In the electron double slit experiment, why is it not, in principle, possible to know which slit the electron went through by the electron's effect on the surrounding EM field? As the electron travels, it would disturb the field, and presumably this disturbance measured at a point far from the experiment would be slightly different depending on whether it went through one slit or the other. So couldn't you (in principle) measure the disturbance in the field at a far enough distance such that the electron hits the detector before the disturbance is measured?'
There have been experiments with single photons at the time which display interference even after knowing which slit the photon went through.
I do not think that the disturbance of the field by a single electron would be detectable, including correlated with which slit it went through. Possibly a variant of the experiment in the link, using entangled electrons might show the same effect, interference even if the path is known.
This is what one expects from quantum mechanics because the fringe pattern is dependent on the probability, i.e. the square of the wave function and there is no reason to believe that a particle is spread out over all its wave function, but a lot of experimental consistent results that tell us it is just a probability to find the whole particle at a specific point on the screen.
In my opinion, the experiments that destroy the interference pattern when attempting to check the slit the particle went through is due to the introduction of a new wavefunction that differed a lot from the undisturbed one. After all we are talking quantum mechanics. Any change in the boundary conditions reflects in the mathematical formula of the wavefunction.