# Talk:Algebraic renormalization

## Reviewer A report

Report on the contribution "Algebraic Renormalization" to Scholarpedia:

The present article, which is written by known experts of the field, provides a nice introduction to and overview of algebraic renormalization. It describes the context, the ingredients, the procedure and the potential applications. Moreover it includes a certain number of references for further reading. Thus it should prove useful to get an idea of the whole subject. I have some general remarks on the abstract (definition of the subject) and on the choice of references as well as some comments on a certain number of details (including some typos which I listed below for convenience).

The abstract (in particular the very first line "Algebraic Renormalization deals with...") does not clearly define the approach of algebraic renormalization and its distinctive features (e.g. as compared to other approaches). This point is only elucidated in the final paragraphs of the introduction from which one can extract/conclude "In short, the aim is to prove that the full perturbative quantum theory has the same invariance properties as the classical theory. Becchi, Rouet and Stora (BRS) found a general iterative scheme (later on called algebraic renormalization) which reduces the proof of the preservation of symmetries at the level of the radiative corrections (or the determination of all possible breakings, the so-called 'anomalies') to the study of the cohomology of the underlying group."

Another point which does not seem to be explicitly stated in the abstract and in text is that the considered models are Lagrangian models: for Hamiltonian models (or first order actions) the field content and symmetry content are somewhat different (and the relationship with the Lagrangian symmetries may be non-trivial, in particular in the presence of diffeomorphisms) even if cohomological methods based on BRST-invariance can also be applied in this context to investigate certain aspects (textbook of Henneaux and Teitelboim). It should be useful to clarify this point for the sake of the reader and maybe to comment on these aspects.

Concerning the choice of references : some of the pioneering work and some dedicated monographs are cited, but others are not.

- Missing pioneering work is :

- Articles by BRS - Reference 'T' of BRST, which is now accessible on the arXiv:

- arXiv:0812.0580 [pdf, ps, other]
- Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism
- I.V. Tyutin Comments: 22 pages, typos corrected,
- preprint of P.N. Lebedev Physical Institute, No. 39, 1975

- Article of WZ where the consistency conditions for anomalous terms were found

- Article of Bouchiat, Iliopoulos and Meyer on the absence of anomalies in the electroweak model.

- Useful complementary references for further reading should be :

- For the applications to the standard model,...:

- arXiv:hep-ph/9907426 [pdf, ps, other]
- Practical Algebraic Renormalization
- P.A. Grassi, T. Hurth, M. Steinhauser
- Journal-ref: Annals Phys. 288 (2001) 197-248

- For anomalies:

- R. A. Bertlmann
- Anomalies in Quantum Field Theory,
- (International Series of Monographs on Physics) Clarendon Press; New Ed edition (2 Nov 2000)

- Some overview on BV, e.g.

- arXiv:hep-th/9109036 [pdf, ps, other]
- Batalin-Vilkovisky Lagrangian Quantisation
- Antoine Van Proeyen
- Comments: 16 pages (Talk given at the "Strings and Symmetries 1991" conference, may 20-25, Stony Brook.)

- Concerning the details:

- Homogenize the authors (identical) addresses

- The abstract says *"... the Power Counting Theorem and the Quantum Action Principle, valid in the axiomatic framework of general quantum field theory."*
The 'axiomatic framework' of QFT is generally considered to be given
by the Wightman/Haag-Araki axioms (textbooks of Streater-Wightman and
Haag),
but this does not seem to be meant here (otherwise precise references
should be given).

- The introduction starts with *"Since the times of Einstein, Heisenberg and Wigner, a physical system is basically defined by giving the set of variables which describe it and the symmetries its *
dynamics must obey.":
the role of Einstein for classical theories is
clear and so is the one of Wigner
for quantum theories. Although Heisenberg also made pioneering
contributions for symmetries (spontaneous symmetry breaking, conformal
symmetry in particle physics), his role seems to be less clear here,
e.g. when compared to Weyl who realized the importance of gauge
symmetries, the unification of fundamental symmetries, the role of
group representations,...).

- Third line of introduction *"The task is then..."* : the task for what ???

- *"The scope of this note is..."* : this sentence obviously represents
a new paragraph.

- *"... vacuum expectation values of time ordered products of fields"*:
since the preceding discussion concerns both classical and quantum
field theory, it should be useful to refer presently to 'field
operators'.

- It should be clearer if the paragraph *"The symmetries of the theory are expressed by..."* started with *"In quantum field theory..."*

- Paragraph *"There are however various..."* : *"Guage theories"* should read *"Gauge theories"*

- Paragraph *"Becchi, Rouet and Stora (BRS) then found,..."*: the
terminology *"solution of the cohomology of its symmetry group"* seems
somewhat unclear. (Maybe 'study of the cohomology of the underlying
symmetry group' is clearer.)

- Sequel of the phrase *"has the same properties as the classical theory"*: means *"has the same symmetry properties as..."* ?

- Paragraph *"In short, the aim of the algebraic renormalization procedure. is to prove"* : no point in the middle of the sentence.

- Paragraph *"The vertex funcional or quantum effective action [] is defined by \cite{sibold-scholarpedia}"* : correct spelling of
'functional' and put the right link for the given reference

- Reference *" Batalin IA and Vilkovisky GA,1983)"*: parantheses missing.

- Paragraph *"Joining together the gauge and matter fields with the ghosts into the set of fields [] , we can rewrite (6)})"*: correct
parentheses for (6)

- Reference *"Haag, Rudolf (1992). Local Quantum Physics. Springer-Verlag., ."* : correct point.

- Reference *"Brenneke, F and Dütsch, M (2009). The Quantum Action Principle in the Framework of Causal Perturbation Theory. in Quantum Field Theory - Competitive Models, Birkhäauser. "*:
the first author's name is 'F. Brennecke' and the complete reference reads
*Brennecke, F. and Duetsch M. (2009): The Quantum Action Principle in the framework of Causal Perturbation Theory, in "Quantum Field Theory: Competitive Models", B. Fauser, J. Tolksdorf and E. Zeidler, eds., (Birkhäuser Verlag), [arXiv:0801.1408]*

Quite generally I would recommend to homogenize the references (e.g. drop the author's first names), as well as to spell check the whole article in order to correct other typos.

### Reviewer B report

I think that the article in its present form meets Scholarpedia’s scopes, in particular concerning synthesis and clearness. Thus it could be published as it is.

However I think that this is a good occasion for putting into clear terms the meaning of the Quantum Action Principle (QAP) which was introduced by Schwinger in his Brandeis Lectures and is a basic tool of renormalization but still considered kind of fancy mathematical concept by the majority of people using QFT.

In the present text QAP is introduced in the seventh paragraph of the introduction and discussed in the second section considering two distinct situations: first the case of (linear and non-linear) field transformations (Eq. (3)) and secondly that a parameter change (Eq.(5)).

The introduction refers to “any variation of the quantum effective action due to variation of the fields….”. I think that, albeit anticipating at this point the reference to Sibold’s article, what is missing and should be inserted in this paragraph is the idea of the differential character of QAP. Of course this point clearly appears in section 2, however I think that in the above mentioned sentence of the introduction one should insert “infinitesimal” in front of “variation” and “transformation”.

Concerning the distinction between the mentioned situations, either field, or parameter change, I think that a short remark concerning the situation in the classical theory could be useful. I mean that the inserting the field transformation: $\phi\to\phi+cP[\phi]$ into the classical action one gets a new action where $c$ appears as a new parameter and it is generally true that the $c$-derivative of the action is an integrated local functional which, for $c=0$ has dimension “d+D” as in Eq.(3). If the transformation is dimensionally homogeneous , in much the same way as in the case of any other parameter the local functional has dimension D, as in Eq.(5). This remark unifies the distinct situations, at least in the classical limit. Concerning the full quantum case, at least in perturbation theory, everybody is ready to accept the idea that things remain almost the same, but one is not allowed to compute the insertion $\Delta$ as a derivative of the classical action.

A final comment concerns the term of cohomology which is really an awful mathematical term; there are many cohomologies which are studied in the mathematical literature, including BRS cohomology. In the present case what is meant is just that, given a nilpotent linear operator transforming into one another the elements of a sequence of finite dimensional linear spaces, the kernel of the operator in a given space contains, but in general does not coincide with the image of the operator acting on the former element of the sequence of spaces. The difference between kernel and image is what is important for renormalization and is called “cohomology” for short. I think that this point could be mentioned in a footnote.

I apologize for the above comments which, of course, recall perfectly known facts to the authors and, I repeat, these are just suggestions that the authors are free to use as they like. I just remind “ infinitesimal” and few misprints as e.g. classical on top of page 4.