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I know of two fascinating and perfectly reasonable explanations of renormalization. However, I'm having difficulty reconciling the two.

The first is to say that when we initially write down a Lagrangian we have parameters which a priori, there is no reason to believe they are exactly equal to the parameters that we are familiar with (mass, coupling, etc.). To account for this we write our parameters as, e.g., $ \lambda _{ bare} = \lambda _{ phys} + \delta \lambda $. We then calculate what $ \delta \lambda $ are assuming a proper definition of what we mean by $ \lambda _{ phys } $. Now we are able to make good predictions because we know precisely what we mean by the coupling, "$ \lambda $".

The alternative explanation (laid out very nicely in this paper) is that when we calculate loop contributions we are assuming that we know the physics to arbitrarily high energies. This is not true, but luckily, physics is independent of the details of the short distance physics. For this reason it doesn't matter how you treat the short distance physics and the easiest thing to do is to introduce point interactions that cancel the divergent contributions. However, using point interactions isn't a requirement but just a way to parametrize our ignorance and since the low energy physics will be independent of the details, we don't need to worry about the fact that our model for the high energy physics is most likely completely wrong.

While I am sure both these interpretations are correct (and equivalent), I am having trouble seeing the connection. The first one seems to say that we were simply misidentifying parameters and the second argues that we are modeling the high energy physics in a convenient way. Can someone make the connection more clear?

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In the second approach, you do have the option to treat the short distance physics in a variety of ways. Choosing point interactions will give you some particular values for $ \lambda _{ bare}$ and some particular rules about how to calculate $\delta \lambda $, and if you choose a different "UV completion" you will get a different value and a different rule.

This is very easy to see because once you have decided what kind of short distance interactions you want to include (that were not there initially), then you need to add the corresponding terms to the lagrangian. If these terms have the same form as the existing terms they will combine to give you different values of the couplings $ \lambda _{ bare}$.

Of course, all these choices should leave $\lambda _{ phys}$ unchanged.

If you think about it, the two methods that you have written are not really different but in reality 1 is the implementation of the ideas of 2.

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