What is the solution to the self-reference problem in fields?

[nomadreid]

This is a well-known problem, the solution I have read a long time ago, but apparently never completely understood, and now am not sure where to look to try again. This concerns the action of a field, say an electric field around an electron, for definitiveness. The electric field affects all charged particles with a strength according to the inverse square law. So far, classical. However, one of the charged particles in the field is that electron itself, and since the distance from it to itself is zero, we get an infinite strength; also one could see it as accumulating an exponentially growing number of virtual carrier particles. Since this does not happen, how does this "self-reference" work itself out? I presume perturbation theory works itself into this somewhere.
Thanks in advance.

 

[Demystifier]

Basically, it works like this. You assume that the electron is small but not exactly pointlike, which replaces infinite quantities by large but finite ones. Then you rename certain not directly measurable quantities in the theory such that the measurable ones do not depend on the exact size of the electron. This procedure is called renormalization.

 

[nomadreid]

Thanks, Demystifier. That is a big help.