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

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Summary: Absolute Reference for Quantum Chaos
Review: 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.

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Summary: Gutzwiller's Chaos in Classical and Quantum Mechanics
Review: 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.

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Summary: The Father of the Quasiclassical Approach to Quantum Chaos
Review: 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!

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Summary: We need more books on the subject
Review: 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.