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12 of 12 people found the following review helpful
on May 1, 2006
Chaichian and Demichev (Vol-I) present a sampling of topics on the mathematical aspects of path integration. There is a second volume which then covers heavier topics (so to speak) such as gravity and quantum field theory. The comments here apply ONLY to the first volume.

For the regular-Joe physicist (such as myself) who actually use path integrals in the "real world", there exists a perpetual inferiority complex about just how much of what we do (and that applies to almost all of mathematical physics) is "rigorously justified". At the same time, all the humdrum axioms and lemmas and proofs and gobbledygook notation is bearable to this group for maybe..what... two-three minutes, maybe?

So, when I first learned that these authors were going to treat this topic with an audience from earth in mind, I was pretty excited.

I was also very interested in the fact that they actually devote the first half of the presentation to the application of path-integrals to the topics of random movement which is of interest to (myself and) most applications outside of the academia. The second half of the book is devoted to the applications to non-relativistic quantum mechanics.

Having frustratedly given up on a multitude of articles on the topic of mathematical aspects of path integration after the first few paragraphs, I, nonetheless, had the nagging feeling about just how deep can they go and still keep the presentation interesting to a physicist? The larger question is: just who the intended audience is? After all, the snobby "rigor" types won't even consider a tutorial format as legitimate, and those who just don't care about the rigor of the underlying mathematics, well, just don't care.

In addition to those like myself who are anxious to know just enough about the rigor, it turns out, there is a fourth group: those who have sadly never heard about path integrals and want to learn just enough for intelligent conversation. I now believe that this book is (or at least, should be) intended for the latter group. Indeed, the preface itself says (p. ix par.5) "The book is intended for those who are familiar with the basic facts from classical and quantum mechanics". Alas it does not say exactly what these audience should expect to get out of this presentation. The preface does say (ibid. par.2) "This book expounds the fundamentals of path integrals...and their numerous applications in... physics". At that early juncture, one would wonder, just how do they intend to deliver on such a ambitious claim in only 320 pages; and it turns out that they don't.

I skimmed the first half with excitement, since the treatment of the stochastic movement by path integrals, is never properly collected in any one place that I've seen (with the qualified exception of a small book by Wiegel-1986). The coverage is fairly broad but never deep. Each short section touches on the main ideas and outlines the computation. This is not necessarily a bad thing in order to inexorably cover maximum ground. What I especially liked were all the worked example problems which would serve the basis for some form of deeper understanding. If it weren't for these, methinks, the novice reader would hardly retain any of the material presented in (what is, at best) an outline fashion.

The biggest disappointment in the section on stochastic movement was the implicit assumption of movement in dense media (hence the ubiquitous Wiener measure) in _all_ computations. I would have liked to see a mention of the fact that it is NOT always true that the variance of displacement is proportional to the first power of time-interval. This misconception is one that has misled the financial industry for nearly a century. I would have liked to see Feynman's approach to stochastic movement (a la Ch.12 Feynman&Hibbs) which elegantly shows a case where the variance goes like the cube of time-interval. Even in cases where the variance is linear in time, it only becomes thus, for times much longer than the interval between scatterings. Even in section 1.2.9 where the calculation starts free of the Wiener measure, the authors are anxious to go the large N regime where the path-integral once-again contains the Wiener measure.

Back to general observations on the book: I found the presentation felt much like the samples of music tracks for CD's for sale on Amazon: a few bars and just as it is getting enticing, it's off to another topic. That's not so bad per se; what makes it frustrating is that the authors do not say where to get the full version. This, despite the fact that the sections seem like they have been cut and pasted (and often abridged) from other sources. Indeed that jibes with the fact that these pages were once the authors' lecture notes. I too would prepare lecture just that way, but there is a long way from printing lecture notes to writing a book, much less a treatise as claimed in par.2 of the preface. If it is true that the sections were paraphrased or lifted from various sources, why not just give those sources and let the reader pursue the topic further? That alone would have made this book worthwhile as a compilation. For this shortcoming I fault the editor at IOP as much as the authors; It is the editor's responsibility to ensure that a book is more than just a fancy print-out of notes by the typical physics-types who by far don't know clear writing from a brick. But I'm really sorry to say that it gets worse.

The business of the missing citations started out as merely annoying. Until I noticed that certain cut-and-pastes are literal lifting of material from one of the sources on the topic I am familiar with. The book by Kleinert contains nearly everything any physicist needs to know about path integrals. It seems that these authors agree, alas, a little too well!!

For example start with the _close_ similarity of equations 2.268, 2.269, and 2.270 (p.165) compared to those of 2.153, 2.154, and 2.155 of Kleinert (3rd ed. section 2.3.2. as of Apr-28-06). While the sequence is only "nearly" identical and the text is paraphrased, the "auxiliary frequency" trick introduced immediately following these equations (unnumbered on top of p. 166 vs. 2.156 Kleinert Apr-28-06 p.113) is identical even in the text preceding it. Since this trick is not found anywhere else in the literature (that I've seen), it stands to good reason for the authors to cite Kleinert. The near identical sequence of calculations continues through 2.2.74 completing the cloning of the section from Kleinert.

There are more examples: pages 168 and 169 of the book seem copied of Kleinert section 2.4 and 2.4.1 on the Gelfand-Yaglom method, without proper citation.

Then there is the nearly identical sequence, in the book's section 2.4.1 starting with eqn 2.4.15 compared to Kleinert's section 6.3 eqn 6.51, and the figure 2.3 compared to Kleinert Fig 6.3.

Conversely, on pg 269 in the discussion on the Coulomb potential in 3D, the authors choose not to follow Kleinert, and to solve the path-integral using a "midpoint prescription". But then it begs the question: why not use some other choice: a post-point or pre-point. The best advice would have been to follow Kleinert's (ch.13) non-holonomic mapping technique. But that's just my preference.

I imagine that similar "issues" relative to other published sources (with which I'm not familiar ) may well exit, as I had first surmised any lecture notes would contain. The problem is that proper citations are missing particularly in cases where novel methods or results have been copied or paraphrased, firstly as proper practice of publication, and secondly, for the sake of the reader should s/he wish to pursue the topic in more detail.

The second half (as in the first half) proceeds in the same outline-esque, fast pace through major topics. As already alluded, this style's merits depend subjectively on the needs and the tastes of the reader.

The first half of the book, being devoted to random movement (Wiener's idea) contains good tutorials on how to move between the path integral approach and the traditional differential-eqn approach to stochastic movement. I can't help but think that the students and applicators of path integral never seem to get quite past the psychological need to show that what they are doing is legit, and really the same as the more traditional differential-eqn approach. The first half of the book serves this particular need well, at least in the cases of the problems typically discussed in standard texts such as the diffusion equation and slightly more complex variations thereof.

Let me cut to the chase here. To learn about path-integrals (for a physicist's purposes) one need only to own two books, the original lecture notes by Feynman (as written and edited by Hibbs) and the Kleinert's 3rd edition, as follow-on. Alas, the former is now out of print (a crime if you ask me), but I have been badgering Dover's editors to reprint it. It also contains many errors and typos; you can get a list of the corrections from me by email (write mathematicus at yahu). Kleinert's book is continually being expanded and corrected by him. He's been known to share individual chapters with other physicists in electronic format, look for him in Berlin.

Finally, when deciding to write a book on a topic where so many distinguished texts exist, the new author should ask himself what is it that I am going to say, or what new point of view am I going to present that is new or different from the existing body of literature. It seems plain that neither the authors nor the editor asked this question before generating the book in its present form. At the present price I am hard pressed to recommend it.
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3 of 3 people found the following review helpful
on February 12, 2009
This two-volume set on path integration contains a multitude of topics neatly organized in chapters and sections with helpfully descriptive titles (even sometimes paraphrasing the punchline of a given section). A previous reviewer notes that there should be more references, in particular to Kleinert's book and in the section on path integrals for Brownian motion, and while I am not very familiar with Kleinert's book, it is true that some sections of Chaichian/Demichev also bear a little too close a resemblance to Barry Simon's Functional Integration and Quantum Physics. Chaichian/Demichev's main merit is that it collects in one place many topics which can be hard to find outside primary sources, like path integration on a curved background, Batalin-Fradkin-Vilkovisky procedure for path integration of systems with constraints, and the correspondence between operator ordering rules and phase space path integral skeletonizations. The treatment of each topic is indeed compact, but arguments which are merely sketched in the text often appear as exercises with hints, and in the introductory comments which open major sections, several references to other books or to papers are usually given. I agree, though, that the authors should have been more careful to avoid lifting sentences from other sources when they compiled their lecture notes into a text. I gave it 5 stars before noticing this issue. Having noticed, I would subtract a star.
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on October 15, 2014
This review is for both books, i.e. vols 1&2. I think the books are very good, and a great exposition of path integrals in physics. The authors present the material in a very logical and well organized way, with individual, more or less self-contained chapters on applications of path integrals to 4 different topics. I think there should be more physics books like this, that focus on one subject and then describe the applications to different areas of physics. The audience for this book in my opinion is professional physicists, either researchers or graduate students. I don't think it's meant as a popular exposition for a general audience, since it goes into details of many derivations, with many mathematical equations.
Concerning what a previous reviewer mentioned. I don't have Kleinert's book on Path Integrals, so I can't comment on who wrote what first, and if anyone copied anyone else. But what I can tell you that very often in physics there is a logical way to present some derivations, and that people often come up with it independently, or both have an older source. For example there are hundreds of books on Quantum Field Theory, and many of them derive some topics in an almost identical way. So if in a work of 2 volumes there are a few similar equations, which are not even particularly important to the book, that doesn't have to mean anything.
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