Plain TeX, 80K, 1 figures autogenerated as gvnn.ps. Use a postscript
printer and compile into postscript with DVIPS. If one does not use a
postscript printer the figures cannot be printed: the paper can still be
printed provided one changes the parameter \driver on line 8 from 1 to
5.
BODY
%TO PRINT THE POSTCRIPT FIGURES THE DRIVER NUMBER MIGHT HAVE TO BE
%ADJUSTED. IF the 4 choices 0,1,2,3 do not work set in the following line
%the \driver variable to =5. Setting it =0 works with dvilaser setting it
%=1 works with dvips, =2 with psprint, =3 with dvitps, (hopefully).
%Using =5 prints incomplete figures (but still understandable from the
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\newcount\driver \driver=1 %%%this is the value to set!!!
%%% the values =0,1 have been tested. The figures are automatically
%%% generated.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FORMATO
\newcount\mgnf\newcount\tipi\newcount\tipoformule
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\tipi=2 %uso caratteri: 2=cmcompleti, 1=cmparziali, 0=amparziali
\tipoformule=1 %=0 da numeroparagrafo.numeroformula; se no numero
%assoluto
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% INCIPIT
\ifnum\mgnf=0
\magnification=\magstep0\hoffset=0.cm
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\overfullrule=10pt
%
%%%%%GRECO%%%%%%%%%
%
\let\a=\alpha \let\b=\beta \let\g=\gamma \let\d=\delta \let\e=\varepsilon
\let\z=\zeta \let\h=\eta \let\th=\vartheta\let\k=\kappa \let\l=\lambda
\let\m=\mu \let\n=\nu \let\x=\xi \let\p=\pi \let\r=\rho
\let\s=\sigma \let\t=\tau \let\iu=\upsilon \let\f=\varphi\let\ch=\chi
\let\ps=\psi \let\o=\omega \let\y=\upsilon
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\let\P=\Pi \let\Si=\Sigma \let\F=\Phi \let\Ps=\Psi \let\O=\Omega
\let\U=\Upsilon
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%% Numerazione pagine
%%%%%%%%%%%%%%%%%%%%% NUMERAZIONE PAGINE
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\def\ora{\oramin }
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%
%%%%%%%%%%%%%%%%% EQUAZIONI CON NOMI SIMBOLICI
%%%
%%% per assegnare un nome simbolico ad una equazione basta
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%%% nelle appendici, \Eqa(...) o \eqa(...):
%%% dentro le parentesi e al posto dei ...
%%% si puo' scrivere qualsiasi commento;
%%% per assegnare un nome simbolico ad una figura, basta scrivere
%%% \geq(...); per avere i nomi
%%% simbolici segnati a sinistra delle formule e delle figure si deve
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%%% \BOZZA. Sinonimi \Eq,\EQ,\EQS; \eq,\eqs; \Eqa,\Eqas;\eqa,\eqas.
%%% All' inizio di ogni paragrafo si devono definire il
%%% numero del paragrafo e della prima formula dichiarando
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%%% bisogna porre \numfig=1 (il numero delle figure non contiene la sezione.
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%%% viene letto all'inizio, se gia' presente. E' possibile citare anche
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%%% corrispondente file .aux; basta includere all'inizio l'istruzione
%%% \include{nomefile}
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\global\newcount\numfig
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\global\advance\numfor by 1
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\let\Eqas=\Eqa
\let\eqas=\eqa
%%%%%%%%%
%\newcount\tipoformule
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% %assegnato
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\input #1.aux \closein13 \fi}
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\input \jobname.aux \closein14 \fi
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CARATTERI %%%%%%%%%%%%%%
\newskip\ttglue
%%cm semplificato
\def\TIPI{
\font\ottorm=cmr8 \font\ottoi=cmmi8
\font\ottosy=cmsy8 \font\ottobf=cmbx8
\font\ottott=cmtt8 %\font\ottosl=cmsl8
\font\ottoit=cmti8
%%%%% cambiamento di formato%%%%%%
\def \ottopunti{\def\rm{\fam0\ottorm}% passaggio a tipi da 8-punti
\textfont0=\ottorm \textfont1=\ottoi
\textfont2=\ottosy \textfont3=\ottoit
\textfont4=\ottott
\textfont\itfam=\ottoit \def\it{\fam\itfam\ottoit}%
\textfont\ttfam=\ottott \def\tt{\fam\ttfam\ottott}%
\textfont\bffam=\ottobf
\normalbaselineskip=9pt\normalbaselines\rm}
\let\nota=\ottopunti}
%%%%%%%%
%% am
\def\TIPIO{
\font\setterm=amr7 %\font\settei=ammi7
\font\settesy=amsy7 \font\settebf=ambx7 %\font\setteit=amit7
%%%%% cambiamenti di formato %%%
\def \settepunti{\def\rm{\fam0\setterm}% passaggio a tipi da 7-punti
\textfont0=\setterm %\textfont1=\settei
\textfont2=\settesy %\textfont3=\setteit
%\textfont\itfam=\setteit \def\it{\fam\itfam\setteit}
\textfont\bffam=\settebf \def\bf{\fam\bffam\settebf}
\normalbaselineskip=9pt\normalbaselines\rm
}\let\nota=\settepunti}
%%%%%%%
%%cm completo
\def\TIPITOT{
\font\twelverm=cmr12
\font\twelvei=cmmi12
\font\twelvesy=cmsy10 scaled\magstep1
\font\twelveex=cmex10 scaled\magstep1
\font\twelveit=cmti12
\font\twelvett=cmtt12
\font\twelvebf=cmbx12
\font\twelvesl=cmsl12
\font\ninerm=cmr9
\font\ninesy=cmsy9
\font\eightrm=cmr8
\font\eighti=cmmi8
\font\eightsy=cmsy8
\font\eightbf=cmbx8
\font\eighttt=cmtt8
\font\eightsl=cmsl8
\font\eightit=cmti8
\font\sixrm=cmr6
\font\sixbf=cmbx6
\font\sixi=cmmi6
\font\sixsy=cmsy6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\font\twelvetruecmr=cmr10 scaled\magstep1
\font\twelvetruecmsy=cmsy10 scaled\magstep1
\font\tentruecmr=cmr10
\font\tentruecmsy=cmsy10
\font\eighttruecmr=cmr8
\font\eighttruecmsy=cmsy8
\font\seventruecmr=cmr7
\font\seventruecmsy=cmsy7
\font\sixtruecmr=cmr6
\font\sixtruecmsy=cmsy6
\font\fivetruecmr=cmr5
\font\fivetruecmsy=cmsy5
%%%% definizioni per 10pt %%%%%%%%
\textfont\truecmr=\tentruecmr
\scriptfont\truecmr=\seventruecmr
\scriptscriptfont\truecmr=\fivetruecmr
\textfont\truecmsy=\tentruecmsy
\scriptfont\truecmsy=\seventruecmsy
\scriptscriptfont\truecmr=\fivetruecmr
\scriptscriptfont\truecmsy=\fivetruecmsy
%%%%% cambio grandezza %%%%%%
\def \eightpoint{\def\rm{\fam0\eightrm}% switch to 8-point type
\textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
\textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei
\textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
\textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
\textfont\itfam=\eightit \def\it{\fam\itfam\eightit}%
\textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}%
\textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}%
\textfont\bffam=\eightbf \scriptfont\bffam=\sixbf
\scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}%
\tt \ttglue=.5em plus.25em minus.15em
\setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}%
\normalbaselineskip=9pt
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\scriptfont\truecmr=\sixtruecmr
\scriptscriptfont\truecmr=\fivetruecmr
\textfont\truecmsy=\eighttruecmsy
\scriptfont\truecmsy=\sixtruecmsy
}\let\nota=\eightpoint}
\newfam\msbfam %per uso in \TIPITOT
\newfam\truecmr %per uso in \TIPITOT
\newfam\truecmsy %per uso in \TIPITOT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%Scelta dei caratteri
%\newcount\tipi \tipi=0 %e' definito all'inizio
\newskip\ttglue
\ifnum\tipi=0\TIPIO \else\ifnum\tipi=1 \TIPI\else \TIPITOT\fi\fi
\def\didascalia#1{\vbox{\nota\0#1\hfill}\vskip0.3truecm}
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DEFINIZIONI VARIE
%
\def\media#1{\langle{#1}\rangle} \let\0=\noindent
\def\guida{\leaders\hbox to 1em{\hss.\hss}\hfill}
\def\tende#1{\vtop{\ialign{##\crcr\rightarrowfill\crcr
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\def\tto{{\Rightarrow}}
\def\pagina{\vfill\eject}\def\acapo{\hfill\break}
\def\fra#1#2{{#1\over#2}}
\def\st{\scriptstyle}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%LATINORUM
\def\etc{\hbox{\it etc}}\def\eg{\hbox{\it e.g.\ }}
\def\ap{\hbox{\it a priori\ }}\def\aps{\hbox{\it a posteriori\ }}
\def\ie{\hbox{\it i.e.\ }}
\def\fiat{{}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%DEFINIZIONI LOCALI
\def\V#1{\vec#1}\let\dpr=\partial\let\ciao=\bye
\let\io=\infty\let\ig=\int
\def\={{ \; \equiv \; }}
\def\Im{{\rm\,Im\,}}\def\Re{{\rm\,Re\,}}
\def\lis#1{{\overline #1}}
\let\0=\noindent
\def\*{\vskip0.3cm}
\def\\{\hfill\break}
\numsec=1\numfor=1
%%%%%%%%%%% GRAFICA %%%%%%%%%
%
% Inizializza le macro postscript e il tipo di driver di stampa.
% Attualmente le istruzioni postscript vengono utilizzate solo se il driver
% e' DVILASER ( \driver=0 ), DVIPS ( \driver=1) o PSPRINT ( \driver=2);
% o DVITPS (\driver=3)
% qualunque altro valore di \driver produce un output in cui le figure
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%
%\newcount\driver \driver=1
%\ifnum\driver=0 \special{ps: plotfile ini.pst global} \fi
%\ifnum\driver=1 \special{header=ini.pst} \fi
\newdimen\xshift \newdimen\xwidth
%
% inserisce una scatola contenente #3 in modo che l'angolo superiore sinistro
% occupi la posizione (#1,#2)
%
\def\ins#1#2#3{\vbox to0pt{\kern-#2 \hbox{\kern#1 #3}\vss}\nointerlineskip}
%
% Crea una scatola di dimensioni #1x#2 contenente il disegno descritto in
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% Il file #4.pst contiene le istruzioni postscript, che devono essere scritte
% presupponendo che l'origine sia nell'angolo inferiore sinistro della
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%
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\line{} \hbox{ \hskip\xshift \vbox to #2{\vfil
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\special{ps: plotfile #4.ps} \special{ps::[end]grestore} \fi
\ifnum\driver=1 #3 \special{psfile=#4.ps} \fi
\ifnum\driver=2 #3 \ifnum\mgnf=0
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\ifnum\mgnf=1
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\ifnum\driver=3 \ifnum\mgnf=0
\psfig{figure=#4.ps,height=#2,width=#1,scale=1.}
\kern-\baselineskip #3\fi
\ifnum\mgnf=1
\psfig{figure=#4.ps,height=#2,width=#1,scale=1.2}
\kern-\baselineskip #3\fi
\ifnum\driver=5 #3 \fi
\fi}
\hfil}}}
%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%DEFINIZIONI LOCALI
\def\AA{{\V A}}\def\aa{{\V\a}}\def\bv{{\V\b}}\def\dd{{\V\d}}
\def\ff{{\V\f}}\def\nn{{\V\n}}\def\oo{{\V\o}}
\def\zz{{\V z}}\def\FF{{\V F}}\def\xx{{\V x}}
\def\yy{{\V y}} \def\q{{q_0/2}}\let\lis=\overline\def\Dpr{{\V\dpr}}
\def\mm{{\V m}}
\def\ff{{\V\f}}\def\zz{{\V z}}\def\mb{{\bar\m}}
\def\CC{{\cal C}}\def\II{{\cal I}}
\def\EE{{\cal E}}\def\MM{{\cal M}}\def\LL{{\cal L}}
\def\TT{{\cal T}}\def\RR{{\cal R}}
\def\sign{{\rm sign\,}}
\def\={{ \; \equiv \; }}\def\su{{\uparrow}}\def\giu{{\downarrow}}
\let\ch=\chi
\def\PP{{\cal P}}
\def\bb{{\V\b}}
\def\Im{{\rm\,Im\,}}\def\Re{{\rm\,Re\,}}
\def\nn{{\V\n}}\def\lis#1{{\overline #1}}\def\q{{{q_0/2}}}
\def\atan{{\,\rm arctg\,}}
\def\pps{{\V\ps{\,}}}
\let\dt=\displaystyle
\def\NN{{\cal N}}
\def\DD{{\cal D}} \def\2{{1\over2}}
\def\txt{\textstyle}\def\OO{{\cal O}}
\def\FF{{\cal F}}
%\def\igb{{\ig \kern-9pt\raise4pt\hbox to7pt{\hrulefill}}}
\def\igb{
\mathop{\raise4.pt\hbox{\vrule height0.2pt depth0.2pt width6.pt}
\kern0.3pt\kern-9pt\int}}
\def\MM{{\cal M}}\def\mm{{\V\m}}
\def\acapo{\hfill\break}
\def\tst{\textstyle}
\def\st{\scriptscriptstyle}\def\fra#1#2{{#1\over#2}}
\let\\=\noindent
\def\*{\vskip0.3truecm}
%%%%%%%%%%%
\catcode`\%=12\catcode`\}=12\catcode`\{=12
\catcode`\<=1\catcode`\>=2
\openout13=gvnn.ps
\write13<%%BoundingBox: 0 0 240 170>
\write13<% fig.pst>
\write13
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\write13<70 90 punto >
\write13<120 60 punto >
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\write13
\closeout13
\catcode`\%=14\catcode`\{=1
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\overfullrule=0.pt
%%%%%%%%%%%%%%%%%%%%%%
%\input fiat
%\BOZZA
\footline={\hss\tenrm\folio\hss}
\fiat
\centerline{\bf PERTURBATION THEORY
\footnote{${}^?$}{\nota
\it keywords:\rm\ Classical mechanics, Quantum field theory, Solid state
physics, Statistical mechanics}}
\ifnum\mgnf=0\*\relax\else\vglue1.5truecm\fi
\centerline{ Giovanni Gallavotti
\footnote{${}^*$}{\nota Notes in margin to the {\it Mathematical Physics
towards the XXI century}
conference, University of Negev, Beer Sheva, 14--19 march, 1993.
This text is archived in the archive
$mp\_arc@math.utexas.edu$, \# 93-???; copies can also be
obtained by e-mail from the author: $gallavotti@vaxrom.infn.it$.\
Permanent address: Dipartimento di Fisica, Universit\`a La Sapienza,
P.le Moro 2; 00185, Roma, Italia}}
\*
\hbox{}\hfill {\it The seed ye sow, another
reaps}\footnote{${}^!$}{\nota
P. Shelley: {\it Song to the men of England.}}
\*
{\bf \S1 Mathematical Physics, in general.}
\*
It is quite difficult to find a subject to discuss as afterthought about
a conference on {\it Mathematical Physics towards the $XXI$ century}. I
will describe some personal viewpoints and discuss a recent work (with
some variations with respect to the previous versions) to stress some of
the points made.
Should I say which is the legacy of Mathematical Physics to the next
century I would point to a subject which has been left out of the talks
by most of the speakers: the understanding and formalization of
perturbation techniques for problems quite different from classical
mechanics (and for classical mechanics as well).
It is of course still true that the so called "non perturbative techniques"
are more impressive and glamorous; they also look deeper. And there are
other aspects of Mathematical Physics that have received a large impulse in
this century.
Let me begin by stressing the similarities between the perturbative and
non perturbative techniques in Mathematical Physics. What I see as
common to the two is what I consider a very basic aspect of Mathematical
Physics which, although coming from the $XIX$ century, has become very
clear during the $XX$ century: the attempt, sometimes successful, to put
on a really rational (\ie mathematically rigorous) basis the fundamental
problems of Physics.
%
It is not an overstatement to claim that many important results came
out of attempts to set in a clear form problems which were too
empirically treated. For instance the Dirac formulation of quantum
mechanics is comparable to the Lagrange and Hamilton formalization of
classical mechanics. The latter made available to everybody the possibility
of computing in an unambigous way the solution to mechanics problems which
were accessible only to few people, who managed to dominate the rather
non systematic state of the matter at the time. After Dirac's work
"everybody" could work out Atomic Physics computations and predictions.
The attempt to axiomatize quantum field theory and renormalization
theory (by Wightman and the Z\"urich school, [SW]) led first ("as a
byproduct") to the clear formulation of the major problems of
Statistical Mechanics (Ruelle, Fisher and the Moscow school, [R]): and
generated a great number of new results and a deep understanding of the
phenomena of phase transition. Thus opening way to the renormalization
group theory of the scale invariance and universality (Widom, Kadanoff,
Fisher, Wilson, [Ma], [P]) which, eventually, brought a good understanding
of renormalization theory and quantum field theory themselves.
Turbulence is a phenomenon which was typically only empirically described
and scarsely "objectively analyzed". Attempting at its precise definition
produced surpringly new results and points of view (Lorenz, Ruelle, [R2],
[ER]): and nowadays everybody knows about "chaos": which was essentially
unknown only twenty years ago.
General relativity is even more exceptional as it is an example (the
only one I know) of a theory born as a mathematically well formulated
theory, [W].
All the above examples, which are not exhaustive and only reflect my
personal interests, are remarkable: but I see many colleagues (and I have
seen many more of them in the past) saying that all of the above was not so
important. The real problems would have been, or had been, indeed solved
without need of refined mathematics or of precise formulations. This
attitude is quite erroneous in my view at least as a general statement, as
the examples of the theory of chaos or of general relativity show, in a clear
way, to those who wish to see.
It is nevertheless true that most progress was achieved before a precise
mathematical formulation: for instance the Schr\"odinger equation came
before Dirac's work. Schwinger, Feynman and Dyson understood
renormalization theory before the work of the mathematical physicists (Hepp,
[H], Glimm, Jaffe, Nelson, Guerra, Spencer, [Si], [E]). In the same way as
the work of Newton, D'Alembert, Laplace, Gauss, Euler
%
\footnote{${}^1$}{\nota I stop the list for lack of space.}
%
preceded analytical mechanics.
The reason I see the mathematical formulations as important, even in
fields in which "progress was already achieved", is that the rigorization
attempts had a "democratic nature": they made, and make, accessible to
the vast majority ideas and methods that were (obviously) perfectly
understood by only a few scientists. And, if science has to advance, it
needs "democracy", \ie many scientists who control what they are doing.
Mathematical Physics in this century had mostly the role of making
accessible results that were very deep and difficult to understand
because the scientists who developed them found no time (or no need) to
explain them in a form that could be widely understood. The
mathematical language is really universal, as Galileo explicitly noted,
and is very well suited for the transmission of knowledge.
\*\*
{\bf \S2 Fundamental, exact, non perturbative, and constructive methods.}
\*
One can distinguish roughly four aspects of Mathematical Physics. The
first I will call "fundamental": it consists in the analysis of the
general structure of various problems, and of their proper mathematical
formulation.
For instance the formalisms of statistical mechanics (general notion of
equilibrium states, of thermodynamic limit, of phase transitions, of
ground states, of correlations), field theory (general notion of
relativistic quantum field, of interaction) or fluid mechanics (general
notion of regular and chaotic motions). In this conference the above
aspects have been stressed and discussed quite in detail.
I consider the "fundamental" aspects as essential, in the same sense as
hamiltonian mechanics is essential for classical mechanics. It is true
that one could just ignore the formalisms: but at least in the case of
hamiltonian mechanics hardly anybody would dare saying that it is
useless.\footnote{${}^2$}{\nota
although the book of Laplace could easily provide arguments in support of
such thesis.}
%
But others could still insist that the formalisms are useless (it is not
uncommon to meet people who refuse to try to understand the general
theory of chaos as emerged from the work of Ruelle on the grounds that
it is a formalism while "science deals with concrete problems"). The
reason I am not too impressed by such statements is that Mathematical
Physics has been able to go far beyond the formalisms in all the
examples I quoted. The formalism has been used to set a frame into
which to formulate precisely concrete problems to be solved, {\it and
their solutions}. I would, in fact, also agree in considering of little
interest a formalism which did not attack (and solve, at least
partially) some physical problems.
A way to make use of a formalism is by discussing in its frame the
properties of concrete models. As some models can be studied "exactly";
and this brings up a second aspect of Mathematical Physics.
This century will leave the legacy of many exactly soluble models which
play, and will continue to play, the role of the basic mechanical models
like harmonic oscillators, central forces and two body problems, rigid
body, vibrating string, heat equation, linear waves): they clarify the
meaning of the formalism and provide examples of physical significance as
well as firm comparison terms for checks of approximate theories and
methods. I just mention the work of Onsager on the Ising model, or the work
of Lieb, Yang and Baxter on the ground state of the Heisenberg model, [B].
Such models, and many others (Schwartzschild and other metrics in general
relativity, [W], Thirring model in field theory, Luttinger model in solid
state Physics, one dimensional dynamics of hard rods, [ML], Korteveg De
Vries equation, Toda Lattice \etc) are not only non trivial and illustrative
of rather complicated behaviour but they have provided important
new ideas in probability theory, PDE's and other fields.
Another way of making use of the formalism is via the so called "non
perturbative methods": they consist in studies of concrete models for
which no exact solution can be provided. One tries to obtain
informations about some general properties by using various special
features of the models. An example of this is the theory of the
thermodynamic limit or the theory of statistical ensembles for systems
with short range forces in statistical mechanics (Fisher, Ruelle, [R]).
More specific (non exhaustive) examples are the existence of
thermodynamics for systems with electromagnetic forces (Dyson, Lenard,
Lieb, Thirring, Fefferman, [L], [Fe]), the theory of coexistence in
the Ising model (Russo, Aizenman, Higuchi, [Ai],[Hi]), the theory of
percolation (Russo, Kesten, [Gr]), the theory of the critical point in
ferromagnets (Aizenman, [Ai]), structure of "large atoms" (Lieb, [L2]),
existence of phase transitions in some quantum statistical mechanics
models (Dyson, Lieb, Simon, [DLS]), theory of "incommensurate crystals"
(Aubry, Mather, [Au],[M]).
Very often the above general "non perturbative" methods are based on
the use of convexity inequalities or of other classes of inequalities
(GKS, FKG, \etc). Therefore very often (though not always) the results
are "non constructive": \ie they are intrinsically unable to provide
estimates for the speed at which some limits are reached, or for the
value of various constants whose existence is proved. They describe
general, non trivial, properties which are very useful to know,
particularly to check approximate theories results.
The fourth aspect are the fully constructive results for specific models,
based on approximations by convergent series expansion to quantities that,
by general arguments, can be shown to exist. Such theories, usually called
"pertubative theories", are often considered "too technical" and the word
"perturbative" has very often a ironic and negative connotation. But they
are certainly among the greatest achievements of the subject and essentially
they provide the only cases in which the "problem" is completely solved. The
latter aspect should not be underestimated: in fact there have been
examples of problems "solved" by general theorems of "fundamental nature"
but whose solution has been subject to critique. In fact one of the dangers
of Mathematical Physics is that, in order to solve a poblem, it has to
formulate it in a precise mathematical way: but this can lead to subtle
changes of the problem itself or to too strict a formulation of it. So that
the solution achieved by one might not be regarded as such by others. This
has certainly damaged the image of research in Mathematical Physics in many
respects (for instance claims on supposed "theorems" on non existence of
crystals only threw discredit on the field).
A very illustrious example of a "theorem" which, although mathematically
correct and of great importance, missed completely the point is the generic
non integrability of pertubations of integrable hamiltonian systems and in
particular of the three body problem, [G2]. This theorem of Poincar\`e has
been poorly interpreted and referred to only by quotations for about half a
century: until it was shown that a slight modification of the statement gave
rise to a non trivial and very interesting class of results (the KAM theory,
see [G2]) which was stating the truth of almost opposite properties.
Another important example is the "triviality theorem" of scalar field
theories in space time dimension $4$, ([Fr],[Ai]): this theorem is valid
under some very strong assumptions, so far, and skeptics may interpret
it as saying that the assumptions are too strong or inappropriate. It
is nevertheless very often quoted as a theorem established in full
generality, see [GR].
Unfortunately the attitude to quote the "theorems" of Mathematical Physics
out of context and as absolute truths, proving or disproving some
fundamental results, is still quite widespread: mostly because non
specialists try to use Mathematical Physics acritically refusing to recall
that theorems need assumptions and the assumptions have a physical meaning
which is not always entirely conveyed by the words which are used to state
them. So that a technical analysis of the assumptions is always necessary
and useful. The above attitude is unfortunately taken even inside
Mathematical Physics itself: specialists in one area tend to quote
acritically results concerning other areas: I am afraid that this will not
be cured by us and it will be a problem also for the next century.
In some sense only "positive" results should really matter. But "no go"
results are also in some sense positive results. And sometimes positive
results might be of little physical interest. Even if they come in a
totally constructive form, with all desirable error or remainder
estimates as they always come out of perturbation theoretic approaches.
Nevertheless perturbation theory is usually non trivial,
and I will illustrate some of its achievements which I find
remarkable not only for their intrinsic interest but for the surprising
unity of methods of solution of different problems, that they make evident.
Another legacy that we shall certainly leave to the next century is our
unwillingness of considering seriously the work done in other fields even
inside the domain of Mathematical Physics. Hence we shall certainly not be
able to find a way to avoid that results found in some area will reappear,
derived under different words but in essentially identical manner, or
regarded as open problems in other areas (or even in the same area): I am
not thinking here of plagiarism (which is of little interest except from a
moral point of view, totally ridiculous in our present world except on
individual grounds), but just of honest ignorance. I am always surprised
when I meet such cases: and I am afraid that I have myself shown this
behaviour. But this might in fact be a "good thing": it is more likely that
the results will not be forgotten if they are duplicated enough many times:
the main works of greek science were lost because there were not enough
copies of them.
%
\*\*
{\bf\S3 A review of perturbation theory methods.}
\*
Perturbation theory arises when one studies a problem dependent on a
parameter $\e$, and for $\e=0$ the problem is exactly soluble (at least in
some aspects). Until recently a perturbation treatment meant finding a
series in $\e$ convergent or asymptotic for small $\e$, whose sum was equal
or asymptotic to a quantity of predeclared interest, for $\e$ small.
The simplest examples of perturbation theory dealt with the problems of
celestial mechanics and in the last century not much attention was devoted
to the actual convergence of the series, the main problem being whether
they could be defined as formal power series (\ie their coefficients were
well defined). In fact Poincar\`e proved that some of the most used series
in celestial mechanics could not possibly be convergent or even defined to
all orders and that they were, at best, asymptotic (see [G2]).
The convergence problem never seemed to bother the theoretical physicists:
perhaps because good enough practical results seemed to come out of the
series, if regarded as asymptotic. Or perhaps because the series met in the
theory of atoms were actually convergent, (see [K],[RS],[T]).
The question of convergence of the series started playing a role when
it was discovered that the series expansions for quantum field theory
were not even well defined, at least not from a naive viewpoint.
Renormalization theory provided a way to define them properly, but it
also generated the puzzle of how proper the definition really was:
whether it was a fundamental definition or just an arbitrary
prescription. This was a particularly pertinent question as it was in
any event clear that in most cases the series could not be convergent
but at best they could be asymptotic.
The first series to be studied was precisely that arising in the theory
of perturbations of integrable hamiltonian systems: this led to the
theorem of Kolmogorov on quasi periodic hamiltonian motions.
In the same years the virial series for statistical mechanics systems with
short range forces was proved to converge for small enough density (Morrey,
[Mo], Groeneveld, Penrose, Ruelle, [R]: independently in an arc of a
decade).
In my opinion the latter was the key stone for the future developments:
it was, I remember, for many a great surprise that such series
describing the behaviour of systems with infinitely many degrees of
freedom could be "proved" convergent. The technique soon found
applications in other fields, for instance to the theory of phase
transitions and to the study of the pure phases and of phase coexistence
(started by Dobrushin, Minlos, Sinai, [S]).
At the end of the sixties the situation was mature for attempting at the
proof of convergence, or asymptoticity, of the formal series generated
by renormalization theory in quantum field theory, say for the
computation of the Schwinger functions. The work on the foundations had
permitted to define precisely the quantities to be studied. The series
giving formally their values had been mathematically proven to be well
defined, at least in simple cases, (Hepp, [H]): a fact that the
discoverers of renormalization theory certainly had clear enough to
consider a formal proof unnecessary. The asymptoticity proof came from
the work by Glimm, Jaffe, Spencer, [E], and it was achieved by showing
the convergence of another series in terms of {\it functions} of the
expansion parameter $\e$, which were singularly depending on $\e$, but
in a controlled way. The difficult part was to set up the appropriate
convergent expansion, whose convergence turned out to be, essentially, a
consequence of the same ideas leading to the convergence of the virial
series.
{\it I will never understand why such result, which among other things
solved the question of whether a relativistic quantum theory with non
trivial $S$--matrix was possible, \ie showed the compatibility of non
trivial quantum mechanics and special relativity (at least for some
models and in ``low'' space time dimension, (\ie $3$)), has been very often
ignored and sometimes considered irrelevant: the introduction of an
important later paper provides us with a rather typical example, see
\rm [PW].}
The above work was the last major one before the new ideas generated by
the renormalization group started playing a role, [Po],[G3]; but a
typical tool of the renormalization group is already present: \ie the
proof of asymptoticity (or convergence) of a perturbation series by
checking convergence of series in other quantities with controlled
(albeit singular) analyticity properties in terms of the expansion
parameter $\e$.
This method can be called "proof of asymptoticity by resummations" and
is a classic mathematical method. What is not classic and non trivial is
its combination with the theory of convergence of the virial series,
known nowadays as {\it cluster expansion}, [E],[G2].
The cluster expansion led also to the solution of some old problems, like
the proof of existence of the Debye screening in classical systems
interacting at large distance by an electrostatic Coulomb potential
(Brydges, [Br]).
Thus it is interesting to see that the resummation techniques could all
be considered part of the same basic idea rooted in the {\it
renormalization group} approach to the critical point in statistical
mechanics. The approach is particularly interesting as it shows that the
really basic idea of {\it asymptotic freedom} which is one of the really
new ingredients in the theory of renormalization and which had escaped
to the founding fathers, is the idea which allows us to build the
rigorous proofs of asymptoticity of the perturbation expansion of
various field theories.
In fact the renormalization group can be viewed as a method for resumming
formal series: this was realized through various works on the rigorous
mathematical meaning of the renormalization group "calculations" (see the
review papers [CE], [Po], [G3], [GK1]) in which the "old" results were
rederived and a host of new ones (among them see [FS]: for the Anderson
localization problem,
%
\footnote{${}^3$}{\nota a remarkable new method, non perturbative, for
this theory has been presented at this conference, see [AM].}
%
[GK2] for the theory of the Gross--Neveu model, [BGPS] for the theory of
the ground state of spin $0$ one dimensional Fermi gases; just to
mention a few among them). Other applications are promising to come, see
[A],[Sh],[Po2],[W2]. All the above problems have one feature in common:
namely they have some scaling property playing a key role. However a
fully unified treatment is not yet available, although it seems that we
possess all the necessary knowledge.
\*\*
%
{\bf\S4 An example of modern perturbation theory: twistless invariant
tori in hamiltonian mechanics.}
%
\*\numsec=4\numfor=1
%
The resummation is not always possible, or not always known: in particular
the asymptoticity of the (well defined) perturbation expansions cannot be
treated rigorusly in important cases such as $4$ dimensional field theories
like quantum electrodynamics or the standard model of elementary particles.
More or less hidden scaling properties are the origin of similarities
of problems of perturbation theory in statistical mechanics and field
theory and other apparently unrelated problems like the Feigenbaum
cascades in the theory of maps of the interval (see [F], [CE2], [ER]), or
the KAM theory or the invariant tori breakdown theory of Kadanoff, Mc
Kay, Aubry, Mather, ([Mc],[AL],[M]). Or the theory of regularity of the
solutions of the Navier Stokes equations in $3$ dimensions, see the
review [G4].
I will describe here the connection between the Kolmogorov theorem on the
existence of quasi periodic motions and the resummations of perturbation
expansions used in quantum field theory. The basic idea goes back to
Eliasson: the idea has a particularly simple application as the series
studied turns out to be a convergent series because of some remarkable
cancellations. Although a part of the problem is therefore trivial (the
series being convergent rather than asymptotic) what is left in non trivial
enough to be interesting and to show the relation with the renormalization
group methods in quantum field theory.
The connection might have been suspected already from the relation
between the KAM theory and the renormalization group methods, pointed
out in various independent works ([G2], [Mc], [ED]). I find it
nevertheless quite surprising that the KAM problem can in fact be
formulated as a field theory problem, in which interesting scaling
properties appear, and treated as such.
The explanation of the field theory model given below requires, to be
understood, some familiarity with the theory of the dipole gas and of
the sine gordon model. But the analysis following it is an independent
and self contained proof of the KAM theorem, not requiring any knowledge
of field theory or statistical mechanics. The field theory model linked
to the KAM theorem was pointed out to me by G. Parisi, who showed me
what I should have done in my attempts to interpret field
theoretically my version of Eliasson's ideas.
To establish a connection with field theory consider first a vector
field $\V F_\pps$, with zero average, defined on the torus $T^l$ with free
propagator given by the operator $(\oo_0\cdot\Dpr)^2$, instead of the
usual laplacian, with $\oo_0\in R^l$ being a vector verifying a strong
diophantine condition, see below.
Consider the function of the field corresponding to the "action potential"
$V(\V F)=\e\ig f(\pps+\V F_\pps)\,d\pps$, where the $f$ is an even
trigonometric polynomial.
The partition function $Z$ is then formally expressed in terms of the
gaussian process $P(d\V F)$ generated by the differential operator
$(\oo_0\cdot\V\dpr)^2$ on the functions $\V F_\pps$ with zero average:
it is the functional integral of the functional $\exp V(\V F)$.
And the Schwinger function $\V h(\pps)= \media{\V F_\pps}$ is the
formal average of $\V F_\pps$ with respect to the measure $Z^{-1}P(d
\V F) e^{V(\V F)}$.
I shall show below that the evaluation of $\V h(\pps)$ via perturbation
theory, if performed {\it neglecting all Feynman diagrams which are nor
tree diagrams}, gives exactly the perturbation series solution to the
problem of finding an invariant KAM torus for the model considered below,
(that I call Thirring model, or rotators model, see \equ(2) below).
It was pointed out to me by G. Parisi that the approximation to $\V
h(\pps)$ obtained by evaluating $\V h(\pps)$ via a perturbation expansion in
$\e$, in which only the Feynman diagrams with tree structure (\ie no loops)
are retained, yields the solution to the equation: $(\oo_0\cdot\Dpr)^2\V
h=-\e\V f'(\pps+\V h(\pps))$, where $\V f'$ denotes the gradient of the
function $f$. Below I show that the latter equation is the equation that
has to be solved to determine the KAM tori for the model introduced in the
following equation (1). Hence a field theoretic interpretation of the KAM
theorem will easily follow. A further connection with field theory was
pointed out to me by A. Berretti who remarked that the tree approximation
is exact in mean field theory: therefore the model provided by Parisi
solved in the mean field approximation gives rise to a non trivial equation
whose solution is equivalent to the Thirring model.
The above formal field theoretic interpretation is interesting, but it
has the drawback of being "an approximation" and one can ask whether one
can find a field theory whose one point Schwinger function is exactly, at
least formally, the function $\V h(\pps)$ describing the invariant tori
of the Thirring model.
This can be done as follows (a procedure probably known in field
theory): let $\V F^\pm_\pps$ be two complex vector fields, with zero
average on the torus $T^l$, and suppose that their free propagator is
$\media{F^+_{i,\pps}F^+_{j,\pps'}}=0=\media{F^-_{i,\pps}F^-_{j,\pps'}}$
while $\media{F^+_{i,\pps}F^-_{j,\pps'}}=\d_{ij}S(\pps-\pps')$ where $S$
has Fourier transform equal to $(\oo_0\cdot\nn)^{-2}$. The fields $\V
F^\pm$ can be realized by considering two independent real vector fields
$\V \F^q$, $q=1,2$, with free propagator $\d_{ij} S(\pps-\pps')$. One
simply sets: $\V F^\pm=\V \F^1\pm i\V\F^2$.
Then consider the Schwinger function:
%
$$\V h(\pps)=\fra{\ig P(d\V F) \,\V F^+_\pps\,
e^{\e\ig\V F^-_{\pps'}\cdot\Dpr
f(\pps'+\V F^+_{\pps'})\,d\pps'}} {\ig P(d\V F)\,
e^{\e\ig\V F^-_{\pps'}\cdot\Dpr f(\pps'+\V F^+_{\pps'})\,d\pps'}}
\Eq(1)$$
%
and it is easy to check that the linearity in $\V F^-$ of the potential
implies that only the tree diagrams of the perturbation expansion of
\equ(1) do not vanish. And the tree diagrams {\it have the same value} as
the ones of the previously considered, single field $\V F$, field theory.
Hence $\V h(\pps)$ is a (formal) sum of the tree diagrams for \equ(1).
One should not think that the functional integrals in \equ(1) are
easy to define rigorously. In fact the fields $\V F^\pm$ are complex
valued and therefore the argument of the exponentials is unbounded: so
that summability is by no means obvious. The KAM theorem discussed below
can be interpreted as a theory of the above functional integral and as a
proof of the convergence of the perturbation expansion for it.
To avoid references to field theory I begin by formulating from scratch
the problem of the KAM theorem version that I will discuss. It will be
a particularization of the Eliasson method, [El], for KAM tori to a
special model: the Thirring model. This is a system of rotators
interacting via a potential. It is described by the hamiltonian:
%
%
$$\fra12 J^{-1}\AA\cdot\AA\,+\,\e f(\aa)\Eq(2)$$
%
where $J$ is the (diagonal) matrix of the inertia moments,
$\AA=(A_1,\ldots,A_l)\in R^l$ are their angular momenta and
$\aa=(\a_1,\ldots,\a_l)\in T^l$ are the angles describing their
positions: the matrix $J$ will be supposed non singular; but we only
suppose that $\min_{j=1,\ldots,l}J_j=J_0>0$, and {\it no assumption} is
made on the size of the {\it twist rate} $T=\min J_j^{-1}$: the results
will be uniform in $T$ (hence the name ``twistless'' that can be given
to the above tori). We
suppose $f$ to be an even trigonometric polynomial of degree $N$:
%
$$f(\aa)=\sum_{0<|\nn|\le N} f_\nn\,\cos\nn\cdot\aa, \qquad
f_\nn=f_{-\nn}\Eq(3)$$
%
We shall consider a "rotation vector" $\oo_0=(\o_1,\ldots,\o_l)\in R^l$
verifying a {\it strong diophantine property}
with dophantine constants $C_0,\t,\g,c$; this means that:
%
$$\eqalign{
1)\kern1.truecm&
C_0|\oo_0\cdot\nn|\ge |\nn|^{-\t},\kern3.5cm\V0\ne\nn\in Z^l\cr
2)\kern1.truecm&
\min_{0\ge p\ge n}\big|C_0|\oo_0\cdot\nn|-\g^{p}\big|>\g^{n+1}\qquad
{\rm if}\ n\le0,\ 0<|\nn|\le (\g^{n+c})^{-\t^{-1}}\cr}
\Eq(4)$$
%
and it is easy to see that the {\it strongly diophantine vectors} have
full measure in $R^l$ if $\g>1$ and $c$ are fixed and if $\t$ is fixed
$\t>l-1$: we take $\g=2,c=3$ for simplicity; note that 2) is empty if
$n>-3$ or $p1$:
%
$${\V\o}_0\cdot\V\dpr\,H^{(k)}_j=-
\sum_{m_1,\ldots,m_l\atop|\mm|>0}\fra1{\prod_{s=1}^l m_s!}
\dpr_{\a_j}\, \dpr^{m_1+\ldots+m_l}_{\a_1^{m_1}\ldots\a_l^{m_l}} f(\oo_0
t)\cdot{\sum}^*\prod_{s=1}^l\prod_{j=1}^{m_s} h^{(k^s_j)}_s(\oo_0 t)\Eq(7)$$
%
where the $\sum^*$ denotes summation over the integers $k^s_j\ge1$
with: $\sum_{s=1}^l\sum_{j=1}^{m_s}k^s_j=k-1$.
The trigonometric polynomial $\V h^{(k)}(\pps)$ will be completely
determined (if possible at all) by requiring it to have $\V0$ average
over $\pps$, (note that $\V H^{(k)}$ has to have zero average over
$\pps$). For $k=1$ one easily finds:
%
$$\tst\V h^{(1)}(\pps)=-\sum_{\nn\ne\V0}
\fra{iJ^{-1}\nn}{(i\oo_0\cdot\nn)^2}f_\nn\,e^{i\nn\cdot\pps}\Eq(8)$$
%
Suppose that $\V h^{(k)}(\pps)$ is a trigonometric polynomial of degree
$\le k N$, odd in $t$, for $1\le k< k_0$. Then we see immediately that
the r.h.s. of \equ(7) is odd in $t$. This means that the r.h.s. of
\equ(7) has zero average in $t$, hence in $\pps$, and the second of
\equ(7) can be solved for $k=k_0$. It yields an even function $\V
H^{(k_0)}(\pps)$ which is defined up to a constant which, however, must
be taken such that $\V H^{(k_0)}(\pps)$ has zero average, to make
$\oo\cdot\Dpr h^{(k)}_j=J_j^{-1} H^{(k)}_j$ soluble. Hence the equation
for $\V h^{(k)}$ can be solved (because the r.h.s. has zero average)
and its solution is a trigonometric polynomial in $\pps$, odd if $\V
h^{(k)}$ is determined by imposing that its average over $\pps$
vanishes.
Hence the \equ(8) provide an algorithm to evaluate a formal power series
solution to our problem. It has been remarked, [El], see also [G1],
that \equ(7) yields a {\it diagrammatic expansion} of $\V h^{(k)}$. We
simply "iterate" it until only $h^{(1)}$, given by \equ(8), appears.
Let $\th$ be a tree diagram: it will consist of a family of "lines" (\ie
segments) numbered from $1$ to $k$ taken from a ``box'' containing $k$
lines with a label, {\it number label}, going from $1$ to $k$ (\ie
distinguishable), arranged to form a (rooted) tree diagram as in the
figure:
%
\insertplot{240pt}{170pt}{%gvnn.tex
\ins{-35pt}{90pt}{\it root}
\ins{25pt}{110pt}{$j$}
\ins{60pt}{85pt}{$v_0$}
\ins{55pt}{115pt}{$\nn_{v_0}$}
\ins{115pt}{132pt}{$j_{1}$}
\ins{152pt}{120pt}{$v_1$}
\ins{145pt}{155pt}{$\nn_{v_1}$}
\ins{110pt}{50pt}{$v_2$}
\ins{190pt}{100pt}{$v_3$}
\ins{230pt}{160pt}{$v_5$}
\ins{230pt}{120pt}{$v_6$}
\ins{230pt}{85pt}{$v_7$}
\ins{230pt}{-10pt}{$v_{11}$}
\ins{230pt}{20pt}{$v_{10}$}
\ins{200pt}{65pt}{$v_4$}
\ins{230pt}{65pt}{$v_8$}
\ins{230pt}{45pt}{$v_9$}
}{gvnn}
\kern1.3cm
\didascalia{fig. 1: A tree diagram $\th$ with
$m_{v_0}=2,m_{v_1}=2,m_{v_2}=3,m_{v_3}=2,m_{v_4}=2$ and $m=12$,
$\prod m_v!=2^4\cdot6$, and some labels. The line numbers,
distinguishing the lines, are not shown.}
To each vertex $v$ we attach a "mode label" $\nn_v\in Z^l,\,|\nn_v|\le N$
and to each branch leading to $v$ we attach a "branch label"
$j_v=1,\ldots,l$. The order of the diagram will be $k=$ number of vertices
$=$ number of branches (the tree root will not be regarded as a vertex).
We imagine that all the diagram lines have the same length (even though
they are drawn with arbitrary length in fig. 1). A group acts on the
set of diagrams, generated by the permutations of the subdiagrams having
the same vertex as root. Two diagrams that can be superposed by the
action of a transformation of the group will be regarded as identical
(recall however that the diagram lines are numbered, \ie are regarded as
distinct, and the superpositon has to be such that all the labels of the
diagram match: \ie the branch label, the mode label {\it and} the number
label). Trees diagrams are regarded as partially ordered sets of
vertices (or lines) with a minimal element given by the root (or the
root line): as usual the order relation will be denoted $\le$ and, in
general, not all pairs of vertices will be comparable. We shall imagine
that each branch carries also an arrow pointing to the root (``gravity''
direction, opposite to the order).
We define the "momentum" entering $v$ as $\nn(v)=\sum_{w\ge v}\nn_w$. If
from a vertex $v$ emerge $m_1$ lines carrying a label $j=1$, $m_2$
lines carrying $j=2$, $\ldots$, it follows that \equ(7) can be
rewritten:
%
$$\V h^{(k)}_{\nn j}=\fra1{k!}
{\sum}^*\prod_{v\in\th}\fra{(-i J^{-1}\nn_v)_{j_v}\,
f_{\nn_v}\prod_{s=1}^l(i\nn_v)^{m_s}_s}{(i\oo_0\cdot\nn(v))^2}
\Eq(9)$$
%
with the sum running over the diagrams $\th$ of order $k$ and with
$\nn(v_0)=\nn$; and the combinatorics can be checked from \equ(7), by
taking into account that we regard the diagram lines as all different
(to fix the factorials). Basically the introduction of the number labels
increases the number of trees, but at the same time they acquire all the
same combinatorial weight $k!^{-1}$, instead of the more complicated
$\prod m_v!^{-1}$ that one would expect from \equ(7).
The ${}^*$ recalls that the diagram $\th$ can and will be supposed such
that $\nn(v)\ne\V0$ for all $v\in\th$ (by the above remarked parity
properties). Note that \equ(9) is implied by the corresponding (6.23)
of [G1]: one can check that the two formulae coincide (by summing over
what in [G1] are called the ``fruit values''). The theory in [G1] is in
fact a little more general, although it is really applied to the same
Thirring model.
There are other diagrams, however, which we would like to eliminate.
They are the diagrams with nodes $v',v$, with $v'0$.
Denoting $T$ a cluster of scale $n$ let $m_T$ be the number of
resonances of scale $n$ contained in $T$ (\ie with incoming lines of
scale $n$), we have the following inequality, valid for any diagram
$\th$:
%
$$N_n\le\fra{4k}{E\,2^{-\e n}}+\sum_{T, \,n_T=n}(-1+m_T)\Eq(10)$$
%
with $E=N^{-1}2^{-3\e},\e=\t^{-1}$. This ``combinatorial'' inequality is
a version of Brjuno's lemma: a proof can be found in [G1].
Consider a diagram $\th^1$ we define the family $\FF(\th^1)$ generated
by $\th^1$ as follows. Given a resonance $V$ of $\th^1$ we detach the
part of $\th^1$ above $\l_V$ and attach it successively to the points
$w\in\tilde V$, where $\tilde V$ is the set of vertices of $V$
(including the endpoint $w_1$ of $\l_V$ contained in $V$) outside the
resonances contained in $V$. Note that all the lines $\l$ in $\tilde V$
(\ie contained in $V$ and with at least one point in $\tilde V$)
have a scale $n_\l\ge n_V$.
For each resonance $V$ of $\th^1$ we shall call $M_V$ the number of
vertices in $\tilde V$. To the just defined set of diagrams we add the
diagrams obtained by reversing simoultaneously the signs of the vertex
modes $\nn_w$, for $w\in \tilde V$: the change of sign is performed
independently for the various resonant clusters. This defines a family
of $\prod 2M_V$ diagrams that we call $\FF(\th_1)$. The number
$\prod 2M_V$ will be bounded by $\exp\sum2M_V\le e^{2k}$.
It is important to note that the definition of resonance is such that
the above operation (of shift of the vertex to which the line entering
$V$ is attached) does not change too much the scales of the diagram
lines inside the resonances: the reason is simply that inside a
resonance of scale $n$ the number of lines is not very large being
$\le\lis N_n\=E\,2^{-n\e}$.
Let $\l$ be a line, in a cluster $T$, contained inside the resonances
$V=V_1\subset V_2\subset\ldots$ of scales $n=n_1>n_2>\ldots$; then the
shifting of the lines $\l_{V_i}$ can cause at most a change in the size
of the propagator of $\l$ by at most $2^{n_1}+2^{n_2}+\ldots< 2^{n+1}$.
Since the number of lines inside $V$ is smaller than $\lis N_n$ the
quantity $\oo\cdot\nn_\l$ of $\l$ has the form
$\oo\cdot\nn^0_\l+\s_\l\oo\cdot\nn_{\l_V}$ if $\nn^0_\l$ is the momentum
of the line $\l$ "inside the resonance $V$", \ie it is the sum of all
the vertex modes of the vertices preceding $\l$ in the sense of the
line arrows, but contained in $V$; and $\s_\l=0,\pm1$.
Therefore not only $|\oo\cdot\nn^0_\l|\ge 2^{n+3}$ (because $\nn^0_\l$
is a sum of $\le \lis N_n$ vertex modes, so that $|\nn^0_\l|\le N\lis
N_n$) but $\oo\cdot\nn^0_\l$ is "in the middle" of the diadic interval
containing it and by \equ(4) does not get out of it if we add a quantity
bounded by $2^{n+1}$ (like $\s_\l\oo\cdot\nn_{\l_V}$). Hence no line
changes scale as $\th$ varies in $\FF(\th^1)$, if $\oo_0$ verifies \equ(4).
{\it This implies, by the strong diophantine hypothesis on $\oo_0$,
\equ(4), that the resonant clusters of the diagrams in $\FF(\th^1)$ all
contain the same sets of lines, and the same lines go in or out of each
resonance (although they are attached to generally distinct vertices
inside the resonances: the identity of the lines is here defined by the
number label that each of them carries in $\th^1$). Furthermore the
resonance scales and the scales of the resonant clusters, and of all the
lines, do not change.}
Let $\th^2$ be a diagram not in $\FF(\th^1)$ and construct
$\FF(\th^2)$, \etc. We define a collection
$\{\FF(\th^i)\}_{i=1,2,\ldots}$ of pairwise disjoint families of
diagrams. We shall sum all the contributions to $\V h^{(k)}$ coming
from the individual members of each family. This is the {\it
Eliasson's resummation}.
We call $\e_V$ the quantity $\oo\cdot\nn_{\l_V}$ associated with the
resonance $V$. If $\l$ is a line in $\tilde V$, see above, we can imagine to
write the quantity $\oo\cdot\nn_\l$ as $\oo\cdot\nn^0_\l+\s_\l\e_V$, with
$\s_\l=0,\pm1$. Since $|\oo\cdot\nn_\l|> 2^{n_V-1}$ we see that the
product of the propagators is holomorphic in $\e_V$ for
$|\e_V|<2^{n_V-3}$.
%
\footnote{${}^2$}{\nota In fact
$|\oo\cdot\nn^0_\l|\ge 2^{n+3}$ because $V$ is a resonance; therefore
$|\oo\cdot\nn_\l|\ge 2^{n+3}-2^{n+1}\ge 2^{n+2}$ so that $n_V\ge n+3$.
On the other hand note that $|\oo\cdot\nn^0_\l|> 2^{n_V-1}-2^{n}$ so
that $|\oo\cdot\nn_\l^0+\s_\l\e_V|\ge 2^{n_V-1}-2^{n}-2^{n_V-3}\ge
2^{n_V-1}-2^{n_V-3}\ge 2^{n_V-2}$, for $|\e_V|< 2^{n_V-3}$.}
%
While $\e_V$ varies in such complex disk the quantity
$|\oo\cdot\nn_\l|$ does not become smaller than $2^{n_V-1}-
2\,2^{n_V-3}\ge2^{n_V-2}$. Note the main point here: the quantity
$2^{n_V-3}$ will usually be $\gg 2^{n_{\l_V}}$ which is the value
$\e_V$ actually can reach in every diagram in $\FF(\th^1)$; this can be
exploited in applying the maximum priciple, as done below.
It follows that, calling $n_\l$ the scale of the line $\l$ in $\th^1$,
each of the $\prod 2 M_V\le e^{2k}$ products of propagators
of the members of the family $\FF(\th^1)$ can be bounded above by
$\prod_\l\,2^{-2(n_\l-2)}=2^{4k}\prod_\l\,2^{-2n_\l}$, if regarded as a
function of the quantities $\e_V=\oo\cdot\nn_{\l_V}$, for $|\e_V|\le
\,2^{n_V}$, associated with the resonant clusters $V$. This even
holds if the $\e_V$ are regarded as independent complex parameters.
By construction it is clear that the sum of the $\prod 2M_V\le e^{2k}$
terms, giving the contribution to $\V h^{(k)}$ from the trees in
$\FF(\th^1)$, vanishes to second order in the $\e_V$ parameters (by the
approximate cancellation discussed above). Hence by the maximum
principle, and recalling that each of the scalar products in \equ(9) can
be bounded by $N^2$, we can bound the contribution from the family
$\FF(\th^1)$ by:
%
$$\left[\fra1{k!}\Big(\fra{f_0 C_0^2 N^2}{J_0}\Big)^k 2^{4k} e^{2k}
\prod_{n\le0}2^{-2nN_n}\right]\left[\prod_{n\le0}\prod_{T,\,n_T=n}
\prod_{i=1}^{m_T}\,2^{2(n-n_{i}+3)}\right]\Eq(11)$$
%
where:
%
\acapo
1) $N_n$ is the number of propagators of scale $n$ in $\th^1$ ($n=1$
does not appear as $|\oo\cdot\nn|\ge1$ in such cases),\acapo
2) the first square bracket is the bound on the product of
individual elements in the family $\FF(\th^1)$ times the bound $e^{2k}$
on their number,
%
\acapo
3) The second term is the part coming from the maximum principle, applied
to bound the resummations, and is explained as follows.
%
\acapo
i) the dependence on the variables $\e_{V_i}\=\e_i$ relative to
resonances $V_i\subset T$ with scale $n_{\l_{V_i}}=n$ is holomorphic for
$|\e_i|<\,2^{ n_i-3}$ if $n_i\=n_{V_i}$, provided $n_i>n+3$ (see above).
\acapo
%
ii) the resummation says that the dependence on the $\e_i$'s has a
second order zero in each. Hence the maximum principle tells us that
we can improve the bound given by the first factor in \equ(11) by the
product of factors $(|\e_i|\,2^{-n_i+3})^2$ if $n_i>n+3$. If $ n_i\le
n+3$ we cannot gain anything: but since the contribution to the bound
from such terms in \equ(11) is $>1$ we can leave them in it to simplify
the notation, (of course this means that the gain factor can be
important only when $\ll1$).
Hence substituting \equ(10) into \equ(11) we see that the $m_T$ is taken
away by the first factor in $\,2^{2n}2^{-2n_{i}}$, while the remaining
$\,2^{-2n_i}$ are compensated by the $-1$ before the $+m_T$ in \equ(10),
taken from the factors with $T=V_i$, (note that there are always enough
$-1$'s).
So that the product \equ(11) is bounded by:
%
$$\fra1{k!}\,(C_0^2J_0^{-1}f_0 N^2)^k e^{2k}2^{4k}2^{6k}
\prod_{n=-\io}^0\,2^{-4 n k E^{-1}\,2^{\e n}}\le \fra1{k!}\, B_0^k\Eq(12)$$
%
with $B_0$ suitably chosen.
To sum over the trees we note that, fixed $\th$ the collection of
clusters is fixed. Therefore we only have to multiply \equ(12) by the
number of diagram shapes for $\th$, ($\le 2^{2k}k!$), by the number of
ways of attaching mode labels, ($\le (3N)^{lk}$), so that we can bound
$|h^{(k)}_{\nn j}|$ by \equ(6).
\*
{\it Remark:}
It is interesting to remark that we know, from the proof of [Th] of the KAM
theorem for the model \equ(2) (for instance, and most directly, see [Th]),
that there is a canonical transformation with generating function having the
form $\F(\AA',\aa)=(\AA'-\AA_0)\cdot\aa+(\AA'-\AA_0)\cdot \V
g(\aa)+\g(\aa)$ with suitable analytic functions $\V g(\aa)$ and $\g(\aa)$,
changing coordinates from $(\AA,\aa)$ to $(\AA',\pps)$, and giving the
invariant torus, \equ(5), the form:
%
$$\AA=\AA_0+\Dpr_\aa \g(\aa),\kern1.5cm \pps=\aa+\V g(\aa)\Eq(13)$$
%
Comparing with \equ(5), and with the equation $(\oo_0\cdot\Dpr_\pps)^2\V
h(\pps)=-\e\Dpr_\aa f (\pps+\V h(\pps))$ (meaning that \equ(5) is an
invariant torus and generating \equ(7)) we see that $\V g(\aa)=-\V
h(\pps)$ and $\Dpr_\aa \g(\aa)=\V H(\pps)=(\oo_0\cdot\Dpr_\pps) \,J\V
h(\pps)$ if $\aa=\pps+\V h(\pps)$.
This implies that $\Dpr_\aa \g(\aa)=-\e(\oo_0\cdot\Dpr_\pps)^{-1}
\Dpr_\aa f(\aa)$, if $\aa=\pps+\V h(\pps)$,
and it would be nice to see which identities among
tree values are equivalent to the last relation (\ie to the identity of
the second order cross derivatives of $\g(\aa)$).
%
\*
{\bf\S5 Some problems that will be left over.}
%
\*
The following are a few problems that have interested me and many of us
which I believe (having no reason to believe the contrary) will be left
over to the next century:
1) develop a perturbation theory that applies to the simplest dynamical
problems in statistical mechanics: like a perturbation expansion in the
density of the transport coefficients. Only "recently" it has been
realized that the classical expansions were simply not well defined, (\ie
had infinite coefficients), if the computation was attempted and resummation
schemes were proposed, [CD].
2) develop a theory of the (incompressible) Navier Stokes equation, like
some constructive theorem for its solution in $d=3$ dimensions, see
[G4]. This seems to be the same as understanding a way to extend the
theory of the $d=2$ case, based on the vorticity conservation. No
mathematical theory of the $d=3$ Navier Stokes equation incorporates the
fact that, for $0$ viscosity, there should be vorticity conservation in
the form of Thomson's theorem. The best known results, [CKN], from the
point of view of existence and regularity are based only on the energy
conservation: or more mathematically on the equation obtained by taking
the scalar product of the velocity field with the difference of the two
sides of the equation, and setting this equal to $0$ (with the pressure
being given by the well known formula): \ie on just too little
information, see also [G4].
3) is it possible to construct (\ie prove existence of) a relativistic
scalar field theory in dimension $4$ which has an asymptotic series for
the Schwinger functions given by the renormalized series for the $\l \f^4$
model? see [GR].
4) can one understand the Bose condensation? definition, existence and
basic properties, see [ADG].
5) can a Fermi liquid be normal at $T=0$, in $2$ or more dimensions? see
[A]. In fact definition, existence, and basic properties of the ground
state of Fermi systems are poorly understood, particularly if $d\ge2$.
6) can one show the existence of a liquid solid phase transition in
statistical mechanics one component systems interacting via short range
translation and rotation invariant interactions?
7) can developed, stationary, turbulence be understood more
quantitatively in terms of the observed scaling properties and in
conformity with the phenomenological theory of Kolmogorov?
\penalty-200
\*\*
{\bf Acknowledgements}: I am grateful to the organizing committe for the
invitation to participate to the meeting and to contribute a paper for the
proceedings. I thank A. Berretti and G. Parisi for the suggestions
described above. And G. Benfatto, S. Miracle Sol\'e, G. Gentile for
their critical comments on \S4.
\*
\penalty-200
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\vskip0.5truecm
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\ciao
ENDBODY