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1) It seems to me that renormalization has the status of a *prescription* for extracting a finite number from an infinite one. It cannot be justified except that this prescription leads to agreement with experiment. Is this a correct understanding?

2) It seems to me that we first identify the cutoff dependent terms in loop diagrams and the prescription is to just drop them. We then figure out what counter term to add to the Lagrangian to have the effect of dropping the above cutoff dependent terms. If these counter terms are of a form that existed in the original Lagrangian, we deem the theory renormalizable. Is this right? If so, is the only purpose of finding the required counter terms to figure out if the theory is renormalizable? If we already knew that the theory was renormalizable, could we just drop the cutoff dependent terms and forget about counter terms?

3) In the presentation in our class, the loop amplitude was divided into a cutoff dependent "infinite part" and a finite part. It was stated that this division has some arbitrariness: we can move a finite amount from one part to the other. A couple of examples were given: in one of them (phi ^3 in 3+1 dimensions, 1 loop) the arbitrariness in this division was stated to be up to a constant. In another example (phi^3 in 5+1 dimensions, 1 loop), the arbitrariness was stated to be up to a quadratic in momentum. I don't understand why the arbitrariness in the former (say) is limited to just a constant. Can't we add any momentum dependent term to infinity and still get infinity? Does the problem occur as momentum tends to infinity?

Thanks!