The renormalization group, as embodied in the Callan-Symanzik equation, has had a profound impact on field theory and particle physics. You've heard from Ramamurti Shankar and Michael about some of its impact on condensed matter physics. I'd like to tell you my view of what the renormalization group has meant to practising condensed matter physicists. What have we learned from renormalization theory? We learned that the detailed physics of matter at microscopic length scales and high energies is irrelevant for critical phenomena. Many different microscopic theories lead to exactly the same physical laws at a critical point. As Michael Fisher explained, one can even make precise quantitative predictions about certain ‘universal’ critical exponents without getting the microscopic physics right in detail. What is important is symmetry, conservation laws, the range of interactions, and the dimensionality of space. The physics of the diverging fluctuations at a critical point, which take place on scales of a micron or more, that's 10 4 angstrom, is largely ‘decoupled’ from the physics at angstrom length scales. This story about scaling laws at a critical point tells us something about the meaning of a ‘fundamental physics’. Fundamental physics is not necessarily the physics of smaller and smaller length scales, to the extent that these length scales decouple from the physics that we're interested in at the moment. To elaborate on this point, I'd like to refer to a short paper, which influenced me a lot as a graduate student, that Ken Wilson wrote in 1972.
The renormalization group, as embodied in the Callan-Symanzik equation, has had a profound impact on field theory and particle physics. You've heard from Ramamurti Shankar and Michael about some of its impact on condensed matter physics. I'd like to tell you my view of what the renormalization group has meant to practising condensed matter physicists. What have we learned from renormalization theory? We learned that the detailed physics of matter at microscopic length scales and high energies is irrelevant for critical phenomena. Many different microscopic theories lead to exactly the same physical laws at a critical point. As Michael Fisher explained, one can even make precise quantitative predictions about certain ‘universal’ critical exponents without getting the microscopic physics right in detail. What is important is symmetry, conservation laws, the range of interactions, and the dimensionality of space. The physics of the diverging fluctuations at a critical point, which take place on scales of a micron or more, that's 10 4 angstrom, is largely ‘decoupled’ from the physics at angstrom length scales. This story about scaling laws at a critical point tells us something about the meaning of a ‘fundamental physics’. Fundamental physics is not necessarily the physics of smaller and smaller length scales, to the extent that these length scales decouple from the physics that we're interested in at the moment. To elaborate on this point, I'd like to refer to a short paper, which influenced me a lot as a graduate student, that Ken Wilson wrote in 1972.
The nature and origins of renormalization group ideas in statistical physics and condensed matter theory are recounted informally, emphasizing those features of prime importance in these areas of science in contradistinction to quantum field theory, in particular: critical exponents and scaling, relevance, irrelevance and marginality, universality, and Wilson's crucial concept of flows and fixed points in a large space of Hamiltonians. Contents Foreword 1 Introduction 2 Whence came renormalization group theory? 3 Where stands the renormalization group? 4 Exponents, anomalous dimensions, scale invariance and scale dependence 5 The challenges posed by critical phenomena 6 Exponent relations, scaling, and irrelevance 7 Relevance, crossover, and marginality 8 The task for renormalization group theory 9 Kadanoff's scaling picture 10 Wilson's quest 11 The construction of renormalization group transformations: the epsilon expansion 12 Flows, fixed points, universality and scaling 13 Conclusions Acknowledgments Selected bibliography Appendix A Asymptotic behavior Appendix B Unitarity of the renormalization group Appendix C Nature of a semigroup Foreword It was a pleasure to participate in the Colloquium cosponsored by the Departments of Philosophy and of Physics at Boston University “On the Foundations of Quantum Field Theory.” In the full title, this was preceded by the phrase: “A Historical Examination and Philosophical Reflections,” which set the aims of the meeting. Naturally, the participants were mainly high-energy physicists, experts in field theories, and interested philosophers of science.
Abstract Mean-field theory is usually taken as a first step toward understanding critical phenomena, providing an overview that reveals qualitative behaviour of physical quantities. However, it is necessary to proceed beyond the mean-field theory to better understand the situation, both qualitatively and quantitatively, when fluctuations play vital roles leading to exponents that cannot be explained by dimensional analysis, thus introducing anomalous dimensions. The present chapter explains the basic concepts of the renormalization group and scaling theory, which allow us to analyze critical phenomena with fluctuations systematically taken into account. The essential step in a renormalization group calculation consists of establishing recursion relations between the parameters defining the Hamiltonian of the system. These recursion or renormalization group equations define a flow with well-defined fixed points. Details other than the values of the relevant operators have no influence on the critical exponents and this represents universality.
Field theory and condensed matter physics Condensed matter physics is a very rich and diverse field. If we are to define it as being “whatever gets published in the condensed matter section of a physics journal,” we would conclude that it ranges from problems typical of material science to subjects as fundamental as particle physics and cosmology. Because of its diversity, it is sometimes hard to figure out where the field is going, particularly if you do not work in this field. Unfortunately, this is the case for people who have to make decisions about funding, grants, tenure, and other unpleasant aspects in the life of a physicist. They have a hard time figuring out where to put this subject which is neither applied science nor dealing with the smallest length scales or the highest energies. However, the richness of the field comes precisely from its diversity. The past few decades have witnessed the development of two areas of condensed matter physics that best illustrate the strengths of this field: critical phenomena and the quantum Hall effect. In both cases, it was the ability to produce extremely pure samples which allowed the discovery and experimental study of the phenomenon. Their physical explanation required the use of new concepts and the development of new theoretical tools, such as the renormalization group, conformal invariance, and fractional statistics.
Continuous field models were originally systematically elaborated (and still are) in the context of particle physics. Most relevant to the study of critical phenomena is the derivation of a renormalization group flow, characterized by a set of Callan–Symanzik equations for Green functions, with regular coefficients derived from perturbation theory. This is supplemented by the idea due to Fisher and Wilson of an expansion in powers of the deviation from the strict renormalizability dimension, i.e. four in the case of the ϕ 4 model, but can also be presumed to work directly in the physical dimension three (and even possibly two). We devote this chapter to a general presentation and a survey of some applications. An appendix gives a short introduction to multicritical phenomena. The Lagrangian and dimensional analysis Introduction We want to investigate universal critical properties. A discrete lattice model appears as a regularizing intermediate stage, which allows a precise meaning to be given to the functional integrals of the field theory. Following the scheme suggested by the mean field approximation, one is tempted to start directly from a continuous model, in which the lattice is replaced by a d -dimensional continuous Euclidean space, and also, even for models with a discrete symmetry, the dynamical variables are replaced by continuous fields. From the original formulation, we just retain the idea of a cutoff factor ∧ large with respect to all momenta, which ensures that some possibly divergent expressions remain finite.
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.