\documentstyle[12pt]{article}
\input psfig
\begin{document}
\title{PERTURBATIVE\ ANALYSIS\ OF\ CHERN-SIMONS\
FIELD\ THEORY\ IN\ THE\ COULOMB\ GAUGE}
\author{Franco Ferrari \thanks{The work of F. Ferrari
has been supported by the European Union, TMR Programme, under
grant ERB4001GT951315} $^{,a}$  and Ignazio Lazzizzera
$^{b,c}$\\ \\
$^a$ {\it LPTHE \thanks{Laboratoire associ\'e No. 280 au CNRS} ,
Universit\'e Pierre er Marie Curie--PARIS VI and} \\
{\it Universit\'e Denis Diderot--Paris VII,
Boite $126$, Tour 16, $1^{er}$ \'etage,}\\
{\it 4 place Jussieu,
F-75252 Paris CEDEX 05, FRANCE.} \\
$^b$ {\it Dipartimento di Fisica, Universit\'a di Trento, 38050 Povo (TN),
Italy.}\\
$^c$ {\it INFN, Gruppo Collegato di Trento, Italy.}
}

%\date{October 96}
\maketitle
\vspace{-4.5in} \hfill{Preprint PAR-LPTHE 96--44, UTF 387/96} \vspace{4.4in}
\begin{abstract}
In this paper we analyse the perturbative aspects of
Chern--Simons field theories in the Coulomb
gauge.
We show that in the perturbative expansion of the Green functions
there are neither
ultraviolet not infrared
divergences.
Moreover, 
all the radiative corrections are zero at any loop order. 
Some problems connected with the Coulomb gauge fixing, like the appearance
of spurious singularities in the computation of the Feynman diagrams,
are discussed and solved.
The regularization used here for the spurious singularities 
can be easily applied also to the Yang--Mills case, which is affected
by similar divergences.
\end{abstract}
\section{Introduction}


In the recent past, the Chern--Simons (C-S) field
theories \cite{jao,hageno} have been intensively studied in connection
with several
physical and mathematical applications \cite{csapps,ienkur}.
A convenient gauge fixing for these theories is provided by the Coulomb gauge.
As a matter of fact, despite of the presence of nontrivial
interactions in the gauge fixed action, the calculations become
considerably simpler than in the covariant gauges and a perturbative
approach is possible also on non-flat manifolds \cite{ffprd}.
Moreover, the dependence on the time in the Green functions is trivial,
so that the C--S field theories can be treated in practice as two
dimensional models.
Starting from the seminal works of refs. \cite{hageno,hagent} and \cite{jat}, 
the Coulomb gauge has been already applied in a certain number of
physical problems involving C--S based models
\cite{ienkur}, \cite{vo}--\cite{bcv}, but still remains
less popular than the covariant and axial gauges.
One of the main reasons is probably the fact that there are
many
perplexities concerning the use of
this gauge fixing,  in particular in the case of the four dimensional
Yang--Mills theories \cite{taylor}--\cite{leiwil}.
Recently, also the consistency of the C--S field theories in the
Coulomb gauge has been investigated using various techniques \cite{ffprd,
devone, cscgform, frig},
but so far a detailed perturbative analysis in the non-abelian case
is missing.
To fill this gap, the radiative corrections of the Green functions
are computed here at any loop order and it is shown
that they vanish identically.
No regularization is needed for the ultraviolet and infrared
divergences since, remarkably, they do not appear in the amplitudes.
The present result agrees with the previous analysis
of \cite{frig}, in which the
commutation relations between the fields are proved to be trivial
using the
Dirac's canonical approach to constrained systems.
It is important to notice that the absence of any quantum
correction despite of the presence of nontrivial self-interactions in the
Lagrangian
is a peculiarity of the Coulomb gauge that cannot be totally
expected from the fact that the theories under consideration
are topological, as finite renormalizations of the fields and
of the coupling constants are always possible.
For instance, in the analogous case of the covariant gauges,
only the perturbative
finiteness of the C--S amplitudes has been shown \cite{csformal}
in a regulatization
independent way exploiting BRST techniques \cite{ss}.
Indeed, a finite shift of the C--S coupling constant has been observed
in the Feynman gauges by various authors \cite{shift, alr}.


The material presented in this paper is divided as follows.
In Section 2 the C--S field theories with $SU(n)$ gauge group are quantized
using the BRST approach. The Coulomb gauge constraint
is weakly imposed and
the proper Coulomb gauge is recovered suitably
choosing the gauge fixing parameter.
The singularities
that may appear in the perturbative calculations are studied in details.
Ultraviolet divergences are predicted by the naive power counting,
but it will be shown in Section 3 that they are absent in
the perturbative expansions
of the Green functions.
Still there are spurious singularities, which arise because the
propagators are undamped in the time direction. They are completely
removed with the introduction of a cut off
in the zeroth components of the momenta.
In Section 3, the quantum contributions to the $n-$point
correlation functions are derived
at all orders in perturbation theory.
The one loop case is the most difficult, as
nontrivial cancellations occur among different Feynman diagrams.
To simplify the calculations, a crucial observation is proved, which
drastically
reduces their number.
The total contribution of the
remaining diagrams is shown to vanish
after some algebra. The gluonic $2-$point function
requires some care and it is treated separately.
At two loop, instead, any single Feynman diagram is identically zero.
The reason is that, in order to build such diagrams,
some components of the propagators and of the vertices are required,
which are missing in the Coulomb gauge.
At higher orders, the vanishing of the Feynman diagrams
is proved by induction in the loop number $N$.
Finally, in the Conclusions some open problems and future developments are
discussed.
\section{Chern-Simons Field Theory in the Coulomb Gauge: Feynman Rules and
Regularization}

The C--S action in the Coulomb gauge looks as follows:
\begin{equation}
S_{CS}=S_0+S_{GF}+S_{FP}
\label{action}
\end{equation}
where 

\begin{equation}
S_0=\frac s{4\pi }\int
d^3x\epsilon ^{\mu \nu \rho }\left( \frac
12A_\mu ^a\partial _\nu A_\rho ^a-\frac 16f^{abc}A_\mu ^aA_\nu ^bA_\rho
^c\right)  \label{csaction}
\end{equation}

\begin{equation}
S_{GF}=\frac {is}{8\pi \lambda }\int d^3x\left( \partial
_iA^{a\,i}\right) ^2  \label{gf}
\end{equation}

and 

\begin{equation}
S_{FP}=i\int
d^3x\,\overline{c}^a\partial _i\left( D^i\left[
A\right] c\right) ^a  \label{fp}
\end{equation}

In the above equations $s$ is a dimensionless coupling constant and the
vector fields $A_\mu^a$
represent the gauge potentials. Greek letters $\mu,\nu,\rho,\ldots$
denote space--time indices, while the first latin letters $a,b,c,\ldots=
1,\ldots,N^2-1$ are used for
the color indices of the $SU(n)$ gauge group
with structure constants $f^{abc}$.
The theory is considered on the flat space-time
$\mbox{\bf R}^3$ equipped with the standard euclidean metric
$g_{\mu\nu}=\mbox{\rm  diag}(1,1,1)$.
The total antisymmetric
tensor $\epsilon^{\mu\nu\rho}$ is defined by the convention
$\epsilon^{012}=1$.
Finally,
$$D_\mu^{ab} \left[ A\right] =\partial _\mu \delta ^{ab}-f^{abc}A_\mu ^c$$
is  the covariant derivative and
$\lambda $ is an arbitrary gauge fixing parameter.

In eq. (\ref{action}) the Coulomb gauge constraint is weakly imposed
and the proper Coulomb gauge fixing\footnotemark{}\footnotetext{
From now on, middle latin letters like $i,j,k,\ldots=1,2$ will indicate
space indices.}, given by:
\begin{equation}
\partial _iA^{a\,i}=0  \label{gaugefix}\qquad\qquad\qquad i=1,2
\end{equation}
is recovered setting $\lambda=0$ in eq. (\ref{gf}).

The partition function of the CS field theory
described by eq. (\ref{action}) is: 
\begin{equation}
Z=\int DAD\overline{c}Dce^{iS_{CS}}  \label{partfunct}
\end{equation}
and it is invariant under the BRST transformations listed below:
\begin{eqnarray}
\delta A_\mu ^a &=&\left( D_\mu \left[ A\right] \right) ^a  \label{brst} \\
\delta \overline{c}^a &=&\frac s{4\pi \lambda }\partial _iA^{a\,i}  \nonumber
\\
\delta c^a &=&\frac 12f^{abc}c^bc^c  \nonumber
\end{eqnarray}
%The complex number $i$ ($i^2=-1$) combines with the $i$'s appearing in eqs.
%\ref{gf} and \ref{fp} to give the right signs with which the Faddeev-Popov
%and gauge fixing functionals should appear in the partition function
%in the Euclidean case. On the other side, the pure C--S term $S_0$
%of eq. \ref{csaction} is metric invariant and retains the $i$




From (\ref{action}), it is possible to derive the Feynman rules of C--S
field theory in the Coulomb gauge.
The components of the gauge field
propagator
$G_{\mu \nu }^{ab}(p)$ in the Fourier space
are given by: 
\begin{equation}
G_{jl}^{ab}(p)=
-\delta ^{ab}\frac{4\pi \lambda }s\frac{p_ip_l}{{\mbox{\rm\bf p}}^4}
\label{gjl}
\end{equation}

\begin{equation}
G_{j0}^{ab}(p)=
\delta ^{ab}
\left(
\frac{4\pi }s\epsilon _{0jk}\frac{p^k}{\mbox{\rm\bf p}^2}-
\frac{4\pi \lambda }s
\frac{p_jp_0}{\mbox{\rm\bf p}^4}
\right)
\label{gjo}
\end{equation}

\begin{equation}
G_{0j}^{ab}(p)=
-\delta ^{ab}
\left( 
\frac{4\pi }s
\epsilon _{0jk}
\frac{p^k}{\mbox{\rm\bf p}^2}+
\frac{4\pi \lambda }s\frac{p_0p_j}{\mbox{\rm\bf p}^4}\right) 
\label{goj}
\end{equation}

\begin{equation}
G_{00}^{ab}(p)=
-\delta^{ab}
\frac{4\pi \lambda }s
\frac{p_0^2}{\mbox{\rm\bf p}^4}
\label{goo}
\end{equation}
with $\mbox{\rm\bf p}^2=p_1^2+p_2^2$, while the
ghost propagator $G_{gh}^{ab}(p)$ reads as follows:
\begin{equation}
G_{gh}^{ab}(p)=\frac{\delta ^{ab}}{\mbox{\rm\bf p}^2}  \label{ggh}
\end{equation}
Finally, the three gluon vertex and the ghost-gluon vertex
are respectively given by:
\begin{equation}
V_{\mu _1\mu _2\mu _3}^{a_1a_2a_3}(p,q,r)=-\frac{is}{3!4\pi }(2\pi
)^3f^{a_1a_2a_3}\epsilon ^{\mu _1\mu _2\mu _3}\delta ^{(3)}(p+q+r)
\label{aaa}
\end{equation}
and
\begin{equation}
V_{\mathrm{gh\thinspace }i _1}^{a_1a_2a_3}(p,q,r)=-i(2\pi )^3\left(
q\right) _{i_1}f^{a_1a_2a_3}\delta ^{(3)}(p+q+r)  \label{acc}
\end{equation}
In the above equation we have only given the spatial components of the
ghost-gluon vertex.
From eq. (\ref{fp}), it is in fact easy to realize that
in the Coulomb gauge
its temporal component is zero.

At this point, a regularization should be introduced in order to handle the
singularities that may arise in the computations of the Feynman diagrams.
The potential divergences are
of three kinds.

\begin{enumerate}
\item  Ultraviolet divergences (UV). The naive power counting gives the
following degree of divergence $\omega (G)$ for a given Feynman diagram $G$: 
\begin{equation}
\omega (G)=3-\delta -E_B-\frac{E_G}2  \label{napoco}
\end{equation}
with \footnotemark{}\footnotetext{We use here the same notations of ref.
\cite{itzu}}

\begin{enumerate}
\item  $\delta =$ number of momenta which are not integrated inside the loops

\item  $E_B=$ number of external gluonic legs

\item  $E_G=$ number of external ghost legs
\end{enumerate}

Eq. (\ref{napoco}) shows that UV divergences are possible in the
two and three point functions, both with gluonic or ghost
legs. Moreover, there is also a possible logarithmic divergence in the case
of the four point interaction among two gluons and two ghosts.
In principle, we had to introduce a regularization for these divergences
but in practical calculations this is not necessary.
As a matter of fact, we will see in Section 3 that there are no
UV divergences in the quantum corrections of the Green functions.

\item  Infrared (IR) divergences. 
In the pure C--S field theories \cite{hageno} there are no problems
of infrared divergences.
As a matter of fact, it can be seen from the Feynman rules written above that
the IR behavior of the gluonic propagator is
very mild ($\sim \frac 1{|\mbox{\rm\bf p}|}$). The
potentially more dangerous
IR singularities due to the ghost propagator are screened by the presence of
the external derivative in the ghost--gluon vertex (\ref{acc}).
However, we notice that IR divergences appear
in the interacting case.
For instance, in three dimensional quantum electrodynamics coupled
with a C--S term, the IR divergences have been discussed in refs.
\cite{jao,jat}. 

\item  Spurious divergences.
These singularities appear because the propagators
(\ref{gjl})--(\ref{ggh}) are undamped in the time direction and are
typical of the Coulomb gauge.
To regularize spurious divergences
of this kind, 
it is
sufficient to introduce a cutoff $\Lambda _0>0$ in the domain of integration
over the variable $p_0$: 
\begin{equation}
\int_{-\infty }^\infty dp_0\rightarrow \int_{-\Lambda _0}^{\Lambda _0}dp_0
\label{spureg}
\end{equation}
The physical situation is recovered in the limit $\Lambda _0\rightarrow
\infty $.
\end{enumerate}

\noindent 
As we will see, this regulatization does not cause
ambiguities in the evaluation of the
radiative corrections at any loop order.
In fact, the integrations over the temporal components of the
momenta inside the loops turn out to be trivial and do not interfere
with
the integrations over the spatial components.

\section{Perturbative Analysis}

In this Section we compute the $n-$point
correlation functions of C--S field theories at any loop order.
To this purpose, we choose for simplicity
the proper Coulomb gauge, setting
$\lambda=0$ in eq. (\ref{gf}).
In this
gauge the gluon-gluon propagator has only two nonvanishing components: 
\begin{equation}
G_{j0}(p)=
-G_{0j}(p)=
\delta^{ab}
\frac{4\pi }s
\epsilon _{0jk}\frac{p^k}{
\mbox{\rm\bf p}^2}  \label{gjopcg}
\end{equation}
The presence of $p_0$ remains confined in the vertices
(\ref{aaa})--(\ref{acc})
and it is trivial because it is concentrated in the Dirac $\delta $%
--functions expressing the momentum conservations. As a consequence,
the CS field theory can be considered as a two dimensional model.

First of all we will discuss the one loop calculations.
The following observation greatly reduces the number of diagrams to be
evaluated:

\begin{description}
\item[Observation:] 
Let $G^{(1)}$ be a one particle irreducible (1PI) Feynman diagram containing
only one closed loop. Then all the
internal lines of $G^{(1)}$ are either ghost or gluonic lines.
\end{description}

To prove the above observation, we notice that the only way to have
a gluonic line preceding or following a ghost line inside a loop 
is to exploit the ghost--gluon  vertex (\ref{acc}).
Thus, if a one loop diagram $G^{(1)}$ with both gluonic and ghost
legs exists, the situation illustrated in fig. \ref{figtw} should occur,
in which at least one gluonic tree diagram
$T_{\nu _1\mu _2...\mu _{n-1}\nu _n}$ is connected to the rest of $G^{(1)}$
by gluing two of its legs, those carrying the indices $\nu_1$ and
$\nu_2$ in the figure, to two ghost--gluon vertices
$V_{\mathrm{gh\thinspace }\nu_1}$ and
$V_{\mathrm{gh\thinspace }\nu_n}$.
At this point, we recall that these vertices have only spatial components
$V_{\mathrm{gh\thinspace }i_1}$ and
$V_{\mathrm{gh\thinspace }i_n}$, $i_1,i_2=1,2$.
As a consequence, since the contractions between gluonic legs are
performed with the propagator (\ref{gjopcg}), it is clear that the necessary
condition for which the whole diagram $G^{(1)}$ does not vanish
is that $\nu_1=\nu_n=0$. On the other side, this is not possible, as
it is shown by fig.
(\ref{figtr}). In fact, because of the presence of an
$\epsilon^{\mu\nu\rho}$ tensor in the gluonic vertex (\ref{aaa}), the most
general gluonic tree diagrams with $n$ legs
$T_{\nu _1\mu _2...\mu _{n-1}\nu _n}$  must have at least
$n-1$ spatial indices in order to be different from zero.
This proves the observation.
\begin{figure}
\vspace{1.5truein}
\special{psfile=fig2rrr.eps hscale=61 vscale=61 hoffset=0 voffset=0}
\vspace{0.25in}
\caption{The figure shows the only possible way in which a tree diagram
$T_{\nu_1\mu_2\ldots\mu_{n-1}\nu_n}$ with
$n$ gluonic legs can be glued to another tree
diagram containing also ghost legs in order to build a
one loop diagram with mixed ghost and gluonic internal lines.}
\label{figtw}
\vspace{1.9truein}
\special{psfile=fig3.eps hscale=61 vscale=61 hoffset=0 voffset=0}
\vspace{0.25in}
\caption{This  figure shows that in an arbitrary tree diagram
$T_{\nu_1\nu_2\ldots\nu_{n-1}\nu_n}$
constructed in terms of the
gauge fields propagator (\ref{gjopcg})  and the
three gluon vertex (\ref{aaa}), only one component in the
space-time indices $\nu_i$,
$i=1,\ldots,n$, can be temporal.}
\label{figtr}
\end{figure}
An important consequence is that,
at one loop, the only non--vanishing diagrams occur when all the external
legs are gluonic. Hence we have to evaluate only the diagrams describing the
scattering among $n$ gluons. 

This can be done as follows. First of all, we
consider the  diagrams
with internal gluonic lines.
After suitable
redefinitions of the indices and of the momenta, it is possible to see that
their total contribution is given by:
\begin{eqnarray}
V_{i_1...i_n}^{a_1...a_n}
\left(
1;p_1,...,p_n
\right)=
C\left[ -i
\left(2\pi
\right)^3
\right]^n
\frac{n!(n-1)!}2\delta^{(2)}
(\mbox{\rm\bf p}_1+...+
\mbox{\rm\bf p}_n) &&  \label{vone} \\
f^{a_1b_1^{\prime }c_1^{\prime }}f^{a_2b_2^{\prime }b_1^{\prime
}}...f^{a_nc_1^{\prime }b_{n-1}^{\prime }}\int d^2\mbox{\rm\bf q}_1
\frac{\left[
q_1^{i_1}...q_n^{i_n}+q_1^{i_2}\ldots q_j^{i_{j+1}}
\ldots q_{n-1}^{i_n}q_n^{i_1}\right] }{
\mbox{\rm\bf q}%
_1^2...\mbox{\rm\bf q}_n^2} &&  \nonumber
\end{eqnarray}
where $C=\left( 2\Lambda _0\right) ^{2n}$ is a finite constant coming from
the integration over the zeroth components of the momenta and
\begin{equation}
\begin{array}{cccc}
q_2= & q_1+p_1+p_n+p_{n-1}+ & \ldots & +p_3 \\ 
\vdots & \vdots&\ddots&\vdots \\ 
q_j=&q_1+p_1+p_n+p_{n-1}+&\ldots&+p_{j+1} \\ 
\vdots & \vdots&& \\ 
q_n=&q_1+p_1&&
\end{array}
\label{qform}
\end{equation}
for $j=2,\ldots,n-1$.
As it is possible to see from eq. (\ref{vone}),
the only nonvanishing components of $V_{\mu
_1...\mu _n}^{a_1...a_n}\left( 1;p_1,...,p_n\right) $ are those for which $\mu
_1=i_1,$ $\mu _2=i_2,...,\mu _n=i_n$, i. e. all tensor indices
$\mu_1,\ldots,\mu_n$
are
spatial.

The case of the Feynman diagrams containing ghost internal lines
is more complicated. After some work, it is possible to
distinguish two different contributions to the Green functions
with $n$ gluonic legs:
\begin{eqnarray}
V_{i_1...i_n}^{a_1...a_n}\left( 2a;p_1,...,p_n\right)=-C\left[ -i\left(
2\pi \right) ^3\right]^n\frac{n!(n-1)!}2
&&  	\nonumber \\
\delta^{(2)}(\mbox{\rm\bf p}_1+...%
\mbox{\rm\bf p}_n)
f^{a_1b_1^{\prime }c_1^{\prime }}f^{a_2b_2^{\prime }b_1^{\prime
}}...f^{a_nc_1^{\prime }b_{n-1}^{\prime }}\int d^2\mbox{\rm\bf q}_1\frac{%
q_1^{i_1}...q_n^{i_n}}{\mbox{\rm\bf q}_1^2...
\mbox{\rm\bf q}_n^2}\label{vtwoa}&&
\end{eqnarray}
and 
\begin{eqnarray}
V_{i_1...i_n}^{a_1...a_n}\left( 2b;p_1,...,p_n\right) =C\left( -1\right)
^{n-1}\left[ -i\left( 2\pi \right) ^3\right] ^n\frac{n!(n-1)!}2 &&
\nonumber \\
\delta ^{(2)}(\mbox{\rm\bf p}_1+...\mbox{\rm\bf p}_n)
f^{a_1b_1^{\prime }c_1^{\prime
}}f^{a_2b_2^{\prime }b_1^{\prime }}...f^{a_nc_1^{\prime }b_{n-1}^{\prime
}}\int d^2\mbox{\rm\bf q}'_1\frac{(q_1')^{i_1}...(q_n')^{i_n}}
{(\mbox{\rm\bf q}_1')^2...
(\mbox{\rm\bf q}_n')^2} &&
\label{vtwob}
\end{eqnarray}
where the constant $C$ is the result of the integration over the
zeroth components
of the momenta and it is the same of eq. (\ref{vone}). 
Apart from an overall sign, eqs. (\ref{vtwoa}) and (\ref{vtwob}) differ
also by the
definitions of the momenta. In (\ref{vtwoa}) the variables $q_2,...,q_n$ are in
fact given by  eq. (\ref{qform}). In eq. (\ref{vtwob}) we have instead: 
\begin{equation}
\begin{array}{cccc}
q_2'=&q_1'+p_1&& \\ 
\vdots & \vdots&& \\ 
q_j'=&q_1'+p_1+&\ldots&+p_{j-1} \\ 
\vdots & \vdots&\ddots&\vdots \\  
q_n'=&q_1'+p_1+&\dots&+p_{n-1}
\end{array}
\label{qformb}
\end{equation}
for $j=2,\ldots,n-1$.

To compare eq. (\ref{vtwob}) with  (\ref{vone}) and (\ref{vtwoa}) we
perform the change of variables
\begin{equation}
q_1=-q_1'-p_1
\label{shift}
\end{equation}
in eq. (\ref{vtwob}).
Exploiting eq. (\ref{shift}) and
the relation $p_1+...+p_n=0$, we obtain:
\[
V_{i_1...i_n}^{a_1...a_n}\left( 2b;p_1,...,p_n\right) = 
-C[-i(2\pi)^3]^n
\frac{n!(n-1)!}2f^{a_1b_1^{\prime
}c_1^{\prime }}f^{a_2b_2^{\prime }b_1^{\prime }}...f^{a_nc_1^{\prime
}b_{n-1}^{\prime }}
\]
\begin{equation}
\delta ^{(2)}(\mbox{\rm\bf p}_1+...+\mbox{\rm\bf p}_n) 
\int d^2\mbox{\rm\bf q}_1
\frac{q_n^{i_1}q_1^{i_2}\ldots q_j^{i_{j+1}}\ldots q_{n-1}^{i_n}}
{(\mbox{\rm\bf q}_1')^2...
(\mbox{\rm\bf q}_n')^2}\label{finvtb}
\end{equation}
where the variables $q_2,\ldots,q_n$ are now defined as in eq. (\ref{qform}).
At this point we can sum 
eqs. (\ref{vone}),
(\ref{vtwoa}) and (\ref{finvtb}) together.
It is easy to realize that the total result is zero, i. e.:

\begin{equation}
V_{i_1...i_n}^{a_1...a_n}\left( 1;p_1,...,p_n\right)
+V_{i_1...i_n}^{a_1...a_n}\left( 2a;p_1,...,p_n\right)
+V_{i_1...i_n}^{a_1...a_n}\left( 2b;p_1,...,p_n\right) =0 \label{finres}
\end{equation}

Still, it is not possible to conclude from eq. (\ref{finres}) that
there are no radiative corrections at one loop in
C--S field theory.
Let us remember in fact that
eq. (\ref{finres})  has been obtained from eq. (\ref{vtwob}) after performing
the shift of variables
(\ref{shift}). This could be dangerous
if there are unregulated
divergences.
However, it is not difficult to verify that
each of the integrals appearing
in the right hand sides of eqs. (\ref{vone}), (\ref{vtwoa})
and (\ref{vtwob}) is IR and UV finite for $n\ge 3$.
Only the case $n=2$ needs some more care.
Summing together eqs. (\ref{vone}), (\ref{vtwoa}) and (\ref{finvtb})
for $n=2$, we obtain the following result:
%\[
%V_{ij}^{ab}\left( 1;p_1,p_2\right) =-\left( 2\pi \right) ^6\left(
%2\Lambda _0\right) ^2N\delta ^{ab}\delta ^{(2)}(\mbox{\rm\bf p}_1+
%\mbox{\rm\bf p}%
%_2)
%\]
%\[
%\int d^2\mbox{\rm\bf q}
%\frac{\left[ q_{i}\left( q+p_1\right)_{j}+\left(
%q+p_1\right)_{i}q_{j}\right] }{\mbox{\rm\bf q}^2\left(
%\mbox{\rm\bf q}+%
%\mbox{\rm\bf p}_1\right) ^2} 
%\]
%\[
%V_{ij}^{ab}\left( 2a;p_1,p_2\right) =\left( 2\pi \right) ^6\left(
%2\Lambda _0\right) ^2N\delta ^{ab}\delta ^{(2)}(\mbox{\rm\bf p}_1+
%\mbox{\rm\bf p}%
%_2)\int d^2\mbox{\rm\bf q}
%\frac{q_{i}\left( q+p_1\right)_{j}}{\mbox{\rm\bf q}%
%^2\left( \mbox{\rm\bf q}+\mbox{\rm\bf p}_1\right) ^2} 
%\]
%\[
%V_{ij}^{ab}\left( 2b;p_1,p_2\right) =\left( 2\pi \right) ^6\left(
%2\Lambda _0\right) ^2N\delta ^{ab}\delta ^{(2)}(\mbox{\rm\bf p}_1+
%\mbox{\rm\bf p}%
%_2)\int d^2\mbox{\rm\bf q}\frac{q_{i}\left( q+p_1\right)_{j}}
%{\mbox{\rm\bf q}%
%5^2\left( \mbox{\rm\bf q}+\mbox{\rm\bf p}_1\right) ^2} 
%\]
%
\[
V_{ij}^{ab}\left( 1;p_1,p_2\right) +V_{ij}^{ab}\left(
2a;p_1,p_2\right) +V_{ij}^{ab}\left( 2b;p_1,p_2\right) = 
\]
\begin{equation}
\left( 2\pi \right) ^6\left( 2\Lambda _0\right) ^2N\delta ^{ab}\delta
^{(2)}(\mbox{\rm\bf p}_1+\mbox{\rm\bf p}_2)
\int d^2\mbox{\rm\bf q}\frac{\left[ q_{i}
(p_1)_{j}-q_{j}(p_1) _{i}
\right] }
{\mbox{\rm\bf q}^2\left( \mbox{\rm\bf q}+\mbox{\rm\bf p}_1\right) ^2}
\label{cru}
\end{equation}
where we have put $q_1'=q_1=q$.
As we see, the integrand appearing
in the rhs of (\ref{cru})
is both IR and UV
finite.  Moreover, 
a simple computation shows that 
the integral over $\mbox{\rm\bf q}$
is zero without the need of the shift (\ref{shift}).
As a consequence, there are no
contributions to the Green functions at one loop.

Now we are ready to consider the higher order corrections.
At two loop, a general Feynman diagram $G^{(2)}$ can be obtained
contracting two legs of a tree diagram $G^{(0)}$ with
two legs of a one loop diagram $G^{(1)}$.
As previously seen, the latter have only gluonic
legs and their tensorial indices are all spatial.
Consequently, in order to perform the contractions by means of the propagator
(\ref{gjopcg}),  there should exist one component of 
$G^{(0)}$ with at least two temporal indices,
but this is impossible. To convince
oneself of this fact, it
is sufficient to look at fig. (\ref{figtr}) and related comments.
The situation does not improve
if we build $G^{(0)}$  exploiting also the ghost-gluon vertex
(\ref{acc}), because it has no temporal component.
As a consequence, all the Feynman graphs vanish identically at two loop order.
Let us notice that it is possible to verify their vanishing
directly, since
the number of two loop diagrams
is relatively small in the Coulomb gauge and one has just to contract the
space-time indices without performing the integrations over the internal
momenta.
However, this
procedure is rather long and will not be reported here.

Coming to the higher order computations, we notice that
a diagram with $N+1$ loops $G^{(N+1)}$
has at least
one subdiagram $G^{(N)}$
containing $N-$loops.
Supposing that $G^{(N)}$ is identically equal to
zero because it cannot be constructed with the
Feynman rules (\ref{ggh})--(\ref{acc}) and (\ref{gjopcg}), also $G^{(N+1)}$
must be zero.
As we have seen above, there are no Feynman diagrams for $N=2$.
This is enough to prove by induction that
the C--S field theories have  no radiative corrections
in the Coulomb gauge
for any value of $N$.

\section{Conclusions}

In this paper we have proved with explicit computations that the C--S field
theories do not have quantum corrections in the Coulomb gauge.
At two loop order and beyond, this is a trivial consequence of the
fact that it is impossible to construct nonzero Feynman diagrams
starting from the vertices and propagators given in eqs.
(\ref{ggh})--(\ref{acc}) and (\ref{gjopcg}).
At one loop, instead, nontrivial cancellations occur
between the different diagrams.
We have also seen that the perturbative
expansion of the Green functions is not affected by
UV or IR divergences.
Only the spurious singularities are present, which are related to the fact
that the propagators are undamped in the time direction.
They are similar to the singularities observed in the four dimensional
Yang--Mills field theories
\cite{taylor}, but in the C--S case appear in a milder form.
In fact, after the regularizarion (\ref{spureg}), their
contribution at any loop order reduces to
a factor in the radiative
corrections and does not influence
the remaining calculations.
Therefore, the results obtained here are regularization
independent.
Moreover, the vanishing of the quantum contributions described in Section 3
is a peculiarity of the Coulomb gauge that does not strictly depend from the
fact that the C--S field theories are topological.
An analogous situation occurs in the light cone gauge in the presence of a
boundary. In that case, radiative corrections arise due to the interactions
of the fields with the boundary, but each Feynman diagram corresponding to
these interactions vanishes identically \cite{empi}.

In summary, our study indicates that the Coulomb gauge is a
convenient and reliable gauge fixing, especially in the
perturbative applications of C-S field theory.
Let us remember that, despite of the fact that the theory does non
contain degrees of freedom, the perturbative calculations
play a relevant role, for instance in the computations of knot invariants
\cite{alr}, \cite{witten}--\cite{axelrod}.
Contrary to what happens using the covariant gauges,
where it becomes more and
more difficult to evaluate the radiative corrections
as the loop number increases \cite{alr,gmm,chaichen}, 
in the Coulomb gauge
only the tree level contributions to the Green
functions survive. This feature is
particularly useful in the
case of non-flat manifolds, where the momentum representation does not exist.
For instance, Feynman rules analogous to those given in
eqs. (\ref{gjl})--(\ref{acc})
have been derived also on the compact Riemann surfaces
\cite{ffunp}.
In the future, besides the applications in knot theory, we plan to extend
our work also to C--S field theories with non-compact gauge group, in order
to include also the theory of quantum gravity in $2+1$ dimensions.
Moreover, most of the pathologies that seem to afflict the four dimensional
gauge field theories, like spurious
and infrared divergences, are also present in the C--S field
theories,
but in a milder form. As a consequence, the latter can be considered
as a good laboratory in order to study their possible remedies.
For example, it would be interesting
to apply to the Yang--Mills case
the regularization (\ref{spureg}) introduced here for the spurious
singularities.
Let us
notice that a different regularization
has been recently proposed in \cite{leiwil}.
Finally, the present analysis is limited to the pure
C--S field theories and more investigations
are necessary for the interacting case.
Until now, only the models based on abelian C--S field theory
have been studied in details, in particular
the
so-called Maxwell-Chern-Simons field theory, whose consistency
 in the Coulomb gauge
has been checked with several tests \cite{devone}.
%A physical application of our results,
%which is currently under consideration, is the
%investigation of the statistics of fermionic and bosonic matter
%fields interacting with nonabelian C--S theories at high temperatures
%\cite{higtemp}. Other interesting applications are $(2+1)$
%quantum gravity and the calculation of the new link invariants
%from C--S field theories quantized
%on Riemann surfaces, whose existence has been formally shown
%in \cite{cotta}. In these latter two cases,
%the possibility offered by the Coulomb gauge of performing explicit
%calculations also on non--flat space--times \cite{ffprd}
%can be exploited.

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\end{document}
Chern--Simons (C--S) field theories \cite{jao}--\cite{witten},
until now only a few calculations
have been performed in the Coulomb gauge \cite{cgcalc}
after the seminal works of refs. \cite{hageno, jat,hagent}.
One of the reasons is that, at least in the case of the
Yang--Mills field
theories, several perplexities arise concerning the use of this gauge
\cite{taylor}--\cite{chetsa}.
The motivations of the present work come from the advantages
offered
by the Coulomb gauge with respect to the more popular covariant and
axial gauges \cite{witten, hagt}, as already emphasized in \cite{ffprd,frig}.
First of all, the C--S theories in this gauge can be regarded
as two dimensional models. In fact, there are no time derivatives
in the gauge fixed action and
the explicit dependence on the time variable in the propagators
and vertices
is factorized in a Dirac $\delta-$function.
As a consequence,
the Feynman rules simplify considerably and perturbative
calculations are possible also in the case of
space--times admitting non--flat spatial sections \cite{ffprd}.
Moreover,
the problems afflicting the Yang--Mills field theories in the Coulomb
gauge, like
the appearance of spurious singularities and infrared divergences
in the higher loop calculations \cite{taylor,chetsa}, are
present in the
C--S field theories \cite{hageno} in a milder form.
Therefore, the latter
provide an ideal laboratory in order to
understand the more complicated Yang--Mills case.

In this paper we continue our investigations of pure C--S field theories
started in ref. \cite{frig}, where
the consistency of the
Coulomb gauge has been checked
exploiting the Dirac's canonical
approach to constrained systems.
With respect to ref. \cite{frig},
the Coulomb gauge
fixing is weakly imposed, so that
the proper Coulomb gauge is recovered
in a suitable limit of the  gauge fixing parameter.
The main result reported here is the perturbative computation of the
$n-$point correlation functions at any loop order.
In agreement with the conclusions of ref. \cite{frig}, we show that all the
radiative corrections vanish identically.
We notice that the absence of quantum corrections despite of the presence of
non-trivial self-interactions is a peculiarity of the Coulomb gauge, which 
cannot be entirely
expected from the fact that the theories under consideration
are topological.
Indeed, finite renormalizations of the fields and of the
coupling constants are always possible.
For instance, in the analogous case of the covariant gauges, where
self-interactions are also present, only the perturbative finiteness of the
C--S amplitudes has been shown \cite{csformal} using BRST techniques
\cite{ss}.
It is important to stress at this point that our calculations are
regularization independent.
At order one loop, in fact, a close inspection to the expressions of the
Green functions which are supposed to have divergent contributions
from the naive power counting
shows the absence of any ultraviolet and infrared singularity.
Roughly speaking, these are replaced by the spurious singularities
in the time variables discussed in ref. \cite{ffprd}, which are easier
to treat.
We show that the sum of all diagrams contributing to a given Green function
vanish identically without the need of introducing any regularization.
At higher loops, the situation is simpler because
the single Feynman diagrams are identically
zero after contracting the internal space--time indices
inside the loops.
The main reason of these cancellations
is that in the Coulomb gauge some components of the
propagators and vertices are missing.

Summarizing, our analysis indicates that the Coulomb gauge is a very
convenient and reliable gauge fixing in all perturbative applications
of the C--S field theories.
As a matter of fact, the self-interactions between the fields are sufficiently
non-trivial to justify a perturbative treatment of the theory and,
on the other side,
the calculations are drastically simplified with respect to
the covariant gauges.
For example,
explicit calculations in the Feynman gauge
exist until now only at the two-loop order \cite{cspert}.
In the Coulomb gauge, instead, it has been shown here that
only the tree level amplitudes survive.
In this way, there is no need to integrate
over the internal loop variables.
This is a great
advantage, particularly
the case of non-flat space--times, where the momentum representation does not
exist. Finally, the C--S field theories provide a good testing
ground for studying the Coulomb gauge in a simplified situation.
For instance, the regularization proposed here for the spurious divergences
could also be applied to the four dimensional Yang--Mills
field theories. 


In fact, we notice that, to obtain the final expression
(\ref{finvtb})
of $V_{i_1...i_n}^{a_1...a_n}\left( 2b;p_1,...,p_n\right)$ 
starting from eq. (\ref{vtwob}), a shift in the variable
$q_1'$ has been performed inside
a potentially divergent integral. As a matter of fact, the
superficial degree of divergence (\ref{napoco}) predicts that there are UV
divergences in the radiative corrections of the propagator and of the
vertices (\ref{aaa}) and (\ref{acc}). Thus the shift (\ref{shift})
can lead to possible anomalies.
However this is not the case, as it is easy to check.
The reason is that the single Feynman diagrams shown in figs. (\ref{figfo})
and (\ref{figfi}) are separately divergent for $n=2,3$, but their
sum is finite.
Let us treat for instance
the two point function. From eqs.  we obtain in the special case $n=2$ the following
contributions (here we omit the regularization \ref{pauvil} for simplicity): 

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\vfill\eject

\begin{center}
FIGURE CAPTIONS
\end{center}

\vspace{1cm}

\begin{description}
\item[\tt 1)]  A possible set of branch cuts on the complex sphere for the $%
Z_n$ symmetric algebry curves. The cuts appear symmetrically on the sheets
composing the curve.\\

\item[\tt 2)]  A possible covering of the curve $\Sigma$ in two sets $%
\Sigma_N$ and $\Sigma_N$. Only the part of the contour $\gamma$ which lies
on the $i-$th sheet is showed.\\

\item[\tt 3)]  An alternative form of the two sets $\Sigma_N$ and $\Sigma_S$%
. $\Sigma_S$ is disconnected in $n$ pieces lying on the different sheets. In
the figure only the piece belonging to the $i-$th sheet has been given.
\end{description}

\end{document}


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\begin{document}
\title{Dirac Quantization of the Chern--Simons Field Theory in the Coulomb
Gauge}

\date{September 1996}
\maketitle
\vspace{-3.5in} \hfill{Preprint PAR-LPTHE 96-38, UTF 386} \vspace{3.4in}
\begin{abstract}
In this letter the
Chern--Simons field theories are studied in the Coulomb gauge
using the Dirac's canonical formalism for constrained
systems.
Contrary to the customary procedure used in dealing with these theories,
we first work out the constraints and then quantize, replacing the
Dirac brackets with quantum commutators. The results obtained are interesting.
The Chern--Simons field theories become two
dimensional models with no propagation along the time direction.
Moreover, we prove that,
despite of the presence of non-trivial self-interactions in the
gauge fixed functional, the commutation relations between the fields
are trivial at any order in perturbation theory
in the absence of couplings with matter fields.
If these couplings are present, instead, the commutation relations become
rather
involved, but it is still possible to study their main properties and to show
that they vanish at the tree level.
\end{abstract}

\section{Introduction}
In this letter we investigate the Chern--Simons field
theories \cite{csft,witten}
with gauge group $SU(n)$ in the Coulomb gauge
using the Dirac's formalism for constrained
systems \cite{dirac,hrt}.
As it happens in the case of the more popular covariant gauges,
also in this gauge the C-S functional contains self-interactions,
but the Feynman rules simplify considerably, even in the case
of space-times admitting non-flat spatial sections \cite{ffprd}.
Another advantage of the Coulomb gauge is that the gauge fixed action
does not contain time derivatives, so that the
C--S theory becomes
in practice a two-dimensional model.

Despite of
many physical and mathematical applications of the
Chern--Simons (C--S)  field theories \cite{witten,csappl},
until now only a few calculations
have been performed in the Coulomb gauge \cite{ffprd,cgincs}.
The reason is that,
at least in the case of the Yang--Mills field theories,
several perplexities arise concerning the use of this
gauge \cite{taylor,leibbrandt,chetsa}.
In this respect, the simpler Chern-Simons models
provide an important laboratory in order to study the problems
of the Coulomb gauge and their possible remedies.
For example, the ambiguities in the Feynman integrals
pointed out in \cite{taylor}
can be
connected to the absence of time derivatives
in the action \cite{ffprd}.  Moreover,
a simple recipe to regularize such ambiguities to
all orders in perturbation theory
has been proposed and successfully tested
in first order calculations \cite{ffprd}.
Nevertheless, a detailed investigation of the consistency of the C--S field
theories in the Coulomb gauge
at all perturbative orders is still missing. To fill this gap,
we exploit in this letter the formalism of
Dirac's canonical approach to constrained systems \cite{dirac,hrt}.

In our analysis,
the computation of
all the constraints
given by the Hamiltonian procedure and by the consistency conditions
is straightforward, apart from some subtleties that are also present
in the covariant gauges \cite{linni}.
Only the derivation of the final Dirac brackets requires  some
care with distributions.
The commutation
relations (CR's) between the fields obtained here
are rather involved. At a first sight, this is surprising in topological
field theories  with vanishing
Hamiltonian and without degrees of freedom. However, at
least in the case in which there are no interactions with matter fields,
this contradiction is only apparent. As a matter of fact,
taking into account the Gauss law
and the Coulomb gauge fixing, it is shown that the commutation
relations between the gauge fields vanish identically
at any perturbative order as expected.
In this way we discover that the Chern--Simons field theories in the Coulomb
gauge are not only perturbatively finite as has been checked
in the covariant gauges \cite{covgau},
but also free.
This is  not a priori evident, because in the Coulomb gauge the 
C--S functional contains non--trivial self--interaction terms.



The material presented in this paper is divided as follows.
In Sectio 2 the C--S field theories with 



impose them
in the strong sense computing the final Dirac brackets.
Putting
\[
\chi_1^a=\text{\prova G}^a\qquad\qquad\qquad
\chi_2^a=
\partial_i A^{i,a}
\]
with $\alpha,\beta=1,2$, we have for
any two observables $A({\bf x})$ and
$B({\bf y})$ \footnotemark\footnotetext{In the following, the time
variable will be omitted from our equations.}:
\[
\left\{ A^a({\bf x}),B^b({\bf y})\right\}^{*}=\left\{ A^a({\bf x}),B^b({\bf y}%
)\right\}-
\]
\begin{equation}
\sum_{\alpha,\beta=1}^2\sum_{c,d}
\int d^2{\bf x}^{\prime }d^2{\bf y}^{\prime
}\left\{ A^a({\bf x}),\chi_\alpha^c({\bf x}^{\prime })\right\}(
C^{-1})^{\alpha\beta,cd}({\bf x}^{\prime },{\bf y}^{\prime })
\left\{ \chi ^{\beta,d}(%
{\bf y}^{\prime }),B^b({\bf y})\right\}   \label{dbndef}
\end{equation}
The matrix 
$(C^{-1})^{\alpha\beta,cd}({\bf x},{\bf y})$ denotes the inverse of the
$2\times 2$ matrix $C_{\alpha\beta}^{ab}({\bf x},{\bf y})=\{\chi^a_\alpha
({\bf x}),\chi^b_\beta({\bf y})\}$.
After some manipulations and remembering that the gauge potentials
satisfy the Coulomb gauge constraint, we obtain:
\[
{\bf C}^{ab}({\bf x},{\bf y})=\left(
\begin{array}{c c }
0 & -D^{ab}_i({\bf x})\partial_{\bf x}^i\delta({\bf x}-{\bf y})\\
D^{ab}_i({\bf x})\partial_{\bf x}^i\delta({\bf x}-{\bf y})  & 0\\
\end{array}\right)
\]
To invert the above matrix, it
is convenient to introduce the function $\text{\prova D}^{cb}
({\bf x},{\bf y})$, defined by the following equation \cite{schwinger}:
\begin{equation}
D^{ac}_i({\bf x})\partial_{\bf x}^i\text{\prova D}^{cb}({\bf x},{\bf y})=
\delta^{ab}\delta({\bf x}-{\bf y})\label{dstorta}
\end{equation}
Supposing that
the Green function $\text{\prova D}^{ab}({\bf x},{\bf y})$ has a sufficiently
good behavior at infinity, it is easy to prove that
\begin{equation}
({\bf C}^{-1})^{ab}({\bf x},{\bf y})=\left(
\begin{array}{c c }
0 & \text{\prova D}^{ab}({\bf x},{\bf y})\\
-\text{\prova D}^{ab}({\bf x},{\bf y})  & 0\\
\end{array}\right)\label{invnew}
\end{equation}
After imposing the constraints (\ref{glaw}) and (\ref{coulombgauge}) in the
strong sense, the Hamiltonian $\check H_{CS}$ vanishes, but the commutation
relations (CR's) between the fields remain complicated.
From eqs. (\ref{dbndef}) and (\ref{invnew}), in fact,  the basic DB's
between the canonical variables $A_i^a$ have the following form:
\[
\left\{A_i^a({\bf x}),A_j^b({\bf y})\right\}^*=
\frac{4\pi}{s}\delta^{ab}\epsilon_{ij}\delta({\bf x}-{\bf y})-
\]
\begin{equation}
\epsilon_{ik}\partial_{\bf x}^kD_j^{bc}({\bf y})\text{\prova D}^{ac}
({\bf x},{\bf y})-
\epsilon_{kj}D_i^{ac}({\bf x})\partial_{\bf y}^k\text{\prova D}^{cb}
({\bf x},{\bf y})\label{maincomrel}
\end{equation}
Let us study the main properties of the above DB's. First of all, they
are antisymmetric as expected:
\begin{equation}
\{ A_i^a({\bf x}), A_j^b({\bf y})\}^*=-
\{ A_j^b({\bf y}), A_i^a({\bf x})\}^*\label{antisym}
\end{equation}
The antisymmetry of the right hand side of eq. (\ref{maincomrel}) is not
explicit, but can be verified with the help of the relation:
\begin{equation}
\label{propsym}
\text{\prova D}^{ab}({\bf x},{\bf y})=\text{\prova D}^{ba}({\bf y},{\bf x})
\end{equation}
The above symmetry of the Green function $\text{\prova D}^{ab}({\bf x},{\bf
y})$ in its arguments is a consequence of the selfadjointness of the
defining equation (\ref{dstorta}) \cite{schwinger}.
Moreover, the CR's (\ref{maincomrel}) are consistent
with the Coulomb gauge. As a matter of fact, it is easy to prove that:
$$\{ A_i^a({\bf x}), \partial^jA_j^b({\bf y})\}^*=
\{ \partial^iA_i^a({\bf x}), A_j^b({\bf y})\}^*=0$$
The case of a Chern--Simons field theory with abelian gauge group $U(1)$ is
particularly instructive in order to understand the meaning of the CR's
(\ref{maincomrel}). Let $U_\mu$ denote the abelian gauge fields.
Then the Lagrangian (\ref{lagrangian}) reads:
$$L_{CS}={s\over 8\pi}\epsilon^{\mu\nu\rho}U_\mu\partial_\nu U_\rho$$
It is now possible to decompose the gauge potentials $U_i$, $i=1,2$ into
transverse and longitudinal components:
$$U_i=\epsilon^{ij}\partial_j\varphi+\partial_i \rho$$
where $\varphi$ and $\rho$ are two real scalar fields.
Exploiting the Coulomb gauge condition it turns out that $\rho=0$.
The canonical momenta are given by:
$$\pi^i=\frac{s}{8\pi}\epsilon^{ij}U_j$$
As a consequence,
from the Gauss law $\partial_i\pi^i=0$, we obtain the relation $\partial_i
\partial^i\varphi=0$. This implies that $\varphi=0$ and thus
there is no dynamics in the C--S
field theory as expected.

The CR's (\ref{maincomrel}) must be consistent with that fact.
Indeed, in the abelian case it is easy to derive the Green function
$\text{\prova D}({\bf x},{\bf y})$ solving eq. (\ref{dstorta}). The result is:
\begin{equation}
\text{\prova D}({\bf x},{\bf y})=-{1\over 2\pi} {\rm log}|
{\bf x}-{\bf y}|\label{dsabelian}
\end{equation}
Substituting the right hand side of the above equation in (\ref{maincomrel}),
we obtain:
$$[U_i(t,{\bf x}),U_j(t,{\bf y})]=0$$
so that the fields do not propagate as expected.
To conclude the discussion of the abelian case, let us notice that
eqs. (\ref{lagdet}) and (\ref{sfour})--(\ref{fixb}) admit only the
trivial solutions $U_0=\lambda_\mu=B=0$ in agreement with the fact that,
in absence of couplings with matter fields,
the C--S theory is topological and there
are no degrees of freedom. \smallskip
In the nonabelian case the situation is analogous, but the equations
of motion of the constraints become nonlinear and can in general be solved
only using a perturbative approach. The relevant equations determining the
fields $A_i^a(z)$, with $i=1,2$, are given by:
\begin{equation}
F_{12}^a=\partial_1A_2^a-\partial_2A_1^a-gf^{abc}A_1^bA_2^c\label{glclone}
\end{equation}
and 
\begin{equation}
\partial_1A_1^a-\partial_2^aA_2^a=0\label{cgclone}
\end{equation}
With respect to eq. (\ref{lagrangian}),
we have introduced here the new coupling
constant $g^2=\frac{8\pi}{9s}$ and the fields $A_\mu$ have been rescaled
in such a way that the new action becomes:
$$
L=\epsilon^{\mu \nu \rho }\left( A_\mu ^a\partial _\nu A_\rho
^a-gf^{abc}A_\mu ^aA_\nu ^bA_\rho ^c\right)
$$
In the following, we will also suppose that $g$ is so small that
a perturbative treatment of the C--S field theory makes sense.
Under this hypothesis, the fields $A_i^a$ can be expanded in powers of $g$:
$$A_i^a(x)=\sum_{n=0}^\infty g^nA_i^{a (n)}(x)$$
where, from eqs. (\ref{glclone}) and (\ref{cgclone}), the $A_i^{a (n)}$'s
satisfy the following equations:
$$\partial_1A_2^{a(0)}-\partial_2A_1^{a(0)}=0\qquad\qquad\qquad
\partial_1A_1^{a(0)}+\partial_2A_2^{a(0)}=0$$
and
\begin{equation}
\partial_1A_2^{a(n)}-\partial_2A_1^{a(n)}-gf^{abc}
A_1^{b(n-1)}A_2^{c(n-1)}=0\qquad\qquad\qquad n=1,\ldots,\infty
\label{hoone}
\end{equation}
\begin{equation}
\partial_1A_1^{a(n)}+\partial_2A_2^{a(n)}=0
\qquad\qquad\qquad n=1,\ldots,\infty
\label{hotwo}
\end{equation}
Assuming that the gauge fields vanish at infinity, the solution of the above
equations at the zeroth order is
\begin{equation}
A_1^{a(0)}(t,{\bf x})=A_2^{a(0)}(t,{\bf x})=0\label{cvd}
\end{equation}
as shown in the abelian case.
Moreover, from eq. (\ref{hoone}), it turns out that $A_1^{a(n)}(t,{\bf x})=0$
for $n=1,\ldots,\infty$, so that all the field configurations solving
eqs. (\ref{glclone})--(\ref{cgclone}) vanish identically.
Pure gauge solutions obtained performing gauge transformations
are not allowed because, at least within
perturbation theory, the Coulomb gauge fixes the gauge freedom completely.
As a consequence, the right hand side
of (\ref{maincomrel}) is equal to zero. Indeed, due to (\ref{cvd}), the
Green function $\text{\prova D}^{ab}({\bf x},{\bf y})$
is given by:
$$\text{\prova D}^{ab}({\bf x},{\bf y})=-\delta^{ab}{1\over 2\pi} {\rm log}|
{\bf x}-{\bf y}|$$
and, substituting in eq. (\ref{maincomrel}), we obtain:
$$\{ A_i^a({\bf x}), A_j^b({\bf y})\}^*=0$$
as expected. Finally, the vanishing of the gauge fields leads to the
trivial solutions $A_0^a=\lambda_\mu^a=B^a=0$  for the Lagrange multipliers
as in the abelian case. To quantize the theory, we have to replace
the Dirac brackets (\ref{maincomrel}) with commutators.
At least in the absence of coupling with matter fields, we obtain
trivial commutation relations between the gauge potentials:
\begin{equation}
\left[A_i^a({\bf x}),A_j^b({\bf y})\right]=0\label{quantumcr}
\end{equation}
%This result shows that
%also in the Coulomb gauge, where self-interactions are present in the
%gauge fixed action, the C-S field theory is a free theory and miraculous
%cancellations should happen at any perturbative order in the
%$n-$point functions.

\section{Conclusions}
In this paper the C--S field theories have been quantized in the Coulomb gauge
within the Dirac's canonical approach to constrained systems.
All the constraints coming from the Hamiltonian procedure and by the
Dirac's consistency requirements have been derived.
As anticipated in the Introduction, the C--S theories become
in this gauge two dimensional
models. Only the fields $A_i^a$, for $i=1,2$, have in fact a dynamics, which
is governed by the commutation relations (\ref{maincomrel}).
If no interactions with matter fields are present, we have shown
that these CR's vanish at all perturbative orders.
Thus the C--S field theories in the Coulomb gauge are not only finite,
but also free.
Work is in progress to verify this result with explicit perturbative
calculations of the correlation functions.
A natural question that arises at this point is if analogous conclusions
can be drawn for the covariant gauges.
For this reason it would be
interesting to repeat the procedure of canonical quantization developed
here 
also in this case.

The situation becomes different
if the interactions with other fields are switched on.
Adding for instance a coupling with a current $J^a_\mu$ of the kind
$\int d^2{\bf x}A_\mu^a J^{\mu,a}$ to the Hamiltonian (\ref{neham}),
it is possible to see that the Gauss law (\ref{glaw}) is modified as follows:
$$D_i^{ab}\pi ^{i,b}+\partial _i\pi
^{i,b}+J_0^a\approx 0$$
Thus eqs. (\ref{cvd}) are no longer valid and we have to consider the
full commutation relations (\ref{maincomrel}).
Remarkably,
they trivially vanish at the zeroth level in the coupling constant $g$.
%For instance, even the Dirac bracket
%$\left\{A_i^a({\bf x}),A_i^a({\bf y})\right\}^*$, where no summation
%in the index $i$ is understood, is not zero.
Moreover, the CR's (\ref{maincomrel})
are perfectly well defined and do not lead
to ambiguities in the quantization of the
C--S models in the Coulomb gauge.
In particular, we have verified here the consistency of (\ref{maincomrel})
with the Coulomb gauge fixing and their antisymmetry under the
exchange of the fields.
%
% work is in progress in checking the
%triviality of the dynamics of the fields in the Coulomb gauge
%by explicit perturbative calculations. At one loop, this has been
%done until now only for the two point function in ref. \cite{ffprd}.
%Moreover, in this paper the Coulomb gauge has been strongly imposed,
%but also the case

\begin{thebibliography}{99}

\bibitem{dirac} P. A. M. Dirac, {\it Lectures in Quantum Mechanics},
Yeshiva University Press, New York 1964.
\bibitem{hrt} A. Hanson, T. Regge and C. Teitelboim, {\it Constrained
Hamiltonian Systems}, Accademia dei Lincei, Roma, 1976 and references
therein.






\bibitem{schwinger} J. Schwinger, {\it Phys. Rev.} {\bf 125} (1962), 1043.
\bibitem{linni} Q.-G. Lin and G.-J. Ni, {\it Class. Quantum Grav.} {\bf 7}
(1990), 1261.
\end{thebibliography}
\end{document}
































