Table Of ContentNew Developments in
Quantum Field Theory
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Volume 364 — Quantum Fields and Quantum Space Time
edited by Gerard 't Hooft, Arthur Jaffe, Gerhard Mack, Pronob K. Mitter,
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Volume 365 — Techniques and Concepts of High-Energy Physics IX
edited by Thomas Ferbel
Volume 366 — New Developments in Quantum Field Theory
edited by Poul Henrik Damgaard and Jerzy Jurkiewicz
Series B: Physics
New Developments in
Quantum Field Theory
Edited by
Poul Henrik Damgaard
Niels Bohr Institute
Copenhagen, Denmark
and
Jerzy Jurkiewicz
Jagellonian University
Cracow, Poland
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PREFACE
Quantum field theory is one of most central constructions in 20th century theo-
retical physics, and it continues to develop rapidly in many different directions. The
aim of the workshop “New Developments in Quantum Field Theory”, which was held
in Zakopane, Poland, June 14-20, 1997, was to capture a broad selection of the most
recent advances in this field. The conference was sponsored by the Scientific and En-
vironmental Affairs Division of NATO, as part of the Advanced Research Workshop
series. This book contains the proceedings of that meeting.
Major topics covered at the workshop include quantized theories of gravity, string
theory, conformal field theory, cosmology, field theory approaches to critical phenomena
and the renormalization group, matrix models, and field theory techniques applied to
the theory of turbulence.
One common theme at the conference was the use of large-N matrix models to
obtain exact results in a variety of different disciplines. For example, it has been known
for several years that by taking a suitable double-scaling limit, certain string theories
(or two-dimensional quantum gravity coupled to matter) can be re-obtained from the
large-N expansion of matrix models. There continues to be a large activity in this area
of research, which was well reflected by talks given at our workshop. Remarkably, large-
N matrix models have very recently – just a few months before our meeting – been
shown to have yet another deep relation to string theory. This time the connection goes
through the so-called M-theory, which can loosely be thought of as a unifying theory
of strings. Also this very recent subject was covered at our workshop. At the very last
moment Yuri Makeenko had to cancel his participation. He fortunately agreed to send
his contribution to this volume.
The understanding of the rôle M-theory plays for the different string theories
originates in some remarkable results concerning duality that have been uncovered
within the last 2-3 years. While so-called T-duality of string theory has been known
for years, it is now being seen in a new light, and also other kinds of dualities have
been found. Simultaneously, exact or approximate dualities have been shown to be
properties of certain highly non-trivial supersymmetric quantum field theories in four
dimensions. Both these dualities, their origin in string theory, as well as direct analyses
of T-duality in the σ-model language were discussed at the meeting.
Another recent application of large-N matrix model techniques has been in the
description of certain exact features of field theories with spontaneous chiral symmetry
breaking (such as Quantum Chromodynamics). A recent flurry of activity has revealed
a number of surprising universal aspects of such quantum field theories, related to the
spectrum of the Dirac operator. At the meeting new and impressive Monte Carlo results
from lattice gauge theory simulations were presented. They appeared to be in complete
v
agreement with the theoretical predictions. Also other aspects of this computational
framework of matrix models were discussed at the meeting, for example in connection
with the behavior at finite temperature, or in the limiting case of no chiral symmetry
breaking.
One final, and also surprising, application of large-N matrix models which was
covered at the workshop concerns the derivation of exact results in the theory of tur-
bulence. Enlightening lectures were also given on the use of quantum field theory
techniques in general to solve problems related to turbulence, and on the application
of magnetohydrodynamics on cosmological scales.
As testified by this volume, numerous other topics were discussed at our workshop.
It left the participants with the distinct impression that despite the long history of the
field, we are now witnessing an extremely fruitful period of developments in quantum
field theory.
We take this opportunity to thank Yu. Makeenko, A. Polychronakos and J.F.
Wheater for serving on the international advisory committee. Very special thanks go
to M. Praszalowicz and B. Brzezicka for their tireless help both before and during the
workshop, and to P. Bialas, Z. Burda, and P. Jochym for much assistance. We would
in particular like to thank Z. Burda for his help in preparing this volume.
Copenhagen and Cracow
Poul H. Damgaard
Jerzy Jurkiewicz
vi
CONTENTS
LECTURERS
The Structure of 2D Quantum Space-Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
J. Ambjørn
Scaling Laws in Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
K. Gaw dzki
Field Theory as Free Fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
A. Carlini and J. Greensite
Center Dominance, Center Vortices, and Confinement . . . . . . . . . . . . . . . . . . . . . . . . . 47
L. Del Debbio, M. Faber, J. Greensite and Š. Olejník
Duality and the Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
P.E. Haagensen
Unification of the General Non-Linear Sigma Model
and the Virasoro Master Equation . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
J. de Boer and M. Halpern
A Matrix Model Solution of the Hirota Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
V.A. Kazakov
Lattice Approximation of Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . 113
J. Kijowski and Gerd Rudolph
Three Introductory Talks on Matrix Models of Superstrings . . . . . . . . . . . . . . . . . . 127
Y. Makeenko
New Developments in the Continuous Renormalization Group . . . . . . . . . . . . . . . . . . 147
T.R. Morris
vii
Primordial Magnetic Fields and Their Development
(Applied Field Theory) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
P. Olesen
Towards Matrix Models of IIB Superstrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
P. Olesen
Quantum Mechanics of the Electric Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
A. Staruszkiewicz
Universal Fluctuations in Dirac Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
J. Verbaarschot
Determination of Critical Exponents and Equation
of State by Field Theory Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
J. Zinn-Justin
SEMINAR SPEAKERS
Collective Dynamics of a Domain Wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
H. Arod
Path Space Formulation of the BFV Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
K. Bering
Surplus Anomaly and Random Geometries . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . 251
P. Bialas, Z. Burda and D. Johnston
Topological Contents of 3D Seiberg-Witten Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
B. Broda
Free Strings in Non-Critical Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
M. Daszkiewicz, Z. Hasiewicz and Z. Jaskólski
Seiberg-Witten Theory, Integrable Systems and D-Branes . . . . . . . . . . . . . . . . . . . . . . 279
A. Marshakov
Microscopic Universality in Random Matrix Models of QCD . . . . . . . . . . . . . . . . . . . . 287
S.M. Nishigaki
New Developments in Non-Hermitian Random Matrix Models . . . . . . . . . . . . . . . . . 297
R.A. Janik, M.A. Nowak, G. Papp and I. Zahed
Potential Topography and Mass Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .315
M. Kudinov, E. Moreno and P. Orland
viii
Past the Highest-Weight, and What You Can Find There . . . . . . . . . . . . . . . . . . . . . . . . 329
A.M. Semikhatov
The Spectral Dimension on Branched Polymer Ensembles . . . . . . . . . . . . . . . . . . . . . . . . 341
T. Jonsson and J.F. Wheater
Solving the Baxter Equation in High Energy QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
J. Wosiek
Participants .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
ix
