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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.

Actual computations of fixed points and eigenvalues usually involve approximations, often crude ones, except for a very limited number of simple cases such as the one-dimensional Ising model of the previous chapter. In real- and momentum-space renormalization group theory, there are no general prescriptions to systematically improve the degree of the approximation with a modest amount of effort. There are established methods to systematically improve precision, but they usually need a large amount of numerical calculations. The scope of the present chapter is modest as we limit ourselves to basic examples, including the epsilon expansion about the Gaussian fixed-point of the Landau-Ginzburg-Wilson model. Finally, the last section illustrates the extension of the renormalization group framework to study quantum phase transitions.

This chapter presents the essential steps of the proof of the renormalizability of a simple scalar field theory: the φ⁴ field theory in d = 4 dimensions. However, all the fundamental difficulties of renormalization theory are already present in this particular example and it will eventually become clear that the extension to other theories is not difficult. The elegant presentation of Callan (Les Houches 1975) is followed, which allows renormalizability and renormalization group (Callan–Symanzik) equations to be proved at once. This presentation is specially suited to the chapter's general purpose since a large part of this work is devoted to applications of renormalization group (RG). Moreover, it emphasizes already at this technical level the equivalence between renormalizability and the existence of a renormalization.

This chapter begins with the formal derivation of QCD as a gauge theory based on the symmetry group SU(3). It then describes the basic reactions of lepton-lepton, lepton-hadron, and hadron-hadron scattering in the language of QCD, and derives the born-level expressions for the basic QCD cross-sections. The chapter addresses the issue of ultraviolet divergences and renormalization. It discusses the renormalization group equations and the issue of infrared-safety in the context of the QCD improved parton model. The treatment of soft gluons then leads naturally to the various hadronization models, which form the basis of the Monte Carlo models discussed in the following chapter.

This chapter discusses the quantum anomaly associated with the scale transformation of space-time coordinates, or the transformation generally called the Weyl transformation. In flat space-time, this anomaly is related to the renormalization group and the calculation of the $β$ function in the renormalization group equation is related to the calculation of the Weyl anomaly. In other words, the renormalization group equation is regarded as an expression of the Weyl anomaly in terms of Green’s functions. The calculation of the one-loop functions in QED and QCD by means of the Jacobians for the Weyl symmetry is illustrated. The Weyl anomalies in curved space-time are briefly explained. An improved finite energy-momentum tensor in renormalizable theory is also mentioned based on an analysis of the Weyl anomaly.

This chapter introduces the concept of renormalization by minimal subtraction, within the framework of dimensional regularization. It first discusses the structure of renormalization constants and renormalization group functions β(g), η(g), η2(g), and then shows that it is specially simple in the minimal subtraction scheme. It performs explicit calculations at two-loop order first in the simple one-component φ⁴ field and then in an N-component field theory with a general four-field interaction. These calculations will be useful for the theory of Critical Phenomena. Finally, to give an example of a theory involving fermions, the renormalization group (RG) functions are calculated at one-loop order in a theory containing fermions interacting through a Yukawa-like interaction with a scalar boson: the Gross–Neveu–Yukawa model.

The standard model of non-relativistic quantum electrodynamics describes non-relativistic quantum matter, such as atoms and molecules, coupled to the quantized electromagnetic field. Within this model, this chapter reviews basic notions, results, and techniques in theory radiation. It describes the key technique in this area — the spectral renormalization group. The review is based on joint works with Volker Bach and Jürg Fröhlich and with Walid Abou Salem, Thomas Chen, Jérémy Faupin, and Marcel Griesemer. A brief discussion of related contributions is given at the end of these lectures.

This chapter presents a brief history of the origin and development of quantum field theory, and of the evolution of the interpretation of renormalization and the renormalization group which has led to our present understanding. This history has two aspects: one directly related to the theory of fundamental interactions that describes physics at the microscopic scale; and another one related to the theory of phase transitions in macroscopic physics and their universal properties. That two so vastly different domains of physics have required the development of the same theoretical framework is extremely surprising. It is one of the attractions of theoretical physics that such relations can sometimes be found.

This chapter describes a general approach to the renormalization group (RG) close to ideas initially developed by Wegner and Wilson, and based on a partial integration over the large-momentum modes of fields. This RG takes the form of functional renormalization group (FRG) equations that express the equivalence between a change of a scale parameter related to microscopic physics and a change of the parameters of the Hamiltonian. Some forms of these renormalization group equations (RGE) are exact and one then also speaks of the exact renormalization group.

This chapter discusses one of the most fertile manifestations of scale separation in field theory: the Wilson operator product expansion (OPE), which provides a precise characterization of the short-distance asymptotics of field theory amplitudes in terms of factorized products of ‘short-’ and ‘long-’distance terms. The useful application of the OPE in particular processes depends on the presence and structure of mass singularities in the relevant amplitudes. The role of the renormalization group in studying high-energy (or short-distance) behaviour is outlined.

Interacting many-particle systems may undergo phase transitions and exhibit critical phenomena in the limit of infinite system size, while the precursors of these phenomena are studied in the theory of finite-size scaling. After surveying the basic notions of phases, phase diagrams, and phase transitions, this chapter focuses on critical behaviour at a second-order phase transition. The Landau-Ginzburg theory and the concept of scaling prepare readers for an elementary introduction to the concepts of the renormalization group, followed by an introduction into the field of quantum phase transitions where quantum fluctuations take over the role of thermal fluctuations.

Chapter 1 features lectures that review the formalism of renormalization in quantum field theories with special regard to effective quantum field theories. While renormalization theory is part of every advanced course on quantum field theory, for effective theories some more advanced topics become particularly important. These topics include the renormalization of composite operators, operator mixing under scale evolution, and the resummation of large logarithms of scale ratios. The lectures from this course thus set the basis for any systematic study of the techniques and applications of effective field theories and offer an introduction for the reader to the content within this book.

This chapter introduces the Ginzburg-Landau functional and provides a general description of fluctuation thermodynamics within the framework of the functional integration over the fluctuation fields approach. The method, in its harmonic approximation, is applied to the effect of fluctuations on heat capacity and magnetization of a superconductor at the critical temperature. The criterion of its validity (Ginzburg-Levanyuk criterion) is derived. An extension of the theory on close vicinity of transition is demonstrated within the framework of the renormalization group approach. The shift in critical temperature by fluctuations of the order parameter and electromagnetic fields is obtained.

As an introduction to the physics of phase transitions and critical phenomena, this chapter explains a number of basic and fundamental ideas such as phases, phase transitions, phase diagrams, universality, and critical phenomena. Especially important is the concept of order parameter, a quantity that measures the degree of asymmetry in the broken symmetry phase. Intuitive accounts are given to the concepts of coarse-graining, and scale and renormalization group transformations, which are powerful, systematic tools to analyze critical behaviour of macroscopic systems. Also explained are several spin and lattice gas model systems, on the basis of which phase transitions and critical phenomena will be studied.

This chapter explains how the asymptotic renormalization group equations (RGE), introduced without too much justification in Section 10.6, can be proved within the framework of statistical (or quantum) field theory. The proof is based on the methods of perturbative statistical field theory introduced in Chapter 12, and a few assumptions that it is thus possible to clarify. The discussion is restricted to theories with an Ising type symmetry and the field s has only one component. Generalization to models with N-component fields and O (N) symmetry is simple.

The RG theory, as applied to Critical Phenomena, has been developed by Kadano, Wilson, Wegner, and many others. This chapter first describes the basic renormalization group ideas in a somewhat abstract and intuitive framework. The formulation will lack precision and the arguments will be largely heuristic. The importance of fixed points in hamiltonian space will be stressed. The special role of gaussian models and their universal properties will be related to the existence of fixed point, the gaussian fixed point. It is shown that the RG equations which appear as a consequence of the necessity of renormalization of local field theories are directly connected with the abstract RG equations introduced in Section 25.1. Universality in the theory of critical phenomena is thus directly related to the property that local field theories are insensitive to the short distance structure, and physics can, therefore, be described by renormalized correlation functions. Conversely, in the statistical sense, QFTs are always close to criticality and their existence, beyond perturbation theory, relies, from the abstract RG point of view, on the presence of IR fixed points in hamiltonian space.

This chapter describes the concept of fractals and their self-similarity, including fractals with complex dimensions. It shows how these geometric and mathematical objects enable one to codify the information contained in the precursory patterns before large stock market crashes. The chapter first considers how models of cooperative behaviors resulting from imitation between agents organized within a hierarchical structure exhibit the announced critical phenomena decorated with “log-periodicity.” It then examines the underlying hierarchical structure of social networks, critical behavior in hierarchical networks, a hierarchical model of financial bubbles, and discrete scale invariance. It also discusses a technique, called the “renormalization group,” and a simple model exhibiting a finite-time singularity due to a positive feedback induced by trend following investment strategies. Finally, it looks at scenarios leading to discrete scale invariance and log-periodicity.