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[en] The aim of relativistic quantum mechanics is to describe the finer details of the structure of atoms and molecules, where relativistic effects become nonnegligible. It is a sort of intermediate realm, between the familiar nonrelativistic quantum mechanics and fully relativistic quantum field theory, and thus it lacks the simplicity and elegance of both. Yet it is a necessary tool, mostly for quantum chemists. Pilkuhn's book offers to this audience an up-to-date survey of these methods, which is quite welcome since most previous textbooks are at least ten years old. The point of view of the author is to start immediately in the relativistic domain, following the lead of Maxwell's equations rather than classical mechanics, and thus to treat the nonrelativistic version as an approximation. Thus Chapter 1 takes off from Maxwell's equations (in the noncovariant Coulomb gauge) and gradually derives the basic aspects of Quantum Mechanics in a rather pedestrian way (states and observables, Hilbert space, operators, quantum measurement, scattering,. Chapter 2 starts with the Lorentz transformations, then continues with the Pauli spin equation and the Dirac equation and some of their applications (notably the hydrogen atom). Chapter 3 is entitled 'Quantum fields and particles', but falls short of treating quantum field theory properly: only creation/annihilation operators are considered, for a particle in a box. The emphasis is on two-electron states (the Pauli principle, the Foldy--Wouthuysen elimination of small components of Dirac spinors, Breit projection operators. Chapter 4 is devoted to scattering theory and the description of relativistic bound states. Chapter 5, finally, covers hyperfine interactions and radiative corrections. As we said above, relativistic quantum mechanics is by nature limited in scope and rather inelegant and Pilkuhn's book is no exception. The notation is often heavy (mostly noncovariant) and the mathematical level rather low. The central topic is the description of atoms and molecules, including relativistic effects. The author fulfils this program in a reasonable way and offers a valuable tool to the targeted audience. I am not overly enthusiastic about the end result, but I might be prejudiced. Clearly, going further would require the full power of quantum field theory, but this is clearly beyond the scope of the book. (book review)

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DOI: 10.1088/0305-4470/37/4/B01; Available online at http://stacks.iop.org/0305-4470/37/1463/a4_4_b01.pdf or at the Web site for the Journal of Physics. A, Mathematical and General (ISSN 1361-6447) http://www.iop.org/; Book review of Relativistic Quantum Mechanics by H M Pilkuhn, 2003 Heidelberg: Springer-Verlag, ISBN 3-540-43666-9; Country of input: International Atomic Energy Agency (IAEA)

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[en] This is a book about the modelling of complex systems and, unlike many books on this subject, concentrates on the discussion of specific systems and gives practical methods for modelling and simulating them. This is not to say that the author does not devote space to the general philosophy and definition of complex systems and agent-based modelling, but the emphasis is definitely on the development of concrete methods for analysing them. This is, in my view, to be welcomed and I thoroughly recommend the book, especially to those with a theoretical physics background who will be very much at home with the language and techniques which are used. The author has developed a formalism for understanding complex systems which is based on the Langevin approach to the study of Brownian motion. This is a mesoscopic description; details of the interactions between the Brownian particle and the molecules of the surrounding fluid are replaced by a randomly fluctuating force. Thus all microscopic detail is replaced by a coarse-grained description which encapsulates the essence of the interactions at the finer level of description. In a similar way, the influences on Brownian agents in a multi-agent system are replaced by stochastic influences which sum up the effects of these interactions on a finer scale. Unlike Brownian particles, Brownian agents are not structureless particles, but instead have some internal states so that, for instance, they may react to changes in the environment or to the presence of other agents. Most of the book is concerned with developing the idea of Brownian agents using the techniques of statistical physics. This development parallels that for Brownian particles in physics, but the author then goes on to apply the technique to problems in biology, economics and the social sciences. This is a clear and well-written book which is a useful addition to the literature on complex systems. It will be interesting to see if the use of Brownian agents becomes a standard tool in the study of complex systems in the future. (book review)

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DOI: 10.1088/0305-4470/36/49/B05; Available online at the Web site for the Journal of Physics. A, Mathematical and General (ISSN 1361-6447) http://www.iop.org/; Book review of 'Brownian Agents and Active Particles: Collective Dynamics in the Natural and Social Sciences' by F Schweitzer, 2003 Heidelberg: Springer-Verlag, ISBN 3-540-43938-2; Country of input: International Atomic Energy Agency (IAEA)

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Journal of Physics. A, Mathematical and General; ISSN 0305-4470; ; CODEN JPHAC5; v. 36(49); p. 12348

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[en] The discovery of the geometric phase is one of the most interesting and intriguing findings of the last few decades. It led to a deeper understanding of the concept of phase in quantum mechanics and motivated a surge of interest in fundamental quantum mechanical issues, disclosing unexpected applications in very diverse fields of physics. Although the key ideas underlying the existence of a purely geometrical phase had already been proposed in 1956 by Pancharatnam, it was Michael Berry who revived this issue 30 years later. The clarity of Berry's seminal paper, in 1984, was extraordinary. Research on the topic flourished at such a pace that it became difficult for non-experts to follow the many different theoretical ideas and experimental proposals which ensued. Diverse concepts in independent areas of mathematics, physics and chemistry were being applied, for what was (and can still be considered) a nascent arena for theory, experiments and technology. Although collections of papers by different authors appeared in the literature, sometimes with ample introductions, surprisingly, to the best of my knowledge, no specific and exhaustive book has ever been written on this subject. The Geometric Phase in Quantum Systems is the first thorough book on geometric phases and fills an important gap in the physical literature. Other books on the subject will undoubtedly follow. But it will take a fairly long time before other authors can cover that same variety of concepts in such a comprehensive manner. The book is enjoyable. The choice of topics presented is well balanced and appropriate. The appendices are well written, understandable and exhaustive - three rare qualities. I also find it praiseworthy that the authors decided to explicitly carry out most of the calculations, avoiding, as much as possible, the use of the joke 'after a straightforward calculation, one finds...' This was one of the sentences I used to dislike most during my undergraduate studies. A student is inexperienced in such matters and needs to look at details. This book is addressed to graduate physics and chemistry students and was written thinking of students. However, I would recommend it also to young and mature physicists, even those who are already 'into' the subject. It is a comprehensive work, jointly written by five researchers. After a simple introduction to the subject, the book gradually provides deeper concepts, more advanced theory and finally an interesting introduction and explanation of recent experiments. For its multidisciplinary features, this work could not have been written by one single author. The collaborative effort is undoubtedly one of its most interesting qualities. I would definitely recommend it to anyone who wants to learn more on the geometric phase, a topic that is both beautiful and intriguing. (book review)

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DOI: 10.1088/0305-4470/36/49/B01; Available online at the Web site for the Journal of Physics. A, Mathematical and General (ISSN 1361-6447) http://www.iop.org/; Book review of 'The Geometric Phase in Quantum Systems' by A Bohm, A Mostafazadeh, H Koizumi, Q Niu and J Zwanziger, 2003 Heidelberg: Springer-Verlag, ISBN 3-540-00031-3; Country of input: International Atomic Energy Agency (IAEA)

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Journal of Physics. A, Mathematical and General; ISSN 0305-4470; ; CODEN JPHAC5; v. 36(49); p. 12345

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[en] In our recently published paper 'Chaos in Bohmian quantum mechanics' we criticized a paper by Parmenter and Valentine (1995 Phys. Lett. A 201 1), because the authors made an incorrect calculation of the Lyapunov exponent in the case of Bohmian orbits in a quantum system of two uncoupled harmonic oscillators. After our paper was published, we became aware of an erratum published by the same authors (Parmenter and Valentine 1996 Phys. Lett. A 213 319) that recognized the error made in their previous calculations. The authors realized that, when correctly calculated, 'aperiodic trajectories with well defined boundaries...have vanishing Lyapunov exponents', i.e., they are not chaotic. We want to supplement our paper with a reference to this erratum. The generic calculation of Lyapunov exponents in Bohmian quantum systems remains an original contribution of our paper (section 2). (addendum)

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S0305-4470(06)23031-2; Available online at http://stacks.iop.org/0305-4470/39/8563/a6_26_021.pdf or at the Web site for the Journal of Physics. A, Mathematical and General (ISSN 1361-6447) http://www.iop.org/; Country of input: International Atomic Energy Agency (IAEA)

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[en] It is difficult not to be amazed by the ability of the human brain to process, to structure and to memorize information. Even by the toughest standards the behaviour of this network of about 10

^{11}neurons qualifies as complex, and both the scientific community and the public take great interest in the growing field of neuroscience. The scientific endeavour to learn more about the function of the brain as an information processing system is here a truly interdisciplinary one, with important contributions from biology, computer science, physics, engineering and mathematics as the authors quite rightly point out in the introduction of their book. The role of the theoretical disciplines here is to provide mathematical models of information processing systems and the tools to study them. These models and tools are at the centre of the material covered in the book by Coolen, Kuehn and Sollich. The book is divided into five parts, providing basic introductory material on neural network models as well as the details of advanced techniques to study them. A mathematical appendix complements the main text. The range of topics is extremely broad, still the presentation is concise and the book well arranged. To stress the breadth of the book let me just mention a few keywords here: the material ranges from the basics of perceptrons and recurrent network architectures to more advanced aspects such as Bayesian learning and support vector machines; Shannon's theory of information and the definition of entropy are discussed, and a chapter on Amari's information geometry is not missing either. Finally the statistical mechanics chapters cover Gardner theory and the replica analysis of the Hopfield model, not without being preceded by a brief introduction of the basic concepts of equilibrium statistical physics. The book also contains a part on effective theories of the macroscopic dynamics of neural networks. Many dynamical aspects of neural networks are usually hard to find in the existing textbook literature, so that this discussion will be very much appreciated. The book is of an exceptionally high quality in all aspects. In my view, the style of presentation and the inclusion of recent aspects of the topic alone make the book a welcomed addition to the existing literature. It is well structured and the material covered was chosen with care. While focusing on quantitative aspects of the subject, the authors adopt a comprehensive style of presentation, being precise, but not pedantic. The student who is not familiar with the field might find the breadth of the book overwhelming at first, but will soon appreciate its pedagogical value. All mathematical derivations are performed and explained step by step for the student to follow, and they are illustrated by many concrete examples and results from computer simulations in well-presented and clear figures. If a student wants to get his hands on the mathematical tools of neural networks theory then this book is a good place to learn from. A set of instructive and valuable exercises complements each chapter (hints are given, but maybe it would have been nice to provide additional brief sample solutions in an appendix). I very much enjoyed the outlook sections at the end of each of the five parts, putting the material covered into its historical context and providing further references. In summary, students of a quantitative discipline will find in this book a clear and self-contained introduction to the subject, lecturers might use it to design postgraduate courses, and finally it will provide a valuable reference for researchers working in the area. This book can be expected to be an asset for all types of readers, even if they already own a book on neural networks. Anyone with a serious interest in the theoretical aspects of the field would be making a mistake not to have a copy on their shelves. (book review)Primary Subject

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Available online at http://stacks.iop.org/0305-4470/39/3849/a6_14_b01.pdf or at the Web site for the Journal of Physics. A, Mathematical and General (ISSN 1361-6447) http://www.iop.org/; Book review of 'Theory of Neural Information Processing Systems' by A C C Coolen, R Kuehn and P Sollich, 2005 Oxford University Press, 569pp, ISBN 0-19-853023-4; Country of input: International Atomic Energy Agency (IAEA)

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[en] This book by Nino Boccara presents a compilation of model systems commonly termed as 'complex'. It starts with a definition of the systems under consideration and how to build up a model to describe the complex dynamics. The subsequent chapters are devoted to various categories of mean-field type models (differential and recurrence equations, chaos) and of agent-based models (cellular automata, networks and power-law distributions). Each chapter is supplemented by a number of exercises and their solutions. The table of contents looks a little arbitrary but the author took the most prominent model systems investigated over the years (and up until now there has been no unified theory covering the various aspects of complex dynamics). The model systems are explained by looking at a number of applications in various fields. The book is written as a textbook for interested students as well as serving as a comprehensive reference for experts. It is an ideal source for topics to be presented in a lecture on dynamics of complex systems. This is the first book on this 'wide' topic and I have long awaited such a book (in fact I planned to write it myself but this is much better than I could ever have written it!). Only section 6 on cellular automata is a little too limited to the author's point of view and one would have expected more about the famous Domany-Kinzel model (and more accurate citation!). In my opinion this is one of the best textbooks published during the last decade and even experts can learn a lot from it. Hopefully there will be an actualization after, say, five years since this field is growing so quickly. The price is too high for students but this, unfortunately, is the normal case today. Nevertheless I think it will be a great success! (book review)

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DOI: 10.1088/0305-4470/37/40/B01; Available online at the Web site for the Journal of Physics. A, Mathematical and General (ISSN 1361-6447) http://www.iop.org/; Book review of 'Modeling Complex Systems' by N Boccara, Heidelberg: Springer-Verlag (2004), 397pp, ISBN 0-387-40462-7; Country of input: International Atomic Energy Agency (IAEA)

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[en] This addendum states the relationship between our recent paper and Mackay et al's work on the negative refraction in gyrotropic chiral media, and emphasizes the advantages of their work that we did not mention in our recent paper. (addendum)

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S0305-4470(06)32221-4; Available online at http://stacks.iop.org/0305-4470/39/15057/a6_48_013.pdf or at the Web site for the Journal of Physics. A, Mathematical and General (ISSN 1361-6447) http://www.iop.org/; Country of input: International Atomic Energy Agency (IAEA)

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Journal of Physics. A, Mathematical and General; ISSN 0305-4470; ; CODEN JPHAC5; v. 39(48); p. 15057

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AbstractAbstract

[en] The stated goal of this book is to fill a perceived gap between undergraduate texts on critical phenomena and advanced texts on quantum field theory, in the general area of renormalization methods. It is debatable whether this gap really exists nowadays, as a number of books have appeared in which it is made clear that field-theoretic renormalization group methods are not the preserve of particle theory, and indeed are far more easily appreciated in the contexts of statistical and condensed matter physics. Nevertheless, this volume does have a fresh aspect to it, perhaps because of the author's background in fluid dynamics and turbulence theory, rather than through the more traditional migration from particle physics. The book begins at a very elementary level, in an effort to motivate the use of renormalization methods. This is a worthy effort, but it is likely that most of this section will be thought too elementary by readers wanting to get their teeth into the subject, while those for whom this section is apparently written are likely to find the later chapters rather challenging. The author's particular approach then leads him to emphasise the role of renormalized perturbation theory (rather than the renormalization group) in a number of problems, including non-linear systems and turbulence. Some of these ideas will be novel and perhaps even surprising to traditionally trained field theorists. Most of the rest of the book is on far more familiar territory: the momentum-space renormalization group, epsilon-expansion, and so on. This is standard stuff, and, like many other textbooks, it takes a considerable chunk of the book to explain all the formalism. As a result, there is only space to discuss the standard φ

^{4}field theory as applied to the Ising model (even the N-vector model is not covered) so that no impression is conveyed of the power and extent of all the applications and generalizations of the techniques. It is regrettable that so much space is spent on rather oversimplified and unrelated models in the first part of the book that, in the end, the reader is left breathless on the threshold of the really interesting material. Despite the earlier emphasis on the application of renormalization ideas in dynamics, in the end the full power of the field-theoretical approach is not applied to the obvious arena of dynamic critical phenomena (where there certainly is currently a gap in the literature) but to the Navier--Stokes equations. The development of the book is somewhat illogical in places. Mean field theory interrupts discussion of block spin methods and scaling arguments---the two are distinct approaches. The Callan--Symanzik equation is introduced before Feynman diagrams are explained, so that there is a hiatus before the actual results for the critical exponents can be found. I think this book is too broad in some respects, and too limited in others, to be a really useful textbook for a course on renormalization methods. Those who have learned these ideas either from field theory, or from nonlinear systems, will find it more rewarding for the sections covering the topics with which they are less familiar. For this reason alone, the book should at least find a place on most library reference shelves. (book review)Primary Subject

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DOI: 10.1088/0305-4470/37/19/B02; Available online at the Web site for the Journal of Physics. A, Mathematical and General (ISSN 1361-6447) http://www.iop.org/; Book review of 'Renormalization Methods - A Guide For Beginners' by W D McComb, Oxford: Oxford University Press (2003), ISBN 0-19-850694-5; Country of input: International Atomic Energy Agency (IAEA)

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[en] Quantum Noise is advertised as a handbook, and this is indeed how it functions for me these days: it is a book that I keep within hand's reach, ready to be consulted on the proper use of quantum stochastic methods in the course of my research on quantum dots. I should point out that quantum optics, the target field for this book, is not my field by training. So I have much to learn, and find this handbook to be a reliable and helpful guide. Crispin Gardiner previously wrote the Handbook of Stochastic Methods (also published by Springer), which provides an overview of methods in classical statistical physics. Quantum Noise, written jointly with Peter Zoller, is the counterpart for quantum statistical physics, and indeed the two books rely on each other by frequent cross referencing. The fundamental problem addressed by Quantum Noise is how the quantum dynamics of an open system can be described statistically by treating the environment as a source of noise. This is a general problem in condensed matter physics (in particular in the context of Josephson junctions) and in quantum optics. The emphasis in this book in on the optical applications (for condensed matter applications one could consult Quantum Dissipative Systems by Ulrich Weiss, published by World Scientific). The optical applications centre around the interaction of light with atoms, where the atoms represent the open system and the light is the noisy environment. A complete description of the production and detection of non-classical states of radiation (such as squeezed states) can be obtained using one of the equivalent quantum stochastic formulations: the quantum Langevin equation for the field operators (in either the Ito or the Stratonovich form), the Master equation for the density matrix, or the stochastic Schroedinger equation for the wave functions. Each formulation is fully developed here (as one would expect from a handbook), with detailed instructions on how to go from one to the other. The development of the topic is precise and well-organized. The derivations are written out in sufficient detail, without frustrating comments like 'it can be shown that'. The book is not quite self-contained, because it relies on the Handbook of Stochastic Methods for some background material (notably the issue of Ito versus Stratonovich). Still, one could very well use this book as a text for a course, supplying the background material to the students in some other form. Quantum Noise is now in its third edition. The second edition was a major expansion, including applications to laser cooling and quantum information processing. The third edition is a relatively minor upgrade, consisting mainly of pointers to recent literature. If you own the second edition, you might well skip this upgrade. If you do not yet own the book, or are still at edition 1, then I would enthusiastically recommend acquiring this handbook, regardless of whether you work in quantum optics or in another field of quantum physics. As I did, you might well find a new tool to attack your favourite problem. (book review)

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Available online at http://stacks.iop.org/0305-4470/38/7595/1.pdf or at the Web site for the Journal of Physics. A, Mathematical and General (ISSN 1361-6447) http://www.iop.org/; Book review of 'Quantum Noise' by C W Gardiner and P Zoller, 2004 Heidelberg: Springer, ISBN 3-540-22301-0; Country of input: International Atomic Energy Agency (IAEA)

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[en] The Gross-Pitaevskii equation, named after one of the authors of the book, and its large number of applications for describing the properties of Bose-Einstein condensation (BEC) in trapped weakly interacting atomic gases, is the main topic of this book. In total the monograph comprises 18 chapters and is divided into two parts. Part I introduces the notion of BEC and superfluidity in general terms. The most important properties of the ideal and the weakly interacting Bose gas are described and the effects of nonuniformity due to an external potential at zero temperature are studied. The first part is then concluded with a summary of the properties of superfluid He. In Part II the authors describe the theoretical aspects of BEC in harmonically trapped weakly interacting atomic gases. A short and rather rudimentary chapter on collisions and trapping of atomic gases which seems to be included for completeness only is followed by a detailed analysis of the ground state, collective excitations, thermodynamics, and vortices as well as mixtures of BECs and the Josephson effect in BEC. Finally, the last three chapters deal with topics of more recent interest like BEC in optical lattices, low dimensional systems, and cold Fermi gases. The book is well written and in fact it provides numerous useful and important relations between the different properties of a BEC and covers most of the aspects of ultracold weakly interacting atomic gases from the point of view of condensed matter physics. The book contains a comprehensive introduction to BEC for physicists new to the field as well as a lot of detail and insight for those already familiar with this area. I therefore recommend it to everyone who is interested in BEC. Very clearly however, the intention of the book is not to provide prospects for applications of BEC in atomic physics, quantum optics or quantum state engineering and therefore the more practically oriented reader might sometimes wonder why exactly an equation is termed useful or important. (book review)

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DOI: 10.1088/0305-4470/36/37/701; Available online at http://stacks.iop.org/0305-4470/36/9797/a337b1.pdf or at the Web site for the Journal of Physics. A, Mathematical and General (ISSN 1361-6447) http://www.iop.org/; Authors of book reviewed: L Pitaevskii and S Stringari; Oxford: Oxford University Press (2003); ISBN 0-19-850719-4; ; Country of input: International Atomic Energy Agency (IAEA)

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