Random matrices are widely and successfully used in physics for almost 60-70 years, beginning with the works of Dyson and Wigner. Although it is an old subject, it is constantly developing into new areas of physics and mathematics. It constitutes now a part of the general culture of a theoretical physicist. Mathematical methods inspired by random matrix theory become more powerful, sophisticated and enjoy rapidly growing applications in physics. Recent examples include the calculation of universal correlations in the mesoscopic system, new applications in disordered and quantum chaotic systems, in combinatorial and growth models, as well as the recent breakthrough, due to the matrix models, in two dimensional gravity and string theory and the non-abelian gauge theories. The book consists of the lectures of the leading specialists and covers rather systematically many of these topics. It can be useful to the specialists in various subjects using random matrices, from PhD students to confirmed scientists.

This is a monograph on geometrical and topological features which arise in quantum field theory. It is well known that when a chiral fermion interacts with a gauge field we have chiral anomaly which corresponds to the fact that divergence of the axial vector current does not vanish. It is observed that this is related to certain topological features associated with the fermion and leads to the realization of the topological origin of fermion number as well as the Berry phase. The role of gauge fields in the quantization procedure has its implications in these topological features of a fermion and helps us to consider a massive fermion as a soliton (skyrrnion). In this formalism chiral anomaly is found to be responsible for mass generation. This has its relevance in electroweak theory where it is observed that weak interaction gauge bosons attain mass topologically.

We are often told that quantum phenomena demand radical revisions of our scientific world view and that no physical theory describing well defined objects, such as particles described by their positions, evolving in a well defined way, let alone deterministically, can account for such phenomena. The great majority of physicists continue to subscribe to this view, despite the fact that just such a deterministic theory, accounting for all of the phe­ nomena of nonrelativistic quantum mechanics, was proposed by David Bohm more than four decades ago and has arguably been around almost since the inception of quantum mechanics itself. Our purpose in asking colleagues to write the essays for this volume has not been to produce a Festschrift in honor of David Bohm (worthy an undertaking as that would have been) or to gather together a collection of papers simply stating uncritically Bohm's views on quantum mechanics. The central theme around which the essays in this volume are arranged is David Bohm's version of quantum mechanics. It has by now become fairly standard practice to refer to his theory as Bohmian mechanics and to the larger conceptual framework within which this is located as the causal quantum theory program.

Particles with fractional statistics interpolating between bosons and fermions have attracted considerable interest from mathematical physicists. In recent years it has emerged that these so-called anyons have rather unexpected applications, such as the fractional Hall effect, anyonic excitations in films of liquid helium, and high-temrperature superconductivity. Furthermore, they are discussed also in the context of conformal field theories. This book is a systematic and pedagogical introduction that considers the subject of anyons from many different points of view. In particular, the author presents the relation of anyons to braid groups and Chern-Simons field theory and devotes three chapters to physical applications. The book, while being of interest to researchers, primarily addresses advanced students of mathematics and physics.

This book deals with quantum physical calculations of anomalous scattering of photons by bound electrons. The main attention is paid to studies of the process of photon scattering in the energy range of deep levels of such many-electron systems as atoms, atomic ions and simple molecules. The book is written for scientists and graduate students, working in the field of atomic and molecular spectroscopy, physics of plasmas, lasers, solid state, ionizing radiation and astrophysics. It can also be used as additional textbook for a special topic of quantum mechanics.

In this book, quantum mechanics is developed from the outset on a relativistic basis, using the superposition principle, Lorentz invariance and gauge invariance. Nonrelativistic quantum mechanics as well as classical relativistic mechanics appear as special cases. They are the sources of familiar names such as "orbital angular momentum", "spin-orbit coupling" and "magnetic moment" for operators of the relativistic quantum formalism. The theory of binaries, in terms of differential equations, is treated for the first time in this book. These have the mathematical structure of the corresponding one-body equations (Klein–Gordon for two spinless particles, Dirac for two spinor particles) with a relativistically reduced mass. They allow the calculation of radiative corrections via the vector potential operator.

In addition to traditional topics, this book includes: selective measurements, Wigner's Theorem of symmetry transformations, generators of quantum transformations, supersymmetry, details on the spectra of Hamiltonians and stability of quantum systems, Bose-Fermi oscillators, coherent states, hyperfine structure of the H-atom for any angular momentum, the non-relativistic Lamb shift, anomalous magnetic moment of the electron, Ramsey oscillatory fields methods, measurement, interference and the role of the environment, the AB effect, geometric phases, including non-adiabatic and non-cyclic, Schrödinger's cat and quantum decoherence, quantum teleportation and cryptography, quantum dynamics of the Stern-Gerlach effect, Green functions, path integrals, including constrained dynamics, quantum dynamical principle and variations, systematics of multi-electron atoms, stability of matter, collapse of "bosonic matter" and the role of spin, intricacies of scattering, quantum description of relativistic particles for any spin and mass, spinors, helicity, the Spin and Statistics Theorem. It also contains numerous problems some of which are challenging enough for research.

Single-molecule studies constitute a distinguishable category of focused - search in nanoscience and nanotechnology. This book is dedicated to the - troduction of recent advances on single-molecule studies. It will be illustrated that studying single molecules is both intellectually and technologically ch- lenging, and also o?ers vast potential in opening up new scienti?c frontiers.

This edition combines the earlier two volumes on Quantum Mechanics of Atoms and Molecules and on Quantum Mechanics of Large Systems, thus including in a single volume the material for a two-semester course on quantum physics. Since this volume is already quite heavy, I could not include many new results which show how lively the subject is. I just want to mention that inequality (IV:4. 1. l. 1) has been sharpened by T. Weidl by a factor 2][ and the difficult problem 1 of (III:4. 6) has been solved by A. Martin. I have to thank N. Ilieva for the devotion in preparing this new edition. Vienna, November 2001 Walter Thirring Preface to the Second Edition: Quantum Mechanics of Atoms and Molecules Ever since the first edition of this volume appeared in 1980 quantum statistical mechanics has florished. Innumerable results in many areas have been obtained and it would require a series of volumes to do justice to all of them. On the other hand the first edition was already rather crowded with many details so it would not be overburdened any more. Thus I added only one chapter on quantum ergodic theory where one can get the main notions across without too much pain. Nevertheless many subjects treated in the book had splendidely developed ever since and the only way out I could see is to add some recent references which the interested reader can consult.

Gauge Theory of Weak Interactions treats the unification of electromagnetic and weak interactions and considers related phenomena. First, the Fermi theory of beta decay is presented, followed by a discussion of parity violation, clarifying the importance of symmetries. Then the concept of a spontaneously broken gauge theory is introduced, and all necessary mathematical tools are carefully developed. The "standard model" of unified electroweak interactions is thoroughly discussed including current developments. The final chapter contains an introduction to unified theories of strong and electroweak interactions. Numerous solved examples and problems make this volume uniquely suited as a text for an advanced course. This third edition has been carefully revised.

The fundamental goal of physics is an understanding of the forces of nature in their simplest and most general terms. Yet there is much more involved than just a basic set of equations which eventually has to be solved when applied to specific problems. We have learned in recent years that the structure of the ground state of field theories (with which we are generally concerned) plays an equally funda­ mental role as the equations of motion themselves. Heisenberg was probably the first to recognize that the ground state, the vacuum, could acquire certain prop­ erties (quantum numbers) when he devised a theory of ferromagnetism. Since then, many more such examples are known in solid state physics, e. g. supercon­ ductivity, superfluidity, in fact all problems concerned with phase transitions of many-body systems, which are often summarized under the name synergetics. Inspired by the experimental observation that also fundamental symmetries, such as parity or chiral symmetry, may be violated in nature, it has become wide­ ly accepted that the same field theory may be based on different vacua. Practical­ ly all these different field phases have the status of more or less hypothetical models, not (yet) directly accessible to experiments. There is one magnificent ex­ ception and this is the change of the ground state (vacuum) of the electron-posi­ tron field in superstrong electric fields.

This book presents a new approach to understanding the foundation of quantum physics through the "quantum wave model" hypothesis. It addresses some of the key challenges in the current quantum theory, including the conflict between quantum mechanics and relativity, and offers a comprehensive solution to many of the existing mysteries in the field. By proposing that the vacuum is a dielectric medium and quantum particles are quantized excitation waves of the vacuum, the book provides a clear physical interpretation of wave-particle duality and explains the physical basis of energy, momentum, and mass.

Cybernetical physics borrows methods from both theoretical physics and control engineering. It deals with the control of complex systems is one of the most important aspects in dealing with systems exhibiting nonlinear behavior or similar features that defy traditional control techniques. This book fully details this new discipline.

Our DMV Seminar on 'Classical Nonintegrability, Quantum Chaos' intended to introduce students and beginning researchers to the techniques applied in nonin­ tegrable classical and quantum dynamics. Several of these lectures are collected in this volume. The basic phenomenon of nonlinear dynamics is mixing in phase space, lead­ ing to a positive dynamical entropy and a loss of information about the initial state. The nonlinear motion in phase space gives rise to a linear action on phase space functions which in the case of iterated maps is given by a so-called transfer operator. Good mixing rates lead to a spectral gap for this operator.

In the last few years, multiscale methods have lead to spectacular progress in our understanding of complex physical systems and have stimulated the development of very refined mathematical techniques. At the same time on the experimental side, equally spectacular progress has been made in developing experimental machinery and techniques to test the foundations of quantum mechanics. In view of this progress, this volume is very timely; it is the first text totally devoted to multiscale methods as applied to various areas of physics and to the relative developments in mathematics.