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When we try to regulate a divergent integral, we introduce another parameter, say $\lambda$ and then compute the integral. We finally take a limit (either $\lambda \rightarrow 0, \infty $) to restore the original situation. I have seen two outcomes of it.

  1. On taking this limit, the divergences reappear. This happens in QED a lot. These divergences have then to be absorbed in the definition of physical (measurable) masses and charges in terms of the bare (unphysical) masses and charges.

  2. On taking these limits, divergences do not appear and everything is fine. An example would be the Fourier transform of the Coulomb potential worked out here.

I want to know what goes on mathematically which makes these two situations different. Why do infinities reappear in the first case and not in the second? I am aware of Cauchy principal value integration if one wants to explain using this concept.

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  • $\begingroup$ The example you give of a Fourier transform of a Coloumb potential is not in QED, is it? As a rule-of-thumb, regularization of integrals in quantum field theory is part of a renormalization process and will involve changing bare quantities into fully-dressed ones. $\endgroup$
    – user245141
    Oct 22 '19 at 15:55
  • $\begingroup$ So you are asking why infinities appear in QED and not in some other calculation? $\endgroup$
    – puppetsock
    Oct 22 '19 at 19:47
  • $\begingroup$ @puppetsock I want to ask what is the mathematical reason that sometimes the introduction of the parameter $\lambda$ can cure the divergences and sometimes it can't. $\endgroup$ Oct 22 '19 at 23:05
  • $\begingroup$ In principle, if you introduce a regulator and the take the limit wehere it goes away, you should recover the initial expression. If that was really divergent, the end result will be as well. However, you may have singular terms of the type $0\times\text{infinity}$ (this happens e.g with anomalies). Here, a proper regularisation may give a finite result. $\endgroup$
    – Toffomat
    Aug 25 at 10:25
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The regularization in a renormalizeable theory does not cure the divergences. The idea is only that it pushes them to the next order of perturbation.

So you first get divergences when you first do a loop because you must integrate over all possible momentum values.

So what you do is renormalize the parameters of the loop. In QED this is the photon is mass zero, the mass of the electron, and the charge of the electron. And you use gauge symmetry. You wind up pushing the divergence to the two loop level. To do two loop calcs you push the divergences to three loops. And so on.

The important thing is that you don't introduce new divergent quantities, or require new parameters. QED has that one vertex, a photon and an electron. At each loop level you have got only the same set of parameters and gauge symmetry. You don't, for example, wind up introducing a new vertex or a new interaction or a new parameter. There are a set of invariance theorems that prove that this works order by order.

This means that, order by order, you can compare your calculations to observations. And, to date, QED is still sitting pretty right in the middle of the data. If a theory could be smug, QED would be pretty smug.

General relativity is a contrast. If you renorm the one loop calcs you introduce a new vertex. That means a new parameter, the "charge" at that vertex, in other words the strength of that vertex. At two loops you introduce another parameter from another vertex. So you lose the ability to make predictions because you keep getting new parameters.

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