PREFACE vii

down-to-earth calculations that lead to experimentally verifiable results are given

scant attention. Actually, I would suggest that the reader might study Quantum

Fields and Strings more profitably after reading the present book, as the real focus

there is on more advanced topics.

There is another book about quantum fields written by a mathematician, Tic-

ciati’s Quantum Field Theory for Mathematicians [121]. In its general purpose it

has some similarity to the present book, but in its organization, scope, and style it

is quite different. It turned out not to be the book I needed in order to understand

the subject, but it may be a useful reference for others.

The foregoing paragraphs should explain why I thought there was a gap in the

literature that needed filling. Now I shall say a few words about what this book

does to fill it.

First of all, what are the prerequisites? On the mathematical side, the reader

needs to be familiar with the basics of Fourier analysis, distributions (generalized

functions), and linear operators on Hilbert spaces, together with a couple of more

advanced results in the latter subject — most notably, the spectral theorem. This

material can all be found in the union of Folland [48] and Reed and Simon [94],

for example. In addition, a little Lie theory is needed now and then, mostly in

the context of the specific groups of space-time symmetries, but in a more general

way in the last chapter; Hall [62] is a good reference for this. The language of

differential geometry is employed only in a few places that can safely be skimmed by

readers who are not fluent in it. On the physical side, the reader should have some

familiarity with the Hamiltonian and Lagrangian versions of classical mechanics,

as well as special relativity, the Maxwell theory of electromagnetism, and basic

quantum mechanics. The relevant material is summarized in Chapters 2 and 3, but

these brief accounts are meant for review and reference rather than as texts for the

novice.

As I mentioned earlier, quantum field theory is built on a very broad base of

earlier physics, so the first four chapters of this book are devoted to setting the stage.

Chapter 5 introduces free fields, which are already mathematically quite nontrivial

although physically uninteresting. The aim here is not only to present the rigorous

mathematical construction but also to introduce the more informal way of treating

such objects that is common in the physics literature, which offers both practical

and conceptual advantages once one gets used to it. The plunge into the deep waters

of interacting field theory takes place in Chapter 6, which along with Chapter 7

on renormalization contains most of the really hard work in the book. I use some

imagery derived from the Faust legend to describe the necessary departures from

mathematical rectitude; its significance is meant to be purely literary rather than

theological. Chapter 8 sketches the attractive alternative approach to quantum

fields through Feynman’s sum-over-histories view of quantum mechanics, and the

final chapter presents the rudiments of gauge field theory, skirting most of the

quantum issues but managing to derive some very interesting physics nonetheless.

There are several ways to get from the starting line to the goal of calculating

quantities with direct physical meaning such as scattering cross-sections. The path

I follow here, essentially the one pioneered by Dyson [25], [26], is to start with free

fields, apply perturbation theory to arrive at the integrals associated to Feynman

diagrams, and renormalize as necessary. This has the advantages of directness and

of minimizing the amount of time spent dealing with mathematically ill-defined