Undergrad What is the point of regularization?

[AndrewGRQTF]

Take for example dimensional regularization. Is it correct to say that the main point of the dimensional regularization of divergent momentum integrals in QFT is to express the divergence of these integrals in such a way that they can be absorbed into the counterterms? Can someone tell me what the definition of the verb "regularize" is?

Also, is it true that the conditions required to be able to use the LSZ formula, like the pole of the exact propagator being at the physical mass and it having residue one, take care of the divergences, so that renormalization is not completely artificial? Can one argue the point of view that counterterms are introduced to satisfy the LSZ conditions, and that they are not meant to cancel any divergences, but end up doing so miraculously?

 

[DarMM]

Take for example dimensional regularization. Is it correct to say that the main point of the dimensional regularization of divergent momentum integrals in QFT is to express the divergence of these integrals in such a way that they can be absorbed into the counterterms?

Yes, it allows you to isolate the divergent terms in a way that respects gauge symmetry.

Can someone tell me what the definition of the verb "regularize" is?

Basically "make finite".

Also, is it true that the conditions required to be able to use the LSZ formula, like the pole of the exact propagator being at the physical mass and it having residue one, take care of the divergences, so that renormalization is not completely artificial?

As far as we can tell yes.

Can one argue the point of view that counterterms are introduced to satisfy the LSZ conditions, and that they are not meant to cancel any divergences, but end up doing so miraculously?

Yes. There would be a shift between terms in the Lagrangian and their physical counterparts regardless, i.e. $e_0$ in the QED Lagrangian would not be the same as $e$ the physical charge. This occurs even in QM for the anharmonic oscillator with Lagrangian:
$$\mathcal{L} = \frac{m\dot{q}^{2}}{2} - \frac{kq^{2}}{2} - \frac{\lambda_0 q^{4}}{4!}$$
Where the physical $\lambda$ is not the same as the $\lambda_0$ in the Lagrangian and one needs to renormalize to attain the latter. It just so happens this also cures divergences in many QFTs.