be enough to calculate infrared observables, they should i.e. be enough to match the result of the calculation made within the deeper theory that include shorter wavelengths as well. The matching of observables with and without the UV mode is possible, generically, only if the effective couplings change with the length one is probing, the exceptions being the exceptional theories called CFT’s that diplay scale-invariant behavior. Stressing the role of perturbation theory is reductive, in my opinion, or perhaps I am just leaning too much towards the pragmatic point of view that one is always working with just a limited set of wavelengths.
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That’s a fair point of view.
I think that the idea that we shouldn’t need higher wavelengths to match IR observables, while intuitive to people with physics experience, needs more unpacking for a general audience. People tend to expect there to be a single “true theory” behind everything, so explaining that it’s still useful to think about the world in terms of low-energy theories takes some setup. You’re right that it isn’t really a perturbative question per se, that hadn’t really crossed when writing this but it’s a legitimate point. I do think that one needs some justification of why some particular quantity is “the coupling” and that’s cleanest to present when it’s the thing you’re perturbing in. I’m unfamiliar with how running coupling shows up in a lattice context, for example.
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