Renormalization group in statistical mechanics and mechanics: gauge symmetries and vanishing beta functions

Two very different problems that can be studied by renormalization group me thods are discussed with the aim of showing the conceptual unity that renor malization group has introduced in some areas of theoretical Physics. The t wo problems are: the ground state theory of an one-dimensional quantum Ferm i liquid and the existence of quasi periodic motions in classical mechanica l systems close to integrable ones. I summarize here the main ideas and sho w that the two treatments, although completely independent of each other, a re strikingly similar. (C) 2001 Elsevier Science B.V. All rights reserved.