In this chapter, we introduce and explain one of the fundamental tools in the study of dynamic critical phenomena, namely dynamic perturbation theory. Inevitably, large parts of Chapter 4 need to be rather technical. Complementary to the straightforward iterative method for the solution of non-linear Langevin equations presented in Section 3.2.3, we now describe the more elegant and efficient field-theoretic techniques. Yet both for the derivation of general properties of the perturbation series and for the sake of practical calculations, the response functional and the following elaborations on Feynman diagrams, cumulants, and vertex functions prove indispensable. As is the case with every efficient formalism, once one has become acquainted with the Janssen–De Dominicis response functional for the construction of dynamic field theory, and therefrom the perturbation expansion in terms of vertex functions, it serves to save a considerable amount of rather tedious work. For the purely relaxational O ( n )-symmetric models A and B, we derive the fluctuation-dissipation theorem within this formalism, and then systematically construct the perturbation series and its diagrammatic representation in terms of Feynman graphs. In order to reduce the required efforts to a minimum, the generating functionals first for the cumulants, and then, motivated by Dyson's equation for the propagator, for the one-particle irreducible vertex functions are examined. As an example, the relevant vertex functions for models A and B are explicitly evaluated to two-loop order. Furthermore, alternative formulations of the perturbation expansion are discussed, and the Feynman rules are given in both the frequency and the time domain. The results of this chapter provide the foundation for the renormalization group treatment in the subsequent Chapter 5.
In this chapter, we introduce and explain one of the fundamental tools in the study of dynamic critical phenomena, namely dynamic perturbation theory. Inevitably, large parts of Chapter 4 need to be rather technical. Complementary to the straightforward iterative method for the solution of non-linear Langevin equations presented in Section 3.2.3, we now describe the more elegant and efficient field-theoretic techniques. Yet both for the derivation of general properties of the perturbation series and for the sake of practical calculations, the response functional and the following elaborations on Feynman diagrams, cumulants, and vertex functions prove indispensable. As is the case with every efficient formalism, once one has become acquainted with the Janssen–De Dominicis response functional for the construction of dynamic field theory, and therefrom the perturbation expansion in terms of vertex functions, it serves to save a considerable amount of rather tedious work. For the purely relaxational O ( n )-symmetric models A and B, we derive the fluctuation-dissipation theorem within this formalism, and then systematically construct the perturbation series and its diagrammatic representation in terms of Feynman graphs. In order to reduce the required efforts to a minimum, the generating functionals first for the cumulants, and then, motivated by Dyson's equation for the propagator, for the one-particle irreducible vertex functions are examined. As an example, the relevant vertex functions for models A and B are explicitly evaluated to two-loop order. Furthermore, alternative formulations of the perturbation expansion are discussed, and the Feynman rules are given in both the frequency and the time domain. The results of this chapter provide the foundation for the renormalization group treatment in the subsequent Chapter 5.
In this central and essential chapter, we develop the dynamic renormalization group approach to time-dependent critical phenomena. Again, we base our exposition on the simple O ( n )-symmetric relaxational models A and B; the generalization to other dynamical systems is straightforward. We begin with an analysis of the infrared and ultraviolet singularities appearing in the dynamic perturbation expansion. Although we are ultimately interested in the infrared critical region, we first take care of the ultraviolet divergences. Below and at the critical dimension d c = 4, only a finite set of Feynman diagrams carries ultraviolet singularities, which we evaluate by means of the dimensional regularization prescription, and then eliminate via multiplicative as well as additive renormalization (the latter takes into account the fluctuation-induced shift of the critical temperature). The renormalization group equation then permits us to explore the ensuing scaling behavior of the correlation and vertex functions of the renormalized theory upon varying the arbitrary renormalization scale. Of fundamental importance is the identification of an infrared-stable renormalization group fixed point, which describes scale invariance and hence allows the derivation of the critical power laws in the infrared limit from the renormalization constants determined in the ultraviolet regime. This program is explicitly carried through for the relaxational models A and B. For model B, we derive a scaling relation connecting the dynamic exponent to the Fisher exponent η . The critical exponents ν , η , and z are computed to first non-trivial order in an ∊ expansion around the upper critical dimension d c = 4.
At present, the only general method available for gaining knowledge from the fundamental principles about the dynamics of a system is the perturbative study. According to Feynman, as described in Chapter 4, instead of formulating quantum theory in terms of operators, the canonical formulation, for calculational purposes quantum dynamics can conveniently be formulated in terms of a few simple stenographic rules, the Feynman rules for propagators and interaction vertices. In Chapters 4 and 5, we showed how to arrive at the Feynman rules of diagrammatic perturbation theory for non-equilibrium states starting from the Hamiltonian defining the theory. The feature of non-equilibrium states, originally carried by the dynamical indices, could be expressed in terms of two simple universal vertex rules for the RAK-components of the matrix Green's functions. We are thus well acquainted with diagrammatics even for the description of non-equilibrium situations. However, for the situations studied using the quantum kinetic equations in Chapters 7 and 8, only the Dyson equation was needed, i.e. the self-energy, the 2-state one-particle irreducible amputated Green's function. No need for higher-order vertex functions was required, and the full flourishing diagrammatics was not put into action. In this chapter we shall proceed the other way around. We shall show that the diagrammatics of a physical theory, including the description of non-equilibrium states, can be obtained by simply stating quantum dynamics, the superposition principle, as the two exclusive options for a particle: to interact or not to interact ! From this simple Shakespearean approach we shall construct the Feynman diagrammatics of non-equilibrium dynamics.
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.
In the preceding chapter, we have introduced several levels for the mathematical description of stochastic dynamics. We now use the kinetic Ising models introduced in Section 2.3.3 to formulate the dynamic scaling hypothesis which appropriately generalizes the homogeneity property of the static correlation function in the vicinity of a critical point, as established in Chapter 1. The dynamic critical exponent z is defined to characterize both the critical dispersion and the basic phenomenon of critical slowing-down. As a next step, and building on the results of Section 2.4, a continuum effective theory for the mesoscopic order parameter density, basically the dynamical analog to the Ginzburg–Landau approach, is constructed in terms of a non-linear Langevin equation. The distinction between dissipative and diffusive dynamics for the purely relaxational kinetics of either a non-conserved or conserved order parameter field, respectively, defines the universality classes A and B. Following the analysis of these models in the Gaussian approximation, they also serve to outline the construction of a dynamical perturbation theory for non-linear stochastic differential equations through direct iteration. In general, however, the order parameter alone does not suffice to fully capture the critical dynamics near a second-order phase transition. Additional hydrodynamic modes originating from conservation laws need to be accounted for as well. The simplest such situation is entailed in the relaxational models C and D, which encompass the static coupling of the order parameter to the energy density. Further scenarios emerge through reversible non-linear mode couplings in the Langevin equations of motion.
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.
At present, the only general method available for gaining knowledge from the fundamental principles about the dynamics of a system is the perturbative study. This consists in dividing the Hamiltonian into one part representing a simpler well-understood problem and a nontrivial part, the effect of which is studied order by order. The expressions resulting from perturbation theory quickly become unwieldy. This chapter deals with propagators. It introduces the retarded and advanced propagator and describes their perturbation theoretic structure in a potential in terms of diagrams. A convenient method of representing perturbative expressions by diagrams was invented by Feynman. Besides the appealing aspect of representing perturbative expressions by drawings, the diagrammatic method can also be used directly for reasoning and problem solving. The easily recognizable topology of diagrams makes the diagrammatic method a powerful tool for constructing approximation schemes as well as exact equations that may hold true beyond perturbation theory.