Main Theorem of Renormalization, but in physicist-speak

[AndreasC]

I have been reading a few things about the mathematical formulation of perturbative QFT, specifically in terms of the Stuckelberg-Petermann RG, the Gell-Mann-Low RG, and their difference. Unfortunately I lack the mathematical background to understand these things in depth, and I'm having a little bit of trouble in connecting all that to what we are usually taught in QFT. Just so we can get on the same page, here is a link to nLab that brings up some of these things and relevant references: https://ncatlab.org/nlab/show/renormalization+group

From what I understand so far, we have the Stuckelberg-Petermann RG, which is a true group, and its members are all the vertex redefinitions (which I understand to be reparametrizations of, say, the mass, or the charge, etc) that take you to a different renormalization scheme. This last part I'm not so sure what it means in physicist-speak. Is it related to field reparametrizations in the following familiar form?

$$ \phi_R = Z_R \phi $$

Now about the GLRG. It is claimed that scaling the S matrix gives a renormalization of the theory, so there exists a relevant vertex redefinition as per the Main Theorem. Unfortunately I don't understand how this renormalization corresponds to what I know in the first place, or what exactly their idea of scaling is, so I don't understand what this statement concretely means either. What I know is that Gell-Mann and Low method is about choosing an arbitrary renormalization point, and when you work out the counterterms, you end up with a relation for the renormalized g so that the bare charge does not depend on the parameter μ. It's a bit hard for me to concretely relate it to the mathematical approach. Any help would be greatly appreciated!

Edit: Perhaps even better to get us all on the same page is the PF insight page on the same issue: https://www.physicsforums.com/insights/newideaofquantumfieldtheory.renormalization/

Unfortunately, it also requires a bunch of background that I do not possess to really understand, and it also doesn't connect it to the usual, counterterm manipulation/reparametrization methods we are taught...

 

[A. Neumaier]

Maybe you'll like the explanations in my tutorial

Renormalization without infinities–an elementary tutorial​

 

[AndreasC]

Maybe you'll like the explanations in my tutorial

Renormalization without infinities–an elementary tutorial​

Thank you, I will look into it immediately when I have the time after all the Easter celebrations today!

 

[AndreasC]

Maybe you'll like the explanations in my tutorial

Renormalization without infinities–an elementary tutorial​

I skimmed it, and while it is interesting, I didn't find anything that directly clears up what I was trying to understand. What I don't really understand is how the formulation in terms of the Main Theorem corresponds to the typical formulation of Green functions etc. What exactly is meant by "renormalization" of the S matrix in that context? What would it mean expressed in terms of Green's functions etc? I'm trying to figure out first of all if it is all supposed to connect a reparametrization of variables such as mass, charge etc to a normalization choice for the fields.

 

[A. Neumaier]

I'm trying to figure out first of all if it is all supposed to connect a reparametrization of variables such as mass, charge etc to a normalization choice for the fields.

Of course. The details depend on the renormalization scheme (and the level of abstractness of the theoretical framework) but the principle is always the same, as apparent from my renormalization tutorial.

In general, renormalization means to use a singular formal parameterization (the renormalization prescription) of the coupling constants (such as bare mass and bare charge in QED) by new parameters and to find a formally equivalent formulation of the theory to be renormalized in these new parameters, such that one can determine the parameters from suitable conditions (the renormalization conditions) relating the new parameters to physical quantities (such as the physical mass and charge in QED) without encountering UV infinities.

You might also be interested in my Insight article on Causal Perturbation Theory - aka Epstein-Glaser (re)normalization -, where it is shown how to avoid infinities completely from the start, in a much less technical way than in Urs Schreiber's treatment of the same topic.

 

[AndreasC]

Of course. The details depend on the renormalization scheme (and the level of abstractness of the theoretical framework) but the principle is always the same, as apparent from my renormalization tutorial.

In general, renormalization means to use a singular formal parameterization (the renormalization prescription) of the coupling constants (such as bare mass and bare charge in QED) by new parameters and to find a formally equivalent formulation of the theory to be renormalized in these new parameters, such that one can determine the parameters from suitable conditions (the renormalization conditions) relating the new parameters to physical quantities (such as the physical mass and charge in QED) without encountering UV infinities.

You might also be interested in my Insight article on Causal Perturbation Theory - aka Epstein-Glaser (re)normalization -, where it is shown how to avoid infinities completely from the start, in a much less technical way than in Urs Schreiber's treatment.

I see. I have read your insight btw, I think it's very informative and overall I've benefited a lot from your answers in various sites.

So, let's say I wanted to "translate" the theorem:
$$ \hat{S} = S \circ Z $$
To, say, two point Green functions in momentum space (propagators), with the following field renormalization:
$$ φ = Z^{1/2}_{R} φ_R$$
Then I take it to mean that, if m and a are the parameters for the initial theory and m', a' are the renormalized ones, then you have:
$$ D_R(p; m', a') = Z_{R}^{(-1)} D(p; m', a')$$
Which comes from the normalization of the functions, then you can find in a unique way functions that relate m', a' to m, a so that:
$$ Z_{R}^{(-1)} D(p; m', a') = D(p; m(m',a'), a(m',a') $$
And therefore:
$$D_{R}(p; m', a') = D(p; m(m',a'), a(m',a')) $$

Correct me if I'm wrong in this understanding. Now suppose we start with, say, the On Shell scheme, and we try to renormalize the electron propagator, in which case we can find the propagator to have a pole in the physical mass and residue 1. Suppose we want to instead set the self energy equal to zero to some arbitrary point $p^2=μ^2$. Then the residue around the pole is no longer 1, but R. So this is the renormalized propagator, but we can relate it to the on shell propagator if only we change the parameters in the on shell case. I don't know if I've got the right picture...

 

[A. Neumaier]

$$D_{R}(p; m', a') = D(p; m(m',a'), a(m',a')) $$

This looks right to me.

Now suppose we start with, say, the On Shell scheme, and we try to renormalize the electron propagator, in which case we can find the propagator to have a pole in the physical mass and residue 1.

mass $m$

Suppose we want to instead set the self energy equal to zero to some arbitrary point $p^2=μ^2$. Then the residue around the pole is no longer 1, but R.

... but $Z_R$, and the pole is now at $m(m',a')$.

 

[AndreasC]

Alright, I see... So in the case of the Gell-Mann-Low "group", the idea is that making the "scale transformation", which could be renormalizing the self energy to an arbitrary μ^2 instead of the physical mass, you get a new renormalization, so as per the Main Theorem, there is a unique way to reparametrize, which of course depends on Μ, hence the sliding coupling constant, right?

 

[vanhees71]

One approach, the old-fashioned one, is that the physics (aka the S-matrix) cannot depend on the arbitrarily chosen renormalization scheme, and this independence leads to the various renormalization-group equations, depending on the choice of the renormalization scheme (on-shell scheme, mass-independent schemes, etc.).

You find a discussion from a less mathematically rigorous point of view here:

https://itp.uni-frankfurt.de/~hees/publ/lect.pdf (Sects. 5.11-5.12)

The more modern approach is a la Wilson and more in the spirit of many-body theory and coarse-graining/resolution of the ("macroscopic") observables.

 

[A. Neumaier]

Alright, I see... So in the case of the Gell-Mann-Low "group", the idea is that making the "scale transformation", which could be renormalizing the self energy to an arbitrary μ^2 instead of the physical mass, you get a new renormalization, so as per the Main Theorem, there is a unique way to reparametrize, which of course depends on Μ, hence the sliding coupling constant, right?

Yes.

The more modern approach is a la Wilson and more in the spirit of many-body theory and coarse-graining/resolution of the ("macroscopic") observables.

But Wilson's RG is only a semigroup - one cannot undo coarsening, hence there are no inverses.

 

[vanhees71]

That's of course true.