If we study the history of science we see produced two phenomena which are, so to speak, each the inverse of the other. Sometimes it is simplicity which is hidden under what is apparently complex; sometimes, on the contrary, it is simplicity which is apparent, and which conceals extremely complex realities. Henri Poincaré Devoting only a short chapter to the renormalization group (RG) is a challenge owing to the wealth of both fundamental concepts and computational tools associated with this term. The RG is indeed encountered in many different domains of theoretical physics, ranging from quantum electrodynamics to second-order phase transitions to fractal growth and diffusion processes – one should rather speak of renormalization group S ! We refer to Brown (1993) and Fisher (1998) for an historical account, to Goldenfeld (1992) and Lesne (1998) and references therein for an overview of the domains of application and variants of RGs. We here emphasize that the RG is a way, maybe the most successful and with no doubt the most systematic and constructive, to derive effective low-dimensional descriptions that capture large-scale and/or long-time behavior . The RG can be extended far beyond the specific scope of critical phenomena, to an iterated multiscale approach allowing the construction of robust and minimal macroscopic models describing the universal large-scale features and asymptotics of a complex system. This generalized viewpoint brings out the close logical and even technical connections that bridge, within a unified framework, perturbative RG for singular series expansions, spin-block RG and momentum-shell RG for critical phenomena, RG for the asymptotic analysis of differential and partial differential equations, and probabilistic RG for the derivation of statistical laws and limit theorems.
If we study the history of science we see produced two phenomena which are, so to speak, each the inverse of the other. Sometimes it is simplicity which is hidden under what is apparently complex; sometimes, on the contrary, it is simplicity which is apparent, and which conceals extremely complex realities. Henri Poincaré Devoting only a short chapter to the renormalization group (RG) is a challenge owing to the wealth of both fundamental concepts and computational tools associated with this term. The RG is indeed encountered in many different domains of theoretical physics, ranging from quantum electrodynamics to second-order phase transitions to fractal growth and diffusion processes – one should rather speak of renormalization group S ! We refer to Brown (1993) and Fisher (1998) for an historical account, to Goldenfeld (1992) and Lesne (1998) and references therein for an overview of the domains of application and variants of RGs. We here emphasize that the RG is a way, maybe the most successful and with no doubt the most systematic and constructive, to derive effective low-dimensional descriptions that capture large-scale and/or long-time behavior . The RG can be extended far beyond the specific scope of critical phenomena, to an iterated multiscale approach allowing the construction of robust and minimal macroscopic models describing the universal large-scale features and asymptotics of a complex system. This generalized viewpoint brings out the close logical and even technical connections that bridge, within a unified framework, perturbative RG for singular series expansions, spin-block RG and momentum-shell RG for critical phenomena, RG for the asymptotic analysis of differential and partial differential equations, and probabilistic RG for the derivation of statistical laws and limit theorems.
The method of the renormalization group (RG) provides theorists with powerful and efficient tools to explore interacting theories, often in regimes where perturbation theory fails. Motivating our discussion with two introductory examples drawn from a classical and a quantum theory, we will first become acquainted with the RG as a concept whereby nonlinear theories can be analyzed beyond the level of plain perturbation theory. With this background, we will then proceed to discuss the idea and practice of RG methods in more rigorous and general terms, introducing the notion of scaling, dimensional analysis, and the connection to the general theory of phase transitions and critical phenomena. Finally, to conclude this chapter, we will visit a number of concrete implementations of the RG program introduced and exemplified on a number of canonical applications. In Chapter 5 φ 4 -theory was introduced as an archetypal model of interacting continuum theories. Motivated by the existence of nonlinearities inherent in the model, a full perturbative scheme was developed, namely Wick contractions and their diagrammatic implementation. However, from a critical perspective, one may say that such perturbative approaches present only a limited understanding. Firstly, the validity of the φ 4 -action as a useful model theory was left unjustified, i.e. the φ 4 -continuum description was obtained as a gradient expansion of, in that case, a d -dimensional Ising model. But what controls the validity of the low-order expansion? Indeed, the same question could be applied to any one of the many continuum approximations we have performed throughout the first chapters of the text.
As discussed in the previous chapter, the phenomenological theory of critical phenomena finds a sound basis in the logical sequence of universality, scaling, relevant and irrelevant variables. The difficult problem of a large number of degrees of freedom strongly correlated within the coherence distance, which diverges at criticality, is thereby, in principle, reduced to the determination of the homogeneous form of the relevant response functions and of the evaluation of a few critical indices related to the relevant parameters, whereas the details specifying each system do not matter. In order to give a microscopic foundation to universality and to the scaling theory, an exact transformation is needed, which changes into a scaling transformation asymptotically near the critical point. This is the renormalization group (RG) approach, recently presented in a huge number of reviewpapers and books.Historically, the first comprehensive overview was given by the sixth volume of the series edited by Domb and Green (1976) and by the classic books by Patashinskij and Pokrovskij (1979) and by Ma (1976). Along with scaling theory, the field-theoretic renormalization group (RG) approach (Gell- Mann and Low (1954); Bogoliubov and Shirkov (1959); Bonch-Bruevich and Tyablikov (1962)) was introduced in critical phenomena (Di Castro and Jona-Lasinio (1969)) by showing how the RG equations generalize the universality relations in the sense that they relate one model system to another by varying the coupling and suitably rescaling the other variables and the correlation functions. At the same time a detailed analysis of the pertubative diagrammatic structure of the correlation functions together with the use of Ward identities yielded a renormalized theory (Migdal (1969); Polyakov (1968, 1969)). By iterating the RG transformation the coupling disappears from the equations near to the critical point, the homogeneous form of the static order-parameter correlation function is obtained and a microscopic definition of the critical indices is given. The scaling picture then acquires a theoretical basis. Wilson (Wilson (1971a, b); Wilson and Kogut (1974)) gave a great contribution to the understanding of the physics underlying the RG procedure and to the explicit calculation of the critical indices. He proposed a simple mathematical realization of the other very physical idea of universality due to Kadanoff of grouping together degrees of freedom on the scale s associated with larger and larger cells.
This book provides an elementary introduction to the notions of continuum limit and universality in statistical systems with a large number of degrees of freedom. The existence of a continuum limit requires the appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature. In this context, the book emphasizes the role of gaussian distributions and their relations with the mean field approximation and Landau′s theory of critical phenomena. The book shows that quasi-gaussian or mean-field approximations cannot describe correctly phase transitions in three space dimensions. The book assigns this difficulty to the coupling of very different physical length scales, even though the systems we will consider have only local, that is, short range, interactions. To analyze the unusual situation, a new concept is required: the renormalization group, whose fixed points allow understanding the universality of physical properties at large distance, beyond mean-field theory. In the continuum limit, critical phenomena can be described by quantum field theories. In this framework, the renormalization group is directly related to the renormalization process; that is, the necessity to cancel the infinities that arise in straightforward formulations of the theory. The book discusses the renormalization group in the context of various relevant field theories. This leads to proofs of universality and to efficient tools for calculating universal quantities in a perturbative framework. Finally, the book constructs a general functional renormalization group, which can be used when perturbative methods are inadequate.
This special issue of Journal of Physics A: Mathematical and General is a collection of articles dealing with the application of renormalization group methods to a wide range of problems in physics. Most articles are based on contributions presented at the 6th International Conference, Renormalization Group 2005 (RG 2005), held at the University of Helsinki from 30 August to 3--September 2005. This was the sixth of a series of conferences on this topic, the first three of which took place in 1986, 1991 and 1996 at the Joint Institute for Nuclear Research in Dubna, Russia. The fourth (RG99) was held in 1999 in Taxco, Mexico, and the fifth (RG02) in 2002 in Tatranska Strba, Slovakia. During the second half of the 20th century a number of important and very successful new paradigms such as scale-dependent (`running') coupling constants, scaling, and universality became firmly established in physics. To a large extent, these advances were due to the development of renormalization group ideas and their successful implementation as powerful calculational schemes to tackle problems with many length scales. While the concept of the renormalization group had originally been introduced in elementary particle physics, its modern form eventually emerged from the study of problems in condensed matter physics, such as critical phenomena and the Kondo effect. This in turn has significantly reshaped our understanding of field theories in high energy physics, and produced powerful computational tools for dealing with difficult many-length-scale problems in distinct fields of physics, ranging from elementary particle and condensed matter physics to hydrodynamic turbulence and growth phenomena in nonequilibrium physics. The history of the renormalization group, characterized by frequent exchanges of ideas between elementary particle and condensed matter physics, bears testimony to the enormous benefits to which such exchanges can lead. The main aim of the series of renormalization group conferences has been to bring together scientists with backgrounds and expertise in the application of the renormalization group to problems in broadly different areas of physics, offering regular events for continuing such fruiful exchanges of ideas. We hope that the present collection of contributions to quantum field theory, mathematical and methodological aspects of the renormalization group, statistical mechanics, condensed matter physics, and stochastic dynamics conveys this spirit. We feel it amply demonstrates the breadth of activity of research based on the renormalization group and its potential for future work. On behalf of the organizers of the conference, we would like to thank the sixty participants of the conference and the members of the Advisory Board for their contributions. Financial and organizational support of the University of Helsinki, the Academy of Finland, the Finnish Society of Sciences and Letters and the Naval Academy is gratefully acknowledged. We are very grateful to all authors who contributed to this special issue. Finally, we owe special thanks to all who assisted in the preparation of it, in particular to the staff of Journal of Physics A: Mathematical and General .
The renormalization-group (RG) method discussed in this chapter has assumed a pivotal role in the modern theory of critical phenomena. It attempts to relate the partition function of a given system to that of a “similar system” with decreased degrees of freedom through a process referred to as renormalization. Exactly how these degrees of freedom are removed from the system, what we mean by a “similar” system, and how successive systems are coupled to one another are essentially the questions we take up in the introductory treatment given in this chapter. The RG method is a topic with large scope and found widely disseminated in an extensive physics literature on the topic; however, it is seldom found in engineering journals. Our purpose here is to try and make sense of some basic ideas with the RG approach so that it is more accessible to this wider community. For this we often rely upon some prior exposes of the subject in more specialized settings [1, 2, 3, 5]. In its complete sense, the RG method has only been made to work, at least analytically, for a few simple statistical-mechanical models. But aside from these numerical results, many important and quite general insights about critical phenomena can be developed from studying this approach to the problem, especially the central role played by length scale as a factor in describing the phenomenology. These ideas have significantly enhanced our understanding of ideas like scale invariance, universality classes, relevant scaling fields (as opposed to irrelevant ones), Hamiltonian renormalization, and so on; these and related concepts lie at the center of modern discourse on the subject. The essential concepts of the approach can be well illustrated using the Ising system since, with this model, lattice spins are fixed in space, which makes the analytical work quite transparent. This approach, called real space renormalization, is the RG method studied in this chapter.
Abstract Chapter 5 first recalls the importance of the concept of scale decoupling in physics. It then emphasizes that quantum field theory and the theory of critical phenomena have provided two examples where this concepts fails. To deal with such a situation, a new tool has been invented: the renormalization group. In the framework of effective quantum field theory, a perturbative renormalization group has been formulated. Its implementation has led to the discovery of fixed points as zeros of beta functions, and calculations of critical exponents of a class of macroscopic phase transitions in the form of Wilson–Fisher epsilon or fixed dimension expansions. These expansions being divergent, they could summed by methods based on the Borel transformation and the determination of the large order behaviour of perturbation theory.