Table Of ContentProgress in Physics
Vol. 7
Editedby
A. Jaffe, G. Parisi,
and D. Ruelle
Springer Science+Business Media, LLC
Scaling and
Self-Similarity
in Physics
Renormalization
in Statistical Mechanics
and Dynamics
Jiirg Frohlich, editor
1983 Springer Science+Business Media, LLC
Editor:
JürgFröhlich
DepartmentofTheoreticalPhysics
ETH-Höriggerberg
8093Zürich,Switzerland
LibraryofCongressCatalogingin PublicationData
Scalingand self-similarity inphysics.
(Progressinphysics ;v.7)
1.Renormalizationgroup. 2.Statisticalmechanics.
3.Fieldtheory. 4.Dynamics. 5.Phasetransformations
(Statisticalphysics) 6.CriticalPhenomena (Physics)
I.Fröhlich,Jürg. II.TitIe:Self-similarity,
Ill.Series:Progressinphysics (Boston, Mass.) ;v.7.
Oc20.7.R43S33 1983 530.1 83-15788
Cll--KurztitelaufuahmederDeutschenBibliothek
Scalingand self-similarityin physics:
renormalizationinstatist. mechanicsand
dynamicsled.byJürg Fröhlich.-Boston;
Basel;Stuttgart:Birkhäuser, 1983.
(Progressinphysics;Vol.7)
NE:Fröhlich,Jürg [Hrsg.];GT
Allrightsreserved.No partofthispublicationmay bereproduced,storedina
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eSpringerScience+BusinessMediaNewYork 1983
Ursprünglicherschienenbei BirkhäuserBoston, Inc. 1983.
Softcoverreprintofthe hardcover 1st edition 1983
ISBN978-1-4899-6764-0 ISBN978-1-4899-6762-6 (eBook)
DOI 10.1007/978-1-4899-6762-6
987654321
TABLE OF CONTENTS
FOREWORD 1
PART I : Equilibrium statistical mechanics and field theory
Large Fluctuations of Random Fields and
Renormalization Group: Some Perspeutives,
G. Jona-Lasinio 11
The Berezinskii-Kosterlitz-Thouless Transition
(Energy - Entropy Arguments and Renormalization
in Defect Gases), J.FrBhlich and T.Spencer 29
Interface and Surface Tension in Ising Model,
C.-E.Pfister 139
Iterated Mayer Exnansions and their Application to
Coulomb Gases, J.Z.Imbrie 163
Rigorous Results on the Critical Behavior in
Statistical Mechanics, M.Aizenman 181
Non-Perturbativ6 Methods for the Study of Massless
Models, J.-R.Fontaine 203
Rigorous Renormalization Group and Asymptotic
Freedom, K.Gaw~dzki and A.Kupiainen 227
On Infrared Superrenormalization, J.Magnen and
R.Seneor 263
4
Ultraviolet Stability in Field Theory The ~3 Model,
T.Balaban 297
PART II : Dynamical Systems
Renormalization Group Analysis for Dynamical Systems,
P.Collet and H.Koch 323
v
vi
Bowen's Formula for the Hausdorff Dimension of
Self-Similar Sets, D.Ruelle 351
Perturbation Theory for Classical Hamiltonian
Systems, G.Gallavotti 359
FOREWORD
The .articles collected in this book have grown out of
a series of seminar talks held at the Institut des Hautes
Etudes Scientifiques. Bures-sur-Yvette. between spring 1981
and summer 1982. These talks were presented by people who.
during that period. happened to be visitors at the Institut
or worked in the Paris area. Most of them were (and still
are) thinking about problems in theoretical and mathemati
cal physics related. in a general way. to ideas of scaling
and self-similarity; problems which nowadays are often
studied with the help of what has become known under the
name of "renormalization group methods". Their ideas and
their results illustrate in which way mathematical physics
has interacted with one of the big themes of present-day
theoretical physics. To present a brief status report about
where these interactions have led us. so far. is a guiding
idea behind this volume.
During the past ten to twenty years. various concepts
of scaling. self-similarity and renormalization have as
sumed a well defined shape and proven to provide a very
productive framework to think about a surprisingly large
class of physical problems. Powerful computational and
analytical tools have emerged from them. Most readers will
be familiar with one or another among many excellent text
books and review articles devoted to scaling and renormal
ization in the theory of critical phenomena in statistical
mechanics and of differentiable dynamical systems. [Ref
erences to some of these texts may be found in several
articles in this book.] Numerous theoretical physicists
1
2
have contributed very important insights and results to
these fields. Thanks to their original and persistent ef
forts we now do have some considerable qualitative and
quantitative understanding of "continuous transitions and
critical phenomena", not only in equilibrium statistical
mechanics and quantum field theory, but also in dynamics
and in the theory of disordered systems.
Precisely because very many people have contributed a
large numer of substantial result s to the subject addressed
in this book, it unfortunately turned out to be necessary
to apply rather arbitrary, but strict criteria to choose
authors of articles for the present book. Such criteria
are usually unsatisfactory. For I still believe in a basic
unity of theoretical and mathematical physics. The criteria
which we finally adopted are not intended to express any
value judgments. [This is no attempt towards "eine Umwer
tung aller Werte".] Rather , they were imposed upon us by
convenience and by consideration of what was within our
modest possibilities.
They are:
1. Every author was supposed to have spent time or given
a seminar talk at the Institut des Hautes Etudes Scien
tifiques during the period between spring 1981 and summer
1982.
2. Only contributions were accepted which describe
mathematically rigorous results related to the main theme
of this book.
3. Every article was supposed to be of relevance to
statistical physics and/or dynamical systems theory.
4. Contributions concerning "scaling and self-similar
ity in mathematics" were not invited.
These criteria may explain why many ideas, results and
calculations of enormous theoretical and practical import
ance to critical phenomena in equilibrium statistical mech
anics and dynamics cannot be found in any of the articles
published in this volume. Moreover, they make it clear why
3
most of the original creators of many important ideas and
methods are not among the contributors.
Crit erion 3 explains why "scaling and renormalization in
relativistic quantum field theory", for example, is great ly
underemphasized in this book. This is most regrettable, for
several reasons:
a) Many of the important ideas on "scaling and renorm
alization" actually originated in relativistic quantum
field theory.
b) The discovery of many relations and deep connections,
heuristic and mathematical, between the theory of continuous
transitions and critical phenomena on the one hand and quan
tum field theory on the other hand, and the many uses of
these connections in actual calculations belong to the
great achievements of theoretical physics in recent times
and has had a profound impact on the development of statis
tical physics and quantum field theory. [Various connec
tions and analogies between quantum field theory and fluid
dynamics may eventually turn out to be of comparable signi
ficance.]
c) Quant um field theory is one of a few basic theories
in modern theoretical physics, and it is the most plausible
and most efficient theoretical basis for particle physics.
As such it is of eminent import ance , and it still attracts
many among the most brilliant theoreticians.
The theory of disordered systems is not represented
in this book, at all. This, too, is really quite regret
table; especially since, during the past few years, nice
mathematically rigorous results about disordered systems,
related to the themes of this book, have started to appear.
Moreover, the theory of disordered systems bears quite a
lot of promise for the near - and not so near - future.
Criterion 4 has excluded all developments related to
scaling, self-similarity and renormalization in mathematics
from this book. That there are many and important such de
velopments is presumably clear to many or most readers. In
