I do think that the selection of topics, the language and the mixture of intuitive arguments and mathematical ideas is very nice and serves its purpose - to give the reader a glimpse of an idea what QC is about. Still, I have a feeling that the uninitiated reader may need a few more words of guidance in the introduction to be put on the right track to start with.
Some words along the line
quantum chaos tries to understand the connection between wave dynamics and an underlying ray dynamics; linear wave theories versus non-linear classical mechanics; connections between waves and rays/trajectories such as in optics exists for all wave equations ...
may be helpful in the introduction. Likewise, I find the 2nd paragraph in the introduction ' These two phenomena ...' somewhat vague - I have troubles understanding what the author wants to say.
As a further comment: At the end of the article - a list of topics is given ('Generalisation of the TF'). Would it be possible to give a sentence or two for each point so to guide the less familiar reader?
Reference list: Cahos --> Chaos (Stoeckmann) In general, it may be helpful to give a few more references or names in the text.
It is my great pleasure to review the article "Quantum Chaos" by Martin Gutzwiller. He is one of the founders of the field that made one of the very few most important contributions to this field. I realize the difficulty to write this article so that it can be understood by non-experts. I recommend to accept it but recommend that the author considers some revisions:
1. Figs. 4 and 5 are too detailed for the level of description in this paper and should be omitted.
2. The term "Resurgence Spectroscopy" is too technical for the level of this paper.
3. The article, that is a review, should be balanced. Two important directions are practically not represented: A. The work of Berry and others on the meaning of the Trace Formula B. The exploration of time dependent Hamiltonians, that started with the work of Chirikov, Ford, Casati and Izrailev.
These optional corrections will make this article of much greater value.
Referee's report on review paper on Quantum Chaos by Martin Gutzwiller
It is not an easy task to write a relatively short review on the field of Quantum Chaos, that in addition should be available for many readers. I think that Gutzwiller has succeded quite well on this duty, where naturally several personal choices are made of what to include and what to exclude.
He might have included a little bit more on the connection between periodic orbit theory and random matrix theory, as well as a discussion on time-dependent systems, but all in all I find the paper suitable for its purpose and well written.
Let me just point out a few minor remarks that the author could consider:
1) "Q stands for the physics of atoms and molecules". Well, "small systems like atoms and molecules", could be more appropreate.
2) Why not give the definition on "physical length" in the path integral discussion.
3) In the subsection on "Connecting the quantum spectrum with a semi-classical spectrum" second and third paragraphs aims at an "intuitive interpretation of the TF". Unfortunately, I dont believe the reader not knowing the field would find these sentences very intuitive.
4) In next section it is not clearly stated which system is regular and which is chaotic, so the final sentence "Evidently chaotic motions are much more ..." becomes unclear.
5) The spectrum in Figure 6 is of course not "random".
6) I cannot find figure 12.
7) In the discussion on "Applications to nuclear physics" random matrix theory is a little bit discussed. I think it would be worthwile to at least include the words "random matrix theory". Also the "important conjecture" could be given by name.