Renormalization group. (English) Zbl 0830.58038

Benfatto, Giuseppe; Gallavotti, Giovanni

Physics Notes. 1. Princeton, NJ: Princeton Univ. Press. viii, 142 p., $ 35.00; £25.00/hbk (1995).

Renormalization group was the subject of a Nobel Prize Lecture by K. G. Wilson [Rev. Mod. Phys. 47, 773-840 (1975)]. Obviously it is an important subject that has numerous applications not only in quantum physics but also in almost all branches of physics and in the modern mathematics of dynamical systems. However, as far as the reviewer is aware, no text on group theory describes the theory of renormalization groups. Here we have a moderately priced and not too thick book published by the Princeton University Press and written by two famous scholars which is entirely devoted to renormalization groups. The reader if he is already familiar with the mysteries of the renormalization group will need no commendation, but the reviewer heartily recommends the book to those who know nothing about the subject and wish to learn something about it – after reading the book, such a reader may not be any wiser about the notion of the renormalization group but, the reviewer is sure, he would, nevertheless, be much wiser. The book does not try to define a renormalization group either explicitly or implicitly or to explain the notion either formally or informally. The only illuminating message on these matters is contained in the following sentence: “The notion of renormalization group is not well defined”. To be fair to the authors, the book is not written for mathematicians, nor for mathematical physicists but for a physicist who not only knows what a lattice system with no magnetic field is but can also accept without any further explanation that the Hamiltonian of such a system is given by \[ H (\varphi) = {1 \over 2} p_0^{- d + 2} \sum_{x,y \in \Lambda} J(x - y) (\varphi_x - \varphi_y)^2 + p_0^{-d} \sum_{x \in \Lambda} r \varphi^2_x \] where \(\varphi\) is a real-valued function, \(x\) and \(y\) are integral multiples of the lattice step denoted by \(p_0^{-1}\). The book follows the now well-ingrained fashion on writing works which make a subject mysterious: this, of course, adds to the magic and therefore to the importance of the subject. In this particular task the authors have succeeded enormously.

Reviewer: C.S.Sharma (London)

MSC:

58Z05	Applications of global analysis to the sciences
81-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
81T17	Renormalization group methods applied to problems in quantum field theory
82-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistical mechanics
82B28	Renormalization group methods in equilibrium statistical mechanics

Keywords:

scale invariance; scale covariance; renormalization groups