This book tells mathematicians about an amazing
subject invented by physicists and it tells physicists how a master
mathematician must proceed in order to understand it. Physicists who
know quantum field theory can learn the powerful methodology of
mathematical structure, while mathematicians can position themselves
to use the magical ideas of quantum field theory in “mathematics”
itself. The retelling of the tale mathematically by Kevin Costello is
a beautiful tour de force.
—Dennis Sullivan
This book is quite a remarkable
contribution. It should make perturbative quantum field theory
accessible to mathematicians. There is a lot of insight in the way the
author uses the renormalization group and effective field theory to
analyze perturbative renormalization; this may serve as a springboard
to a wider use of those topics, hopefully to an eventual
nonperturbative understanding.
—Edward Witten
Quantum field theory has had a profound influence on mathematics,
and on geometry in particular. However, the notorious difficulties of
renormalization have made quantum field theory very inaccessible for
mathematicians. This book provides complete mathematical foundations
for the theory of perturbative quantum field theory, based on Wilson's
ideas of low-energy effective field theory and on the
Batalin–Vilkovisky formalism. As an example, a cohomological
proof of perturbative renormalizability of Yang–Mills theory is
presented.
An effort has been made to make the book accessible to mathematicians who
have had no prior exposure to quantum field theory. Graduate students who
have taken classes in basic functional analysis and homological algebra
should be able to read this book.
Readership
Graduate students and research mathematicians interested in
quantum field theory and mathematical physics.