I think that the paper by J.Zinn-Justin gives a good overview of the subject. I like it. However in my opinion some part of the article is a bit too complicated. It is written in the description of the goals of Scholarpedia, that the papers must be understandable for good undergraduate students specialising in the field and for post graduate students specialising in other fields. In my opinion some parts of the article are difficult for students understanding. Now I comment in more details.
- Introduction. After the sentence "The most useful applications to physics.." I would add the words: "In particular path integrals are indespensable for the study of quantum gauge invariant theories which constitute the basis of modern models of elementary particle interactions." In my opinion the successful use of the path (or field) integrals in gauge theories played a crucial role in the acceptance of this method by the physics community. In fact many proofs (like equivalence of different gauges) still are given only in the framework of the path integral method. I agree with the the referee B, that it would be desirable to have also some discussion of the path integrals for constrained systems and in particular for gauge fields. But I assume that such a paper is planned separately.
- Chapters 1,2,3,4 -no revisions.
- Chapter5. In my opinion the section "Gaussian expectation values and Wick's theorem" is not really needed. It is not very useful and only complicates the reading.
- Chapter 6. Barrier penetration in the semi-classical limit. In my opinion this section is too complicated. One should either to drop it, or to make it more understandable for the students.
- Chapter 7. No revisions.
- Chapter 8. The section Grassman path integrals and fermions requires some explanations. One should either refer to (planned?) article, or make the section more detailed, providing the definitions and explanations.
- Chapter 9. I have already written that in my opinion the most useful applications of the path (field) integrals are in the domain of gauge theories. Having in mind that such articles is planned, I still think that one should at least mention such applications.
- Answer to referee A:
A sentence has been added in the introduction concerning gauge theories as well as in the last section. Otherwise, the quantization of gauge theories will be dealt with in a separate article. Still a reference to the article gauge invariance has been added. I disagree that the section on Gaussian measure should be dropped. Indeed, most of field theory calculations rely on perturbation expansion, that is, on reducing the evaluation of path integrals to Gaussian expectation values. Concerning the section on barrier penetration, I shall try to improve its readibility. The section on Grassmann integration, indeed, is certainly not self-contained. A separate article on differentiation and integration in Grassmann algebras is clearly needed. Rewiewer A.
- Reviewer A:
I still think that the chapter "Barrier penetration in the semiclasical limit" is too complicated and drops out from the general spirit of the article. However I think that the author has a right to present his own vision of the subject. So if the author insists I am ready to accept the article in this form. In the chapter concerning the integral over Grassmanian variables, I would wrote more explicitely that the reading of this chapter requires some knowleadge of Grassmanian variables and in particular the definition of the integral over grassmanian variables. Concerning the chapter "Gaussian expectation values and Wick's theorem" I absolutely agree with the author that the perturbative expansion of the path integral is important and should be discussed. I only had in mind that in my opinion the important part of this section may be easily included into the next section "perturbative expansion". Moreover the statement which the author names "Wick's theorem" strictly speaking is not the Wick's theorem, although of course is closely related to it. Finally I wish to note that in my opinion in the list of the literature on the subject in some place the book "Gauge fields. Introduction to quantum theory.", Nauka 1978, Benjamin, 1980, by A.A.Slavnov, L.D.Faddeev, should be included. To my knowleadge it was the first book where the quantum field theory was presented in terms of the path integral method. But I wish to repeat that in my opinion the author may present the subject as he sees it and I am ready to accept the present version as well.
- Answer to reviewer A
A reference to Faddeev and Slavnov has been added and to answer the concerns of the referee, Gaussian expectation values have fused with Pertubative expansion, a reference to initial Wick's theorem added, a clear warning at the beginning of the section on Grassmann path integral has been added (as well as before instanton section together with a link to a future article on semi-classical expansion). I hope this answers the msot serious points raised by the referee. J Z-J
The article explains well the main points of the subject, gives useful examples of actual calculations, and generally prepares the reader to go more fully into the subject. Perhaps I would have given Wiener and Feynman more prominence earlier in the article, for example Feynman's polaron calculation, which, through its elimination of an infinity of oscillator variables has been the framework for many other significant applications (e.g., the Caldeira-Leggett treatment of dissipative systems).
The only place I would not agree with the author (and this is not hard science, but only an opinion) is his statement:
"The most useful applications to physics of the path integral idea involve in fact integrals over fields."
There is a tremendous amount of activity, for example in chemical physics, where the path integral is used as a calculational tool (rather than the more formal tool it constitutes in field theory). See for example the workshop http://www.mpipks-dresden.mpg.de/~qdcpis08/ that just took place at a Max Planck Inst.
One other application, which I consider speculative, but which has drawn a lot of attention, is the use of the path integral for quantizing gravity. I'm not sure of the history but it may go back to Bryce de Witt and I also recall an old paper of Misner laying out a general picture. Also, although I do not know the details, I recently read a Scientific American article in which the authors said thay were using the path integral, summing over geometries they had constructed by pasting together simplexes (and using causality).
All these remarks of mine are to be considered suggestions. The author can let the article stand as it is---it's his article and his taste. But my suggestions may be useful.
Answer to referee C I have modified the offending remark. I certainly agree with the referee that, for instance, the work of Caldeira and Legett is a major importance.
- Reviewer C:
The author has substantially revised his introduction, and I really like it now. The coverage of topics is balanced and, given the need to be concise and brief, a good introduction. I found only one point that could use a bit of elaboration, since so many have erred with respect to it. This is the treatment of a particle in a magnetic field. Limitations on the class of acceptable paths are mentioned, but in practice what people need to know is that when you discretize the time (a.k.a., time-slicing) for a real calculation, it's necessary to use the midpoint rule. That is, the vector potential must be evaluated as A[(x(t)+x(t+epsilon))/2]. This is equivalent to the demand for gauge invariance, which the author does mention, but I would say that it's worth emphasizing. (This is the same issue as in the Ito vs. Stratonovich was of doing Wiener integrals.)
Answer I certainly agree with the reviewer's remark but this point on time-discretization requires, in my opinion a longer discussion to be really useful (see, .e.g., the discussion in my book on path integrals),
The article covers remarkably well many fields, where Path Integrals have been used. However, since the article is the only one on Path Integrals in the Scholarpedia, it would be useful to mention a few missing fields though briefly and giving some more "References" and add to "Further Reading":
- Thus, in the text of the article a small section on
"Path integrals in Quantum Mechanics with Constraints"
and as well one section on
"Path Integral Quantization of Gauge-Field Theories"
could be added, since the largest triumph of the Path Integrals has been in the latter one.
- Also "Path Integral Calculation of Quantum Anomalies" would be useful to mention.
Correspondingly, in the "References" of the article, the following references could be added:
Faddeev-Popov article in PLB 25,29 (1967); Fujikawa's article in PRL(1979);
- Perhaps in the references also the following article by
F. Berezin and M. Marinov in Annals of Physics 104, 336-362 (1977) could be added.
- In the part " Further Reading", the following two books would be useful to add, since they cover many aspects of the Path Integrals which could be impossible to include in the present article. Namely
i) The book by V.N. Popov, Functional Integrals in QFT and Statistical Physics(1983);
ii)The book by M. Chaichian and A. Demichev, Path Integrals in Physics, 2 volumes (2001).
Answer to referee B
I shall add some references, as the referee suggests. Path integrals corresponding to quantization with constraints may be covered by the expected article of quantization of gauge theories. Otherwise, since the present article is already verly long, this topic could be presented in a separate add-on article (provided somebody is prepared to write it. The other topics suggested by the referee, which are more involved, would also deserve separate articles.
I fully agree with the author's view that the mentioned subjects can be covered by other articles in future. The added references are also adequate and in my opinion the article is ready by all criteria.