Renormalization Methods: A Guide for Beginners W. D. McComb Oxford U. Press, New York, 2004. $89.50 (330 pp.). ISBN 0-19-850694-5
Renormalization originated in quantum field theory as a method of removing UV divergences in perturbation expansions. The subsequent development in the 1960s of the renormalization group introduced the novel concept of running couplings, which depend on the energy scale at which they are measured, and led to such groundbreaking discoveries as asymptotic freedom in quantum chromodynamics, for which David Gross, David Politzer, and Frank Wilczek received the Nobel Prize in Physics last year. Yet renormalization methods and the renormalization group probably have had an even more profound impact on condensed matter theory and statistical mechanics than on quantum field theory.
Aside from providing a mathematical framework from which to derive scaling laws and obtain nonclassical critical exponents near continuous phase transitions, in the past three decades, the renormalization group approach has provided a solid conceptual foundation for exploring such paradigmatic notions as universality, relevant degrees of freedom, and fixed points in parameter space. In fact, one may argue that any reduction of a complex interacting system to an effective model described by only a few variables and governed by a small set of control parameters tacitly relies on renormalization ideas. In any such model, “irrelevant” degrees of freedom are somehow integrated out to arrive at a fixed-point theory that then contains “dressed” particles and effective couplings between the remaining degrees of freedom.
W. David McComb, an expert in the field of fluid turbulence, is an avid supporter of the above overarching and almost philosophical view of renormalization. He set for himself the ambitious goal of rendering renormalization techniques and the ideas of the renormalization group accessible to advanced undergraduates and beginning graduate students in physics and neighboring sciences. I admire his courage in attempting to teach, essentially from scratch and in a mere 300 pages, technically demanding topics that encompass field-theoretic formulations based on path integrals and stochastic nonlinear hydrodynamics.
A text is certainly needed to bridge the gap between basic undergraduate course material in statistical mechanics and modern research topics. Excellent classic texts in the field, including H. Eugene Stanley’s Introduction to Phase Transitions and Critical Phenomena (Clarendon Press, 1971) and Shang-keng Ma’s Modern Theory of Critical Phenomena (W. A. Benjamin, 1976), miss the wide applications that the renormalization group has enjoyed more recently. Although those applications are aptly reflected in Nigel Goldenfeld’s Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, 1992), John Cardy’s Scaling and Renormalization in Statistical Physics (Cambridge U. Press, 1996), and Paul Chaikin and Tom Lubensky’s Principles of Condensed Matter Physics (Cambridge U. Press, 1995), most undergraduates and beginning graduate students will find the level of those texts quite demanding. Moreover, the field theory classics such as Daniel Amit’s Field Theory, the Renormalization Group, and Critical Phenomena (McGraw-Hill International Books, 1978) and Jean Zinn-Justin’s Quantum Field Theory and Critical Phenomena (Clarendon Books/Oxford U. Press, 1989) most likely will be even more out of their reach.
In part 1 of Renormalization Methods, McComb introduces the basic ideas of renormalization. He begins on a very elementary level, which implies that he needs to explain fundamental mathematical tools such as Gaussian integrations, Green’s functions, perturbation theory, and high-temperature expansions. His tour de force works quite well. He manages to cover a wide range of topics, which include anharmonic oscillators, Debye–Hückel screening, fractals, percolation, mean-field theory, dynamical systems, and diagrammatic representations and renormalization in quantum field theory. However, I find it unfortunate that Lev Landau’s Fermi-liquid theory for interacting electron systems is not mentioned at all, because it so beautifully illustrates the renormalization paradigm. In addition, I noticed that quantum mechanics receives a somewhat rough treatment in chapter 1: Erwin Schrödinger’s surname is misspelled, the equation named in his honor is stated incorrectly, and observables are not properly represented by operators.
Parts 2 and 3 of McComb’s book describe, on a generally accessible level, the technical framework and ideas of the renormalization group for applications in statistical mechanics—from the standard topic of equilibrium critical phenomena to nonlinear stochastic dynamics. The concise treatment manages to cover the essentials appropriately, although certainly a more careful proofreading could have eliminated a few lapses. For example, I find it important to state that the renormalization procedure constitutes a semi-group and that the Ising model does display “net magnetism,” namely, paramagnetism, at high temperatures. I also certainly do not understand what the author is trying to convey in chapter 7 with the misleading statement that “the Ising model is equivalent to an assumption that there are no correlations.”
But more important, in the chapter on the field-theory approach, McComb fails to explain the connection between UV divergences and the infrared singularities that are physically relevant for critical phenomena. I warmly welcome the inclusion of noisy hydrodynamics and stochastic differential equations of the Langevin type. However, McComb discusses neither the Einstein relation, which in thermal equilibrium connects the white-noise correlation strength with the relaxation rate, nor the crucial impact the functional form of the stochastic force correlator may have on the scaling properties of nonequilibrium systems. Given that McComb does introduce many of the required tools in field theory, I would have found it beneficial had he included an exposition of the path-integral representation of stochastic processes and developed from there the dynamic perturbation expansion, rather than using the somewhat cumbersome iteration of the equations of motion.
My feeling is that the author’s own research expertise has influenced his choice of topics, perhaps too much. Stochastically driven Navier–Stokes equations and turbulence cascades do not represent the simplest, most accessible examples of dynamic scaling. I would have first discussed the noisy Burgers equation, which McComb does mention, and simple relaxational kinetics, and then ventured into, for example, the critical dynamics of isotropic ferromagnets.
In summary, McComb manages to convey the essence of the renormalization group philosophy to uninitiated readers. The scope of his text is admirable; however, I see his book as only partially successful in explaining the required formalism in a way that would allow careful students to proceed with their own detailed calculations. In that respect, I would probably still recommend to true beginners The Theory of Critical Phenomena (Oxford U. Press, 1993) by James J. Binney and coauthors, which should be supplemented with the more advanced texts mentioned earlier. Nevertheless, Renormalization Methods should be an excellent source of material for anyone who plans to lead advanced undergraduates and first-year graduate students beyond the standard course material toward current research topics. I shall certainly keep the book in close reach when preparing my classes.
