How important are electromagnetic tidal effects in QFT? Can they be used to determine whether a particle is point-like? I just did a back-of-the-envelope calculation, which surprised me. I calculated the difference in acceleration (due to repelling like-charges) experienced by two sides of an electron the size of the classical electron radius, when placed one angstrom from another electron. I used purely classical formulas:
$$F = m_{e}\Delta a ~=~ \frac{k_e q_e^2}{r^2} - \frac{k_e q_e^2}{(r+r_e)^2},$$ 
Where $m_e$, $q_e$ are the mass and charge of the electron, $r_e$ is the classical electron radius, and $\Delta a$ is the difference in acceleration (the tidal effect) between the two sides of the electron. 
Using $r = One\,\, Angstrom = 10^{-10}m$,  I get:
$$\Delta a ~=~ 1.5 \times 10^{18} m/s^2.$$ 
In other words, assuming I didn't make a mistake, the electromagnetic tidal effect is enormous!
This brings up some questions:  


*

*Would this effect be measurable if the electron were not point-like (or far smaller than $r_e$)?  

*Can we prove a particle is nearly point-like by considering electromagnetic tidal effects like the above?  

*Are these sorts of effects studied or considered at all in QFT?
 A: This is not a good way of determining if the electron is pointlike because you are using classical forces. If the electron is in the ground state, the force can't do anything to it, because it might not be able to mix the electron with th excited state.
It is analogous to a molecule. If you apply a force to a molecule where the work done over a molecular radius is much much less than the excitation energy of the molecule, the molecule will respond to the force, but it won't jiggle. It can't jiggle, because the jiggling requires energy equal to the splitting between the ground state and the excited state.
Any structure of the electron will show up as excited states of the electron, or electron dissociation. A while ago, people suspected that the muon and tau are evidence of electron sub-structure, but today, people just think this is a sign of something like E8 string theory.
A: What you are discussing is elastic scattering of electrons on electrons.
Experimental high energy physics has been measuring the form of the particles through elastic scatterings: electrons on protons, muons on protons, pions on protons etc with all the possible combinations. That is how it was found that the proton has an extent and is not a point particle. The extent is parametrized by the electric  and or magnetic form factor. That is what gives us the experimental size of a nucleus and also of a proton and generally molecules. (Form factors are similar in concept as getting the crystal structure of a crystal by observing x-ray scattering symmetries and using Fourier transforms.). 
What @dmckee said in his comment is correct. The experimental limit for  the size of an electron is way beyond any classical estimate of the size so your numbers cannot be correct.
The experimental limits on the decay of the electron also exclude its being composite.
The fact that the limits on the electron dipole moment are so  stringent also indicates a "point" particle.
