I'm not sure I'd say it's a breakthrough since theorists have expected it all along. And current treatments of this all have a key limitation, that they are using non-relativistic QM. Which in practice works fine since all the experiments we can currently do in this area are well within the non-relativistic domain.
A real breakthrough would be a relativistic quantum theory that included gravity as well as the other interactions. But that's probably still some way off.
Yeah, sometimes one has to be optimistic. I'm aware of the limitations of using non-relativistic QM but it seemed to me that this experimental connection, if confirmed, would encourage or even force relativistic work to center stage.
Essentially, both theoretical and experimental practitioners could now say 'we've now a concrete stating point'. On reflection, I wouldn't expect a full GR explanation to come easy. Still, it's a start.
The simple answer is that it's right there in the Hamiltonian, and the Hamiltonian is the central operator in QM, the one that determines time evolution. The fact that the EM potential appears there has been known almost as long as QM itself. Much of the recent QM experimentation in gravitational fields has been making use of recent technological advances to verify, what theorists have expected all along, that the gravitational potential acts just like the EM potential in the Hamiltonian.
It's a while since I last looked at Aharonov and Bohm's '59 paper but if I recall your point about the Hamiltonian is covered there. My understanding is that in this paper the key difference from the earlier work to which you also refer is that their new solution to the Hamiltonian now involves a phase factor.
The point I should have made was that I wasn't thinking so much about the mathematical explanation of the Aharonov-Bohm effect but in more general terms where perhaps this new experimental work may reinvigorate interest in the subject and in related areas.
Whilst QFT provides us with an exquisitely accurate mathematical account for the purposes of calculation, it says little about the underlying physics per se. Thus, it seems to me that we stil have a limited understanding about the nature of say virtual particles, ZPE, etc. and essentially no understanding of why the electric, magnetic and fine structure constants and others are the values they are.
Research into the Aharonov-Bohm effect, could eventually lead to a deeper more fundamental understanding of the subject although, given past history, I fully accept that coming up with a major breakthrough in the near future is probably unlikely. (It's even more unlikely that we'll resolve the constants problem anytime soon, if at all.)
It seems to me that the most significant aspect of this work is that we now have more than just a theoretical framework that connects EM and gravity/the gravitational field at a QM level (or seemgly so). Tenuous it may be but it seems like a good start.
For my part, I still worry about why, say, the electric and magnetic constants have the value they do or why our understanding of ZPE is seemingly at odds with reality given the ludicrous value of the calculated zero-point radiation of the vacuum. But then, this is more about philosophy than it is about physics.
That comment is short, succinct and understandable, it's the best summary of the physics involved that I've heard in so few words.
It seems to me that we don't put sufficient emphasis into considering potentials. Viewing things from the aspect of potentials I reckon changes one's perspective, and I think the Aharonov-Bohm effect is a case in point, without doing so we easily come aground.
However, the Aharonov-Bohm effect still perplexes me more than I'd like - in QM, why exactly does potential have observable effects? Why is it that a phase shift occurs in the wave function of a charged particle in the near vicinity of a solenoid even though both magnetic and electric fields are negligible? (I'm either not convinced by explanations that I've heard to date or thst I don't fully understand them.)
Whilst my understanding of the Aharonov-Bohm effect is deficient in this regard, it nevertheless seems to me that if the effect can also be detected in a gravitational field context then we've likely found a profound and deeply relevant connection between EM fields/relatively and gravity. If verified, then we'd have to consider it a breakthrough in our understanding of what up until now has been an intractable problem.
The key point is that we've now experimental evidence for such a connection and that's really good news.