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
which is like determining the distance
from the Empire State Building to the Eiffel Tower to within a millimeter.)
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