\documentstyle[epsf,12pt]{article}                  
\begin{document}
\def\salto{\par\vskip .5cm}
\date{}
\author{C. BECCHI}
\title{INTRODUCTION TO BRS SYMMETRY}
%\REF{10}
\maketitle

\salto
\centerline{\it Dipartimento di Fisica, Universit\`a di Genova,}
\centerline{\it Istituto Nazionale di Fisica Nucleare, Sezione di Genova,}
\centerline{\it  via Dodecaneso 33, 16146 Genova (Italy)}
\salto
\noindent 
This paper contains the lecture notes of a short course on the quantization of gauge
theories. Starting from a sketchy review of scattering theory, the paper describes
the lines of BRST-Faddeev-Popov quantization considering the problem of a
non-perturbative extension of this method. The connection between Slavnov-Taylor
identity and S-matrix unitarity is also discussed.

 \vfill \footnote{Lectures given at
the ETH, Zurich, May  22-24, 1996}

\eject


\def\.{\cdot}
\def\la{\lambda}
\def\s{\sigma}
\def\bs{{\bar\sigma}}
\def\t{\tau}
\def\o{\over}
\def\v{\vec}
\def\a{\alpha}
\def\z{\zeta}
\def\c{\gamma}
\def\b{\beta}
\def\d{\delta}
\def\k{\chi}
\def\f{\phi}
\def\C{\Gamma}
\def\S{{\bf S}}
\def\P{\Psi}
\def\T{\Theta}
\def\La{\Lambda}
\def\O{\Omega}
\def\x{\xi}
\def\n{\eta}
\def\u{\omega}
\def\ub{{\bar \u}}
\def\o{\over}
\def\p{\partial}
\def\ip{\int {d p\o  (2\pi)^4}}
\def\inx{\int d^4 x}
\def\iny{\int d^4 y}
\def\inz{\int d^4 z}
\def\dv{d \vec}
\def\ne{\not=}
\def\+{\bigoplus}
\def\D{\Delta}
\def\fo{{\cal F}_0}
\def\oo{{\cal O}}
\def\fc{{\cal F}_C}
\def\({\left(}
\def\){\right)}
\def\[{\left[}
\def\]{\right]}
\def\l.{\left.}
\def\r.{\right.}

\def\sec{\section}
\def\ss{\subsection}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray}&&}
\def\eea{\end{eqnarray}}
\def\nn{\nonumber \\ &&}
\def\nnn{\nonumber \\ }
\def\acca{\right.\nn\left.}
\def\ber{\begin{array}}
\def\eer{\end{array}}


\def\sp{\par\vfill\eject}
\def\quasisalto{\par\vskip .4cm}
\def\saltone{\par\vskip .7cm}
\def\saltino{\par\vskip .2cm}
\sec{Introduction}
\salto
These lectures begin recalling some general results of scattering theory 
\cite{1}.  The reduction formulae for the S-matrix  are given in terms of the Green
functional in the case of a massive field theory together with The Feynman formula for
the Green functional. Then the analysis comes to  gauge theories for which the concept
of gauge orbit is introduced.  The Faddeev-Popov definition \cite{fp} of a finite
functional measure is given. The BRST external differential operator along the orbits
is introduced together with the full  BRS operator \cite{brs}. The gauge algebra of the
infinitesimal gauge transformations is briefly discussed.
Assuming the existence of a global gauge fixing (no Gribov ambiguity \cite{3})
the Slavnov-Taylor identity and the gauge fixing independence of the
theory is deduced from the BRS invariance of the functional measure.
The extension of the Faddeev-Popov formula to the case of Gribov
ambiguities is briefly discussed together with that of the Slavnov-Taylor
identity.
The Slavnov-Taylor identity is then translated in terms of the proper
functional ( the effective action ) and the extension of the method to the
case of gauge algebras that are closed only modulo the field equations is
discussed.
Limiting for simplicity the study to the case of massive fields the
Slavnov-Taylor identity is applied to the two-point functions, this leads
to the introduction of the BRS symmetry for the asymptotic fields ( in
the form of Kugo and Ojima \cite{ko} )  and to the proof the existence of a
physical Hilbert space in which the S-matrix is unitary.


\sec{The S-matrix}
\salto
In quantum field theory \cite{1} the scattering amplitudes are 
computed by means of the reduction formula. This can be simply written using
the Green functional  generator of the theory that is defined according:
\be Z[j]\equiv e^{iZ_c}=<\O,T\( e^{i\inx \f (x) j(x)}\) \O>\ .\label{1}\ee
where $\f$ and $j$ in general label a set of quantized fields  and
corresponding sources with different Lorentz covariances. $Z_c$ is the
connected functional. The Green functional is computed by means of the Feynman
formula: \be Z[j]=\int d\mu\  e^{i\inx \f (x) j(x)}\ ,\label{2}\ee in terms
of the functional measure $d\mu$ of the theory that is deduced from it bare
action by the heuristic relation; $d\mu=\prod_x d\f (x) e^{i S (\f )}$.

Computing the two-point function according:
\be{\d^2\o\d j_\a(x)\d j_\b(0)}Z_c|_{j=0}\equiv \D^{\a\b}(x)\ ,\label{3}\ee
and excluding for simplicity the presence of massless fields,  we can
separate from $\D$ the asymptotic propagator $\D_{as}$:
\be \D^{\a\b}(x)=\sum_\la\ip{e^{ipx}\o m_\la^2-p^2-i0_+}\C_\la^{\a\b}(p)
+R(x)\equiv \D^{\a\b}_{as}(x)+R(x)\ ,\label{4}\ee
where the Fourier transform of $R$ has no pole in $p^2$. It is clear that the
asymptotic propagator is by no means unique since $\C_\la^{\a\b}$ is defined up to a
polynomial in $p^2$ vanishing at $m_\la^2$; however this lack of uniqueness
does not affect the $\S$ matrix. Then one introduces the asymptotic free fields
$\f_{in}$ with the commutation relations: \be\[\f_{in}^{\a (+)} (x),\f_{in}^{\b (-)}
(0)\]=\sum_\la\ip e^{ipx} \theta (p^0)\d (p^2-m_\la^2)\C^{\a\b}(p)\ ,\label{5}\ee
and the asymptotic wave operator: \be K_{\a\c}(\p)\D^{\c\b}(x)=\d_\a^\b\d (x)\
,\label{6}\ee and one computes the S matrix of the theory according:
\be \S =:e^{\inx\f^\a (x) K_{\a\b}(\p ){\d\o\d j_\b (x)}}:Z  |_{j=0}\equiv 
:e^\Sigma : Z
|_{j=0}\
.\label{7}\ee
\sec{Gauge invariance and BRS symmetry}
\salto
Now we come to the quantization of gauge theories. In this course we shall
disregard the crucial problem of the explicit non-perturbative 
construction of the theory, limiting our analysis to the formal and symmetry
aspects that should allow the construction and characterize the solution of
the full quantum theory. We shall be often concerned with the functional
measure of the theory; avoiding any consideration of its actual definition we
shall indifferently pass from the Minkowskian form
 \be d\mu=\prod_x d\f (x) e^{i S (\f)}\label{mm}\ee to the Euclidean one
 \be d\mu=\prod_x d\f (x)e^{- S (\f )}\ .\label{me}\ee Furthermore, in order to
simplify the notation we shall merge all the labels of the fields into a single
index, thus disregarding the space-time variable. Assuming the usual convention of
summation over repeated indices we shall also often omit the integration
symbol. However one should keep firmly in mind that the fields are local
variables and that locality is considered to play a crucial role in field
theory. For many reasons we shall also avoid discussing many mathematical
aspects that should bring our analysis too far from its purposes.
Let us call ${\cal F}_0$ the field space, that is the configuration space upon
which the gauge theory is constructed. In a gauge theory $\fo$ is fibered by the
gauge orbits ${\cal O}$ that is the set of gauge transforms of a given
configuration.  Considering the
infinitesimal transformations and translating everything in differential
geometry terms ( we are freely
following e.g. \cite{2}), we are given a system of partial differential
operators $\{X\}$ on $\fo$, that we shall label with the index $I$,
 and that in any point of
$\fo$ define a system of tangent vectors to the corresponding orbit. Denoting the
generic field coordinate in $\fo$ by the $\f^\a$,  we can write these operators 
in the form \be X_I=P^\a_I(\f )\p_{\f^\a}\ ,\label{8}\ee
or more explicitly in a pure non abelian case:
\be X_a(x)=\p_\mu{\d\o\d A^a_\mu(x)}-gf_{ab}^c  A^b_\mu(x) 
{\d\o\d A^c_\mu(x)}\ .           \ee 
The system $\{X\}$ is often called the differential
system of the orbits, whose very existence implies:
\be\[X_I,X_J\]=C_{IJ}^K (\f )X_K\ ,\label{9}\ee
that is the complete integrability of $\{X\}$. In the standard situation
the algebra (\ref{9}) is a Lie algebra, the structure functions $C_{IJ}^K$ are
constants. We shall see in the following how (\ref{9}) can be weakened
restricting the integrability condition to the "mass shell", that is modulo
the field equations. We also assume that an $X$-invariant measure be uniquely
defined up to an orbit-independent normalization constant. This we shall call the
vacuum invariant measure, that is associated with the vacuum state of the theory. In
general one is interested in the vacuum correlators of local observables; these
correspond to the integrals of a different class of invariant measures that can be
written as the product of the vacuum measure times  gauge invariant functionals
depending on the field variables corresponding to a suitably localized space-time
domain. A generic invariant measure will be assumed to belong to this class.

 The vacuum functional measure is constant over the orbits
${\cal O}$; in general this makes the functional measure of $\fo$ non integrable
and the Green functional ill defined. 
This difficulty is cured by the
Faddeev-Popov trick. To recall it conveniently let us assume, even that
 if this is in general not the case, that the original
field variables trivialize the fibration; that is let us assume that the set
of fields $\{\f\}$ is decomposed according $\{\x\} $ and $\{\n\}$ where  $\{\x\}
$ are constant along the orbits and $\{\n\}$ are "vertical" coordinates. Then
it is natural to make the measure integrable by multiplying it through an integrable
functional of $\n$ whose integral over ${\cal O}$ corresponding to the above
mentioned $X$-invariant  measure be independent of the orbit ( of $\xi$ ). One
often considers the invariant Dirac $\d_{inv} \[\n-\bar\n\]$:
\be \d_{inv} \[\n-\bar\n\]\equiv\d \[\n-\bar\n\] det |X_I\ \n^J|\ ,\label{10}\ee
 but more generally one can consider its
convolution  with a suitable $\bar\n$-functional.

Notice that the determinant appears in (\ref{10}) since the action of a gauge
transformation does not correspond to a Euclidean transformation on the $\n$
variables. The Faddeev-Popov measure \cite{fp} is obtained by the substitution:
 \be d\mu\rightarrow d\mu\ 
\d_{inv}\[\n-\bar\n\]\ , \label{11}\ee 
The invariant  Dirac measure can be
easily written as a functional Fourier transform.
Introducing two
sets of Grassmann variables $\{\u^I\}$ and $\{\ub_J\}$ that can be simply
identified with the generators of an exterior algebra  and the corresponding
derivatives that we label by $\{\p_{\u ^I}\}$ and by $\{\p_{\ub _J}\}$,
one introduces the Berezin integral: \be\int d\u^I\equiv{1\o\sqrt{2\pi i}}
\p_{\u ^I}\
,\label{11a}\ee for $\u$ and an analogous definition for $\ub$. Then using
the so called Nakanishi-Lautrup multipliers $\{b_J\}$ one can reproduce the
right-hand side of (\ref{10}) in the form: 
\be\int\prod db_I\prod d\u^J\prod d\ub_K\
\ e^{i\[ b_I\n^I- \ub_I\u^JX_J\n^I\]}\ .\label{13}\ee
This formula can be interpreted as an enlargement of the field space $\fo$
with the addition of a set of ordinary fields corresponding to the 
Nakanishi-Lautrup multipliers and of two sets of anticommuting fields
corresponding to the exterior algebra generators. We call $\fc$ the new space. We
also introduce the measure on $\fc$:
\be d\mu_C\equiv d\mu\prod_I db_I\prod_J d\u_J\prod_K d\ub_K \ .\label{mc}\ee


Looking now  into the details of  (\ref{13}), we see that  the
differential operator $\u^IX_I$ appearing in the exponent can be replaced
by its minimal nilpotent extension:
 \be
d_V=\u^IX_I-{1\o2}\u^I\u^J C_{IJ}^K (\f )\p_{\u_K}\ ,\label{15}\ee
 This
operator, that is often called the BRST operator \cite{brs}, is nilpotent due to
(\ref{9}) and to the corresponding Jacobi identity. That is:
 
 \bea\label{16}d_V^2=0 \leftrightarrow\left
\{\ber{l}\u^I\u^J\(X_IX_J-{1\o2} C_{IJ}^K (\f )X_K\)=0;\\ 
\u^I\u^J\u^K\( C_{IJ}^M (\f )C_{MK}^L (\f )-X_IC_{JK}^L (\f )\)=0. 
\eer\right. \label{16a}\eea


 Identifying the system $\{\u\}$ with that of the $X$-left-invariant forms
  we can interpret  the differential operator  $d_V$ as the
vertical exterior differential operator  on $\fo$ that is with the operator on
$\fo$ that in any point is identified with the exterior differential operator on the
orbit. 

Let us now come back to the trivializing coordinates, it is clear that these exist
globally only in very special cases, in particular when the corresponding fibration is
trivial. However in order that (\ref{11}) define a finite measure over $\fo$ it is
sufficient that the
global section of $\fo$ considered as a fibered space be given,
that is that it identify a single point on any orbit. This condition is equivalent
to the existence of a system of local
functionals $\{\Psi (\f)\}$ that we shall continue to label with the index $I$,
for which the Jacobian $det|X_I\Psi^J |$ does not vanish in the points where
$\Psi=\bar \Psi $ for some $\bar\Psi$. 

Assuming this condition, we shall replace in the exponent
in  (\ref{13}) the coordinate $\n^I$ by a generic functional $\Psi^I(\f)$ writing the
exponent as: \be i S_{GF}=i\[ b_I\(\Psi^I-\bar\Psi^I\) -\ub _I d_V \Psi^I\]\ ,\label{17}\ee
The above formula gives the definition of the gauge fixing action $S_{GF}$.

(\ref{17}) can be translated in a simpler form introducing a new exterior
derivatiion $s$ acting on the algebra generated by $\u$ and $\ub$ whose action on $\f$
and $\u$ coincides with that of $d_V$  and:
\be s\ub=b\ \ \ ,\ \ \ sb=0\ , \label{18}\ee
and hence
\be s=d_V+b_I\p_{\ub_I}\ .\label{19}\ee
It it clear that $s$ is nilpotent and that (\ref{17}) is written:
\be S_{GF}= s\[\ub_I\(\Psi^I-\bar\Psi^I\)\]\ ,\label{20}\ee
it is also obvious that $s$ commutes with the physical functional measure $d\mu$.
A further generalization of the measure, that includes also the convolutions 
of  (\ref{20}) with generic functionals of $\bar\Psi$, is obtained extending the
choice of $\Psi$ to $b$, $\u$ and $\ub$-dependent local functionals. In the
following we shall replace $\ub_I\(\Psi^I-\bar\Psi^I\)$ with a generic functional
$\Theta$ carrying the same quantum numbers. In the standard situation $\Theta$ is
a strictly local quadratic functional of $b$; that is it is the space-time integral
of a second order polynomial in $b$, independent of its derivatives; therefore $b$
is an auxiliary field. However there are models, in particular in supergravity, in
which \cite{nie} $b$ corresponds to propagating degrees of freedom that play the
role of extra ghosts. 

A further comment on the auxiliary role of $b$ and hence of $\ub$ is here
necessary. In this study we are tacitly considering the field space $\fc$ finite
dimensional, strictly speaking this is, of course, not true since every field
corresponds to an infinite number of variables; however one assumes that some
mechanism, e.g. asymptotic freedom, renders finite the number of effective
degrees of freedom.\eject With this proviso let us consider:
\bea \int\prod_I db_I\prod_Jd\ub_J\ e^{is\Theta}=lim_{\epsilon \rightarrow 0}
\int\prod_I db_I\prod_Jd\ub_J\ e^{is\Theta-\epsilon\sum_Kb_k^2}
\nn=i\ lim_{\epsilon \rightarrow 0}\int\prod_I db_I\prod_Jd\ub_J\int_0^1dt\
s\(\Theta\) e^{is\Theta t-\epsilon\sum_Kb_k^2}\nn=d_V\[i\ lim_{\epsilon \rightarrow
0}\int\prod_I db_I\prod_Jd\ub_J\int_0^1dt\ \Theta e^{is\Theta
t-\epsilon\sum_Kb_k^2}\] \ ,\label{exact}\eea where we have used the fact that the
Berezin integral of a constant gives zero. (\ref{exact}) shows that the Faddev-Popov
measure corresponds to the insertion of a $d_V$-exact factor into the functional
measure and the fields $b$ and $\ub$ are auxiliary in the sense that they allow the
explicit construction of this factor in local terms. Furthermore we see from 
(\ref{exact}) that the resulting measure on $\fo$ that is:
\be d\mu\int\prod_Kd\u^K\prod_I db_I\prod_Jd\ub_J\ e^{is\Theta}\ee
is an exact top form. It follows that its integral over a compact cycle, such as a
 gauge group orbit of a lattice gauge theory,
vanishes \cite{neu}\footnote{I thank M.Testa for calling my attention to this
reference}. This is due to the fact that the gauge fixing equation $\Psi=\bar\Psi$ has
on a cycle an even number of solutions whose contributions to the above measure cancel
pairwise. However it should be clearly kept in mind that according to the Faddeev-Popov
prescription the functional integral should not cover the whole orbits but only a
compact subset of every orbit containing a single solution of the gauge fixing
equation. To be explicit let us consider the extreme example in which $\fo$ reduces to
a circle, a single $U(1)$ gauge orbit; choosing $\Theta=\ub \ sin\ \varphi$ setting;
$s=i\u\p_{\varphi}+b\p_{\ub}$ and integrating over the whole space, one gets 
\be\oint d\varphi \int db d\u d\ub e^{i b sin\ \varphi +\ub\u cos\ \varphi} =\oint
d\varphi\d\(sin\  \varphi\)cos\ \varphi=0  \ ,\ee
while with the actual prescription, one has:
\be\int_{-\epsilon}^{\epsilon} d\varphi \int db d\u d\ub e^{i b sin\ \varphi 
+\ub\u cos\ \varphi}=1\ .\ee


To conclude this section let us remember \cite{3} that in the case
of covariant and local gauge choices the condition
that $\Psi=\bar\Psi$ define a global section of the orbit space does not hold true,
the situation is less clear for the so called axial gauges, that however suffer even
worst diseases \cite{gau}. We shall see how this difficulty can be overcome in
the situation in which $\fo$ can be divided into a system of cells $U_a$ in which one
can find for every cell a $\Psi_a$ defining a section of the cell.


\sec{The Slavnov-Taylor identity}
\salto

The particular structure of the functional measure allows an immediate proof, up to
renormalization effects, of the Slavnov-Taylor (S-T) identity. That is: for
any measurable functional $\Xi$ :
\be\int d\mu_C \ e^{iS_{GF}} s\ \Xi=0\ .\label{21}\ee
Indeed, using the same arguments as for (\ref{exact}), we get:
\bea\int d\mu_C \ e^{iS_{GF}} s\ \Xi=\int d\mu_C\  s\[ e^{iS_{GF}} \ \Xi\]\nn=
\int d\mu\prod_Id\u^Id_V\int\prod_Jdb_J\prod_Kd\ub_K\ e^{iS_{GF}} \ \Xi=0\
,\label{sti} \eea
since the last expression apparently corresponds to an exact top form whose support,
according to the general prescription, is contained in the integration domain, and
hence that vanishes on the boundaries of this domain.
 Considering again the extreme example of last section consider e.g.:
\bea\int_{-\epsilon}^{\epsilon} d\varphi \int db d\u d\ub e^{i b sin\ \varphi 
+\ub\u cos\ \varphi}s\[\ub A\(\varphi\)\]\nn=
\int_{-\epsilon}^{\epsilon} d\varphi \int db d\u d\ub e^{i b sin\ \varphi 
+\ub\u cos\ \varphi}\[b A\(\varphi\)-i\ub\u A'\(\varphi\)\] \nn=
\int_{-\epsilon}^{\epsilon} d\varphi\p_{\varphi} \int db  e^{i b sin\
\varphi } A\(\varphi\)=0
\ , \eea
since $ \int db  e^{i b sin\ \varphi } A\(\varphi\)$ vanishes at
$\varphi=\pm\epsilon$.


 The identity (\ref{21}) can be interpreted saying that all correlators between 
elements of the image of $s$ and
$s$-invariants  vanish. Indeed, according to its definition given in section 3, the
$s$-invariant functionals can be considered
 to be generic local factors in the invariant measure
$d\mu$.
Considering the $s$ operator as the natural extension of $d_V$, the exterior
derivative corresponding to the gauge transformations, it is natural to
assume as a basic principle of gauge theories the identification of
observables with $s$-invariant functionals. Due to the nilpotency of $s$ this set
contains the image of $s$, whose elements, however, correspond to trivial observables
according to (\ref{21}). Therefore the non-trivial observables belong to the quotient
space of the kernel of $s$ versus its image, that its to the cohomology of $s$.

It remains to verify that the cohomology of $s$ is equivalent to that
of the vertical exterior differential operator $d_V$. Indeed consider the functional
differential operator:
\be \D\equiv-\(b_I\p_{b_I}+\ub_I\p_{\ub_I}\)\ .\ee
Let $P$ be the projector on the
kernel of $\D$, that is on the $\ub$ and $b$-independent functionals. It is apparent
that $\D$ and hence $P$, commutes with $s$; therefore a generic $s$-invariant:
\be \(d_V +b_I\p_{\ub_I}\)X=0\ ,\label{kern}\ee
is the sum of two terms: $PX$ and $(1-P)X$, satisfying:
\be d_VPX=0\quad,\quad s\(1-P\)X=0\ .\ee
In much the same way, an  element of the image of $s$: $Z=sY$, is decomposed
according:
\be PZ=d_VPY\quad,\quad (1-P)Z=s(1-P)Y\ .\ee
Therefore the cohomology of $s$ is the union of that of $d_V$ in the kernel of $\D$
and that of $s$ in the kernel of $P$.
We want to verify that this second contribution is trivial. Indeed, consider the
differential operator: $\ub_I\p_{b_I}$, satisfying:
\be\{\ub_I\p_{b_I},s\}=-\D\ ,\ee
for $X$ $s$-invariant, this yields:
\be s\ub_I\p_{b_I}(1-P)X=-\D(1-P)X\rightarrow 
(1-P)X=-s\D^{-1}\ub_I\p_{b_I}(1-P)X\ ,\ee and hence $(1-P)X$ belongs to the image of
$s$.

  A second consequence of (\ref{21}) is the gauge fixing independence of the
correlators of observables. Indeed let us compare
 the expectation values of the same
$s$-invariant functional $\O$ computed with two   measures corresponding to
choices of $\Theta$  differing by $\d\Theta$.
 To first order in $\d\Theta$ the difference of the expectation values is given by
\be\  i\int d\mu \ e^{iS_{GF}} s\( \d \Theta\) \O =0\ .\label{22}\ee
Of course this implies the independence of the expectation values of the choice of
$\Theta$ in a certain class of measurable functionals. Even in perturbation theory
this is not enough to prove that the expectation values in a renormalizable gauge
coincide with those in a non-renormalizable one.

We now come to the problem of extending the functional measure to the situation in
which the gauge fixing  is defined only locally.
In general the orbit manifold has to be divided into cells, each corresponding to a
different choice of $\T$: every cell in the orbit space corresponds to a cell $U_a$
in $\fo$. Let $\chi_a (\f)$ be a suitable smooth positive function with support in
$U_a$ and and such that the set  $\{\chi\}$ be a partition of unity on the union of
the supports of the gauge-fixed measures $d\mu_C e^{is\T_a}$. This is shown in the
figure where the shaded bands corresponds to the support of the measures and the
circles to the cells.
\begin{figure}
\begin{center}
\leavevmode  
\epsfxsize=6cm
\epsffile{cell.eps}
\end{center}
\end{figure}
  Explicitly a cell will be defined giving its center, that
is a special configuration $\f_a$ (background field), and defining the characteristic
functions $\chi_a$ according:
\be\chi_a(\f)=\theta\(R^2_a-\Vert\f-\f_a\Vert^2\)\prod_b\epsilon\(\Vert\f-\f_b\Vert^2-
\Vert\f-\f_a\Vert^2\)\ ,\ee
where $\theta$ is a smoothed Heavyside function and $\epsilon (x)=\theta (x)-
\theta (-x)$; $\Vert\f-\f_a\Vert$ is the $L^2$ norm of the difference
 $\f -\f_a$. Hints about the values of $R_a$ can be found in \cite{zw}. 

The BRS invariant functional measure corresponding to this
local gauge choice is given by \cite{bi}: \bea  d\mu_C\[\sum_a\chi_ae^{is\T_{a}}-
i^n\sum_{n=1}^{\infty}{(-1)^{n(n-1)\o2}\o n+1}\(s\chi_{a_{1}}...s\chi_{a_{n}}\
\chi_{a_{n+1}}\)_A
\r.\nn\l.\p_{\T}\(\T_{a_{1}}...\T_{a_{n+1}}\)e^{is\T_{a_{1},...,a_{n+1}}}\]\
,\label{1a}\eea 
where we have used the definitions:
\be \p_{\T}\(\T_{a_{1}}...\T_{a_{n}}\)\equiv\sum_{l=1}^n(-1)^{l+1}
\T_{a_{1}}...\check\T_{a_l}...\T_{a_{n}}\ .\label{def1}\ee
and:
\be
e^{is\T_{a_{1},...,a_{n}}}\equiv\int_0^\infty\prod_{i=1}^ndt_i\ \d\(\sum_{j=1}^nt_j-1\)
e^{is\sum_{k=1}^n t_k \T_{a_k}}\ .\label{def2}\ee
It is also assumed that the effective order of multiple cell intersections be finite.
In the Appendix [(\ref{a7})] it is proven that under this hypothesis the measure
(\ref{1a}) satisfies the Slavnov-Taylor identity.

 The lack of gauge invariance of the
characteristic functions of the cells induces new contributions to the measure
localized on the cell (regularized) boundaries. Of course the above measure could be
ill defined if the cells would accumulate around some singularity of $\fo$. This could
perhaps induce instabilities of BRS symmetry in the sense of \cite{fu}.

 Another version of the S-T identity
concerns the Green functional that involves the sources $j_\a$ of $\f^\a$,
$J^I$ of $b_I$, $\bs_I$ of $\u^I$ and $\s^I$ of $\ub_I$ and is defined according:
\be Z\[\ j , J , \bs , \s\ \]\equiv\int d\mu_C \ e^{iS_{GF}} e^{i\ \[j_\a\f^\a+J^I
b_I+\bs_I \u^I+\s^I \ub_I\]}\ .\label{23}\ee 
The mentioned version of the S-T
identity is of course:
 \be \int d\mu_C \ e^{iS_{GF}} s\ e^{i\ \[j_\a\f^\a+J^I
b_I+\bs_I \u^I+\s^I \ub_I\]}=0\ .\label{24}\ee It is  possible, exploiting
the nilpotency of $s$, to translate (\ref{24}) into a functional differential
equation for $Z$; this requires the introduction of further sources for the
non-linear terms appearing in the $s$-transforms of the fields; these are the source
$\c_\a$ for $s\f^\a$ and $\z_I$ for $s\u^I$ that appear in a further factor in the
functional measure: \be d\mu_C \ e^{iS_{GF}}\rightarrow d\mu_C \ e^{iS_{GF}}
 e^{i\ \[\c_\a s\f^\a + \z_I s \u^I\]}=d\mu_C \ e^{iS_{GF}}
 e^{-i\ s\[\c_\a \f^\a - \z_I  \u^I\]} \.\label{25}\ee
Notice that the introduction of the sources for the fields and their variations has
enlarged the functional exterior algebra where our functionals are defined; indeed
in particular $\s , \bs$ and $\c$ are odd elements of this algebra. In the
following formulae many derivatives are in fact anticommuting derivatives, this
induces some obvious change of sign.

Now, inserting the new measure into  (\ref{24}) we get:
\be \int d\mu_C \ e^{iS_{GF}}e^{i\ \[\c_\a s\f^\a + \z_I s \u^I\]}
s\ e^{i\ \[j_\a\f^\a+J^I b_I+\bs_I \u^I+\s^I \ub_I\]}=0\ ,\label{26}\ee 
that is:
\bea \int d\mu_C \ e^{iS_{GF}}e^{i\ \[\c_\a s\f^\a + \z_I s \u^I\]}
\[j_\b s\f^b - \bs_J s \u^J \r.\nn\l. - \s^K b_K\]
 e^{i\ \[j_\a\f^\a+J^I b_I+\bs_I \u^I+\s^I \ub_I\]}=0\ ,\label{27}\eea 
this is equivalent to the first order partial differential equation for the
extended Green functional:
\bea \[j_\b \p_{\c_\b} - \bs_J \p_{\z_J} - \s^K \p_{J^K}\]
\int d\mu_C \ e^{iS_{GF}}e^{i\ \[\c_\a s\f^\a + \z_I s \u^I\]}
 e^{i\ \[j_\a\f^\a+J^I b_I+\bs_I \u^I+\s^I \ub_I\]}\nn\equiv 
\[j_\b \p_{\c_\b} - \bs_J \p_{\z_J} - \s^K \p_{J^K}\] Z=0\ .\label{27a}\eea 
This equation translates the S-T identity in terms of the Green functions.
The same equation holds true for the connected functional:
\be Z_c \[\ j , J , \bs , \s\ , \c , \z\]\equiv -i\  
log Z \[\ j , J , \bs , \s\ , \c , \z\]\ .\label{28}\ee
It is very useful to translate (\ref{28}) for the proper functional \cite{1}.
Perturbatively this is the functional generator of the 1-particle-irreducible
amplitudes and it is generally defined as the Legendre transform of $Z_c$. It is
often called the effective action, although this name is also shared by completely
different objects. Introducing the collective symbol ${\cal J}$ for the field
sources ($  j , J , \bs , \s\ $), ${\cal K}$ for 
the other sources ($ \c , \z $) and $\Phi$
for the fields ($ \f , b , \u , \ub $), one defines the field functional:
\be\Phi\[{\cal J},{\cal K}\]\equiv\p_{{\cal J}}Z_c\[{\cal J},{\cal K}\]- \p_{{\cal
J}}Z_c\[0,0\]\ ,\label{29}\ee then, assuming that the inverse functional ${\cal
J}\[\Phi,{\cal K}\]$ be uniquely defined, one has the proper functional:
\be\Gamma\[\Phi,{\cal K}\]\equiv Z_c\[{\cal J}\[\Phi,{\cal K}\],{\cal K}\]-
\int\(\Phi+\p_{{\cal J}}Z_c\[0,0\]\) {\cal J}\[\Phi,{\cal K}\]\ .\label{31}\ee
It is easy to verify that:
\be\p_{\Phi}\Gamma\[\Phi \[{\cal J},{\cal K}\],{\cal K}\]=\pm{\cal J}\
,\label{pm}\ee and
\be \p_{{\cal K}}\Gamma\[\Phi ,{\cal K}\]|_{\Phi=
\Phi \[{\cal J},{\cal K}\]}= \p_{{\cal K}}Z_c\[{\cal J},{\cal K}\]\ .\ee
Therefore:
\be \p_{\Phi}\p_{\Phi'}\Gamma\[\Phi ,{\cal K}\]|_{\Phi=
\Phi \[{\cal J},{\cal K}\]}=-\[\p_{{\cal J}}\p_{{\cal J}'}Z_c\[{\cal J},{\cal
K}\]\]^{-1}\ . \ee
That is: the second field-derivative of $\Gamma$ gives the full wave operator.
Notice that the $-$ sign in (\ref{pm}) refers to bosonic fields while in the
fermionic case one has the opposite sign.

Using the above identities one can immediately write the S-T identity for the proper
functional:
\be\p_{\f^\a}\C\p_{\c_\a}\C+\p_{\u^I}\C\p_{\z_I}\C+b_I\p_{\ub_I}\C=0\
.\label{stp}\ee
This identity is a crucial tool in many instances, we shall exploit it in the
analysis of unitarity, even more important is however its role in renormalization.
A third interesting application of (\ref{stp}) is the search for generalizations of
the geometrical setting of gauge theories. This is based on the fact that  
(\ref{stp})
is verified by the classical action $S$ of a gauge theory. Indeed the classical
action is the first term in the loop-ordered perturbative expansion of $\C$.

Under the standard assumption of a closed gauge algebra (\ref{9}) $S$ has the
following structure:
\be S=S_{inv}(\f ) +is\[\ub_I\Theta^I\(\Phi\)\]
+\c_\a\u^IP^\a_I(\f)+{1\o 2}\z_IC^I_{JK}\u^J\u^K\ .\label{so}\ee
In the case of renormalizable theories this structure is obliged by
the condition that the dimension of the action be limited by that of space-time.
However the low energy effective actions of more general theories, such as e.g.
supergravity, are free from dimensional constraints; this allows the introduction
of terms of higher degree in the sources $\c$ and $\z$. Disregarding the gauge 
fixing,  setting $\ub=b=0$, let us consider for example:
\be S=S_{inv}+\c_\a\u^IP^\a_I(\f)+{1\o 2}\u^I\u^J\c_\a\c_\b R^{\a\b}_{IJ}+
{1\o 2}\z_IC^I_{JK}\u^J\u^K\ ,\label{s1}\ee
that inserted into (\ref{stp}) gives:
\bea\u^IP_I^\a\p_\a S_{inv}=0\ ,\nn
 \c_\b\u^I\u^J\[P^\a_J\p_\a P^\b_J + R^{\a\b}_{IJ}\p_\a S_{inv}-
{1\o 2}P^\b_K C^K_{IJ}\]=0\ ,\nn
\z_K\u^I\u^J\u^L\[P^\a_I\p_\a C^K_{JL}+ C^K_{MI} C^M_{JL}\]=0\ ,\nn
\c_\a \c_\b\u^I\u^J\u^L\[\p\c P^\b_I R^{\c\a}_{JL}+{1\o2}\(P_I^\c\p_\c
R^{\a\b}_{JL}+R^{\a\b}_{MI}C^M_{JL}\)\]=0\ ,\nn
\c_\a \c_\b\c_\c\u^I\u^J\u^K\u^L\[ R^{\d\a}_{IJ}\ \p_\d  R^{\b\c}_{KL}\]=0\nn
\c_\a \z_K\u^I\u^J\u^K\u^L\[ R^{\d\a}_{IJ}\ \p_\d  C^M_{KL}\]=0\ .\label{bv}\eea
In order to simplify the notation we have written $\p_\a$ instead of $\p_{\f^\a}$.
The first line requires the invariance if the $S_{inv}$ under the physical action of
the differential system ${X}$ in (\ref{8}), the second one replaces the first line in
(\ref{16a}). It is indeed clear that the first term in this line gives 
the commutator of two $X$'s, the second one gives the structure functions of the
algebra while the last one is new; it defines the deviation from a closed algebra
that however, being proportional to the field derivative of the  physical action,
vanishes on the mass shell.
The third and fourth line in (\ref{bv}) follow from the Jacobi identity, the
second line of (\ref{16a}); the last two lines follow from the particular choice
of (\ref{so}) excluding e.g. terms of second order in $\z$.
(\ref{so}) shows the simplest example of the extensions of our method to open
algebras that have been introduced by Batalin and Vilkovisky \cite{4}. 

A very simple example of a mass-shell closed gauge algebra can be found if one tries
to use the BRS algorithm to compute a $n$-dimensional Gaussian integral in polar
coordinates\footnote{This exercise has been suggested by J.Fr\"ohlich}. Let ${\vec
x}$ with components $x_i\ ,\ (i=1,..,n)$ be the variable and $S_{inv}=-{x^2\o2}$
define the invariant measure under the action of the gauge group $O(n)$ corresponding
to the BRS transformations:
\be sx_i=\u_{ij}x_j\, \label{ex1}\ee
where $\u_{ij}$ is the $O(n)$ ghost antisymmetric in its indices and, as usual,
 the sum over repeated indices is understood. The polar coordinate gauge choice
corresponds to the vanishing of $n-1$ components of  ${\vec x}$. This configuration
has a residual $O(n-1)$ invariance and therefore the Jacobian matrix in (\ref{10}) is
highly degenerate. To overcome this difficulty one has to enlarge the BRS structure
adding ghosts for ghosts ($\c_{ij}$); and hence introducing the ghost transformations:
\bea s\u_{ij}=\c_{ij}-\u_{ik}\ \u_{jk}\nn
s\c_{ij}=\c_{ik}\ \u_{jk}-\u_{ik}\ \c_{jk}\ .\label{ex2}\eea
With this choice $s$ is not nilpotent; indeed $s^2xi=\c_{ij}x_j$.
It is mass-shell nilpotent since the "field equations" are: $\p_{x_i}S_{inv}=x_i=0$.
Disregarding the structure of the gauge fixing and introducing a suitable set of
sources, (\ref{so}) is adapted to the present case according:
\be S=S_{inv}+\mu_is x_i +\z_{ij}s\u_{ij}+\eta_{ij}s\c_{ij}+{1\o2}\mu_i\mu_j\c_{ij}\
.\label{ex3}\ee
(\ref{ex3}) satisfies the Slavnov-Taylor identity (\ref{stp}) ( for $b=0$ ) The
 last term in (\ref{ex3}) corresponds to the second order term in $\c$ in
(\ref{so}).

 \sec{Unitarity} \salto
 The first step in the analysis of $\bf S$ unitarity is the study of the
asymptotic propagators and wave operators of a gauge theory \cite{be}. For this we
compute at the origin the second derivative  of (\ref{stp}) with respect
to the space-time Fourier transform ${\tilde \f}^\a (p)$ of $\f^\a$ and $\u^I$;
setting: 
\be \p_{{\tilde \f}^\a (p)}\p_{{}_{\f^\b}}\C [0]\equiv \C_{\a\b}(p)\
,\label{d1}\ee 
\be \p_{\u^I}\p_{{\tilde \c}_\b (p)}\C[0]\equiv V_I^\a (p)\
,\label{d2}\ee 
we get:
\be \C_{\a\b}(p)\ V^\b_I(p)=0\ .\label{trans}\ee This corresponds to the
transversality condition for the vacuum polarization in QED. In much the same way,
taking the second derivative with respect to ${\tilde b}_I (p)$ and $\u^J$, we get
\be \C^I_\a (p) V^\a_J (p)= C^I_J(p)\ ,\label{degene}\ee
 with the definitions:
\be \p_{{\tilde \f}^\a (p)}\p_{{}_{b_I}}\C[0]\equiv  \C^I_\a (p)  \ ,\label{d3}\ee
and:
\be \ \p_{\u^I}\p_{{\tilde \ub}_J (p)}\C[0]\equiv \- C_I^J (p)\
.\label{d4}\ee
We define furthermore:
\be \p_{{\tilde b}_J (p)}\p_{{}_{b_I}}\C[0]\equiv  \La^{IJ} (p)  \ .\label{d5}\ee
 $\C_{\a\b}, \C^\a_I $ and $ \La^J_I$ are elements of the field wave
operator $C_I^J$ of the ghost wave
operator. (\ref{degene}) shows that for a generic choice of
momentum $C$ being a non-degenerate matrix, the field valued vectors with
components $V_I^\a$ are linearly independent. We shall call them longitudinal. One
can complete a
basis of the field space introducing a further set of vectors labelled
by the index $r$ and with components $V_r^\a$.  We shall call these
vectors tranverse. We define:
\be \C^{(T)}_{rs}\equiv  V_r^\a V_s^\b\C_{\a\b}\ ,\label{d6}\ee
and
\be F^I_r\equiv V_r^\a \C^I_\a\ .\label{d7}\ee
Taking into account (\ref{trans}) and (\ref{degene}) we can write the wave
operator in the $\f , b$ space according:
  \be\label{wave}
\left(\ \ \ber{ccc}\ \ \C^{(T)} & 0 & \ \  F \\0  &
  0 & \ \  C \\{\tilde F} & {\tilde C} &\ \  \La  \eer\ \right),\ee
where the first line and column correspond to the transverse field components, the
second ones correspond to the longitudinal components  and the third to
the $b$ field. 
The corresponding asymptotic wave operator is:
  \be\label{waveas}
\left(\ \ \ber{ccc}\ \ \C_{(asy)}^{(T)} & 0 & \ \  F_{(asy)} \\0  &
  0 & \ \  C_{(asy)} \\{\tilde F}_{(asy)} & {\tilde C}_{(asy)} &\ \  \La_{(asy)}
 \eer\ \right).\ee
The asymptotic states correspond to vectors belonging to the kernel of the asymptotic
wave operator; these are given by their amplitudes: $ v^r , w^I , z_J $ satisfying:
\bea \C_{(asy)rs}^{(T)} v^s+ F_{(asy)r}^Iz_I=0\nn
C_{(asy)I}^Jz_J=0\nn
F_{(asy)r}^Iv^r+C_{(asy)J}^Iw^J+\La_{(asy)}^{I J}z_J=0\ .\label{asf}\eea
This system has three different families of solutions:

i) the solutions with: $ w=z=0$ and:
\be \C_{(asy)rs}^{(T)} v^s=0\ ,\ F_{(asy)r}^Iv^r=0\ ,\label{tran}\ee that correspond
to the transverse fields $\f^r_{in}$. Notice that the number of independent solutions
of the first equation is reduced by the second equation acting as a constraint. In
general the number of these constraints coincides with that of the ghost degrees of
freedom.

ii) the solutions with $v=z=0$ and $C_{(asy)J}^Iw^J=0$ that correspond to the
longitudinal components $\f^I_{in}$.

iii) the solutions without trivial components that correspond to the field $b_{in, I}$.

The solutions of the last two kinds are in the same number as and degenerate with
 those of the ghost wave equation.

 In terms of these asymptotic fields and of the ghost ones we can write the
$\S$ matrix in the form (\ref{7}) with: \bea \Sigma
=\f^r_{in}\(\C_{(asy)rs}^{(T)}\p_{j_s}+F_{(asy)r}^I\p_{J^I}\)\p_{_I} + \f^I_{in} 
C_{(asy)I}^J\p_{J^J}\nn +b_{in, I}\(F_{(asy)r}^I\p_{j_r}
 +  C_{(asy)J}^I\p_{j_J}+ \La_{(asy)I J}\p_{J^J}\)\nn
+\(\ub_{in, I} C_{(asy)J}^I\p_{{\bar \sigma}_J} -\ \u_{in}^I
C_{(asy)I}^J\p_{\sigma^J}\) \ .\label{sigma}\eea
Notice that, if the sources are restricted on the mass shell, the S-T identity
is written:
\be\[j_\a V^\a_I\p_{{\bar \sigma}_I} - \s^K \p_{J^K}\] Z=
\[j_I\p_{{\bar \sigma}_I} - \s^K \p_{J^K}\] Z\equiv {\cal S}Z=0\
,\label{stas}\ee
and hence we have:
\be:e^\Sigma:{\cal S}Z|_{{\cal J}=0}=:e^\Sigma\(b_{in, J}  C_{(asy)I}^J
\p_{{\bar \sigma}_I}-\u^I_{in} C_{(asy)I}^K\p_{J^K}\):Z|_{{\cal J}=0}=0\
.\label{60}\ee
Now we introduce a hermitian operator $Q$ \cite{ko}, defined in the asymptotic Fock
space generated by the "in" fields, by the following relations:
\bea Q, \O >=0\quad\ \  , \quad \[Q, \f^r_{in}\]=0\quad \ ,
\quad \[Q, \f^I_{in}\]=i\ \u^I_{in}\quad, \quad  \nn\[Q, b_{in, I}\]=0\quad, \quad 
\{Q, \u_{in}^I\}=0\quad, \quad 
\{Q, \ub_{in, I}\}=i \ b_{in, I}\ .\label{com}\eea
Notice that with our conventions all the asymptotic 
fields are hermitian  except $\ub_{in, I}$
that is anti-hermitian, however the gauge fixing action is hermitian. Notice also
that Q generates a nilpotent transformation on the Fock space. Its kernel is the
subspace generated by the transverse fields $ \f^{(-)r}_{in} $ and by $b_{in,
I}^{(-)}$ and $\u_{in}^{(-)I} $, while its image is the subspace generated by
$b_{in, I}^{(-)}$ and $\u_{in}^{(-)I } $. Therefore the cohomology of $Q$ can be
identified with the subspace of the Fock space generated by  $ \f^{(-)r}_{in} $ .
Notice  that the asymptotic properties of these field components are completely
determined by the invariant part of $\C$ and hence it is expected that they generate
a positive norm space. It is also clear from the definitions (\ref{com}) that the
states in the image of $Q$ have vanishing scalar product with those in  its
kernel, they are in particular zero norm states.
Furthermore the commutator $\[Q,\S\]$ vanishes due to (\ref{60}), indeed:
\bea \[Q,\S\]=:\[Q,e^\Sigma\]:Z|_{{\cal J}=0}
=\nn -i:e^\Sigma\(b_{in, J} 
C_{(asy)I}^J \p_{{\bar \sigma}_I}+
\u^I_{in} C_{(asy)I}^K\p_{J^K}\):Z|_{{\cal J}=0}=0\
.\label{61}\eea
Under the assumption that the measure corresponding to a hermitian Lagrangian
define a "pseudo"unitary $\S$-matrix in the asymptotic Fock space:
\be \S \S^{\dag} =\S^{\dag} \S =I\ ,\ee
 and that
$\C^{(T)}$ correspond to a positive metric transverse space - both assumptions are
obvious in perturbation theory - we conclude that, owing to (\ref{61}) and to  the
above discussed properties of $Q$, the $\S$-matrix is unitary in the physical space
identified with the cohomology of $Q$.

We consider for example an $SU(2)$ Higgs model that involves an iso-triplet of vector
fields ${\vec A}_\mu$, a triplet of Goldstone particles ${\vec \pi}$, and the Higgs
field $\sigma$ that we shall not consider explicitly. Therefore $\f^\a$ stands for
${\vec A}_\mu\ ,\ {\vec \pi}$. We add a further iso-triplet of Nakanishi-Lautrup
multipliers $\vec b$.
In the tree approximation the free lagrangian density is given by:
\be{\cal L}=-{\vec F_{\mu\nu}\cdot\vec F^{\mu\nu}\o 4} + {1\o2}\(\p\vec \pi+gV\vec
A\)^2+\vec b\cdot\(\p\vec A-\rho\vec\pi\)+\a{b^2\o 2}\ ,\ee
From now on we shall disregard the isotopic indices since all the wave operators are
diagonal in the isotopic space.
The wave operator (\ref{d1}) is given by:
 \be\label{hig2}
\left(\ \ \ber{cc}\ \ p^\mu p^\nu-g^{\mu\nu}p^2+g^2V^2g^{\mu\nu} & -igVp^\mu  \\
-igVp^\nu &
 p^2 
 \eer\ \right),\ee
where the first row and line correspond to $A$ and the second ones to $\pi$.
$\C^I_\a$ in (\ref{d3}) corresponds to:
\be\label{hig3}
\left(\ \ \ber{c}\ ip^\mu \\ -\rho
\eer\ \right),\ee
and $V^\a_I$ in  (\ref{d2}) is given by:
\be\label{hig4}
\left(\ \ \ber{c}\ ip^\mu \\ gV
\eer\ \right).\ee
Furthermore we introduce the transverse vectors:
\be V_r^\a=\left(\ \ \ber{c}\ gVg^{\mu\nu}\\ip^\mu
\eer\ \right).\ee
Here the index $r$ corresponds to the suffix $\nu$.
The transverse wave operator is:
\be\C^{(T)}_{rs}=\(g^2V^2-p^2\)g^{\mu\nu}+p^\mu p^\nu \(3g^2V^2+p^2\)\
,\label{hig5}\ee
One finds the corresponding asymptotic states, solutions of the wave equation:
\be \C^{(T)}_{rs}(p)v_s(p)=0\ ,\ee for $p^2=g^2V^2$. These solution span
a three dimensional linear space; due to the second equation in (\ref{tran}) they
correspond to four-vectors  orthogonal to the momentum.
 \sp
\appendix
\sec{Appendix}
In this appendix we prove (\ref{1a}) exploiting (\ref{21}).
The first step will be the proof of two lemmas whose recursive use will lead to
(\ref{1a}).
Let us  consider the set of cells $\{U_a\}$ and the corresponding
partition of unity $ \{\chi_a\}$ and gauge fixing functionals $\{\T_a\}$
We define:
\be \(s\chi_{a_{1}}...s\chi_{a_{n-1}}\ \chi_{a_{n}}\)_A\equiv
\sum_{k=1}^n(-1)^{k-n}\chi_{a_{k}}s\chi_{a_{1}}...s\check\chi_{a_k}...s\chi_{a_{n}}\
,\label{a1}\ee
where the check mark above $\chi_{a_k}$ means that the corresponding term should be
omitted. It is fairly evident that the functional (\ref{a1}) is antisymmetric with
respect to permutations of the indices $\( a_1,..., a_n\)$ and that its support  is
contained in the intersection of the corresponding cells.

It is apparent that:
\be s \(s\chi_{a_{1}}...s\chi_{a_{n-1}}\
\chi_{a_{n}}\)_A=(-1)^{n+1}n\ s\chi_{a_{1}}...s\chi_{a_{n}}\ .\label{a2}\ee
Furthermore, taking into account that $ \{\chi_a\}$ is a partition of unity, one
has: \bea\sum_{a_{n+1}}\(s\chi_{a_{1}}...s\chi_{a_{n}}\ \chi_{a_{n+1}}\)_A
=\sum_{a_{n+1}}\sum_{k=1}^{n+1}(-1)^{k-n-1}\chi_{a_{k}}
s\chi_{a_{1}}...s\check\chi_{a_k}...s\chi_{a_{n+1}}\nn
=s\chi_{a_{1}}...s\chi_{a_{n}}\ ,\label{a3}\eea
indeed only the term with $k=n+1$ contributes to the second member giving the
right-hand side of this equation. Comparing (\ref{a2}) with  (\ref{a3}), we
get:
\be s \(s\chi_{a_{1}}...s\chi_{a_{n-1}}\
\chi_{a_{n}}\)_A=(-1)^{n+1}n\ \sum_{a_{n+1}}
\(s\chi_{a_{1}}...s\chi_{a_{n}}\ \chi_{a_{n+1}}\)_A\ .\label{a4}\ee
Let now $A_{a_{1},...,a_{n+1}}$ be antisymmetric in its indices, one has:
\bea \sum_{a_{1},...,a_{n+1}}A_{a_{1},...,a_{n+1}}\p_{\T}\(\T_{a_{1}}...\T_{a_{n}}\)
e^{is\T_{a_{1},...,a_{n}}}\nn =i{(-1)^n\o
n+1}\sum_{a_{1},...,a_{n+1}}A_{a_{1},...,a_{n+1}}\ s\
\p_{\T}\(\T_{a_{1}}...\T_{a_{n+1}}\) e^{is\T_{a_{1},...,a_{n+1}}} \ .\label{a5}\eea
Indeed, using the identity:
\be\sum_{a_{1},..,a_{n+1}}A_{a_{1},..,a_{n+1}}V_{a_{1},..,a_{n}}={(-1)^n\o
n+1}\sum_{k=1}^{n+1}(-1)^{k+1}\sum_{a_{1},..,a_{n+1}}A_{a_{1},..,a_{n+1}}V_{a_{1},..,
\check a_k,..,a_{n+1}}\ ,\ee
 the left-hand side of (\ref{a5}) is written:
\bea{(-1)^n\o
n+1} 
\sum_{a_{1},...,a_{n+1}}A_{a_{1},..,a_{n+1}}\sum_{k<l=1}^{n+1}
(-1)^{k+l}\T_{a_{1}}..\check \T_{a_{k}}..\check \T_{a_{l}}..\T_{a_{n+1}}\nn
\(e^{is\T_{a_{1},..,\check a_k,..,a_{n+1}}}-e^{is\T_{a_{1},..,\check a_l,..,a_{n+1}}}\)
 \nn =i{(-1)^n\o
n+1} 
\sum_{a_{1},...,a_{n+1}}A_{a_{1},..,a_{n+1}}\sum_{k<l=1}^{n+1}
(-1)^{k+l}\T_{a_{1}}..\check \T_{a_{k}}..\check \T_{a_{l}}..\T_{a_{n+1}}\nn
s\(\T_{a_k}-\T_{a_l}\)e^{is\T_{a_{1},...,a_{n+1}}}\ ,
 \eea
from which one reaches (\ref{a4}).

Let us now consider the extension of (\ref{sti}) to the case of a cell decomposition of
$\fc$, one has:
\bea \int d\mu_C\sum_{a}\chi_{a}e^{is\T_{a}}sX=-\int
d\mu_C\sum_{a_1}s\chi_{a_1}e^{is\T_{a_1}}X\nn=\int d\mu_C\sum_{a_1,a_2}
\(s\chi_{a_{1}}\chi_{a_{2}}\)_Ae^{is\T_{a_1}}X\nn={1\o2}\int d\mu_C\sum_{a_1,a_2}
\(s\chi_{a_{1}}\chi_{a_{2}}\)_A\(e^{is\T_{a_1}}-e^{is\T_{a_2}}\)X\nn=
-{i\o2}\int d\mu_C\sum_{a_1,a_2}
\(s\chi_{a_{1}}\chi_{a_{2}}\)_As\p_{\T}\(\T_{a_{1}}\T_{a_{2}}\)e^{is\T_{a_1,a_2}}X
\ .\label{a6}\eea
In (\ref{a6}) we have used (\ref{a3})  and (\ref{a5}) with $n=1$. From (\ref{a6}) we
have: \bea \int d\mu_C\[\sum_{a}\chi_{a}e^{is\T_{a}}+{i\o2}\int d\mu_C\sum_{a_1,a_2}
\(s\chi_{a_{1}}\chi_{a_{2}}\)_A\p_{\T}\(\T_{a_{1}}\T_{a_{2}}\)e^{is\T_{a_1,a_2}}
\]sX\nn=-{i\o2}\int d\mu_C\sum_{a_1,a_2}
s\(s\chi_{a_{1}}\chi_{a_{2}}\)_A\p_{\T}\(\T_{a_{1}}\T_{a_{2}}\)e^{is\T_{a_1,a_2}}\
.\eea
 Using recursively the same equations we get: 
\bea   d\mu_C\[\sum_a\chi_ae^{is\T_{a}}-
\sum_{n=1}^{m}i^n{(-1)^{n(n-1)\o2}\o n+1}\(s\chi_{a_{1}}...s\chi_{a_{n}}\
\chi_{a_{n+1}}\)_A
\r.\nn\l.\p_{\T}\(\T_{a_{1}}...\T_{a_{n+1}}\)e^{is\T_{a_{1},...,a_{n+1}}}\]sX  \nn=
i^m\ {(-1)^{m(m-1)\o2}\o m+1}\int d\mu_Cs\(s\chi_{a_{1}}...s\chi_{a_{m}}\
\chi_{a_{m+1}}\)_A\nn\p_{\T}\(\T_{a_{1}}...\T_{a_{m+1}}\)s\
e^{is\T_{a_{1},...,a_{m+1}}}X \ .\label{a7}\eea
It is clear that, if the maximum effective number of intersecting cells is $N$ the
right-hand side of (\ref{a7}) vanishes for $m\geq N$.


    \sp
\begin{thebibliography}
{9}

\bibitem{1}
C.Itzykson e J.-B.Zuber, Quantum Field Theory, MacGraw-Hill, New-York 1980. 

\bibitem{fp}
L.D.Faddeev and V.N.Popov, Phys.Letters B25 (1967) 29.

\bibitem{brs}
C.Becchi, A.Rouet and R.Stora,  Phys Letters. B32 (1974) 344.

C.Becchi, A.Rouet and R.Stora,  Commun.Math. Phys. 42 (1975) 127.

L.V.Tyutin, Lebedev preprint FIAN n.39 (1975).

C.Becchi, A.Rouet and R.Stora,  Ann. Phys. 98 (1976) 287

\bibitem{2}
 S.Sternberg, Lectures on Differential Geometry, Prentice Hall
Inc. 1964 

\bibitem{nie}
N.K.Nielsen, Nuclear Phys. B140 (1978) 499.

\bibitem{neu}
H. Neuberger,  Phys Letters. B183 (1987) 337.

\bibitem{3}
V. N. Gribov, ``Instability of non-abelian gauge theories and 
impossibility of choice of Coulomb gauge'', SLAC Translation 176, (1977).

I.M. Singer
Commun.Math.Phys. 60 (1978) 7.

\bibitem{zw}
D.Zwanziger, Nuclear. Phys. B209 (1982) 336.

G.Dell'Antonio and D.Zwanziger, Nuclear. Phys. B326 (1989) 333. 


\bibitem{bi} C. Becchi and C. Imbimbo,
A Lagrangian formulation of Two-Dimensional Topological Gravity and  Cech-De Rham
Cohomology;  in: BRS Symmetry, M.Abe, N.Nakanishi, I.Ojima ed.s,  Univwersal Academy
Press, Tokyo, 1996. 

\bibitem{gau}
see e.g.: P.Gaigg, W.Kummer and M.Schweda (Eds.) Physical and
Nonstandard Gauges, Lecture Notes in Physics, Vol.361,
Springer-Verlag Berlin Heidelberg 1990.

\bibitem{fu}
 K. Fujikawa, Nuclear. Phys. B223 (1983) 218.


\bibitem{4}
I.A.Batalin and G.A.Vilkovisky Nucl.Phys. B234 (1984) 106.


\bibitem{be}
see e.g.:

     C.Becchi, Lectures on the renormalization of gauge theories, in:
     Relativity, Groups and Topology II (les Houches 1983) B.S. DeWitt
     and R.Stora Eds. (Elsevier Science Pub.B.V. 1984). 

\bibitem{ko}
T.Kugo and I.Ojima, Prog. Theor. Phys. 66 (1979) 1.

\end{thebibliography}
\end{document}

