Fabio Books

The theory of C*- and W*-algebras is an extensive one, and some of its results are applied here to study the properties of infinite quantum systems, with particular attention paid to the ergodicity of such systems. The author attempts to take notions from the ordinary measure-theoretic, "classical" theory of ergodicity and apply them to the case of quantum systems. The occurrence of chaos in classical systems is of course well-defined from a mathematical standpoint, and a natural question to ask is whether chaos can occur in quantum systems. At first thought, one might be tempted to assert that it cannot, due to the linearity and unitary of the time evolution in quantum physics. In addition, the non-commutativity of the observables in quantum systems makes a straightforward generalization of the commutative classical situation nontrivial.

There are a few ways of viewing chaos in the classical context, one being the notion of "differentiable" dynamics, and one being "topological" dynamics. The former is more related to the accepted physical notion of chaos, while the latter is more atune to the interest of the mathematician. Another one is measure-theoretic, and fits the needs of both the physicist and the mathematician, the former in the realm of statistical mechanics, and the latter in the study of measure-preserving transformations. All of these viewpoints do have an intersection however, and all are necessary to form a more complete viewpoint of what it means for a time evolution of a system to be chaotic or random in some way. The field of quantum chaos is still an active one, and there is as of this date no definition of it that is acceptable to everyone, both physicists and mathematicians.

This book is a helpful guide to the research in this area, and is fairly up-to-date, in spite of being almost ten years old. The first couple of chapters concentrate on casting classical ergodic theory in the C*- and W*-algebra frameworks. The treatment though is physically motivated, and the author is careful in his definitions and mathematical proofs. He also gives the reader insight as to why infinite systems are needed in the study of chaos, such as the case of a harmonic oscillator in a heat bath. Quasi-periodicity, characteristic of finite systems without a continuous spectrum, would destroy the exponential decay of the correlation functions if the heat bath were not infinite. The author also points out the difference between Bernoulli and K-systems, showing that the Kolomogorov-Sinai entropy is not a complete invariant for K-systems but it is for Bernoulli systems. Topological entropy, not so useful for the physicist but of great use for the mathematician, is discussed in detail. Classical dynamical systems are thought of as abelian C*-algebras with the time evolution modeled by the Z-action of a *automorphism of the algebra. To study the ergodic properties in this context requires the use a normalized state that is invariant under the *automorphism. The author also gives an interesting motivation for the need for W*-algebras. He starts with the Baker map, views it as a 2-dimensional Ising model, its index functions arising from a chessboard lattice. Taking finer and finer chessboard lattices gives an abelian C*-algebra, but using the norm topology results in the C*-algebra being too small. Representing this C*-algebra as a multiplication operators on the Hilbert space of square-integrable functions results in an algebra that is strongly and weakly closed on this Hilbert space, i.e. a W*-algebra.

The remainder of the book deals with the problem of ergodicity and chaos in infinite quantum systems and the corresponding mathematical problems that arise in this context. Some of the more important of these include the generalization to infinite dimensions of the fact that in finite dimensions the algebra of observables always has a representation equivalent to the Fock representation. This generalizes to the Gelfand-Naimark-Segal construction in infinite dimensions, which is unique modulo a unitary map. Another notion that breaks down in infinite dimensions is the Gibbs state, which must be replaced by that of a KMS state. In addition, states on (finite-dimensional) matrix algebras, which satisfy the modular relations at inverse temperature ( = 1), have their generalization to the (infinite-dimensional) W*-algebra case via the Tomita-Takesaki theorem. The author studies some toy models and bosonic and fermionic systems, one interesting one being the non-commutative 2-torus, which gives an algebraic formulation of the quantum Hall effect. For fermionic systems, one sees clearly the effects of noncommutativity in preventing strong topological mixing: they induce correlations that cannot be suppressed in the time evolution of the system.

In addition, the author investigates whether Kolmogorov entropy is meaningful in the noncommutative context. The role of quantum K-systems is particularly interesting here, as they are defined so as to not have quasi-periodicity in their dynamics. The noncommutative Arnold cat map is given as an example of a quantum K-system. The Connes-Narnhofer-Thirring entropy is offered as a generalization of Kolmogorov entropy. The author discusses the problem as a consequence of the difficulty in generalizing the ordinary classical probabilistic notions to this case, since states can be changed when the observable is measured. And as everyone who has to work with the time evolution of operators in say, the Heisenberg picture, noncommutativity can generate more operators. This aggravates the situation considerably in the attempt to carry over classical ergodic theory to the noncommutative case. The existence of quantum correlations causes even more headaches. The solution due to CNT is to start with finite collections of finite dimensional algebras, and construct an entropy functional on them that respects as many of the properties of the classical entropy as possible. This functional is defined in terms of "Abelian models" of these algebras, these being finite dimensional Abelian algebras along with a state and a positive map that when composed with the state gives the state of the original algebra. The resulting CNT-entropy is applied to the case of quasi-free automorphisms and the author then ends the book with a proposal for a noncommutative topological entropy.

Great book for romance fans. Another winner! Fabio IS the sexiest man in the world!

I was fortunate enough to pick this book up from a Borders Express store that was closing. It was on sale for $19.99 and 75% off. I had to have it! I am so happy that I picked it up. The photos are amazing! There are explanations to the photos and maps for each section of the world being covered. It makes me want to get out there and see these ancient ruins. It's blows me away each time I look at the pictures to think that these ancient civilizations built these massive structures without our modern tools.

Over 200 projects and organizations that work with animals and the earth are detailed in the book. You will find a short description of the organization, a listing of the species involved, as well as practical information-requirements, costs, application procedures etc. The website address is also included so you can find the most current information for the organizations you are interested in. You will also find a couple of indexes in the back that will help you organize your list of 'to go on' by cost and location as well the species or groups.
If you have been toying with volunteerism for a vacation alternative, this book will definitely fill you with ideas about where to go and who to help.

If you ever plan a trip to the Holy Land this book is for you. It is full of pictures of ancient places and maps. You can plan your trip ahead of time. For those who have visited it will bring back warm memories. For those who cannot go there the pictures will take you on a virtual trip. The photographs and maps are priceless

I think this is the most complete account of research into the design and implementation of logic based IR systems. Basicaly, everybody who is somebody in logical-probabilistic IR has a paper in this book. I think the editors did a very good job.

Being a lover of philosophy and also a Jew, I found Rabbi Borodowski's interplay of Biblical commentary and the interpretations of the great Jewish philosophers fascinating and enlightening. In many ways, the text introduced me to some of the lesser known philosophers and piqued my interest in studying more of their writings.

The style of writing made even the most difficult passages flow smoothly and quickly. The read made for a couple of enjoyable train-rides to work!

If you like philosophy and religion, and enjoy an in depth analysis focused on a tiny slice in time, this is a book for you.

This makes a very nice coffee table book. Great pictures with a little blurb about each one.

one of these stories (la luna y las ratas) is a masterpiece...but perhaps only for those who know these sort of people well enough to be totally bemused by them....five stars not because all the five stories are worth five stars but because the story above deserves at least twenty....

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I bought this heavy, large-format book new and have never regretted the price, and now you can buy it used for a few dollars.

This book with its long shots and close-up takes you right to the mysterious places of the world. I'm just now looking at the large-format photo of "Petra, the Rose Red City." I never realized how big the church carved in the cliff is!

The book covers ruins from all over the world, including Europe, Africa, Asia, the Americas and Oceana (great photos of the statues on Easter Island). The commentary is well-written and gives the reader valuable insights into the various civilizations.

If you like to curl up with a good book on a cold winter's night, this is the one. The sunny photos make me want to pack up and go traveling. Highly recommended.