Quantum field theory (QFT) is a rich and robust part of mathematical and
theoretical physics, with a wide range of applications in statistical physics,
condensed matter physics, particle physics and string theory. Many (if
not most) of the applications in four space-time dimensions require some
sort of systematic approximations.
In contrast, quantum field theories
in lower dimensions
often display special properties which open the door, at least in
principle, to their exact solution.
In many cases, the methods used to obtain exact solutions
are peculiar to low-dimensional space-times, but the exact
solutions thus obtained can teach us a great deal about more realistic
QFTs in higher dimensions. Thus, lower-dimensional QFTs
provide excellent models for exploring deep theoretical and computational
issues, in addition to their direct applications to physical processes
in reduced dimensions, which include polymers, boundary fluctuations,
spin chains, and quantum Hall systems. Such models are also
vital testing grounds for fundamental theoretical ideas
such as integrability, phase transitions, collective and
non-perturbative quantum behaviour, and non-equilibrium
statistical mechanics. We should also mention the effective dimensional
reduction which occurs in the high temperature limit, or is induced by disorder.
Low-dimensional quantum field theory thus sits at a unique
intersection point of contemporary mathematics and physics, allowing
many of these issues to be addressed rigorously and analytically.
Surprising inter-relations continue to arise, from the ubiquitous
nature of the Bethe ansatz and Calogero models, to the link between
integrable
spin chains and the anomalous dimensions of operators in gauge theories.
Aspects of the latter connection have
been known at least since the 1990s; they have recently
received tremendous attention in connection with N=4 supersymmetric
Yang--Mills theory (in four space-time dimensions) and the AdS/CFT correspondence.
One unifying idea is the Bethe ansatz, whose 75th anniversary we
celebrate this year, and
whose wide-ranging impact in diverse branches
of modern mathematical physics is made abundantly clear in this
special issue. In addition, new ideas and techniques of
significance to both mathematics and physics have emerged: the recent and spectacular progress on
geometrical aspects of phase transitions made possible through the
ideas of stochastic (Schramm) Loewner evolution, or SLE, is a
particularly noteworthy example.
All papers in this special issue are invited, refereed contributions
from leading experts. Many are pedagogical reviews, capturing the current
state of knowledge in a particular subject, putting the subject in
context and outlining open challenges. Others are research papers,
reflecting the rapid flux of new ideas and results in this rich and
exciting field. Together, they represent a broad sample of
the entire field. We would like to take this opportunity to thank all
of the authors for their efforts in making this special issue possible, and
Kazuhiro Hikami for his help in the early stages of this project.
We hope it inspires many new developments!