By Paul Baillon
World Scientific
Hardback: £57
Also available at the CERN bookshop
The theory of differential manifolds is a common substratum of much of our current theoretical descriptions of physical phenomena. It has proved to be well adapted to many branches of classical physics –mechanics, electromagnetism, gravitation – for which it has provided a framework for a precise formulation of fundamental laws. Its use in quantum physics has led to spectacular discoveries associated with the unification of electromagnetic, weak and strong interactions. In this connection, manifolds appear not only in the description of the substratum of these phenomena but also in the description of the phenomena themselves, in terms of the so-called gauge theories.
This mathematical theory constitutes an important body of contemporary mathematics. Baillon’s book, which aims at making the subject accessible to a readership that is rich in a completely different culture, adopts an unconventional expository style. Instead of appealing to intuition based on mathematically non-rigorous images and analogies – a common practice – it insists on providing complete proofs of most of the elementary mathematical facts on which the theory is grounded.
A substantial part of the book is devoted to a detailed description of the necessary mathematical equipment. Applications culminate in an introduction to some delicacies of the electroweak theory, as well as of general relativity.
