That’s really interesting. I’d have to resurrect a lot of rusty university physics and math to follow the argument in detail, but assuming the Princeton attribution is correct, one has to give it credibility.
Of course it’s not something that would be noticed one way or another in normal conductors: the resistive losses would dominate.
@Dr.Strangelove is correct in the phenomena he points out for an electron flow, but the current flowing around a superconducting ring is not a flow of electrons, it is a flow of a bosonic superfluid, which also won’t radiate any energy at DC, but can’t be treated as a statistical collection of individual fermionic charges.
I don’t think that’s really true: if you do have a time-varying current, then your loop is simply an antenna. I don’t have good intuition about the relationship between resistive losses and radiated EM power, but I do know that antennas can be >50% efficient in some cases, so I don’t think it’s the case that you just wouldn’t notice.
The paper I cited claims that the losses aren’t quite zero, just that there’s a constant of (\frac{v}{c})^{2N} to the radiated power. v=1\ cm/s and N is about a mol, so that’s a very, very small value.
Certainly, though it does appear from @peccavi that the situation isn’t quite as “simple” with superconductors.
Really, your comment seemed entirely plausible at first, but I recalled from somewhere that a time-invariant current has a static field and can’t possibly radiate. And yet you have electrons circulating around, which are obviously themselves radiating. That discrepancy seems pretty weird, and indeed it was enough to search for a derivation.
Do you have a high-level (i.e. handwaving) explanation for the difference? Intuitively it seems it should not change the basic argument. Cooper pairs are widely spaced. They’re bosons, so they overlap, but the individual electrons are still going to repel each other. Not that intuition ever works all that well for quantum phenomena.
As an aside, before re-reading the paper in detail, it made me think of the behavior of atoms, and how the electrons do not radiate. We understand this to be because electrons can only emit energy in discrete packets, and some of the electrons are in the ground state, and they’re fermions so they can’t just all fall into the same state. But the paper makes this note:
This problem was first posed (and solved via series expansions without explicit mention of Bessel functions) by J.J. Thomson [4]. He knew that atoms (in what we now call their ground state) don’t radiate, and used this calculation to support his model that the electric charge in an atom must be smoothly distributed. This was a classical precursor to the view of a continuous probability distribution for the electron’s position in an atom.
So in a way, another perfectly good explanation is that because the electron is smoothly smeared out, it is not a concentrated charge whirling about, but rather a smooth, constant current flow. Which we already know does not radiate. You don’t really need to invoke QM at all for the basic explanation.
I’m not a theorist, so any explanation I give is going to be handwavy. 
The condensation into a single ground state and the subsequent correlated activity (for example, the pairs creating current in a superconducting loop have a single, continuous wave function) mean that the physics behind the phenomena (Meissner effect, flux quantization, persistent current, Josephson tunneling, etc.) is found in the collective action, not the statistics of individual particles (or pairs). Flux quantization, in fact, is another collective phenomena (the magnetic flux existing within a superconducting loop due to the persistent current, cannot change, except in units of a flux quanta) that shows “decay” of a persistent superconducting current can’t occur in a linear fashion- any energy lost in current decay has to be larger than a flux quanta’s worth.
I looked around for a good physics paper on persistent current, but the phenomena is so embedded in the field that I couldn’t find anything that wasn’t a specific look at odd cases (paramagnetic defects in Type II alloys?)
I did find a sort of high level description in Bob Schreiffer’s Nobel lecture (he’s the S in BCS). It’s about as high level as I can find without going to completely phenomenological explanations.
On another note, in my searches, I discovered that there is such a thing as a persistent current in a normal metal ring (at least theoretically). It’s only possible in rings that are small compared to the electron phase coherence distance (so nm size) and the experimental evidence is still being developed.
I think the simplest way to see that there’s no radiation from a constant-current loop is to realize that the loop has no phase.
I do know that antennas can be >50% efficient in some cases,
That’s interesting, I did not think antennas could be anything like 50% efficient. But I suppose if one is tuned very exactly to the driving frequency it could be so: resonance is a curious phenomenon.
The paper I cited claims that the losses aren’t quite zero
As for the losses not being quite zero… maybe a handwaving explanation would be that while the current in a conductor travels at close to the speed of light, the electrons themselves are not moving anything like that fast… ?