Is it convincing to you?

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 \delta-like potential V_0 (x)=c_0\cdot\delta (x) and calculates scattering amplitudes (reflection/transmission amplitudes). In particular, he obtains the “low-energy” formula: T=-i p/c_0. It is just a regular calculation. Everything is physically reasonable and no renormalization is necessary so far. In particular, when the potential coefficient c_0 tends to infinity, the transmission amplitude vanishes and the incident wave is completely reflected. It is comprehensible in case of a positive c_0, but it holds as well for a negative value of c_0 in the “wave” mechanics.

After that, the author considers another interaction potential V_1 (x)=c_1\cdot\delta' (x). This potential gives “undesirable” results. Replacing V_1 (x) with a “regularized” version of this kind V_1 (x)=c_1\cdot[\delta(x+a)-\delta(x-a)]/2a, the author obtains a “regularized” amplitude T=-i a p/c_1 ^2. Again, so far so good. When a\to 0, the transmission amplitude tends to zero too. It is qualitatively comprehensible because each c_1\delta (x)/a grows in its absolute value as in case of |c_0| \to \infty in the problem considered just above. V_1 is a highly reflecting potential. Factually, it separates the region x<0 from the region x >0, like an infinite barrier of a finite width.

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 V_1 because we do not know how to write down something like V_0. As well, it is possible that in experiment one observes a non zero transmission amplitude and renormalization of V_1 “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: T=-i p/c_0. Comparing it with T=-i a p/c_1 ^2, he concludes that they are equivalent if c_0=c_1^2/a. The author denotes c_0 as c_R and calls it a “phenomenological parameter” to compare with experiment via the famous formula T=-i p/c_0. After finding it from experimental data, the author says that a theory with V_1 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 c_1 is “not observable”, I wonder why and for what reason one then proposes such a V_1 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 V_1? This is what I call a theory reformulation. Am I not reasonable?

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