Quantum particles feel the influence of gravitational fields they never touch

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.

Right, it seems that for many the 'potential' worldview is hard to grasp (it was for me too until it drawned on me that it's important).

I blame this on poor training and poor textbooks, they don't emphasize the importance of potentials. Also, we seem to grow up with a 'fields' perspective, electric fields and so on.

Maxwell was on the 'right' path with his original formulation of his equations where potentials were involved. However, when Heaviside reformulated them to the 'vector' view we quickly lost the 'potential' one.

No doubt, Heaviside's formulation is incredibly useful in electrical engineering and eleconics and as you'd know that's how they usually appear in textbooks. Trouble is, outside advanced physics texts the 'potential' view is usually omitted. Educators really need to fix this.

Another problem is that the description of a potential is not up to scratch. All too often we seem to be stuck with highschool physics descriptions - those that involve pith balls. The concept that we never measure absolute energy, but only differences often gets lost when describing potentials.

You'd reckon that after Feynman's well documented whingeing about the fact he wasn't taught about potentials early enough that you'd think by now educators would have had sufficient time to have rewritten their notes but apparently they've not.