Talk:Bogoliubov-Parasiuk-Hepp-Zimmermann renormalization scheme

Reviewer A (John Lowenstein)

In writing this review, I am assuming that the primary readership for this article consists of students and researchers in particle theory and quantum field theory (QFT) who are interested in learning about an effective means for renormalizing Feynman amplitudes, perhaps to further their understanding of the conceptual framework of QFT, or perhaps as a practical tool for calculating multi-loop Feynman amplitudes.

For such readers, the author provides an accurate, well written introduction to the subject of BPHZ renormalization. I recommend its inclusion in Scholarpedia with only minor changes. Below I will make a few suggestions for such changes in the present article ("version 1"), as well as pose a few questions which I believe the typical reader will want to see answered in a "version 2" (which the author should be encouraged to submit at a later date).

First, the suggestions for changes in the current article.

1. In the abstract, replace "Feynman diagrams" with "Feynman amplitudes".

2. In the abstract, include the fact that BPHZ renormalization is based on the systematic subtraction of momentum-space integrals. This is what distinguishes it from other methods of renormalization.

3. In the first paragraph of "The Problem", delete the sentence "According to the laws of quantum mechanics....dealing with." This is true, but not pertinent to the discussion (unless one wants to get into the subject of relating Feynman amplitudes to Hilbert space vectors, which is not done here).

4. In paragraph 4 of "Diagrammatics", replace "Since the integrands are rational functions" with "Since the integrand is a rational function" to agree with singular nouns later in the sentence. Also, replace "by subtracting the first d(gamma) terms of their Taylor expansion" with "by subtracting all terms up to degree d(gamma) in its Taylor expansion" (there may be several momentum variables, hence more than one term of each degree)

5. The last sentence of "Diagrammatics" should read "Using the forest formula together with a specific prescription for going around the poles..."

Now, some more general considerations. The author does a good job of answering the following questions:

(1) What is a Feynman diagram and its associated transition amplitude?

(2) What is renormalization?

(3) What is the general strategy of BPHZ for removing divergences?

(4) How, in very general language, does the scheme work?

(5) How is the BPHZ formalism used to study composite fields, symmetries, and anomalies?

Questions which I believe the typical reader would like to have answered in addition are the following:

(1') What does a BPHZ-subtracted integral look like, in detail, in a simple example (e.g. that of Fig. 9)? In seeing the explicit example (with Zimmermann's special epsilon prescription), he or she will be able to make sense of the statement about absolute convergence. An additional statement can then be made about the convergence (in the sense of distributions) in the limit of vanishing epsilon. This can be stated in language accessible to a typical physics graduate student without going into a lot of detail, but nevertheless emphasizing that BPHZ is a mathematically rigorous method.

(2') Over the years, has BPHZ had an important impact on the fields of quantum field theory (QFT) and high-energy particle physics? Have there been any major successes which stand out when one looks back over the past 40 years? (I would mention at least the BRS formulation of non-Abelian gauge theories.) Some assessment of the history (which the author is in a good position to provide) would be very appropriate in a Scholarpedia article.

(3') Is BPHZ renormalization an essential (or at least, very useful) tool for a theorist who is active today? How is it being used effectively in current research? Has it been superseded by newer methods?

(4') BPHZ is only one of several comprehensive renormalization schemes. What are its advantages and disadvantages relative to the others?

Renormalization is a calculation technique, not something mysterious (EDITOR'S NOTE: this comment is not from an officially invited reviewer.)

The article incorrectly states that field theory is useless without renormalization. In fact, renormalization "only" is a sophisticated calculation technique.

... is a mathematically consistent method of rendering Feynman diagrams finite while maintaining the fundamental postulates of relativistic quantum field theory (Lorentz invariance, unitarity, causality).

It is regularization that makes Feynman diagrams finite. Renormalization, in contrast, is a calculation technique, that also allows to extract cutoff-independent (regularization-independent) quantities.

The rules, one has set up, were too naive.

A regularized field theory contains all required rules. Only normally it cannot be solved in closed form. Instructive exceptions are the large N limit of some N-component field theories. The exact solutions are perfectly OK. That effective quantities must me matched with measured quantities goes by itself.

Reviewer B (C.M. Becchi)

I would like to add two final sub-minor comments: 1-It would be better to use two different symbols concerning fields and concerning test functions (e.g. \varphi for fields and \phi for test functions). 2- The present Scholarpedia status suggest to maximize the number of crossed citations. For a purely informatic paradox the article "BRST Symmetry" does not appear as a link. In order to have this link you could insert immediately after {\bf BRS transformations} : (BRST Symmetry) I think that this would be enough.