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It's been a long time I am trying to accept renormalization in QFT but still I cannot be satisfied.

The usual pedagogical way one introduces renormalization is to cure infinities that arise from perturbative expansions.

Now, I can accept the statement that we are doing the perturbation expansion via the wrong coefficient, say the bare coupling ##g_{0}## . The physical quantity is instead ##g=g_{0}-\delta g## or we are dealing with the wrong mass, ##m=m_{0}-\delta m ## .

One can then fix this, expressing our Lagrangian in terms of counterterms. But this means that the Lagrangian itself is infinite!! How can I believe in a theory based on a Lagrangian that makes no sense?? All considerations we can infer from the Lagrangian, say Noether's theorem, say the gauge symmetry itself, obviously is based on a finite Lagrangian.

In summary, what I see from a QFT book:

1) Define the Lagrangian for a specific theory (assuming of course it is a well-behaved mathematical structure)

2) Using the Lagrangian, extract the symmetries of the theory, impose gauge invariance.

3) Find out that the Lagrangian is indeed infinite. But still retain all considerations made.

What I am looking for is a textbook or notes that considers renormalizationa priori.Starts from general assumptions about QFT (existence of a separable Hilbert space, Poincarè invariance, causality), justifies renormalization, obtain general theorems (e.g. spin-statistics), and then calculates amplitudes and cross-sections for specific theories, without the surprise "oh look it's divergent".

I know also there is the Wilson approach to QFT, which is satisfactory, but I cannot understand its connection to the usual approach. The RG in Wilson's approach is based on the variation of the cut-off, in the usual approach it's based on the variation of the renormalization point.

Thanks

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# QFT counter-terms

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