Preface

This book is an attempt to present the rudiments of quantum field theory

in general and quantum electrodynamics in particular, as actually practiced by

physicists for the purpose of understanding the behavior of subatomic particles, in

a way that will be comprehensible to mathematicians.

It is, therefore, not an attempt to develop quantum field theory in a mathe-

matically rigorous fashion. Sixty years after the growth of quantum electrodynam-

ics (QED) and forty years after the discovery of the other gauge field theories on

which the current understanding of the fundamental interactions of physics is based,

putting these theories on a sound mathematical foundation remains an outstanding

open problem — one of the Millennium prize problems, in fact (see [67]). I have no

idea how to solve this problem. In this book, then, I give mathematically precise

definitions and arguments when they are available and proceed on a more informal

level when they are not, taking some care to be honest about where the problems

lie. Moreover, I do not hesitate to use the informal language of distributions, with

its blurring of the distinction between functions and generalized functions, when

that is the easiest and clearest way to present the ideas (as it often is).

So: why would a self-respecting mathematician risk the scorn of his peers by

undertaking a project of such dubious propriety, and why would he expect any of

them to read the result?

In spite of its mathematical incompleteness, quantum field theory has been an

enormous success for physics. It has yielded profound advances in our understand-

ing of how the universe works at the submicroscopic level, and QED in particular

has stood up to extremely stringent experimental tests of its validity. Anyone with

an interest in the physical sciences must be curious about these achievements, and

it is not hard to obtain information about them at the level of, say, Scientific Amer-

ican articles. In such popular accounts, one finds that (1) interaction processes are

described pictorially by diagrams that represent particles colliding, being emitted

and absorbed, and being created and destroyed, although the relevance of these dia-

grams to actual computations is usually not explained; (2) some of the lines in these

diagrams represent real particles, but others represent some shadowy entities called

“virtual particles” that cannot be observed although their effects can be measured;

(3) quantum field theories are plagued with infinities that must be systematically

subtracted off to yield meaningful answers; (4) in spite of the impression given by

(1)–(3) that one has blundered into some sort of twilight zone, these ingredients

can be combined to yield precise answers that agree exquisitely with experiment.

(For example, the theoretical and experimental values of the magnetic moment of

the electron agree to within one part in

1010,

which is like determining the distance

from the Empire State Building to the Eiffel Tower to within a millimeter.)

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