Mass of a point charge A point charge is defined as an electric charge at a mathematical point with no dimensions(Wikipedia: https://en.wikipedia.org/wiki/Point_particle#Point_charge). Can anything be said about its mass in general if not explicitly specified in a given scenario?
 A: In classical electrodynamics, assuming a point charge to be having a finite charge, the net electrostatic self energy carried by it is given by
$$ \text{Self Energy} = 1/2 \int E^2 dV$$
Upon performing the intergral in three dimensions, since the electric field of a point charge diverges at the origin, therefore the rest mass by the virtue of the electrostatic field (given by $E = mc^2$) diverges. The question is now, how can we kick around a point charge if it has got infinite mass?
The problem is resolved to some extent in QED. If we take a bare mass for, lets say an electron in the Lagrangian, then the first order correction to the self energy is a logarithmic divergent term added to the bare mass. So in order to ensure the validity of the perturbation series as well as the fact that all physical quantities require a physical mass instead of the bare mass, for all purposes the bare mass is not used and instead the physical mass of the electron is used. So the problem is cured by imposing the constraint in QED that the perturbative expansion in terms of Feynmann diagrams should be valid although the mass of the electron is not in terms of the bare mass originally taken but in terms of the experimentally measured physical mass.
