Customer Reviews

Chaos in Classical and Quantum Mechanics (Interdisciplinary Applied Mathematics) (v. 1)

5 star:		(5)
4 star:		(1)
3 star:		(0)
2 star:		(2)
1 star:		(0)

 

 

There are all kinds of physics books. There are books for didactics; nobody actually uses, say, Jackson's book on Classical Electrodynamics for anything but didactics. You can tell the books that are useful for teaching in a classroom by the presence of extensive problem sets. This book has none. So, with this, let me dispose of the notion that you should be able to teach a 'quantum chaos' course based on this book. There are also books which explain in great technical detail all about a given small subject area; Fritz Haake's book on 'Quantum Signatures of Chaos' is a good example of this (though it's really about Random Matrix theory, which is but one quantum signature of chaos; a whole book on just one small area of this subject -covered in a few pages in Gutzwiller's book). Then there are books which are reviews of a subject. Gutzwiller wrote a book which is a review of a subject. It's a very good one.

One could actually go through this, read it, know about the existence of many things, and read up on them as is appropriate for one's research. That is how I used the book. Gutzwiller writes very well, and communicates his enthusiasm for the subject matter. The bibliography is extensive, and you could indeed learn all about anything you needed to by going through the bibliographic references for the details. An actual review which takes you from knowing about Hamilton Jacobi theory and the WKB approximation and covering all the material he presented in every detail needed to understand the whole mess .... I doubt as any such book exists. It would be an encyclopedia. If you simply want to understand the Gutzwiller trace formula starting at that level, there are books which do it reasonably well (Cvitanovic's is a great effort in that direction). This is not such a book; this is a review. A review of a very important body of work. That is this book's purpose, and it serves it admirably well.

As for why Gutzwiller is important; if you think the correspondence principle is interesting, the Gutzwiller trace formula is the main useful way we know of thinking about it in detail. Most physicists think about the area where quantum mechanics becomes classical as a sort of intellectual blur. This is a mistake. This is arguably a much more interesting place to think about than, say, what happened in the first 10^-35 seconds of the big bang. For example, one can actually do experiments in the semiclassical regime. Experiments which only involve optics. There's gold in them there hills, and nobody is looking there. Mesoscopic physics as applied to device physics may eventually force people to think about these issues in detail, but I somehow doubt the device guys are going to reinvent quantum mechanics. Gutzwiller thought about these issues in the 1960s while working on practical problems for IBM. These are the fundamental mysteries, folks: if you're an ambitious young physicist who wants to make a real contribution to science; leave the noodle theory alone, and look into this stuff. Following the herd will get you a nice career, but immortality belongs to the one who plays Einstein or Poincare to Gutzwiller's Minkowski.

Organization and presentation of the ideas of modern chaos theory for classical and quantum world without deeper mathematical proofs in perfect fashion meets in this book. Historical, mathematical and physical origins of the chaos and semiclassical quantization problems have appeared as an introductory. Topics covered with great care, If you like to have a good foundation in this field, this book is absolute must. It served as a guide where to go in quantum chaos. Read it word by word and consult many references in nature of classical, statistical and quantum mechanical, differential geometrical and applied mathematics. Excellent book.

The first thing a non-specialist should do is to hire a reputable consultant or tutor to translate this book into approximately ordinary English, but it would be a bad mistake to miss this book because it is full of remarkable ideas that apply to almost every field. To show you how translation would work, it turns out that doughtnuts (torus, plural tori in math) are key to classical and quantum chaos. You might ask: why doughtnuts, instead of, say, balls (beachballs, tennis balls, microscopic balls, etc.)? Well, let's consider a person's head as a ball. You can comb a person's hair flat on his head, but not without a part or irregularity (eddy) somewhere - see page 33 of the book, which says this is slightly different words. On a doughnut, however, you can comb hair flat without any irregularities. The hair with the directions in which they point are called vectors in mathematics, and all together they form a vector field. Chaos occurs when these doughtnuts get disturbed, say by raising the energy (perturbations). Gutzwiller of IBM New York pioneered the study of the relationship between classical and quantum chaos, but anybody who has read IBM books or manuals know that they usually require heavy translation. If you absolutely can't find a math or physics consultant or tutor, go through Gutzwiller's book page by page until you find something like this which you can understand, and keep reading the comprehensible paragraphs.

There are not many good books on chaos in both classical and quantum mechanics; this one is surely an exception. Gutzwiller is well known to be an expert in the field. However, it's a little bit disappointing that the book may be too "qualitative", that it loses the rigor as appeared in related journals. Afterall, this is a good book.

Gutzwiller is the master and one of the main contributors of quantum chaos. His writing is expansive and insightful. However this books doesn't serve as a textbook by its own, as it is overly concise, short of mathematical rigor and overlooking a lot of aspects in the classical chaos (which I think is fair).

I am used to reading graduate texts where I have to fill in extensive portions of the logic. However, this book does not even begin to explain the results contained in it. It is simply a collection of results, with references to other texts with little explanation (although he does a good job motivating the results).

As I was writing this review, I opened the book to a random page (279) and found the following: "Zaslavski (1977) also starts from the trace formula; he gets a rather explicit expression for the spacing probability P(x) which involves the Kolmogorov entropy..." How did he do this? Well, if you want to know then you have to go read Zaslavski to find out. The entire book is this way.

Maybe this is a great reference for the researcher who already understands the field, but for someone trying to learn it from this book it's worthless.

Today the Gutzwiller trace formula is the standard quasiclassical approach to studies of the spectrum in quantum chaos research.
The most striking property of classics chaos is the sensitive dependence on initial conditions such that neighboring trajectories in phase space separate at an exponential rate. As result the long time behavior of the system is unpredictable. Instead of looking for exponential sensitivity, the main stream of quantum chaos research is currently concentrated on identifying the fingerprints of classical chaos in quantum systems. If one were to identify unique fingerprints of classical chaos in quantum mechanics, one could use these to define quantum chaos.
The starting point for the Gutzwiller quasiclassical analysis is the path integral formulation of the propagator. From this quantum expression one can deduce a formula that contains a sum over all classical periodic orbits. This is the main contribution of Gutzwiller.
Excellent Book!

While the question of connecting quantum theory to classical
mechanics, ODEs, PDEs, etc is a very deep one, and the
author is a main contributor in the subject, the books
lacks of narrative momentum... Readers of, say, QFT a la
Itzkinson-Zuber will be accostumed to this, but I feel a
more focused work on the subject is still to be written.