The notion of a point-like but charged particle is somewhat contradictory: on one hand we imagine it as localized in space (like a neutral particle); on the other hand, such a "particle" is "long-handed", i.e., it is felt far away from it "position". One may think that the charge is not point-like but is a "part" of a complicated, extended system. When you tear the system apart, you feel a resistance force. In case of a spring, this force increases with distance. In case of a shewing gum, it decreases ;-).

In QED there is a Coulomb gauge where the interaction potential is instant 1/r but acts between "waves", not "points". These waves are "overlapping" and we cannot really separate them.

The notion of a point-like but charged particle is somewhat contradictory: on one hand we imagine it as localized in space (like a neutral particle); on the other hand, such a "particle" is "long-handed", i.e., it is felt far away from it "position". One may think that the charge is not point-like but is a "part" of a complicated, extended system. When you tear the system apart, you feel a resistance force. In case of a spring, this force increases with distance. In case of a chewing gum, it decreases ;-).

In QED there is a Coulomb gauge where the interaction potential is instant 1/r but acts between "waves", not "points". These waves are "overlapping" and we cannot really separate them.