Ebook: Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory
- Genre: Physics // Quantum Physics
- Tags: Quantum Physics, Quantum Information Technology Spintronics, Thermodynamics, Statistical Physics Dynamical Systems and Complexity
- Series: Texts and Monographs in Physics
- Year: 1992
- Publisher: Springer-Verlag Berlin Heidelberg
- Edition: 1
- Language: English
- pdf
Simple random walks - or equivalently, sums of independent random vari ables - have long been a standard topic of probability theory and mathemat ical physics. In the 1950s, non-Markovian random-walk models, such as the self-avoiding walk,were introduced into theoretical polymer physics, and gradu ally came to serve as a paradigm for the general theory of critical phenomena. In the past decade, random-walk expansions have evolved into an important tool for the rigorous analysis of critical phenomena in classical spin systems and of the continuum limit in quantum field theory. Among the results obtained by random-walk methods are the proof of triviality of the cp4 quantum field theo ryin space-time dimension d (::::) 4, and the proof of mean-field critical behavior for cp4 and Ising models in space dimension d (::::) 4. The principal goal of the present monograph is to present a detailed review of these developments. It is supplemented by a brief excursion to the theory of random surfaces and various applications thereof. This book has grown out of research carried out by the authors mainly from 1982 until the middle of 1985. Our original intention was to write a research paper. However, the writing of such a paper turned out to be a very slow process, partly because of our geographical separation, partly because each of us was involved in other projects that may have appeared more urgent.
The subject of this book is equilibrium statistical mechanics, in particular the theory of critical phenomena, and quantum field theory. The central theme is the use of random-walk representations as a tool to derive correlation inequalities. The consequences of these inequalities for critical-exponent theory and the triviality question in quantum field theory are expounded in detail. The book contains some previously unpublished results. It addresses both the researcher and the graduate student in modern statistical mechanics and quantum field theory.
The subject of this book is equilibrium statistical mechanics, in particular the theory of critical phenomena, and quantum field theory. The central theme is the use of random-walk representations as a tool to derive correlation inequalities. The consequences of these inequalities for critical-exponent theory and the triviality question in quantum field theory are expounded in detail. The book contains some previously unpublished results. It addresses both the researcher and the graduate student in modern statistical mechanics and quantum field theory.
Content:
Front Matter....Pages I-XVII
Front Matter....Pages 1-1
General introduction....Pages 3-36
Phase transitions and critical points in classical spin systems: A brief survey....Pages 37-52
Scale transformations and scaling (continuum) limits in lattice spin systems....Pages 53-57
Construction of scaling limits: the renormalization group....Pages 59-77
Random walks as Euclidean field theory (EFT)....Pages 79-99
EFT as a gas of random walks with hard-core interactions....Pages 101-115
Random-surface models....Pages 117-178
Front Matter....Pages 179-179
Introduction....Pages 181-187
Random-walk models in the absence of magnetic field....Pages 189-203
Random-walk models in the presence of a magnetic field....Pages 205-211
Factorization and differentiation of the weights....Pages 213-227
Correlation inequalities: A survey of results....Pages 229-272
Front Matter....Pages 273-273
Background material....Pages 275-295
Inequalities for critical exponents....Pages 297-366
Continuum Limits....Pages 367-403
Back Matter....Pages 405-444
The subject of this book is equilibrium statistical mechanics, in particular the theory of critical phenomena, and quantum field theory. The central theme is the use of random-walk representations as a tool to derive correlation inequalities. The consequences of these inequalities for critical-exponent theory and the triviality question in quantum field theory are expounded in detail. The book contains some previously unpublished results. It addresses both the researcher and the graduate student in modern statistical mechanics and quantum field theory.
Content:
Front Matter....Pages I-XVII
Front Matter....Pages 1-1
General introduction....Pages 3-36
Phase transitions and critical points in classical spin systems: A brief survey....Pages 37-52
Scale transformations and scaling (continuum) limits in lattice spin systems....Pages 53-57
Construction of scaling limits: the renormalization group....Pages 59-77
Random walks as Euclidean field theory (EFT)....Pages 79-99
EFT as a gas of random walks with hard-core interactions....Pages 101-115
Random-surface models....Pages 117-178
Front Matter....Pages 179-179
Introduction....Pages 181-187
Random-walk models in the absence of magnetic field....Pages 189-203
Random-walk models in the presence of a magnetic field....Pages 205-211
Factorization and differentiation of the weights....Pages 213-227
Correlation inequalities: A survey of results....Pages 229-272
Front Matter....Pages 273-273
Background material....Pages 275-295
Inequalities for critical exponents....Pages 297-366
Continuum Limits....Pages 367-403
Back Matter....Pages 405-444
....