Up for review: Gauge Fields, Knots and Gravity by John Baez and Javier P. Muniain. Published by World Scientific; ISBN 9810220340 (pbk)
In short: Buy this book immediately!
Gauge Fields, Knots and Gravity is a surprisingly small book, given the hefty title, but its goal is to provide a solid introduction to these subjects, rather than attempt a complete and detailed treatment. The authors state in the preface that they “hope that both physicists who wish to learn more differential geometry and topology, and mathematicians who wish to learn more gauge theory and general relativity, will find this book a useful place to start.” Speaking as a physicist, I can report that they succeeded marvelously, and I further admit to having learned a lot of gauge theory and general relativity as well.
The book is divided into three parts, which you’d be forgiven for expecting are the three advertised topics. Gauge fields and knots are covered in part II, gravity in part III, while part I, under the heading of Electromagnetism, gives easily the best introduction to differential geometry that I have come across. By the end of the first part the reader can understand and appreciate Maxwell’s equations in the simple coordinate-independent form and . The second part takes up the gauge theory aspects of Maxwell’s equations directly, treating fiber bundles, connections, and curvature while working up to the Yang-Mills equation, Chern-Simons theory, and the links to knot theory. Even more ambitious is part III, which (somewhat hurriedly) covers the standard mathematical apparatus of general relativity before moving on to the real goals, the ADM formalism and prospects for quantization in Ashtekar’s new variables.
There are almost surely hundreds of precise textbooks on differential geometry and fiber bundles, many bringing to mind the observation by C.N. Yang that “There are only two kinds of math books. Those you cannot read beyond the first sentence, and those you cannot read beyond the first page.” On the other end of the spectrum are the “[x] for physicists” books which often treat their chosen material intuitively but not precisely enough to be useful in calculating or deriving anything. The chief strength of this book is its ability to do both well, and in a non-cumbersome formalism. Concepts are explained in a clear, easy to read manner and then connected to precise definitions written in a useful formalism. Any one of these three can make for a useful book, but Baez and Muniain set a new standard by offering all three. And over 300 exercises.