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
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