Problem of infinities in QED Other than Renormalization what were the other approaches taken to solve the problem (by physicists of that period)? Why were they not as successful as Renormalization?
 A: One group of those alternative approaches:
Some people have attempted to modify classical electrodynamics, removing singularities before even going to a quantized theory. There are, for example, the nonlinear theory by Born and Infeld (1934) or the higher-order theory by Bopp (1940) and Podolsky (1942). They were less (or even not) successful due to technical difficulties (no superposition in the nonlinear theory, ghosts in the higher-order theory) and no experimental confirmation of the effects in classical field theory.
A: I know of two other approaches, there is Schwinger's source theory. What this exactly does, nobody seems to really know, see the three volume set "Particles, Sources, And Fields" by him. Basically it is QED without the quantum fields. Some claim without infinities.
The other approach is causal pertubation theory. I think this is due to a paper by Epstein and Glaser (1973) and laid out in Scharf's 'finite quantum electrodynamics'. The idea is, that the infinities in pertubation theory come from not being careful with distributions, and if one keeps causality at each step, all results are finite. (No finite claim about the full summation of the pertubation expansion is done, of course).
A: Renormalization is not successful itself, it is not sufficient. One needs to sum up soft mode condtributions too, if we speak of QED.
Renormalization (of mass) in the Classical electrodynamics is not successful either. We content ourselves wwith some approximate equations, as a matter of fact.
