Renormalization theory is still with us and very much alive since its
birth over three decades ago. It has reached such a high level of sophistication
that any book on the subject has to be mathematically rigorous to
do any justice to it. Since the early classic works on quantum electrodynamics,
it has been studied systematically and has become the method for
computations in relativistic quantum field theory. The success of renormalizaztion
has been recorded at least twice in the history of physics
through Nobel prizes, once in 1965 for the work on quantum electrodynamics,
and again in 1979 for the work on the unification of the electromagnetic
and weak interactions. Quantum electrodynamics is in excellent
agreement with experiments, and the unified field theories seem to be
quite promising candidates for a more complete theory. In spite of the
importance of renormalization theory in physics, very few field theorists
seem to know the intricate details of this subject. This is essentially due to
the complex nature of the subject. Accordingly a book on renormalization
theory would be quite justifiable at this stage, if not a matter of urgency.
We define two major lines of work on the subject. The first is to develop
the subtraction formalism, provide its convergence proof, and extract
general and valuable information from the subtractions, all done in a
model-independent way, and is mathematical in nature. The other deals
with model building, “modified” Feynman rules, symmetry principles
and spontaneous symmetry breaking, renormalization group equations,
and operator product expansions, to mention just a few problems. This
second line of work is a rapidly developing branch of research, and there
is still much to be done. The first line of work is well established, and this
book is devoted exclusively to it and deals with the basic facts of renormalization
theory. Perhaps another suitable title for the book would have
been “The Core of Renormalization.” The subtraction scheme we use
has a very simple structure; we were inspired by the ingenious and classic
work of Salam in its formulation. It is carried out in momentum space and
applied directly to the Feynman integrand without ultraviolet cutoffs. A
unifying theorem of renormalization is given that brings us into contact
with other standard approaches of subtractions. In particular, the latter
establishes the long-standing problem of essentially the equivalence of
the paths taken in the ingenious approaches of Salam and Bogoliubov.