Insights Causal Perturbation Theory - Comments

[vanhees71]

The strategy is to never introduce them. The distributions used have the mathematically correct singularities, and these distributions are manipulated in a mathematically well-defined way. Thus infinities cannot appear by design.

Only in the fact that the final results agree with the results of conventional renormalization schemes. The starting point (i.e., the axioms and the first order ansatz) does not refer to anything that would need renormalization.

This is a wrong, unsupported claim. One needs it only if one starts with the ill-defined Dyson series.

How many loops are you using for your QCD calculations?

Well, I guess I'm only wasting your time and I should rather make another attempt to read Scharf's book again, but why is it wrong that you need renormalization in perturbation theory?

In the conventional theory you need to choose a renormalization scheme and the proper vertex functions depend on this choice. The S-matrix elements are independent of the choice (at the order you've calculated them), which is the content of the renormalization group equations. So is causal perturbation theory just a special choice of a renormalization scheme and where in this scheme are the renormalization-group equations hidden?

 

[A. Neumaier]

How many loops are you using for your QCD calculations?

why is it wrong that you need renormalization in perturbation theory?

In the conventional theory you need to choose a renormalization scheme and the proper vertex functions depend on this choice. The S-matrix elements are independent of the choice (at the order you've calculated them), which is the content of the renormalization group equations. So is causal perturbation theory just a special choice of a renormalization scheme and where in this scheme are the renormalization-group equations hidden?

The parameterization of the S-matrix of QED in terms of the physical mass and charge fixes the first order term in $S(g)$ and hence everything, so there is nothing to be renormalized.

But there is some freedom in the construction. It can be used to introduce a redundant parameter at the cost of introducing running coupling constants and more complex formulas. Since the physical electron charge corresponds to a running charge at zero energy, the parameterization of the S-matrix in terms of the physical mass and charge corresponds to a conventional renormalization at zero photon mass.

Scharf writes in the 1995 edition:

(p.260:) It should be remembereded that the vacuum polarization tensor, for other reasons, is normalized by the conditions (3.6.34, 35). Then, it gives no contribution to charge normalization, too. If one assumes a different normalization of $\Pi(k)$, then the coupling constant in $T_1(x)$ and the physical charge are no longer equal. This is the starting point for the renormalization group. This subject will be discussed in Sect. 4.8.

(p.271:) The subject of this section is called renormalizability in other textbooks. The reader will agree that the prefix "re" is of no use here. By renormalization we always mean finite renormalization of an already normalized T-distribution, as discussed in Sect. 3.13, for example.

The redundant parameter would have no effect in the nonperturbative solution. But since the expansion point is different, it leads to different results at each order of perturbation theory. These perturbative results are then related by finite renormalizations in terms of a Stückelberg-Petermann renormalization group. It expresses the charge appearing in the coupling constant - now no longer the experimental charge but running with the energy scale - in terms of the physical mass and charge.

Thus renormalization is finite and optional. Maybe this is special to QED since the free physical parameters have a direct physical meaning.

 

[Tendex]

I'm also new to this approach and haven't got around to getting a copy of Scharf's book yet, so I might be misunderstanding concepts. Acording to your comment in #62 about the moral of the video lecture I linked, it seemed to me that the CPT approach is in some sense opposite to the effective field renormalization group approach that is oriented to the traditional perturbative approach with Feynman propagators plagued with UV divergences, so it looks to me that trying to recover renormalization group equations in it goes against the spirit of CPT, is this so?

 

[A. Neumaier]

I'm also new to this approach and haven't got around to getting a copy of Scharf's book yet, so I might be misunderstanding concepts. Acording to your comment in #62 about the moral of the video lecture I linked, it seemed to me that the CPT approach is in some sense opposite to the effective field renormalization group approach that is oriented to the traditional perturbative approach with Feynman propagators plagued with UV divergences, so it looks to me that trying to recover renormalization group equations in it goes against the spirit of CPT, is this so?

Not really. There are two very different renormalization groups which should not be mixed up. The first one by Wilson is important in nonequilibrium thermodynamics and for condensed but approximate descriptions in terms of composite fields. The second, older one by Stückelberg is the most important one in local quantum field theory and is not related to effective fields but to overparameterization.

• The Wilson renormalization group (actually only a semigroup, but the name has stuck) is based on removing high energy degrees of freedom by repeated infinitesimal coarse graining. It loses information and hence leads to approximate effective field theories and the Wetterich renormalization group equation.
• The Stückelberg-Petermann renormalization group (a true group) expresses the running coupling constant through the Callan-Symanzik renormalization group equation. This group is due to the existence of a redundant mass/energy parameter and has nothing to do with effective fields, as it does not change the contents of the theory, only the perturbative expansion.

The Stückelberg-Petermann renormalization group already appears in the quantum mechanics of an anharmonic oscillator when one wants to relate the perturbation series obtained by perturbing around Hamiltonians describing harmonic oscillators with different frequency. The frequency chosen is arbitrary and hence nonphysical; it is the analogue of the renormalization scale in QFT.

 

[Tendex]

Ok, so in #62 you(and the lecturer) just meant that effective theory in the sense of Stückelberg-Petermann renormalization group was not as nice dealing with perturbative UV divergences as CPT?

Also I believe in particle physics they sometimes mix the philosophy of the Wilsonian RG approach with the perturbative RG in their quest for machines with ever higher energies.

 

[A. Neumaier]

Ok, so in #62 you(and the lecturer) just meant that effective theory in the sense of Stückelberg-Petermann renormalization group was not as nice dealing with perturbative UV divergences as CPT?

Also I believe in particle physics they sometimes mix the philosophy of the Wilsonian RG approach with the perturbative RG in their quest for machines with ever higher energies.

No.

Effective field theory is always in the sense of Wilson, and #62 is only about this. The Wilson RG connects a family of different QFTs accurate at different energies.

Effective field theories deal with UV divergences by changing the problem. By imposing an effective cutoff they simply ignore the full theory and approximate it by something different, sufficient for experimental practice up to a certain energy. Thus they do not need to account for the (in local quantum field theories unavoidable) singularities from which UV divergences arise in the common sloppy treatments.

On the other hand, the Stückelberg-Petermann renormalization group describes a reparameterization of the same field theory, and hence cannot get rid of the physical singularities in the theory. Instead one needs the causal machinery.

 

[vanhees71]

How many loops are you using for your QCD calculations?The parameterization of the S-matrix of QED in terms of the physical mass and charge fixes the first order term in $S(g)$ and hence everything, so there is nothing to be renormalized.

But there is some freedom in the construction. It can be used to introduce a redundant parameter at the cost of introducing running coupling constants and more complex formulas. Since the physical electron charge corresponds to a running charge at zero energy, the parameterization of the S-matrix in terms of the physical mass and charge corresponds to a conventional renormalization at zero photon mass.

Scharf writes in the 1005 edition:The redundant parameter would have no effect in the nonperturbative solution. But since the expansion point is different, it leads to different results at each order of perturbation theory. These perturbative results are then related by finite renormalizations in terms of a Stückelberg-Petermann renormalization group. It expresses the charge appearing in the coupling constant - now no longer the experimental charge but running with the energy scale - in terms of the physical mass and charge.

Thus renormalization is finite and optional. Maybe this is special to QED since the free physical parameters have a direct physical meaning.

Well, the running of the coupling is important in perturbation theory even in QED. It's a much better approximation using tree-level scattering results at high energies using the running coupling than to use the (quasi-)onshell scheme from low-energy QED (the running coupling at a scale around the Z-mass is 1/128 rather than 1/137).

Of course, if at the end causal PT is equivalent to standard PT, there must be a possibility to renormalize, i.e., to change the renormalization scheme. Of course, if you start with a renormalized theory, i.e., finite expressions for the proper vertex functions changing the renormalization scheme is a finite change of the running parameters.

 

[A. Neumaier]

the running of the coupling is important in perturbation theory even in QED. It's a much better approximation using tree-level scattering results at high energies using the running coupling than to use the (quasi-)onshell scheme from low-energy QED (the running coupling at a scale around the Z-mass is 1/128 rather than 1/137).

Yes, but in case of QED it just means introducing an extra parameter that modifies the free theory with respect to which you perturb. It is like changing the free frequency in the perturbation theory of an anharmonic oscillator. Physical results are independent of this choice but perturbative results are not. In the case of a QFT one can introduce even more than one such redundant parameter and then has a multipaameter RG.

Of course, if at the end causal PT is equivalent to standard PT, there must be a possibility to renormalize, i.e., to change the renormalization scheme. Of course, if you start with a renormalized theory, i.e., finite expressions for the proper vertex functions changing the renormalization scheme is a finite change of the running parameters.

And it is, as done by Scharf in the Section on the renormalization group.

 

[A. Neumaier]

The whole point of resummation is that it includes important contributions from all energies. The size of the terms in the power series is completely irrelevant for the behavior of the resummed formulas.

Borel summation is not sufficient because of the appearance of renormalon contributions. The promising approach is via resurgent transseries, an approach much more powerful than Borel summation.

A nice overview is in the recent lecture

• G. Dunne, Resurgent Asymptotics of Hopf Algebraic Dyson-Schwinger Equations (2020)