Volume 1 is a collection of reprinted works of Alexei Zamolodchikov who was a prominent theoretical physicist of his time. It contains his works on conformal field theories, 2D quantum gravity, and Liouville theory. These original contributions of Alexei Zamolodchikov have a profound effect on shaping the fast developing areas of theoretical physics. His ideas are expressed lucidly, such as the recursive relation for conformal blocks and the structure of conformal bootstrap in Liouville theory, including the boundary Liouville theory. These ideas are at the foundation of the subject and they are of great interest to a wide community of physicists and mathematicians working in diverse areas. This volume is a part of the 2-volume collection of remarkable research papers that can be used as an advanced textbook by graduate students specializing in string theory, conformal field theory and integrable models of QFT. It is also highly relevant to experts in these fields. Volume 2 includes Alexei Zamolodchikov's works on non-perturbative methods in 2D quantum field theory, and on integrable models.
Volume 1 is a collection of reprinted works of Alexei Zamolodchikov who was a prominent theoretical physicist of his time. It contains his works on conformal field theories, 2D quantum gravity, and Liouville theory. These original contributions of Alexei Zamolodchikov have a profound effect on shaping the fast developing areas of theoretical physics. His ideas are expressed lucidly, such as the recursive relation for conformal blocks and the structure of conformal bootstrap in Liouville theory, including the boundary Liouville theory. These ideas are at the foundation of the subject and they are of great interest to a wide community of physicists and mathematicians working in diverse areas. This volume is a part of the 2-volume collection of remarkable research papers that can be used as an advanced textbook by graduate students specializing in string theory, conformal field theory and integrable models of QFT. It is also highly relevant to experts in these fields. Volume 2 includes Alexei Zamolodchikov's works on non-perturbative methods in 2D quantum field theory, and on integrable models.
Abstract This book is an introduction to statistical field theory, which is an important subject within theoretical physics and a field that has seen substantial progress in recent years. The book covers fundamental topics in great detail and includes areas like conformal field theory, quantum integrability, S-matrices, braiding groups, Bethe ansatz, renormalization groups, Majorana fermions, form factors, the truncated conformal space approach and boundary field theory. It also provides an introduction to lattice statistical models. Many topics are discussed at a fairly advanced level but via a pedagogical approach. In particular, the book presents in a clear way non-perturbative methods of quantum field theories that have become decisive tools in many different areas of statistical and condensed matter physics, and which are currently an essential foundation of the working knowledge of a modern theoretical physicist.
Quantum field theory (QFT) is a rich and robust part of mathematical and theoretical physics, with a wide range of applications in statistical physics, condensed matter physics, particle physics and string theory. Many (if not most) of the applications in four space-time dimensions require some sort of systematic approximations. In contrast, quantum field theories in lower dimensions often display special properties which open the door, at least in principle, to their exact solution. In many cases, the methods used to obtain exact solutions are peculiar to low-dimensional space-times, but the exact solutions thus obtained can teach us a great deal about more realistic QFTs in higher dimensions. Thus, lower-dimensional QFTs provide excellent models for exploring deep theoretical and computational issues, in addition to their direct applications to physical processes in reduced dimensions, which include polymers, boundary fluctuations, spin chains, and quantum Hall systems. Such models are also vital testing grounds for fundamental theoretical ideas such as integrability, phase transitions, collective and non-perturbative quantum behaviour, and non-equilibrium statistical mechanics. We should also mention the effective dimensional reduction which occurs in the high temperature limit, or is induced by disorder. Low-dimensional quantum field theory thus sits at a unique intersection point of contemporary mathematics and physics, allowing many of these issues to be addressed rigorously and analytically. Surprising inter-relations continue to arise, from the ubiquitous nature of the Bethe ansatz and Calogero models, to the link between integrable spin chains and the anomalous dimensions of operators in gauge theories. Aspects of the latter connection have been known at least since the 1990s; they have recently received tremendous attention in connection with N =4 supersymmetric Yang--Mills theory (in four space-time dimensions) and the AdS/CFT correspondence. One unifying idea is the Bethe ansatz, whose 75th anniversary we celebrate this year, and whose wide-ranging impact in diverse branches of modern mathematical physics is made abundantly clear in this special issue. In addition, new ideas and techniques of significance to both mathematics and physics have emerged: the recent and spectacular progress on geometrical aspects of phase transitions made possible through the ideas of stochastic (Schramm) Loewner evolution, or SLE, is a particularly noteworthy example. All papers in this special issue are invited, refereed contributions from leading experts. Many are pedagogical reviews, capturing the current state of knowledge in a particular subject, putting the subject in context and outlining open challenges. Others are research papers, reflecting the rapid flux of new ideas and results in this rich and exciting field. Together, they represent a broad sample of the entire field. We would like to take this opportunity to thank all of the authors for their efforts in making this special issue possible, and Kazuhiro Hikami for his help in the early stages of this project. We hope it inspires many new developments!
Volume 1: From Brownian Motion to Renormalization and Lattice Gauge Theory. Volume 2: Strong Coupling, Monte Carlo Methods, Conformal Field Theory, and Random Systems. This two-volume work provides a comprehensive and timely survey of the application of the methods of quantum field theory to statistical physics, a very active and fruitful area of modern research. The first volume provides a pedagogical introduction to the subject, discussing Brownian motion, its anticommutative counterpart in the guise of Onsager's solution to the two-dimensional Ising model, the mean field or Landau approximation, scaling ideas exemplified by the Kosterlitz-Thouless theory for the XY transition, the continuous renormalization group applied to the standard phi-to the fourth theory (the simplest typical case) and lattice gauge theory as a pathway to the understanding of quark confinement in quantum chromodynamics. The second volume covers more diverse topics, including strong coupling expansions and their analysis, Monte Carlo simulations, two-dimensional conformal field theory, and simple disordered systems. The book concludes with a chapter on random geometry and the Polyakov model of random surfaces which illustrates the relations between string theory and statistical physics. The two volumes that make up this work will be useful to theoretical physicists and applied mathematicians who are interested in the exciting developments which have resulted from the synthesis of field theory and statistical physics.
Volume 1: From Brownian Motion to Renormalization and Lattice Gauge Theory. Volume 2: Strong Coupling, Monte Carlo Methods, Conformal Field Theory, and Random Systems. This two-volume work provides a comprehensive and timely survey of the application of the methods of quantum field theory to statistical physics, a very active and fruitful area of modern research. The first volume provides a pedagogical introduction to the subject, discussing Brownian motion, its anticommutative counterpart in the guise of Onsager's solution to the two-dimensional Ising model, the mean field or Landau approximation, scaling ideas exemplified by the Kosterlitz-Thouless theory for the XY transition, the continuous renormalization group applied to the standard phi-to the fourth theory (the simplest typical case) and lattice gauge theory as a pathway to the understanding of quark confinement in quantum chromodynamics. The second volume covers more diverse topics, including strong coupling expansions and their analysis, Monte Carlo simulations, two-dimensional conformal field theory, and simple disordered systems. The book concludes with a chapter on random geometry and the Polyakov model of random surfaces which illustrates the relations between string theory and statistical physics. The two volumes that make up this work will be useful to theoretical physicists and applied mathematicians who are interested in the exciting developments which have resulted from the synthesis of field theory and statistical physics.
We review conformal field theory on the plane in the conformal bootstrap approach. We introduce the main ideas of the bootstrap approach to quantum field theory, and how they apply to two-dimensional theories with local conformal symmetry. We describe the mathematical structures that appear in such theories, from the Virasoro algebra and its representations, to BPZ equations and conformal blocks. Examples include Liouville theory, (generalized) minimal models, free bosonic theories, the H3+ model, and the SU2 and SL2(R) WZW models. We also discuss relations between some of these models, and limits of these models when the central charge and/or conformal dimensions tend to particular values.
This book is a course in modern quantum field theory as seen through the eyes of a theorist working in condensed matter physics. It contains a gentle introduction to the subject and therefore can be used even by graduate students. The introductory parts include a derivation of the path integral representation, Feynman diagrams and elements of the theory of metals including a discussion of Landau–Fermi liquid theory. In later chapters the discussion gradually turns to more advanced methods used in the theory of strongly correlated systems. The book contains a thorough exposition of such non-perturbative techniques as 1/N-expansion, bosonization (Abelian and non-Abelian), conformal field theory and theory of integrable systems. The book is intended for graduate students, postdoctoral associates and independent researchers working in condensed matter physics.