I usually don't have much to add to these discussions, but this one is intriguing. A similar question was asked by Chris Rose here at Rutgers:
What does the communication channel "look like" between an electron and proton in a hydrogen atom.
By "look like", he meant bits/second.
Somehow, they are "telling" each other where they are and adjusting their properties accordingly. Rich
--yes, this is an excellent kind of question. I presume "seconds" measured in the center of mass frame of the hydrogen atom. Naively guessing one might try this: if E is the ionization energy of hydrogen (13.6 eV) then the time scale for 1 bit transmit is of the order h/E, where h=Planck constant. This is 3*10^(-16) seconds.
The thing is, no experiment ever directly measures the rest mass of the electron. It measures something else, and then concludes "If <insert assumptions about physics> is true, then the rest mass of the electron is X, with a measurement standard deviation of Y".
The answer to "how small can Y be?" depends crucially on what assumptions about physics are made.... without restrictions on the physics used in the calculation, > there can't be any lower bound to Y.
--ok, fine, I agree; but just choose your favorite chunks of well accepted contemporary physics then find the best bounds you can. Which physical constant has the most intrinsic uncertainty level?