Renormalization group made clearer Sergei Winitzki DRAFT of December 26, 2004 Abstract I attempt to explain the use of renormalization group in quantum field theory from an elementary point of view. I review an elementary quantum-mechanical problem involving renormalization as a pedestrian example of a theory which is inherently ill-defined without a cutoff. After introducing a cut- off, one usually obtains a perturbative expansion that becomes invalid when the cutoff is removed. The renormalization group approach is treated as a purely mathematical technique (the Woodruff-Goldenfeld method) that improves the behavior of non-uniform perturbative expansions. By means of renormaliza- tion, one derives a perturbative expansion which is uniform in the cutoff, and therefore valid in the limit of infinite cutoff. I illustrate the application of this method to singular perturbation problems in ordinary differential equations. 1 Introduction 1.1 Motivation The method of renormalization group (RG) is widely used in theoretical and mathematical physics. Wil- son’s formulation of RG appeals to physical intuition which works well in solid state physics. However, it is more difficult to understand the renormalization of divergent perturbative expansions in quantum field theory. While the details of calculations are clear, one is left with an impression of “black magic” because the logic behind those calculations remains hidden. Textbooks usually present such arguments as “the bare coupling constants are taken to infinity to cancel the divergences,” and “the renormalization group describes the freedom to choose the physical renormalization scale.” I never felt at home with theories con- taining cutoffs and renormalization scales (which are essentially hidden, unobservable parameters chosen for “mathematical convenience”) or cutoff-dependent coupling constants that tend to infinity. In these notes I try to adopt the conservative point of view that infinite quantities are undefined and that the mathematical formulation of a physical theory must be made as clear as possible. Instead of choosing a hidden parameter of a physical theory out of “mathematical convenience,” I would like to formulate the physical theory as a well-defined mathematical problem and then use whatever mathematical method is convenient to solve that problem. In perturbative quantum field theory, physical quantities are found from series expansions in coupling constants 1 , such as a (x, ε)= a 0 (x)+ εa 1 (x)+ ε 2 a 2 (x)+ ..., (1) where a is a physical quantity, x is a parameter on which a depends, and ε is the small parameter of expan- sion. In most cases of interest, some of the coefficients a k (x) are expressed through divergent integrals, for instance one might have a 1 (x)=  ∞ 0 kdk k 2 + x 2 . (2) Thus the perturbative expansion (1) appears to be invalid even for extremely small ε. This troublesome situation may be given two possible interpretations. Either one may assume that the quantity a (x, ε) is actually finite but singular at ε = 0 and therefore does not possess a straightforward series 1 This series is most often divergent and should be understood as an asymptotic expansion in ε. In practical calculations, one never computes more than the first few terms of this series. 1