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in standard approach to QFT, you study fields, S matrix, and you get a perturbative expression. You see that at higher terms you find infinities and so you renormalize.

Now, Weinberg states that the renormalization procedure of mass and fields must be done even with no infinities. I like this approach.

I want to have a general idea of renormalization, starting from the basic assumptions of QFT, without considering infinites or perturbation theory.

So we postulate that for a field theory (scalar for semplicity) it must be [itex]<0|\phi\left(0\right)|1>=1[/itex], where [itex]|1>[/itex] is a one-particle state (its existence is also postulated) and [itex]|0>[/itex] the vacuum. Then we also require that the mass operator [itex]P^{2}[/itex] has the physical mass [itex]m^{2}[/itex] as eigenvalue on [itex]|1>[/itex]. Via the Källén–Lehmann decomposition we see that this [itex]m^{2}[/itex] is an isolated pole of the 2 point Green function.

In order to satisfy these two assumptions (not obvious in an interacting field), fields and mass must be rescaled, or renormalized, so we get a renormalized Lagrangian with the renormalized fields and masses, and containing counterterms. This is so done without having to deal with perturbation theory.

Will this automaticaly induce finite results in perturbation theory? A rescaling of the coupling constant of the theory is missing in this approach, and can this be done in a general non-perturbative way, as just did for mass and fields?

Any reference who treats QFT and renormalization in the most general possibile way are welcome.

Thank you

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# General renormalization

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