Some related questions on Renormalization:
Why is renormalization even necessary? My understanding is that the supposed problem is that the sums of certain amplitudes end up being infinite. But actual observations are made at the level of probabilities, not amplitudes. Once we actually make an observation, all the different possible results of a particular observable "collapse" into a single observed result that occurs with a certain probability and that is always finite. So if we always get a single finite result, why do we care that there are infinities in the equations at the level of amplitudes? The only reason I can think of is that "purifying" the equations of these infinite amplitudes makes it possible for us to make realistic computations and predictions. Is that the practical reason for renormalization? If so, can anyone give an actual example of such a computation/prediction that requires renormalization?
The standard description of what charge renormalization represents physically is that the supposedly infinite "bare charge" of, say, an electron gets screened by an infinite number of virtual positrons, leaving a small finite net observed negative charge. OK, this makes a certain amount of sense (with one qualification I'll get to). However, there doesn't seem to be an equivalently reasonable physical picture for mass renormalization. If the mechanism is similar to that of charge screening, then presumably an infinite positive mass would have to be "screened" by an infinite negative mass, leaving a small positive remainder (or the bare mass would have to be negative and screened by a positive mass). Some sources on renormalization imply that this is what happens; other sources do not. If this is actually what's supposed to happen, what are the specifics of this mechanism, and what possible sense could it make to talk about a "negative mass" given that we never observe any such thing? If some type of "mass screening" is not what's going on here, is there any other easily understood physical explanation?
Even if we can come up with good physical pictures for both charge and mass renormalization, the business of "cancellation of infinities" seems mathematically nonsensical. In ordinary mathematics, you can't do ordinary arithmetic on infinities. If I add 2 infinities of the same order, I don't get something twice as infinite, I get one infinity of the same order: I + I = I. And if I subtract one infinity from another I don't get a finite result, I just get the same infinity. So is there a mathematically reasonable way of describing what happens during this "cancellation of infinities" or are we just presuming that the ordinary mathematical rules don't apply to the physics? And what possible warrant would we have for such a presumption?