I would like to discuss briefly a “proof” of necessity and usefulness of renormalization on a popular example being taught to students of University of Maryland. This example was taken from What is an example of an infinity arising in QFT and an example of a renormalization technique being used to deal with it? The direct pdf file reference is the following: Page on umd.edu
The problem is simple. The author considers first a -like potential and calculates scattering amplitudes (reflection/transmission amplitudes). In particular, he obtains the “low-energy” formula: . It is just a regular calculation. Everything is physically reasonable and no renormalization is necessary so far. In particular, when the potential coefficient tends to infinity, the transmission amplitude vanishes and the incident wave is completely reflected. It is comprehensible in case of a positive , but it holds as well for a negative value of in the “wave” mechanics.
After that, the author considers another interaction potential . This potential gives “undesirable” results. Replacing with a “regularized” version of this kind , the author obtains a “regularized” amplitude . Again, so far so good. When , the transmission amplitude tends to zero too. It is qualitatively comprehensible because each grows in its absolute value as in case of in the problem considered just above. is a highly reflecting potential.
But the author does not like this result. He wants fulfilling a “low-energy theorem” for this potential too. He wants a non zero transmission amplitude! I do not know why he wants this, but I suspect that in “realistic” cases we use interactions like because we do not know how to write down something like . As well, it is possible that in experiment one observes a non zero transmission amplitude and renormalization of “works”. So, his desire to obtain a physical result from an unphysical potential is the main “human phenomenon” happening in this domain. We want right results from a wrong theory. We require them from it! (The second theory is wrong because of wrong guess of potential.)
Of course, a wrong theory does not give you the right results whatever spells one pronounces over it. So he takes an initiative and replaces a wrong result with a right one: . Comparing it with , he concludes that they are equivalent if . The author denotes as and calls it a “phenomenological parameter” to compare with experiment via the famous formula . After finding it from experimental data, the author says that a theory with describes the experimental data.
Is his reasoning convincing to you as a way of doing physics?
If one is obliged to manipulate the calculation results with saying that is “not observable”, I wonder why and for what reason one then proposes such a and insists on correctness and uniqueness of it? Because it is “relativistic and gauge invariant”? Because after renormalization it “works”? And how about physics?
Factually, the renormalized result belongs to another theory (to a theory with another potential). Then why not to find it out from physical reasoning and use instead of ? This is what I call a theory reformulation. Am I not reasonable?