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Quantum entropy and its use
304
Book Reviews
THE
SHAGGY
STEED
OF PHYSICS. MATHEMATICAL BEAUTY PHYSICAL WORLD D. Oliver (Springer, Berlin, xv + 298 pp., 71 figs, 1994)
IN THE
The title of this book draws its inspiration from a story by W. B. Yeats, and the author assures us that the book “is intended for lovers”. I think this means “for everyone”, for are we not all lovers? When one gets down to it, however, one soon discovers that the author is an admirer of mechanics (classical, relativistic, quantum), and, like many before him, is fascinated by symmetries and action principles. This has prompted him to produce a companion volume for the many textbooks students have to study in order to learn these subjects. There are few jokes, no cartoons-just an emphasis on symmetry with a good deal of mathematics. But it is made enjoyable. The electronics expert should be aware of this book, even though it does not bear directly on his subject. There are incidentally 14 pages of illustrated biographies of famous scientists, but I would question whether Dirac died in Miami rather than Tallahassee; or whether Einstein died in 1953 (rather than 1955), as the index informs us. Professor
P. T. Landsberg
Faculty of Mathematical Studies, University of Southampton Southampton SO9 5NH, U.K.
(Springer,
Texts
and
QUANTUM ENTROPY AND ITS USE M. Ohya and D. Petz Monographs in Physics, Heidelberg, 335 pp., ISBN Hardcover, DM 148.00)
3-540-54881-5,
The concept of entropy originated in classical thermodynamics, where it appeared as a rather mysterious mathematical gadget, which served both as a key to the thermal properties of matter and as an indicator of the direction of irreversible processes. Its subsequent microscopic interpretation, by Boltzmann and Planck, as a precise measure of the degree of disorder of the state of a system, removed the mystery and provided an essential connection between microphysics and macrophysics. It also led to ramifications in quantum statistical mechanics, where Von Neumann (1932) generalised the Boltzmann-Planck formula to arbitrary mixed states; in communication theory, where Shannon (1948) related entropy to information; and in theory of classical dynamical systems, where Kolmogorov and Sinai (1959) constructed a theory of dynamical entropy, which provided a measure of the stochasticity, or dissipativity, of a process. In the last twenty years, there have been further radical advances in the theory of quantum mechanical entropy. These have been achieved mainly within an operator algebraic framework for quantum theory, which extends the traditional one of Dirac and Von Neumann in such a way as to reveal certain intrinsic features of complex systems, such as phase structure, that had previously been masked by finite size effects. This algebraic formalism also provides a natural framework for a general theory of quantum entropy, with applications in a variety of areas where quantum mechanics plays an essential role: these include statistical thermodynamics, stochastic processes, non-linear optics and communication theory. The book, Quantum Entropy and its Use, by M. Ohya and D. Petz, is devoted to the mathematical structure of the theory of quantum entropy, within the terms of the operator
Book Reviews
305
algebraic framework. In view of the wide-ranging developments in the subject, there is probably a call for such a book, which brings together the various facets of quantum entropy and related concepts, such as relative entropy, conditional entropy and dynamical entropy. This book is unimbiguously mathematical, containing more than two hundred theorems (or propositions or lemmas), with proofs, in three hundred pages. As a mathematical work. It is well organised, and the proofs are clear. Furthermore, it has a very extensive list of references, covering the literature on the subject at the time the book went to press. On the other hand, it makes little concession to the reader who is not well-versed in operator algebra. Indeed, it provides neither the basic definitions of this area of mathematics, nor any discussion of the advantages it can bring. Furthermore, it does not really provide a discussion of the physical or engineering problems to which quantum entropy is relevant. For example, there is nothing in the book about such questions as irreversibility, order-disorder transitions, or the connection between dynamical entropy and stochasticity. Instead, there is just a succession of definitions and theorems, which could provide the raw materials needed to tackle such questions, but not the insights required for their effective application. I would summarise by saying that this book should be of interest to two classes of readers. The first would consist of mathematicians, who are well-versed in operator algebra and interested in the general structure of the theory of quantum entropy. The second would consist of physicists or engineers, with a working knowledge of these algebras, who know which mathematical properties of entropy they require for some problem: such scientists might well find what they are looking for amid the theorems of this book. On the other hand, scientists who want either to learn about operator algebra, with its relevance to entropy theory, or to understand the physical and engineering problems to which quantum entropy is relevant, should look elsewhere. G. L. Sewell London, U.K.
DESIGN AND FABRICATION OF ACOUSTO-OPTIC DEVICES Editors: A. P. Goutzoulis and D. R. Pape (plus S. V. Kulakov, editor of Russian Contribution) (Marcel1 Dekker, Inc., New York, 512 pp., ISBN 0-8247-8930-X, $165 (subject to possible change), 1994) As the title suggests, this book is intended to cover the design and fabrication of acousto-optic (A-O) devices. It is, in my view, a well-prepared book which covers the basic concepts in addition to containing a number of excellent chapters on the design of this increasingly-useful component. The description of applications is up-to-date and there are excellent chapters on manufacture, plus a first-rate chapter of procedures for the testing of A-O devices. There is a good balance between the coverage of engineering aspects and the theoretical principles. The text is also well illustrated with clear diagrams, useful graphs and helpful photographs. This is one of the better multi-author books which I have reviewed recently, both in view of its contents and the observation that real care appears to have been taken in its structure. As is almost inevitable with such books, however, there are occasional lapses in the editing. For example, there is a description of a test arrangement for A-O tunable filters which occurs before a chapter dedicated to this subject. The few examples which occur, do not, however, detract significantly from the value of the text. Perhaps my only significant reservation regarding the book is its price. At $165 a copy, I was grateful that I received mine as a free reviewer’s issue! Despite the book’s merits, I would
Book Reviews
THE
SHAGGY
STEED
OF PHYSICS. MATHEMATICAL BEAUTY PHYSICAL WORLD D. Oliver (Springer, Berlin, xv + 298 pp., 71 figs, 1994)
IN THE
The title of this book draws its inspiration from a story by W. B. Yeats, and the author assures us that the book “is intended for lovers”. I think this means “for everyone”, for are we not all lovers? When one gets down to it, however, one soon discovers that the author is an admirer of mechanics (classical, relativistic, quantum), and, like many before him, is fascinated by symmetries and action principles. This has prompted him to produce a companion volume for the many textbooks students have to study in order to learn these subjects. There are few jokes, no cartoons-just an emphasis on symmetry with a good deal of mathematics. But it is made enjoyable. The electronics expert should be aware of this book, even though it does not bear directly on his subject. There are incidentally 14 pages of illustrated biographies of famous scientists, but I would question whether Dirac died in Miami rather than Tallahassee; or whether Einstein died in 1953 (rather than 1955), as the index informs us. Professor
P. T. Landsberg
Faculty of Mathematical Studies, University of Southampton Southampton SO9 5NH, U.K.
(Springer,
Texts
and
QUANTUM ENTROPY AND ITS USE M. Ohya and D. Petz Monographs in Physics, Heidelberg, 335 pp., ISBN Hardcover, DM 148.00)
3-540-54881-5,
The concept of entropy originated in classical thermodynamics, where it appeared as a rather mysterious mathematical gadget, which served both as a key to the thermal properties of matter and as an indicator of the direction of irreversible processes. Its subsequent microscopic interpretation, by Boltzmann and Planck, as a precise measure of the degree of disorder of the state of a system, removed the mystery and provided an essential connection between microphysics and macrophysics. It also led to ramifications in quantum statistical mechanics, where Von Neumann (1932) generalised the Boltzmann-Planck formula to arbitrary mixed states; in communication theory, where Shannon (1948) related entropy to information; and in theory of classical dynamical systems, where Kolmogorov and Sinai (1959) constructed a theory of dynamical entropy, which provided a measure of the stochasticity, or dissipativity, of a process. In the last twenty years, there have been further radical advances in the theory of quantum mechanical entropy. These have been achieved mainly within an operator algebraic framework for quantum theory, which extends the traditional one of Dirac and Von Neumann in such a way as to reveal certain intrinsic features of complex systems, such as phase structure, that had previously been masked by finite size effects. This algebraic formalism also provides a natural framework for a general theory of quantum entropy, with applications in a variety of areas where quantum mechanics plays an essential role: these include statistical thermodynamics, stochastic processes, non-linear optics and communication theory. The book, Quantum Entropy and its Use, by M. Ohya and D. Petz, is devoted to the mathematical structure of the theory of quantum entropy, within the terms of the operator
Book Reviews
305
algebraic framework. In view of the wide-ranging developments in the subject, there is probably a call for such a book, which brings together the various facets of quantum entropy and related concepts, such as relative entropy, conditional entropy and dynamical entropy. This book is unimbiguously mathematical, containing more than two hundred theorems (or propositions or lemmas), with proofs, in three hundred pages. As a mathematical work. It is well organised, and the proofs are clear. Furthermore, it has a very extensive list of references, covering the literature on the subject at the time the book went to press. On the other hand, it makes little concession to the reader who is not well-versed in operator algebra. Indeed, it provides neither the basic definitions of this area of mathematics, nor any discussion of the advantages it can bring. Furthermore, it does not really provide a discussion of the physical or engineering problems to which quantum entropy is relevant. For example, there is nothing in the book about such questions as irreversibility, order-disorder transitions, or the connection between dynamical entropy and stochasticity. Instead, there is just a succession of definitions and theorems, which could provide the raw materials needed to tackle such questions, but not the insights required for their effective application. I would summarise by saying that this book should be of interest to two classes of readers. The first would consist of mathematicians, who are well-versed in operator algebra and interested in the general structure of the theory of quantum entropy. The second would consist of physicists or engineers, with a working knowledge of these algebras, who know which mathematical properties of entropy they require for some problem: such scientists might well find what they are looking for amid the theorems of this book. On the other hand, scientists who want either to learn about operator algebra, with its relevance to entropy theory, or to understand the physical and engineering problems to which quantum entropy is relevant, should look elsewhere. G. L. Sewell London, U.K.
DESIGN AND FABRICATION OF ACOUSTO-OPTIC DEVICES Editors: A. P. Goutzoulis and D. R. Pape (plus S. V. Kulakov, editor of Russian Contribution) (Marcel1 Dekker, Inc., New York, 512 pp., ISBN 0-8247-8930-X, $165 (subject to possible change), 1994) As the title suggests, this book is intended to cover the design and fabrication of acousto-optic (A-O) devices. It is, in my view, a well-prepared book which covers the basic concepts in addition to containing a number of excellent chapters on the design of this increasingly-useful component. The description of applications is up-to-date and there are excellent chapters on manufacture, plus a first-rate chapter of procedures for the testing of A-O devices. There is a good balance between the coverage of engineering aspects and the theoretical principles. The text is also well illustrated with clear diagrams, useful graphs and helpful photographs. This is one of the better multi-author books which I have reviewed recently, both in view of its contents and the observation that real care appears to have been taken in its structure. As is almost inevitable with such books, however, there are occasional lapses in the editing. For example, there is a description of a test arrangement for A-O tunable filters which occurs before a chapter dedicated to this subject. The few examples which occur, do not, however, detract significantly from the value of the text. Perhaps my only significant reservation regarding the book is its price. At $165 a copy, I was grateful that I received mine as a free reviewer’s issue! Despite the book’s merits, I would