% This file was generated from
% trial5 f1.eps f2.eps f3.eps f4.eps f5a.eps f5b.eps f6.eps f7.eps f8.eps f9.eps by /public/bin/texpsinclude.
% This file contains included PostScript.
% TeX writes out the included PostScript to files.

% Here are the macros for writing out the included PostScript:
% Macros for dumping included Postscript to files.
% Requires Plain TeX.  Maybe other flavors will work too?
% Jamie Stephens, jamies@math.utexas.edu, 28 Nov 94

% If you're in the UT Math Department, see
% /usr/local/doc/tex/texpsinclude.text and /public/bin/texpsinclude.
% Parts of these files are attached at the end of this file.

% Adapted from Knuth's \answer macro in the TeXbook.

\def\endofps{EndOfTheIncludedPostscriptMagicCookie}
\chardef\other=12
\newwrite\psdumphandle 
\outer\def\psdump#1{\par\medbreak
  \immediate\openout\psdumphandle=#1
  \copytoblankline}
\def\copytoblankline{\begingroup\setupcopy\copypsline}
\def\setupcopy{\def\do##1{\catcode`##1=\other}\dospecials
  \catcode`\\=\other \obeylines}
{\obeylines \gdef\copypsline#1
  {\def\next{#1}%
  \ifx\next\endofps\let\next=\endgroup %
  \else\immediate\write\psdumphandle{\next} \let\next=\copypsline\fi\next}}
\outer\def\closepsdump{
  \immediate\closeout\psdumphandle}

%% 
%% Here's a shell script for automating the use of psdump.tex
%% 

% #!/bin/bash
% 
% PSDUMP=/usr/local/tex/macros/local/psdump.tex
% TEXPSINCLUDEHELP=/usr/local/doc/tex/texpsinclude.help
% 
% if [ $# -eq 0 ]; then
%   echo "Usage: $0 file.tex {files.ps} > output.tex" >&2
%   echo "See $TEXPSINCLUDEHELP for more information." >&2
%   exit 1
% fi
% 
% echo "% This file was generated from"
% echo "% $@ by $0."
% echo "% This file contains included PostScript."
% echo "% TeX writes out the included PostScript to files."
% echo 
% echo "% Here are the macros for writing out the included PostScript:"
% cat $PSDUMP
% echo
% 
% for argument in $@; do
%   if [ $argument != $1 ]; then
%     echo "% Here's the Postscript for $argument:"
%     echo "\message{Writing file $argument}"
%     echo -n "\psdump{$argument}"
%     echo "Including $argument." >&2
%     cat $argument
%     echo "EndOfTheIncludedPostscriptMagicCookie"
%     echo
%     echo "\closepsdump"
%   fi
% done
% 
% echo 
% echo "% Finally, here is $1:"
% 
% cat $1

%%
%% Here's some documentation:
%%

% This file documents how to use the texpsinclude command in /public/bin.

% Here's the problem that texpsinclude solves:

% You have a TeX file called foo.tex that you want to distribute as a
% single TeX file.  The problem is that foo.tex needs two Postscript
% files, bar1.ps and bar2.ps, for embedded figures.  You'd like a single
% TeX file which somehow includes bar1.ps and bar2.ps.  When TeX
% processes foo.tex, TeX should extract bar1.ps and bar2.ps from
% foo.tex.  

% Here's how to do what you want to do:

%  texpsinclude foo.tex. bar1.ps bar2.ps > bigfoo.tex

% (In general: "texpsinclude <texfile> <psfiles> > <outfile>".)  If you
% enter this command, the result is a new file called bigfoo.tex.  The
% file bigfoo.tex contains bar1.ps and bar2.ps.  If you give bigfoo.tex
% to a friend, she can make your document with:

%  tex bigfoo.tex

% This command writes out bar1.ps and bar2.ps, and the command also
% TeX's foo.tex.

% Note: If *you* run tex on bigfoo.tex in the same directory, then TeX
% will write over your .ps files.  Be careful when testing your
% bigfoo.tex.

%% 
%% Here's another way to use psdump.tex
%%

% EXAMPLE (remove the leading % signs to make it work):
%
%\psdump{example.ps}These three lines
%are going be dumped "as is"
%to the file example.ps
%EndOfTheIncludedPostscriptMagicCookie
%\closepsdump

% Here's the Postscript for f1.eps:
\message{Writing file f1.eps}
\psdump{f1.eps}%!PS-Adobe-2.0 EPSF
%%Title: /tmp/xfig-fig004210
%%Creator: fig2dev
%%CreationDate: Tue Jul 23 16:02:35 1996
%%For: panos@linux0 (Panayotis Panayotaros,llave,grad,0994)
%%BoundingBox: 0 0 673 299
%%Pages: 0
%%EndComments
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/col5 {1 0 1 setrgbcolor} bind def
/col6 {1 1 0 setrgbcolor} bind def
/col7 {1 1 1 setrgbcolor} bind def
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/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def
%%EndProlog

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EndOfTheIncludedPostscriptMagicCookie

\closepsdump
% Here's the Postscript for f2.eps:
\message{Writing file f2.eps}
\psdump{f2.eps}%!PS-Adobe-2.0 EPSF
%%Title: /tmp/xfig-fig004264
%%Creator: fig2dev
%%CreationDate: Tue Jul 23 16:24:40 1996
%%For: panos@linux0 (Panayotis Panayotaros,llave,grad,0994)
%%BoundingBox: 0 0 77 75
%%Pages: 0
%%EndComments
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\def\no{\noindent}

\TITLE A DIAGRAMMATIC INTERPRETATION OF CLASSICAL AND SEMICLASSICAL
PERTURBATION THEORY

\AUTHOR 
Panayotis Panayotaros

\FROM
Department of Physics
The University of Texas at Austin
Austin, TX 78712-1081
\ENDTITLE
\vskip 5 em
\ABSTRACT 
We present a diagrammatic interpretation of the Lie-series classical 
perturbation theory and indicate the use of diagrams 
in a normal form calculation from fluid mechanics.  
We also extend the diagrammatic 
interpretation to 
the Rayleigh-Schr\"odinger perturbation 
theory of quantum mechanics.
An advantage of the unified approach is that we 
have systematic expansions in $ \hbar $, so that in 
the $ \hbar \rightarrow 0 $ limit we easily recover the classical 
diagrams.  
\ENDABSTRACT

\SECTION Introduction

In this paper we present a  diagrammatic method for computations 
in the perturbation theory of hamiltonian systems. The method 
can be thought of as a diagrammatic interpretation of 
the Lie-series perturbation theory of
classical mechanics and is based on the properties of 
Poisson brackets. As will become clear our diagrammatic
interpretation is elementary. Moreover, we have found it 
rather efficient in some lengthy computations, 
especially in infinite dimensional hamiltonian systems, 
and that was our original motivation for presenting it. 
We shall indicate the use of diagrams with an example drawn 
from a hamiltonian model of surface gravity waves (water waves). 

Also we will present a quantized version of the diagrams 
using the analogy between the 
Rayleigh-Schr\"odinger perturbation theory
of quantum mechanics and classical Lie-series.
The analogy goes back to the origin of quantum mechanics but is also  
of recent interest especially from the 
point of view of relating small $\hbar $ asymptotics of the perturbed 
eigenvalues to features of the classical phase 
space of a near-integrable system
(see for instance [EGH], [Ba]). 
>From our definition of the quantum graphical rules, it will be clear
that in the limit $ \hbar \rightarrow 0 $ we recover  
classical ones.   

A typical problem in canonical perturbation theory is the behaviour of 
a hamiltonian dynamical system near an elliptic fixed point. 
The problem arises, for instance, in the study of systems of weakly
coupled harmonic oscillators and in non-linear wave equations.
One of the tools used in the local study of such systems is
the Poincare-Birkhoff 
method of successive elimination of the 
lowest order non-linear terms by changes of 
coordinates.
In general
there will be
non-linear terms that can not be eliminated, the resonant terms.
The reduced systems
containing only resonant terms are said to be in 
normal form. Systems in normal form are frequently
more tractable, and generally speaking the method is very 
useful for asymptotic calculations, for example approximating
periodic and quasi-periodic orbits. In cases where the dynamical system
under consideration satisfy extra conditions, normal form 
arguments can be strengthened to proofs of existence of
periodic and quasiperiodic 
orbits. 

For hamiltonian vector fields, since all the information is 
contained in the hamiltonian, the goal of Poincare-Birkhoff method is to 
simplify the hamiltonian by canonical transformations. 
Canonical transformations can be constructed by the the so-called 
Lie-series. In particular, letting $ M$ be the phase
space, $ \{ , \}$ the Poisson bracket 
%on $C^\infty (M) $ 
and $ \chi $ and $ f$ functions on $M$
% $\in C^\infty (M) $, 
we define 
the map  $ Ad_\chi$ 
% :$C^\infty (M) \rightarrow C^{\infty}(M) $
by $ Ad_{\chi}f = \{\chi,f\} $ and its formal exponential
$ (\exp \epsilon Ad_{\chi}) $ by 
%: C^\infty (M) \rightarrow C^{\infty}(M) $
$$ (\exp \epsilon Ad_\chi)f = f + \sum^{\infty}_{k=1}
{ {\epsilon^k}\over k!} ( Ad_\chi)^k  f =
f +\epsilon \{\chi,f\} + {{\epsilon^2}\over2}\{\chi,\{\chi,f\}\} + \ldots
\quad . \EQ(1.1) $$
The map $ \exp \epsilon Ad_\chi$ is the formal time-$\epsilon$ 
map of the vector field with 
hamiltonian  $ \chi $, acting on the functions on $M$.  
The convergence of the series for $ \exp \epsilon Ad_\chi$ 
requires $\chi$ and $f$ 
to be real analytic and $\chi$ or $\epsilon$ sufficiently small.

Let us outline the Poincare-Birkhoff procedure using 
Lie-series (referred to as 
Lie-series perturbation theory), following the version
of Dragt-Finn 
(see [DF]).
Initially we are given a hamiltonian 
$ g = g_0 + \epsilon g_1 $ 
with $ g_0$ the hamiltonian of the linearised motion 
and $ \epsilon g_1 $ the non-linear perturbation ($\epsilon$ is small). 
The first step 
is to try to find a function $ \chi_1 $ such that 
$ (\exp \epsilon Ad_{\chi_1})g $ 
has no terms of order $\epsilon$.
Using the series of \equ(1.1) this means that 
we should have
$$ \{\chi_1, g_0\} + g_1 = 0 \EQ(1.2) $$
or $ Ad_{g_0} \chi_1 = g_1 $.
After $\chi_1$ is found 
the new hamiltonian is 
$(\exp \epsilon Ad_{\chi_1})g = g_0 + \epsilon^2 \tilde g_2 $ with 
$\tilde g_2 $ calculated using \equ(1.1). The procedure can be iterated:
to find a function $ \chi_2 $ 
so that $(\exp \epsilon^2 Ad_{\chi_2}) (\exp \epsilon Ad_{\chi_1})g $
has no terms of order $ \epsilon^2 $ we must solve 
 $ Ad_{g_0} \chi_2 = \tilde g_2 $ 
and so forth 
for higher orders. 
However in general
the kernel of $ Ad_{g_0} $ will be non-empty, and
to solve \equ(1.2) 
we replace $ g_1 $ by its part belonging to the image of $ Ad_{g_0} $
(the non-resonant part), and $\chi_1 $ is chosen accordingly . 
The resonant terms of $g_1$ 
i.e those in the kernel of $ Ad_{g_0} $ can not be eliminated. 
(Note that some of the resonant terms are integrable.)
Similar statements apply as we iterate to eliminate higher order
terms.
After iterating the procedure $ r $ times the transformed hamiltonian 
will contain only the resonant terms (to order $\epsilon^r$). 
If we want to calculate the normal form Hamiltonian up to 
some order we have to evaluate repeated Poisson brackets between
the functions $ \chi_i$ and lowest order terms of the Hamiltonian.
We have found that the manipulations are more efficient using the 
graphical method described below.

Note that there is another variant of the Lie-series method
(see [D] or [C]) in which 
one looks for a function 
$ \chi = \epsilon \chi_1 + {\epsilon}^2 \chi_2 + \cdots 
+{\epsilon}^r \chi_r $ so that  
$ (\exp Ad_{\chi})g $ 
will bring $g$ to normal form up to order $r$. The $\chi_i $ 
are determined recursively,
while the resonant terms are the same.
The two formalisms are equivalent from the point of 
view of asymptotic expansions, but have different 
convergence and numerical stability properties 
(see Appendix C of [LMM]).     

Also we would like to note that recently there has been
been a lot of activity in diagrammatic methods in classical
perturbation theories (see for instance [G]).
These methods are quite different from the ones presented here as 
they concern convergent
series expansions of parameterizations of invariant tori 
of fixed frequencies in    
near-integrable systems.
Our Poincare-Birkhoff approach concerns a whole
neighborhood of the origin and is asymptotic.

\SECTION Diagrammatic interpretation

The diagrammatic interpretation consists of    
a correspondence between diagrams and 
functions on the phase space $ M $, 
and a rule for assigning a 
new graph to the Poisson bracket of two graphs. 

The classical phase space $ M $ will be $ \real^{2d} $ with the 
canonical symplectic structure and corresponding Poisson bracket. 
Given a harmonic oscillator hamiltonian with frequencies 
$ \omega = [ \omega_1, \ldots, \omega_d ] $ we will 
define the complex variables $ a_i $, $a^*_i $ by 
$$ a_i = { 1 \over \sqrt{2} } ( \sqrt{\omega_i} q_i + 
i { p_i \over { \sqrt{\omega_i}}  } ) \quad , \quad 
a^*_i = { 1 \over \sqrt{2} } ( \sqrt{\omega_i} q_i - 
i { p_i \over { \sqrt{\omega_i}}  } )   \quad .  \EQ(2.01)  $$ 
The map $ a(q,p) $ embeds $ M $ canonically to $  \complex^{2d}$ 
with the Poisson bracket $J$ given by
$$ J(f,g)= \{ f, g\} = i \sum_{i=1}^d \left( 
{ {\partial f} \over {\partial a_i}  }
{ {\partial g}  \over {\partial a^*_i} } - 
{ {\partial g} \over {\partial a_i}  }
{ {\partial f}  \over {\partial a^*_i} } \right) \quad . \EQ(2.02)   $$
Hamilton's equations in the complex variables can be written as 
$ \dot a_\gamma = -i { {\partial H} \over {\partial a^*_\gamma} }  $. 

  
We may also consider infinite dimensional analogues of the 
above phase space. For instance the index $ i $ 
could be running over $ \integer^n $ 
or $ \real^n$ with sums replaced by integrals.   
We emphasize that what follows applies also to the infinite case,
where the harmonic oscillator modes correspond to
the sinusoidal plane waves of linear wave equation.

A real analytic observable (e.g. hamiltonian) $ H $ such that 
$ H(0) = 0$ and $ \nabla H(0) = 0 $
can be written as 
%$$ H =  \sum_{ \kappa, \lambda} \omega_{\kappa \lambda }
% a_\kappa a^*_\lambda  +
%\sum_{ \kappa, \lambda, \mu}
%( A_{\kappa \lambda \mu } a_\kappa a_\lambda a_\mu + 
%B_{\kappa \lambda \mu } a_\kappa a_\lambda a^*_\mu + 
%C_{\kappa \lambda \mu  } a_\kappa a^*_\lambda a^*_\mu  +  \EQ(2.1) $$
%$$ D_{\kappa \lambda \mu } a^*_\kappa a^*_\lambda a^*_\mu )
%+ \cdots  $$
$$ H =  \sum_{ \kappa_1, \kappa_2} \omega_{\kappa_1 \kappa_2 }
 a_{\kappa_1} a^*_{\kappa_2}  +
\sum_{ \kappa_1, \kappa_2, \kappa_3}
( A_{\kappa_1 \kappa_2 \kappa_3 } a_{\kappa_1} a_{\kappa_2} a_{\kappa_3} + 
B_{\kappa_1 \kappa_2 \kappa_3 } a_{\kappa_1} a_{\kappa_2} a^*_{\kappa_3} +
\EQ(2.1) $$
$$ C_{\kappa_1 \kappa_2 \kappa_3 } a_{\kappa_1} a^*_{\kappa_2} a^*_{\kappa_3}+
D_{\kappa_1 \kappa_2 \kappa_3 } a^*_{\kappa_1} a^*_{\kappa_2} a^*_{\kappa_3})
+ \cdots  $$
i.e. quartic and higher. 
(The coefficients $\omega_{\kappa \lambda }$, 
$  A_{\kappa \lambda \mu }$ etc. are determined by the problem and typically
$ H $ is real). We now describe the diagrammatic rules. 

1) The set of graphs we consider will be the set $D $ of 
simply connected directed trees 
with indices at each leg (branch). (Legs will meet at vertices.)
Legs connecting two vertices will be referred to as 
internal legs, while legs with a free end will be 
referred to as external legs.
The one-vertex trees obtained by cutting the internal legs of a 
multi-vertex tree $ t $ will be called the one-vertex subtrees of $ t $.
External legs will carry one index. Internal legs will carry 
two indices one for each vertex being connected, so that the  
one-vertex subtrees will be in $ D $. 

2) To describe the correspondence between trees and polynomials,
we start with one-vertex trees.
%The diagrammatic rules are as follows:
We assign to each homogeneous polynomial term 
$$ \sum_{ \kappa_1, \cdots, \kappa_n, \lambda_1, \cdots, \lambda_m}
I_{ \kappa_1 \cdots \kappa_n \lambda_1 \cdots \lambda_m}
a_{\kappa_1} \cdots a_{\kappa_n} a^*_{\lambda_1} \cdots a^*_{\lambda_m} $$ 
a vertex with $ n + m $ legs (branches), one leg corresponding to each of 
the $ a_{\kappa_i} $ and $ a^*_{\lambda_j} $.
On each leg there will be an arrow, pointing into the vertex for legs 
representing the $ a^*_{\lambda_j} $ and outwards from the vertex 
for legs representing the $ a_{\kappa_i} $.
Also, each leg will carry the index of 
the $ a_{\kappa_i} $ or $ a^*_{\lambda_j} $ it represents. 

For example, the    
hamiltonian of \equ(2.1) will be, up to quartic terms, as in Figure 1
\vskip 2 em
\centerline{ \epsfxsize=4 in \epsfbox{f1.eps}  }
\vskip 2 em
\centerline{Figure 1 }
(here the legs are labeled by the sub-index $i$ of the $ \kappa_i$). 

The dots in the vertices represent the coefficients 
$ \omega_{\kappa \lambda}$, $A_{\kappa \lambda \mu } $ etc.
and the summation over the indices. The dots in the vertices
should in general be represented by different symbols, 
for instance boxes, triangles etc. so that we can distinguish
between graphs with the same number of in-going and out-going legs
but different coefficients.  
In actual computations we need very few such symbols.

We now describe multi-vertex trees. 
A multi-vertex tree represents a homogeneous polynomial 
in the variables indexed by the external legs of the tree.
The outgoing and ingoing arrows represent 
the $ a_{\kappa_i} $ and $ a^*_{\lambda_j} $ respectively. 
The coefficient of this polynomial 
will be  the product of the coefficients of 
all the one-vertex subtrees of the multi-vertex tree, 
summed over all the indices appearing in 
external legs and contracted over the pairs of indices representing 
legs that 
connect vertices.  An example of a multi-vertex tree
and its one-vertex sub-trees is shown in Figure 2.
\vskip 2 em
\centerline{ \epsfxsize=2 in \epsfbox{f2.eps}  }
\vskip 2 em
\centerline{Figure 2 } 
The one-vertex
subtrees in Figure 2 represent the homogeneous polynomials 
$$ \sum_{ \alpha_1, \alpha_2, \alpha_3} 
A_{ \alpha_1 \alpha_2 \alpha_3} a_{\alpha_1} a_{\alpha_2} a_{\alpha_3} 
\quad \hbox{and} \quad
\sum_{ \beta_1, \beta_2, \beta_3, \beta_4} 
B_{ \beta_1 \beta_2 \beta_3 \beta_4} 
a^*_{\beta_1} a_{\beta_2} a_{\beta_3} a^*_{\beta_4} $$
respectively. 
The polynomial for the multi-index tree of Figure 2 is  
$$ \sum_{ \alpha_1, \alpha_3, \beta_1, \beta_2, \beta_3,
\lambda }
A_{ \alpha_1 \lambda \alpha_3 } B_{ \beta_1 \beta_2 \beta_3 \lambda}
 a_{\alpha_1} a_{\alpha_2}  
a^*_{\beta_1} a_{\beta_2} a_{\beta_3}  \quad . $$
Note that the polynomial represented by 
multi-vertex tree may be alternatively viewed as a one-vertex tree 
with an appropriate coefficient.


{ \bf Proposition 2.1  } 
With the above conventions, the Poisson bracket $\{ f, g \} $ of two  
trees $f,g  \in D $ is the sum of the graphs obtained by 
joining an outgoing arrow of $f $ to an in-going arrow of $ g $ 
and the trees obtained by joining
an in-going arrow of $ f $ to an outgoing arrow of $ g $. 
The trees obtained this way will also have signs: 
$ (- i) $ for trees obtained by joining an outgoing arrow
of $ f $ to an in-going arrow of $ g $, and $ (+i) $ for trees formed 
by joining an in-going arrow of $ f $ to an outgoing 
arrow of $ g $. An example is in Figure 3.  

{ \it Proof }
Recall that
{ \item{(i)} $ \{ a_\kappa  , a^*_\lambda \} = - i \delta_{\kappa \lambda} ,
\quad \{ a_\kappa  , a_\lambda \} =
\{ a^*_\kappa  , a^*_\lambda \} = 0 $ } 
%$$ \quad (i) \quad  
%\{ a_\kappa  , a^*_\lambda \} = - i \delta_{\kappa \lambda} ,
%\quad \{ a_\kappa  , a_\lambda \} =
%\{ a^*_\kappa  , a^*_\lambda \} = 0 $$ 
for all indices $\kappa$, $\lambda$ ,
and that
{ \item{(ii)} $ \{ \sum_i a_i f_i , \sum_j b_j g_j \} = 
\sum_{i,j} a_i  b_j \{f_i ,  g_j \} $,$ \quad \forall 
 a_i, b_j \in \complex $, $ f_i, g_j  \in C^\infty (M) $ ,} 
{\item{(iii)} $  \{ \prod^n_{i=1} f_i , \prod^m_{j=1} g_j \} = 
\sum^{n,m}_{i, j}  \{ f_i , g_j \}
\prod^n_{r=1 , r \not= i} f_r \prod^m_{s=1 , s \not= j} g_s $, 
$ \quad \forall  f_i ,g_j  \in C^\infty (M) $ ,} 
from the linearity of the bracket and the 
product rule respectively.
Applying (ii) and (iii) to 
$$ F = \sum_{ c_1, \cdots, c_n, c_{n+1}, \cdots, c_{n+m}  }
I_{ c_1 \cdots c_n  c_{n+1}  \cdots  c_{n+m}            }
a_{c_1} \cdots a_{c_n} a^*_{c_{n+1}} \cdots a^*_{c_{n+m} } $$ 
and $ F'$ same as $F $ with the $ I$, $n$ and $m$ primed we have 
$$ \{ F, F' \} = 
 \sum_{i,j} 
\sum_{ \scriptstyle  c_1, \cdots, c_n, c_{n+1}, \cdots, c_{n+m}  \atop
 c_1, \cdots, c_{n'}, c_{n'+1}, \cdots, c_{n'+m'}   }
I_{ c_1 \cdots c_n  c_{n+1}  \cdots  c_{n+m} }
I'_{ c_1 \cdots c_n'  c_{n'+1}  \cdots  c_{n'+m'} }
\{ f_{c_i}, f_{c_j} \} $$ 
$$ \prod^{n+m}_{\alpha = 1, \alpha \neq i}  f_{c_{\alpha}}
\prod^{n'+m'}_{\beta = 1, \beta \neq j}  f_{c_{\beta}} $$
with $  f_{c_{\alpha}}  = a_{c_{\alpha}} $ if $ \alpha < n $, 
$  f_{c_{\alpha}}  = a^*_{c_{\alpha}} $ otherwise, 
and similarly for  $  f_{c_{\beta}} $
and from (i) and the correspondence between trees and polynomials 
we obtain the rule.  \QED

An example is in Figure 3 below.
\vskip 2 em
\centerline{ \epsfxsize=4 in \epsfbox{f3.eps}  }
\vskip 2 em
\centerline{Figure 3 }

{\bf Remark 2.1 } The rules are essentially the same if we use the 
variables $ (q_\gamma , p_\gamma) $ or the well 
known action-angle variables. Assigning inward arrows to the 
$ q_\gamma $ and outward arrows to the $ p_\gamma $ , the rules are the 
same, except that there are no $i$' s 
when we take the bracket of two graphs. 

The diagrammatic notation can be compressed 
by considering diagrams without 
arrows or indices. We shall refer to these as bare trees
and denote the set of bare trees by $\Delta $. 
%(The trees with 
%arrows and indices encountered above will now be referred to as 
%graphs.) 
Each bare tree $ \tau \in \Delta $ will represent a sum of graphs 
obtained by putting different combinations of arrows and indices to $\tau$. 
%In particular:

1) We describe the correspondence between bare trees and sums of trees by 
first discussing one-vertex bare trees. 
Letting $ \mu $, $\nu $ be the respective number
of outgoing and in-going legs of a tree, we 
use a one-vertex bare tree with $n$ legs to represent 
the sum of all one-vertex trees
with different pairs $ \mu $, $\nu $ such that $ \mu +\nu = n $.
For example, the first terms of 
the hamiltonian of \equ(2.1) is in Figure 4.

%\vskip 2 em
%\centerline{ \epsfxsize=3 in \epsfbox{f4.eps}  }
%\vskip 2 em
%\centerline{Figure 4 }

To a multi-vertex bare tree with $ n $ external legs and $ k $ one-vertex 
bare subtrees $ \tau_i $, $ i = 1, \ldots, k$ 
we first assign the sum of un-indexed trees 
(directed trees)
corresponding to the different allowed pairs 
$ (\mu_1, \nu_1),(\mu_2, \nu_2), \ldots, (\mu_k, \nu_k) $ with 
$ \mu_j $, $ \nu_j $ the respective number of outgoing and in-going 
legs of $ \tau_j $. Note that due to the 
presence of the internal legs not all pairs $(\mu_j,\nu_j) $ 
are allowed. Then, to each of the un-indexed graphs we assign the sum 
of all the graphs corresponding to the different combinations 
of indices we can put on the legs. Specifically, given a one-vertex 
un-indexed subgraph $\tau$ with $\mu$ outgoing and $\nu$ in-going legs, 
of which
there are $ \tilde\mu $ outgoing and $\tilde\nu$ in-going legs that 
are internal, these are the combinations of putting $ \tilde\mu$ 
numbered hats over the numbers $ 1,2,\ldots,\mu$ and 
$\tilde\nu$ numbered hats over $ \mu+1, \mu+2,\ldots,\mu+\nu $, 
the hats representing the internal outgoing and in-going legs.     

\vskip 2 em
\centerline{ \epsfxsize=3 in \epsfbox{f4.eps}  }
\vskip 2 em
\centerline{Figure 4 }

{ \bf Proposition 2.1  } The Poisson bracket between 
two bare trees $ \theta $, $ \phi $ is the sum of the 
topologically distinct trees formed by joining 
an external leg of $ \theta $ with an external leg of $ \phi $.
The internal legs must be drawn so that the arrows we put on 
them will point either to the right or the left. Then we can 
give a $(-i)$ sign to arrows pointing to the right, 
$ (+i)$ for arrows to the left. The sign of the whole multi-vertex
graph will be the product of the signs of the internal (directed) legs.

{ \it Proof }
The rules for the Poisson bracket between bare trees are an implication 
of the rules for the 
Poisson bracket between trees. We need to check that 
by adding arrows and indices to bare trees $ \theta$, $\phi$ and 
taking the Poisson bracket of 
the resulting sums of trees we obtain the same
trees as when using the rule for the 
bracket of $ \theta $ and $ \phi $ 
and then adding arrows and indices to obtain the trees. 
This is done by simply noticing 
each tree obtained in one way can be obtained in the other way as well. \QED 

For the $r$-th step of the normal form calculation we need to know 
$ \tilde H_r $, the order $r$ part of  
$ (\exp \epsilon^{r-1} Ad_{S_{r-1}})\ldots(\exp \epsilon Ad_{S_1})H $. 
With $ \tilde H_r $ we can determine the
function $ S_r$ that eliminates the non-resonant part of $ \tilde H_r $
and $ \overline H_r $, the resonant part of 
$ \tilde H_r $ and go to the next step. 
It is easy to express $ \tilde H_r $ in terms of 
$ H_1,\ldots,H_r$, $\overline H_1, \ldots, \overline H_{r-1}$ and
$ S_1, \ldots, S_{r-1}$. If we use one-vertex trees with $ n+2 $ legs  
and with the vertex circled and boxed 
to represent $ S_n$ and $ \overline H_n $ respectively, then  
$ \tilde H_2$, $ \tilde H_3$ and $ \tilde H_4$ are as in 
Figures 5a and 5b.

\vskip 2 em
\centerline{ \epsfxsize=4 in \epsfbox{f5a.eps}  }
\vskip 2 em
\centerline{Figure 5a }

\vskip 2 em
\centerline{ \epsfxsize=4 in \epsfbox{f5b.eps}  }
\vskip 2 em
\centerline{Figure 5b }

\SECTION An application to fluid mechanics

We shall indicate the use of diagrams with a normal form calculation 
for an infinite dimensional hamiltonian system.
In particular, we 
will consider a hamiltonian model of waves on the surface of a fluid 
layer surrounding a gravitating sphere (water waves). 
A detailed description of the model and the computations
can be found in [LP].

The evolution equation for this system can be written as an
infinite analogue of Hamilton's equation. 
The canonical variables are the wave amplitude $ q $ 
and the surface hydrodynamic potential $ p$, both functions on the 
2-sphere, while the hamiltonian (functional) is the total 
energy of the system. 
We expand $ q$ and $ p$ in spherical harmonics $ Y_i $
and denote 
the coefficient of the $i$-th harmonic (or $i$-th mode) by $ q_i$, $ p_i$. 
(The index $i$ runs over the pairs $ (l,m)$ with $ l$ a positive integer,
$ m = -l, -l+1, \ldots, l$.) Using the frequencies of the linearised problem
we also define the complex variables $ a_i$ and $a^*_i$ as in \equ(2.01).
The Poisson bracket is (after a change of coordinates) as in 
\equ(2.02). 
The hamiltonian $H$ can be written as 
$  H = H_0 + \epsilon H_1 + \epsilon^2 H_2 + \ldots $,
with $ H_m$ homogeneous polynomials of degree $ m+2 $ in 
$ a_i$, $a^*_i$ and 
$ \epsilon $ a small dimensionless parameter. Physically, the ratio
of the typical wave amplitude over the thickness of the fluid layer.
The other dimensionless parameter of the problem, 
the ratio $\beta $ of depth over radius is set here to $ O(1)$. 
Graphically $H$ is as in Figure 1.

Our calculation consists of eliminating the terms of 
order $ \epsilon $ and some of the terms of order $ \epsilon^2 $.
It is convenient to use the multi-index notation 
$$  z^k {\overline z}^{\overline k}
= a_1^{k_1} \ldots a_d^{k_d} \ldots 
(a^*_1)^{\overline k_1} \ldots (a^*_d)^{ \overline k_d} \ldots
\quad ,  $$
with $ |k| = k_1 + k_2 + \ldots $ 
 $ |\overline k| = \overline k_1 + \overline k_2 + \ldots $, 
and write 
$ H_m = \sum_{ | \kappa| + | \overline \kappa | = m+2}
A^m_{ | \kappa|, | \overline \kappa |}
z^k {\overline z}^{\overline k} $. 
To find $ S_1$ for which $ \exp \epsilon Ad_{S_1}H $
has no terms of order $ \epsilon $ we are led to
$ \{ S_1, H_0 \} + H_1 = 0 $. Noting that 
$$ \{ {\overline z}^{\overline k}  z^k,  H_0 \} = -i 
(\sum_i \omega_i(k_i - \overline k_i )) 
  {\overline z}^{\overline k} \hat z^k , $$ 
the solution is 
$$ \chi_1 =  \sum_{\kappa + \overline \kappa = 3}
 {   { A^{1}_{\kappa, \overline \kappa}  } \over
  { \sum_i \omega_i(k_i - \overline k_i ) }   } 
 {\overline z}^{\overline k}   z^k , $$ 
i.e. the resonant terms 
are those satisfying $ \sum_i \omega_i(k_i - \overline k_i ) = 0 $. 
Moreover, 
the hamiltonian is spherically symmetric 
and as a consequence
every cubic monomial coupling three modes, say 
$a_{i_1},a_{i_2},a_{i_3} $ in $ H_1 $ 
is proportional to the integral of 
the product of the corresponding harmonics 
$ Y_{i_1}, Y_{i_2}, Y_{i_3} $. 
Thus the resonant triples of modes must also satisfy 
the condition that these integrals do not 
vanish. 
% (we refer to this condition as angular momentum addition rule)
In [LP] we have shown that $ H_1 $ contains no resonant terms
and thus can be eliminated completely. 
Next, we 
eliminate the non-resonant terms of $ \tilde H_2 $, the order 
$ \epsilon^2 $ terms of $ \exp \epsilon Ad_{S_1}H $. 
>From \equ(1.1) we have that 
$ \tilde H_2  = H_2 + {1 \over 2} \{ S_1 , H_1 \} $. 
Writing $ S_1 $ as in Figure 6, we obtain $ \tilde H_2 $ graphically. 
%(since $ \tilde H_2 $ is real we don't need to write them all)
Proceeding as before we find analogous resonance conditions 
(again involving conditions on integrals of spherical harmonics)
for four modes and
we have shown that the resonant terms of $\tilde H_2 $ 
(denoted by $ \overline H_2$)
can only be of the type 
corresponding to diagrams with two outgoing and two in-going legs. 
$ \overline H_2$ is represented in Figure 7.    

\vskip 2 em
\centerline{ \epsfxsize=4 in \epsfbox{f6.eps}  }
\vskip 2 em
\centerline{Figure 6 }

%\vskip 2 em
%\centerline{ \epsfxsize=4 in \epsfbox{f7.eps}  }
%\vskip 2 em
%\centerline{Figure 7 }

The result of this calculation is that 
%the second order normal form $ H_0 + \epsilon^2  \overline H_2 $ 
%is simplified enough that   
we can now easily identify 
families of 
periodic solutions 
of the normal form system with hamiltonian $ H_0 + \overline H_2 $.
These approximate solutions of the full system  
correspond to traveling and standing waves involving one or two modes. 
$ \overline H_2  $ calculated graphically gives us the relation 
between the amplitude and speed of these waves. 
Also, again using $ \overline H_2$, we can identify invariant manifolds 
of solutions involving a finite number of modes. These manifolds 
are in fact invariant to higher order and on them we can find
new families of periodic solutions of the water wave system. 

Note that the resonance conditions can be read from the diagrams.
For instance, the resonant 
terms of a diagram with $m$ outgoing and $n$ outgoing legs will
be those satisfying 
$\omega_{\kappa_1} + \ldots + \omega_{\kappa_m} -  
\omega_{\lambda_1} - \ldots - \omega_{\lambda_n} = 0  $. 
A similar statement can be made about the angular addition rules. 

\vskip 2 em
\centerline{ \epsfxsize=4 in \epsfbox{f7.eps}  }
\vskip 2 em
\centerline{Figure 7 }

\SECTION Semiclassical diagrams

As several authors have noted classical 
perturbation theories in the form of 
Lie-series or the Hamilton-Jacobi theory can be ``quantized'' to yield  
the Rayleigh-Schr\"odinger perturbation theory 
for the spectrum of the hamiltonian operator in quantum mechanics
(see [GP1], [EGH], [BV]). 
Here we would like to show that 
the graphical interpretation can be extended 
to the quantized version of the Lie-series. 

To describe the quantized Lie-series and the diagrammatic
interpretation  
it is sufficient to 
consider the algebra $W $ generated by  
polynomials in 
the $ \hat a_i $ and $  (a^*_j){\hat{}}$, where $ i, j = 1,2 \ldots, d $.
The $ \hat a_i $ and $  (a^*_j){\hat{}}$ are required to satisfy the 
the commutation relations
$$  [ \hat a_i , (a^*_j) {\hat{}} ] = 
- i \hbar \delta_{ij} \quad ,
\quad [ \hat a_i  , \hat a_j ] =
[ (a^*_i) {\hat{}},  (a^*_j) {\hat{}} ] = 0 
\quad , \quad   \forall \quad i, j  = 1,2 \ldots, d   \EQ(4.1) \quad $$
where $ [ \hat f, \hat g] =  \hat f \hat g -\hat g \hat f $.
Also we will consider a quadratic polynomial $ \hat h_0 $ given by
$$   \hat h_0  = 
{ 1 \over 2 } \sum_{i=1}^d \omega_i 
( (a^*_i){\hat{}} \hat a_i +  \hat a_i (a^*_i){\hat{}}) $$
with $ \omega_i \in \real $ and the polynomial
$ \hat h =  \hat h_0 + \epsilon \hat h_1 + \epsilon^2 \hat h_2 + \ldots$,
with  the $\hat h_\mu$ of order $ \mu+2$ in $ a_i$ and $ a^*_i $. 
We will assume, without loss of generality that the 
products appearing in the $ \hat h_\mu $ are 
in ``Wick-ordered'' form, that is, with the $ (a^*_j){\hat{}}$ preceding  
the  $ \hat a_i$.

In quantum mechanics the $ \hat a_i $ and $  (a^*_j){\hat{}}$
are represented as operators in a Hilbert space, 
while $ \hat h_0$ and $ \hat h $ are the harmonic oscillator 
with frequencies $ \omega = [ \omega_1, \ldots, \omega_d] $
and perturbed harmonic oscillator hamiltonian operators respectively. 
In particular, we can take 
the Hilbert space of quantum states to be (the Bargmann space) 
$ F_d (\complex^d) $  of holomorphic functions 
$ \psi:\complex^d \rightarrow \complex $ 
for which
$ \int_{R^{2d}}  |\psi|^2 e^{-|z|^2/\hbar} dqdp $ is finite 
and represent  $ \hat a_i $ and $  (a^*_j){\hat{}}$ by 
$$ \hat a_i \psi = z_i \psi \quad , \quad  
(a_j^*){\hat{ }} \psi = \hbar \partial_{z_j} \psi \quad.  \EQ(4.2)$$
Note however that in the description of the perturbation theory and
the diagrams that follows 
we only need the algebra $W$, the commutation relations
\equ(4.1) and $ \hat h_0$ and $ \hat h $.

The problem of quantum perturbation theory
is the determination of the spectrum of $ \hat h $.
The spectral decomposition of $ \hat h_0$ is known.
>From the algebraic point of view we want to simplify $ \hat h $ 
using transformations that preserve the commutators structure.
In analogy with classical mechanics  
we define the map $ \widehat{Ad}_{\hat g} $ by
$  \widehat{Ad}_{\hat g} \hat f = [\hat g, \hat f ] $ and its 
formal exponential
$$ (\exp \epsilon  \widehat{Ad}_{\hat g} )\hat f = 
\hat f + \epsilon [ \hat g, \hat f] + 
{ 1 \over 2 } \epsilon^2 [ \hat g, [ \hat g, \hat f]] + \ldots  \EQ(4.3) . $$ 
As in the classical case we try to eliminate the $ O(\epsilon)$ 
terms of $ \hat h $, 
order by order, by a successive application of the map
$ \exp \epsilon  \widehat{Ad}_{\hat g} $. 
We start by trying to find $ \hat \chi_1 $ so that 
$ \exp \epsilon \widehat{Ad}_{ \hat \chi_1} \hat h $ has no terms of order 
$ \epsilon $ and we are led to 
$$ [ \hat \chi_1, \hat h_0 ]  + \hat h_1 = 0  \EQ(4.4)  $$
To solve \equ(4.4), observe that for $ \hat h_0$ quadratic we have that
$$ [ 
% \sigma_{ \kappa, \overline \kappa}
  (\hat {\overline z})^{\overline k} \hat z^k, \hat h_0] = 
 -i 
% \sigma_{\kappa, \overline k}
(\sum^d_{i=1} \omega_i(k_i - \overline k_i )) 
 (\hat {\overline z})^{\overline k} \hat z^k,   \EQ(4.5) $$
and recall that (using the multi-index notation)
$$ \hat h_1 = \sum_{n=1}^3 \sum_{\kappa + \overline \kappa = n}
A^{n}_{\kappa, \overline \kappa}
 (\hat {\overline z})^{\overline k}  (\hat z)^k 
%+  
%\hbar \sum_{\kappa + \overline \kappa = 1}
%A^{1,1}_{\kappa, \overline \kappa}
% (\hat {\overline z})^{\overline k}  (\hat z)^k
$$ 
and therefore 
$$ \hat \chi_1 = \sum_{n=1}^3 \sum_{\kappa + \overline \kappa = n}
 {   { A^{n}_{\kappa, \overline \kappa}  } \over
  { \sum_i \omega_i(k_i - \overline k_i ) }   } 
 (\hat {\overline z})^{\overline k}  (\hat z)^k 
%+  
%\hbar \sum_{\kappa + \overline \kappa = 1}
% {   { A^{1,1}_{\kappa, \overline \kappa} } \over
% { \sum_i \omega_i(k_i - \overline k_i ) }   }   
%(\hat {\overline z})^{\overline k}  (\hat z)^k \quad 
$$ 
%Note here that  
%$ [ \hat g, \hat h_0 ] = [ g,  h_0 ]{\hat{}}  $
%for all operators $ \hat g $.
The resonance condition is similar to the classical one, 
and $  \chi_1 $ is chosen accordingly to eliminate the 
non-resonant terms of $ \epsilon \hat h_1 $.
We repeat the procedure to remove the non-resonant terms 
$ \epsilon^2 $ terms  
of $ \exp \epsilon \widehat{Ad}_{ \hat \chi_1} \hat h $
and similarly for higher orders in $\epsilon$.
The algorithm is clearly analogous to the 
classical Lie-series. 
After  $ r $ steps the operator
will be in ``normal form''. 
The relevance of the procedure 
is that if the frequencies are non-resonant to order $r$ , 
i.e. if $ <\omega,  \nu> \neq 0 $ for $ \nu \in \integer^d$, 
$ | \nu | < r + 2 $, 
the transformed operator will be diagonal to $ o(\epsilon^r)$ 
in the basis of the eigenfunctions of $ \hat h_0$. Thus the 
quantized Lie-series algorithm 
is in effect the (non-degenerate) Rayleigh-Schr\"odinger perturbation 
theory.   
In the case of resonance, some off-diagonal terms will remain. These 
are the classical resonant terms plus corrections of $ O(\hbar) $.
A semiclassical version of degenerate  Rayleigh-Schr\"odinger series,
corresponding to the Poincare ``secular'' perturbation  
theory of classical mechanics is also possible along
similar lines (see [Ba], also [GP2]).

The graphical interpretation consists of a rule for assigning 
graphs to polynomial operators and a rule for assigning a graph to  
the commutators of two graphs. 
Graphs will be assigned to Wick-ordered operators only. 
>From the above, 
after Wick-ordering a quantized observable $ \hat h $, 
and provided that we can Wick-order the commutator of two 
Wick-ordered terms, we only have to deal with 
Wick-ordered polynomials only.

The rules are as follows:

1) The set of graphs under consideration will now be denoted by $ D_q$.
An element of $ D_q$ will be 
a directed tree with indices, constructed recursively
in the following way: we take  
$ \kappa $ one-vertex elements $ t_i$ of $ D $ 
(as in chapter 2)
and start by joining $s_1 $ outgoing (ingoing)legs of $ t_1$ to $s$ 
ingoing(outgoing) legs of $t_2$.
Denote this by $ t_1 \cup t_2$. 
At the $i$-th step we join $ s_i$  outgoing (ingoing) legs 
of the $ t_1 \cup \ldots t_i$  to  $ s_i$ 
ingoing(outgoing)
legs of $ t_{i+1} $, 
until $ i= \kappa - 1 $.
The $ s_i $ are arbitrary.  
The external legs will carry one index, while internal legs will 
carry two, one for each vertex being joined. 
Note that $ D \subset D_q $. 

Examples are in Figure 8 below.

\vskip 3 em
\centerline{ \epsfxsize=3 in \epsfbox{f8.eps}  }
\vskip 3 em
\centerline{Figure 8}


2) We first discuss one-vertex trees.
To a Wick-ordered homogeneous polynomial 
$$ \sum_{ \kappa_1, \cdots, \kappa_n, \lambda_1, \cdots, \lambda_m}
I_{ \kappa_1 \cdots \kappa_n \lambda_1 \cdots \lambda_m}
(a^*_{\lambda_1}){\hat{}} \cdots (a^*_{\lambda_m}){\hat{}}
\hat a_{\kappa_1} \cdots \hat a_{\kappa_n} $$ 
we assign a vertex with $ n + m $ 
indexed legs with arrows, 
one outgoing (in-going) leg for each $a_{k_i}$ ($a^*_{k_j}$).
(This rule follows the classical one verbatim, 
with the $ \hat a_i $, $ (a^*_j){\hat{}} $ playing the role
of the  $ a_i $, $  a^*_j $ respectively.)

The rule for assigning polynomials to multi-vertex graphs 
is similar to the one for the classical graphs.
A multi-vertex tree represents a homogeneous polynomial 
in the variables indexed by the external legs of the tree.
The outgoing and ingoing arrows represent 
the $ a_{\kappa_i} $ and $ a^*_{\lambda_j} $ respectively. 
The coefficient of this polynomial 
will be  the product of the coefficients of 
all the one-vertex subtrees of the multi-vertex tree, 
summed over all the indices appearing in 
external legs and contracted over the pairs of indices representing 
legs that 
connect vertices.
Note that now multi-vertex graphs can be multiply connected. 
An example is in Figure 9.

\vskip 3 em
\centerline{ \epsfxsize=4 in \epsfbox{f9.eps}  }
\vskip 3 em
\centerline{Figure 9  }

{ \bf Proposition 4.1  }
We form the commutator $ [ F, G ]$  of two graphs 
$ F$, $G$ by adding all the graphs obtained by joining 
$ n$ in-going arrows of  $F $ to $ n$ outgoing arrows of $G$ and 
$ n$ outgoing arrows of  $F $ to $ n$ in-going arrows of $G$, 
where $ n = 1, 2, 3, \ldots $. 
The graphs will be multiplied by signs and 
powers of $ \hbar$: $ ( -i \hbar)^n $ for graphs obtained by joining
$ n$ outgoing legs from $ F $ to $ n$ in-going of $ G$, and 
$  -( - i \hbar)^n $ for graphs obtained by joining
$ n$ in-going legs from $ F $ to $ n$ outgoing from $ G$.
 
{ \it Proof }
%To see the validity of the graphical rule 
We consider 
the commutator of two operators
$$ \hat F = \sum
A_{ \overline \kappa_1, \cdots, \overline \kappa_{\overline \mu}, 
\kappa_1, \cdots, \kappa_\mu }
(a^*_{\overline \kappa_1}){\hat{}} \cdots 
(a^*_{\overline \kappa_{\overline \mu}}){\hat{}} 
\hat a_{\kappa_1} \cdots \hat a_{\kappa_\mu} $$ and
$$ \hat G = \sum
B_{ \overline \lambda_1, \cdots, \overline \lambda_{\overline \nu}, 
\lambda_1, \cdots, \lambda_\nu }
(a^*_{\overline \lambda_1}){\hat{}} \cdots 
(a^*_{\overline \lambda_{\overline \nu}}){\hat{}} 
\hat a_{\lambda_1} \cdots \hat a_{\lambda_\nu} \quad . $$
%by Wick-ordering $ \hat F \hat G $,  $  \hat G\hat F$ in a systematic way. 
We first 
Wick-order $ \hat F \hat G $ by passing the $ \hat a_{\kappa_i}$ 
to the right of the $ (a^*_{\overline \lambda_j}){\hat{}} $
in a systematic way. 
We order the pairs of the 
sub-indices $(i,j)$ of the 
$\hat a_{\kappa_i}$, $ (a^*_{ \overline \lambda_j}){\hat{}}$
%$  \kappa_i$, $\lambda_j)$ 
lexicographically, i.e. 
$(1,1) \prec (1,2) \prec \ldots \prec (\mu,\overline \nu)$. 
Also we use the multi-index $ \gamma = (i,j) $ and 
for $ \gamma = (a,b)$ let $ I(\gamma)= a$, $ J(\gamma)= b$.
Also $ \Gamma_0 = \{(i,j), i=1,\ldots,\mu, j=1,\ldots,\overline \nu \}$.
Define $ L_{ij} \hat F \hat G$ to be 
$ \hat F \hat G$ with  
$\hat a_{\kappa_i}$ with  $ (a^*_{ \overline \lambda_j}){\hat{}}$
interchanged,  
and $ R_{ij} \hat F \hat G$ to be 
$ \hat F \hat G$ with  
 $\hat a_{\kappa_i}$ and  $ (a^*_{\overline \lambda_j}){\hat{}}$
replaced
by $ [\hat a_{\kappa_i}, (a_{\lambda_j}){\hat{}} ] $.
We write 
$$  \hat F \hat G = L_{1,1}\hat F \hat G + R_{1,1}\hat F \hat G = 
 L_{1,2}L_{1,1}\hat F \hat G +  R_{1,2}L_{1,1}\hat F \hat G +
R_{1,1}\hat F  = \ldots $$ 
i.e. apply $ L_{i,j} $ repeatedly, in the order induced by 
the $ (i,j)$, each time producing a commutator, and so that 
at most one commutator appears in each term. 
The procedure terminates after $ \mu$ steps and we have 
$$   \hat F \hat G = S_1 \hat F \hat G $$
with 
$$  S_1 \hat F \hat G = 
\sum_{\gamma_1 \in \Gamma_1} {\tilde R}_{\gamma_1} 
{ \prod_{\tau_1 \in T_1 } L_{\tau} } 
\hat F \hat G  \quad , $$
where $  \Gamma_1  =\Gamma_0$ and 
$  {\tilde R}_{\gamma_1} =  L_{\gamma_1} $ 
if $ \gamma_1 = \max\{ \gamma \in \Gamma_1\}$,
$ R_{\gamma_1} $ otherwise, and 
$ T_1 \equiv T(\gamma_1 )=
\{ \tau \in \Gamma_1 | \tau \prec \gamma_1  \}$.
We repeat the procedure to terms of $  S_1 \hat F \hat G $ that are not 
Wick-ordered, so as to have at most two commutators. We get 
$  S_2 \hat F \hat G $, and similarly 
$$ \hat F \hat G = S_1 \hat F \hat G = \ldots = S_n \hat F \hat G =\ldots $$
with 
$$  S_n \hat F \hat G = 
\sum_{\gamma_1 \in \Gamma_1} ( \ldots
(\sum_{\gamma_{n-1} \in \Gamma_{n-1}} ( 
\sum_{\gamma_n \in \Gamma_n }  {\tilde R}_{\gamma_n} 
{ \prod_{\tau_n \in T_n} } L_{\tau_n}    )   
 {\tilde R}_{\gamma_{n-1}}
{ \prod_{\tau_{n-1} \in T_{n-1}} }      
L_{\tau_{n-1}}    ) \ldots    $$
$$ \ldots ) {\tilde R}_{\gamma_1}
{ \prod_{\tau_1 \in T_1} } L_{\tau_1} \hat F \hat G $$ 
with
$ \Gamma_p \equiv \Gamma_p(\gamma_{p-1},\ldots,\gamma_1) = 
\{ \gamma \in \Gamma_0  | I(\gamma)>I(\gamma_{p-1}), 
J(\gamma) \neq J(\gamma_{p-1}), \ldots,J(\gamma_{1}) \}   $,  
$ T_p \equiv T(\gamma_p )=
\{ \tau \in \Gamma_p | \tau \prec  \gamma_p  \}    $ and  
$  {\tilde R}_{\gamma_p} =  L_{\gamma_p} $ 
if $ \gamma_p = \max \{ \gamma \in \Gamma_p\}$,
$ R_{\gamma_p} $ otherwise.

Then note that, first   
in $ S_n \hat F \hat G $ we encounter  
all the possible products of commutators 
$ [ \hat a_{\kappa_{\sigma_1}},
(a^*_{ \overline\lambda_{\rho_1}}){\hat{}}] \ldots
[ \hat a_{\kappa_{\sigma_n}},  (a^*_{\overline \lambda_{\rho_n}}){\hat{}}] $
with $ \sigma_1 < \sigma_2 < \ldots < \sigma_n$ and 
$ \rho_j \neq \{ \rho_1, \ldots, \rho_{j-1} \} $, $ j = 1, \ldots, n$,
each such product appearing once
as a summand of $ S_n \hat F \hat G $. 
Moreover, in $  S_{n+1} \hat F \hat G $ all these terms are 
Wick-ordered 
and therefore, from the diagrammatic interpretation 
at $  S_{n+1} \hat F \hat G $ we have all the diagrams with 
$ m $ outgoing legs of $ \hat F $ joined to 
$ m $ in-going legs of $ \hat G $, with $ m = 1, 2, \ldots, n$,  
plus $ \hat F \hat G$ Wick-ordered.

Applying the algorithm to $ \hat G \hat F $ we similarly 
obtain the diagrams with
$ m $ ingoing legs of $ \hat F $ joined to 
$ m $ outgoing legs of $ \hat G $, with $ m = 1, 2, \ldots, n$,  
plus $ \hat F \hat G$ Wick-ordered. 
Considering $\hat F \hat G - \hat G \hat F $ 
we have the proposition \QED 


{\bf Remark 3.1 } Note that comparing the 
last rule with its classical analogue we have 
the classical limit that, graphically
$$  { \lim_{\hbar \to 0}  }  { i \over \hbar } [\hat f, \hat g ]  
= \{f, g \}  $$ 

Note that the quantum ``normal form'' hamiltonian 
obtained after  $r $ steps of the algorithm will contain polynomials
that are multiplied by powers of $ \hbar $. This is clear, for instance 
from the rule for the commutator of two graphs.
Therefore the dependence of the perturbed eigenvalues on $ \hbar $ 
will be explicit. In the case of non-resonance the eigenvalues 
obtained from the quantum Lie-series will be 
those given by the semiclassical EBK values (see [EGH]) 
plus $ O(\hbar) $ corrections.   

We would like to point out that the analogy between the 
classical and quantum Lie-series above can be discussed more fully 
within the framework of pseudodifferential operators, where we 
can consider  
explicit ways to map classical observables $(f)$ to 
linear operators $( \hat f )$, i.e. quantizations (for 
a heuristic introduction see [BS], ch. 5). 
In particular,  
under certain assumptions on the 
classical hamiltonian $ h_0 + \epsilon g$, the frequencies of $ h_0 $ 
(and using the Weyl quantization) we can make error estimates 
for the eigenvalues obtained using the above 
semiclassical expansions, 
and show that the error is small uniformly in $ \hbar $.   
(this follows from [Ba] with some modifications, see also 
[BV] for a different approach).
In our presentation we assume
that the quantization step is already taken, and that the operators
have been Wick-ordered. It is in fact possible to discuss
diagrams for polynomials that are not Wick-ordered, by introducing
a convention that takes into account the order of the $ \hat a_{i}$ 
and $ (a^*_{j}){\hat{}} $, e.g. numbering the external legs.
The advantage of using Wick-ordered polynomials is that 
the ordering of the $ \hat a_{i}$ 
and $ (a^*_{j}){\hat{}} $ is taken into account and the 
correspondence between graphs and polynomials is similar 
to the one for the classical case.

In summary, we have presented a simple graphical 
interpretation of the classical Lie-series algorithm and 
its quantized version, which takes the form of the 
Rayleigh-Schr\"odinger perturbation 
theory. Also we gave an 
application to an asymptotic calculations in 
a problem of classical fields (hydrodynamics).



\SECTION Acknowledgments

I would like to thank Professor de la Llave for encouraging
me to write this note and for several suggestions.  
I would also like to acknowledge partial support from NSF and TARP
grants.

\SECTION REFERENCES


\parindent  = 0 pt

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\end