Proof of perturbative gauge invariance for tree diagrams to all orders
Michael D\"utsch
[2mm] Institut f\"ur Theoretische Physik
Universit\"at Z\"urich
CH-8057 Z\"urich
Switzerland
[2mm]
It is proved that classical BRS-invariance of the Lagrangian implies perturbative gauge invariance for tree diagrams to all orders. The proof applies in particular to the Einstein Hilbert Largrangian of gravity.\\ \\ {PACS.} 11.15.Bt Gauge field theories: General properties of perturbation theory
hep-th0502071.tex
\def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt\hbox{\vrule width.#2pt 
height#1pt \kern#1pt \vrule width.#2pt}\hrule height.#2pt}}}}
\def\d{\partial}
\def\dh{\mathop{\vphantom{\odot}\hbox{$\partial$}}}
\def\dl{\dh^\leftrightarrow}
\def\w{\mathchoice\sqr45\sqr45\sqr{2.1}3\sqr{1.5}3\,} 
\def\=d{\,{\buildrel\rm def\over =}\,}
\def\psq{{\overline{\psi}}}

\documentclass[12pt]{article}
\usepackage{amssymb}\usepackage{amsmath}\usepackage{amsthm}

\renewcommand{\today}{}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\bg}{\begin{gather}}
\newcommand{\eg}{\end{gather}}
\newcommand{\CC}{\mathbb C}
\newcommand{\RR}{\mathbb R}
\newcommand{\ZZ}{\mathbb Z}
\newcommand{\NN}{\mathbb N}
\newcommand{\MM}{\mathbb M}
\newcommand{\supp}{{\rm supp\>}}
\newcommand{\C}[1]{{\mathcal #1}}
\newcommand{\ket}[1]{\lvert#1\rangle}       % Ket-Vektor 
\newcommand{\bra}[1]{\langle#1\rvert}       % Bra-Vektor
\renewcommand{\labelenumi}{(\roman{enumi})}
\newtheorem{prop}{Proposition}
\newtheorem{thm}[prop]{Theorem}
\newtheorem{cor}[prop]{Corollary}
\newtheorem{lemma}[prop]{Lemma}
\newtheorem{definition}{Definition}
%\newcommand{\XX}[1]{* \marginpar{\footnotesize * #1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{Proof of perturbative gauge invariance for tree 
diagrams to all orders}
\author{Michael D\"utsch \\[2mm] 
Institut f\"ur Theoretische Physik\\
Universit\"at Z\"urich\\
CH-8057 Z\"urich, Switzerland\\
{\tt \small duetsch@physik.unizh.ch}\\[2mm]}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\date{}
\maketitle
\begin{abstract}
It is proved that classical BRS-invariance of the Lagrangian implies perturbative gauge invariance 
for tree diagrams to all orders. The proof applies in particular to 
the Einstein Hilbert Largrangian of gravity.\\
\\
{\bf PACS.} 11.15.Bt Gauge field theories: General properties of perturbation theory

\end{abstract}


\tableofcontents
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
One of the greatest challenges of present-day quantum field theory (QFT) is the search 
for a quantum theory of gravity. In this context revolutionary approaches are intensively studied, 
e.g.~non-commutative space-times, string theory and loop quantum gravity. Since this paper
is related to BRS-symmetry \cite{BRS}, we only mention that a BRS-formulation of gravity
was given in \cite{DRM,Dix,TN} and that the structure of the possible anomalies has been 
worked out by cohomological methods, see e.g.~ \cite{BDK,BBH}.

The main result of this paper is much more modest: we prove
{\it perturbative gauge invariance} (PGI) \cite{DHKS,Scharf,S-wiley,G,stora}
(which is a condition in perturbative QFT that is related to BRS-invariance) for gravity,
but our result is restricted to {\it tree diagrams}. Since PGI for tree diagrams
(PGI-tree) is equivalent to PGI in {\it classical} field theory (cf.~Appendix B of
\cite{DF2} and Sect.~2), our result is actually a statement for classical gravity.
However, it is also a justification to use PGI for the construction of a perturbative QFT
for spin-2 gauge fields - a project started in \cite{Schorn,SW,Grillo,S-wiley,G:grav,GS}.

In the latter the requirement of PGI-tree has been used to determine the possible
interactions of massless spin-2
gauge fields \cite{Schorn,SW,S-wiley}. Making a polynomial 
ansatz for the interaction ${\cal L}=\sum_{n=1}^\infty\kappa^n\,{\cal L}^{(n)}$
(where $\kappa$ is the coupling constant), it has been worked out that the most 
general solutions for ${\cal L}^{(1)}$ and ${\cal L}^{(2)}$ agree with the 
corresponding terms of the Einstein-Hilbert Lagrangian of gravity 
up to physically irrelevant terms, see \cite{SW,S-wiley}.
But continuing this procedure to higher orders the amount of computational
work increases strongly and, due to the non-renormalizability of spin-2
gauge fields, one never comes to an end. That is, violations of PGI-tree
can appear to arbitrary high orders and, if PGI-tree can be fulfilled, it is not clear, 
that the general solution for ${\cal L}^{(k)},\>k\geq 3$, agrees with the corresponding 
term of the Einstein-Hilbert Lagrangian. The {\it main purpose of this paper is to prove 
that the Einstein-Hilbert Lagrangian, completed by a gauge fixing and a 
Faddeev-Popov ghost term} (we follow \cite{KO}), {\it yields a solution of
PGI-tree to all orders}.

To a large extent we formulate the proof independently of the model. However,
the applications to massless and massive spin-1 fields (Sects.~4.1 and 4.2)
are only of pedagogical value: for {\it renormalizable models} a violation of PGI-tree
can usually\footnote{The statement is valid for renormalizable models in $4$ 
dimensions with the property that all terms of the interaction ${\cal L}$
are at least of third order in the basic fields.}
appear only up to third order (as can easily be seen by power counting), and
using this fact, PGI-tree has been proved by explicit computation of the lowest orders
for various spin-1 models \cite{DHKS,DS,G}.

By {\it perturbative gauge invariance} we mean the following condition. Let a free 
quantum gauge theory (i.e.~a free Lagrangian ${\cal L}^{(0)}$) and the corresponding 
free BRS-transformation $s_0$ be given and let ${\cal L}^{(0)}$ be 
$s_0$ invariant, that is
$s_0\,{\cal L}^{(0)}=-\d_\mu I^{(0)\,\mu}$ for some field polynomial $I^{(0)\,\mu}$. 
In addition let $j^{(0)\,\mu}$ be the corresponding conserved Noether current,
$\d_\mu j^{(0)\,\mu}_{S_0}=0$, and let $Q$ be the corresponding charge: 
$Q=\int d^3x\,j_{S_0}^{(0)\,0}(x^0,\vec{x})$ (``free BRS-charge''). 
(The lower index $S_0$ signifies always
that we mean the 'on-shell fields', i.e.~the free field equations are valid; for 
a precise formulation see \cite{DF2,DF3} or Appendix A.) 
PGI requires that to an interaction ${\cal L}^{(1)}$ there exists a 
Lorentz vector ${\cal L}_1^{(1)\,\nu}$ and a normalization of the time ordered products 
$T^N$ such that
%%%%%%%%%
\beq
[Q,T_{S_0}^N\bigl({\cal L}^{(1)}(x_1)...{\cal L}^{(1)}(x_n)\bigr)]=i\sum_{l=1}^n\d_\nu^{x_l}\>
T_{S_0}^N\bigl({\cal L}^{(1)}(x_1)...{\cal L}_1^{(1)\,\nu}(x_l)...{\cal L}^{(1)}(x_n)\bigr)\label{PGI}
\eeq
%%%%%%%%%%%%%%%% 
(where $[\,\cdot\,,\,\cdot\,]$ denotes the commutator with respect to the $\star$ product).
The upper index $(1)$ of  ${\cal L}^{(1)}$ signifies that we mean the term of first order 
(in the coupling constant $\kappa$) of the total interaction ${\cal L}=\sum_{n=1}^\infty
\kappa^n\,{\cal L}^{(n)}$ and similarly ${\cal L}_1^{(1)\,\nu}$ is the term of first
order of the total ``$Q$-vertex'' ${\cal L}_1^{\nu}=\sum_{n=1}^\infty
\kappa^n\,{\cal L}_1^{(n)\,\nu}$. Higher order terms ${\cal L}^{(n)}$ (${\cal L}_1^{(n)\,\nu}$
resp.), $n\geq 2$, are absorbed in a finite renormalization $T({\cal L}^{(1)}...{\cal L}^{(1)})
\rightarrow T^N({\cal L}^{(1)}...{\cal L}^{(1)})$ ($T({\cal L}^{(1)}...{\cal L}_1^{(1)\,\nu}...)
\rightarrow T^N({\cal L}^{(1)}...{\cal L}_1^{(1)\,\nu}...)$ resp.) of tree diagrams. This is 
always possible, as shown in Sect.~2.

PGI plays two different roles:
\begin{itemize}
\item [(i)] PGI-tree restricts the interaction ${\cal L}=\sum_{n=1}^\infty\kappa^n\,{\cal L}^{(n)}$
strongly, as mentioned above.
\item  [(ii)] For loop diagrams it is a highly non-trivial (re)normalization condition,
which cannot always be fulfilled, e.g.~in case of the axial anomaly.
\end{itemize}
%Only (i) is relevant for us since this paper deals with tree diagrams only.  

The motivations to require PGI are the following.
\begin{itemize}
\item[(A)] In purely massive theories the adiabatic limit exists \cite{EG1}, i.e.~there is an $S$-matrix
%%%%%%%%%
\begin{equation}
S={\bf 1}+\lim_{g\to 1}\sum_{n=1}^\infty\frac{i^n}{n!}\int dx_1...dx_n\,g(x_1)...g(x_n)\,
T_{S_0}^N\bigl({\cal L}^{(1)}(x_1)...{\cal L}^{(1)}(x_n)\bigr)\ .\label{S}
\end{equation}
%%%%%%%%%%%%%%
Kugo and Ojima \cite{KO1,KO} have shown that in the adiabatic limit the physical
Hilbert space ${\cal H}$ can be expressed in terms of the free BRS-charge $Q$:
%%%%%%%%%%
\begin{equation}
{\cal H}=\frac{{\rm ker}\>Q}{{\rm ran}\>Q}\ .
\end{equation}
%%%%%%%%%%%%%%
PGI implies $[Q,S]=0$ and, hence, $S$ is well defined on ${\cal H}$. That is,
PGI is a sufficient condition for the quantization of purely massive gauge theories and,
as shown in \cite{DSchroer}, it is even almost necessary for this purpose.
\item[(B)] Since PGI is well defined also for theories in which the adiabatic limit 
does not exist, PGI-tree can be used to derive the
interaction ${\cal L}$ for all kinds of gauge theories. Making a polynomial 
ansatz for ${\cal L}$, PGI-tree and
some obvious requirements (e.g.~Lorentz invariance, ghost number zero and in
case of spin-1 fields renormalizability) determine ${\cal L}$ to a far extent. We
recall the highlights (besides the already mentioned derivation of the Einstein-Hilbert
Lagrangian). 
\begin{itemize}
\item The Lie algebraic structure of spin-1 fields needs not to be put in, 
it can be derived in this way \cite{stora}.
\item For non-Abelian massive spin-1 theories it is impossible to satisfy these
requirements for a model with only gauge fields and ghosts (fermionic and 
bosonic). The inclusion of additional {\it physical} scalar fields 
(corresponding to Higgs fields) yields a solution \cite{DS}. 
\end{itemize}
\end{itemize}

In this paper we proceed in the direction opposite to (B): we assume that a Lagrangian
${\cal L}_{\rm total}=\sum_{n=0}^\infty\kappa^n\,{\cal L}^{(n)}$ and a 
BRS-transformation $s=\sum_{n=0}^\infty\kappa^n\,s_n$ (of the interacting fields) 
\cite{BRS} are given and that ${\cal L}_{\rm total}$ is BRS-invariant: 
$s\,{\cal L}_{\rm total}=-\d_\mu I^{\mu}$ for some 
formal power series $I^{\mu}$. We prove that this assumption implies PGI-tree.

Our proof of PGI-tree relies strongly on results which have been found in
\cite{DF}, \cite{BD} and \cite{DF2}. Namely, let us start with the local conservation of the
BRS-current,
%%%%%%%%%%%%%%%%
\begin{gather}
\d_\mu^x\,T_{S_0}^N\bigl(j^{(0)\,\mu}(x),{\cal L}^{(1)}(x_1)...{\cal L}^{(1)}(x_n)\bigr)=\notag\\
-i\sum_{l=1}^n
\d_\nu^{x_l}\,\Bigl(\delta(x-x_l)\>T_{S_0}^N\bigl({\cal L}^{(1)}(x_1)...{\cal L}_1^{(1)\,\nu}
(x_l)...{\cal L}^{(1)}(x_n)\bigr)\Bigr)\label{localcurrconserv}
\end{gather}
%%%%%%%%%%%%%%%% 
where the higher order terms $j^{(n)}\,,\>n\geq 1$, of the total BRS-current
$j^{\mu}=\sum_{n=0}^\infty\kappa^n\,j^{(n)\,\mu}$ are absorbed in a renormalization
$T(j^{(0)},{\cal L}^{(1)}...)\rightarrow T^N(j^{(0)},{\cal L}^{(1)}...)$ of tree diagrams
(see Sect.~2). In Appendix B of \cite{DF} and in \cite{BD} it is proved that by 
smearing out (\ref{localcurrconserv}) with a test function $f(x)$
which satisfies $f\vert_{\bar {\cal O}}=1$, where ${\cal O}$ is an open double cone containing 
$x_1,...,x_n$, one obtains PGI (\ref{PGI}). It has even been shown that 
(\ref{localcurrconserv}) is necessary for PGI
provided the ghost number is conserved (Sect.~4.5.2 of \cite{BD}, related ideas 
are given in \cite{Hurth}). Motivated by these facts we proceed as follows. 
In (non-perturbative) classical field theory we show that BRS-invariance of the 
Lagrangian (for constant coupling) implies local conservation of the BRS-current.
The latter holds also for the perturbative expansion of the classical fields,
i.e.~for the retarded product of classical field theory $R^{\rm class}$. Since
$R^{\rm class}$ agrees with the contribution of the tree diagrams
$R^{\rm tree}$ to the retarded product of QFT \cite{DF1,DF2}, 
we obtain the translation of (\ref{localcurrconserv}) into $R^{N\,{\rm tree}}$,
after a finite renormalization $R\rightarrow R^N$ (of tree diagrams). 
Then, proceeding analogously 
to the step from  (\ref{localcurrconserv}) to PGI (see Sect.~5.2 of \cite{DF2}),
we obtain PGI-tree for $R^{N\,{\rm tree}}$. (Up to third 
order this result is derived also in Appendix B of \cite{DF2}.)
%in perturbative {\it classical field theory} 
%we show that the local current conservation (\ref{dj(g)}) (which
%corresponds to (\ref{localcurrconserv}))
%implies PGI for the classical retarded product\footnote{In classical field 
%theory the {\it commutator} of $Q$ with a retarded product 
%is replaced by the {\it Poisson bracket.}} $R^{\rm class}$ (see (\ref{d_QR}) below), where 
%we use the translation of the procedure (\ref{localcurrconserv})-(\ref{PGI})
%into retarded products (given in Sect.~5.2 of \cite{DF2}). (Up to third 
%order this is explicitly worked out in Appendix B of \cite{DF2}.)
%Moreover, in classical field theory the local current conservation (\ref{dj(g)}) can 
%be derived from the field equations (as verified for concrete models in Sect.~3). Hence, 
%it follows that  $R^{\rm class}$ satisfies PGI. Since $R^{\rm class}$ agrees
%with the contribution of the tree diagrams
%$R^{\rm tree}$ to the retarded product of QFT \cite{DF1,DF2}, 
%and since solely the Poisson bracket $\{Q, \>\cdot\>\}$ contributes  to 
%$\frac{i}{\hbar}[Q,\>\cdot\>]$ (see Lemma 1 in Sect.~2),  it follows PGI  
%for $R^{\rm tree}$. 
Finally we show that PGI-tree is maintained in the transition to the corresponding time 
ordered product $T^N$ (by using results about the counting of powers of $\hbar$ 
given in Sect.~5 of \cite{DF1}).

The paper is organized as follows.
In Sect.~2 we assume that the {\it classical} BRS-current is locally conserved  
(\ref{dj(g)}), and give a model independent proof that this implies PGI-tree.
In Sect.~3 we trace back this assumption to BRS-invariance of the 
Lagrangian for constant coupling, still independently of the model.
In doing so we obtain explicit formulas for the BRS-current $j^\mu$
and the $Q$-vertex ${\cal L}_1^{\nu}$.

We then illustrate the formalism:  for
massless and massive Yang-Mills theories 
we find that our formulas for $j^\mu$ and ${\cal L}_1^{\nu}$ yield results 
which agree with the literature (Sect.~4.1 and 4.2).

Then, we turn to the main objective of this paper: massless spin-2 gauge fields 
(Sect.~4.3). From Kugo and Ojima 
\cite{KO} we recall the BRS-invariance of the Lagrangian of gravity
and show that it fits in our formalism. 
This completes our proof of PGI-tree to all orders for massless 
spin-2 gauge fields. We verify that our formula for $j^{(0)\,\mu}$ 
agrees with the literature also in the spin-2 case. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{From classical current conservation to perturbative gauge invariance
for tree diagrams}\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let ${\cal P}$ be the polynomial algebra generated by the basic classical (off-shell)
fields and their partial derivatives, see Appendix A. In this Sect.~we assume that there are given
\begin{itemize}
\item an action $S_{\rm total}(g)=S_0+S(g)$ with free part $S_0=\hbar^{-1}\,\int dx\,
{\cal L}^{(0)}(x)$ and interacting part 
%%%%%%%%%%%%%
\beq
S(g)=\int dx\,{\cal L}(g)(x)\ ,\quad {\cal L}(g)(x):=\hbar^{-1}\>
\sum_{k=1}^\infty \kappa^k\,(g(x))^k\, {\cal L}^{(k)}(x)\ ,\label{L(g)}
\eeq
%%%%%%%%%%%%%%%
${\cal L}^{(k)}\in {\cal P}\>(\forall k=0,1,2,3,...)$, where $\kappa$ is the 
coupling constant and $g\in{\cal D}(\RR^4)$ is a test function which
switches the interaction;
\item a BRS-current 
%%%%%%%%%%
\beq
j^{\mu}(g)(x):=\sum_{k=0}^\infty \kappa^k\,(g(x))^k\, j^{(k)\,\mu}(x)\ ,\quad
 j^{(k)\,\mu}\in {\cal P}\ ;\label{j(g)}
\eeq
%%%%%%%%%%%
\item and a  $Q$-vertex\footnote{In the terminology of \cite{S-wiley} the defining 
property of a $Q$-vertex ${\cal L}_1^{(1)\,\nu}$ is (\ref{PGI}) for $n=1$.
The fact that we use the word ``$Q$-vertex'' does not mean that we assume this identity 
to hold, it will be part of our conclusion. In addition, in our terminology a  $Q$-vertex
contains also terms of higher orders in $\kappa$.}
%%%%%%%%%%
\beq
{\cal L}_1^{\nu}(g)(x):=\sum_{k=1}^\infty \kappa^k\,(g(x))^{(k-1)}\,
{\cal L}_1^{(k)\,\nu}(x)\ ,\quad {\cal L}_1^{(k)\,\nu}\in {\cal P}\ .\label{L_1(g)}
\eeq
%%%%%%%%%%%
\end{itemize}
Usually ${\cal L}^{(k)},\>j^{(k)\,\mu}$ and ${\cal L}_1^{(k)\,\nu}$ are of 
$(k+2)$-th order in the basic fields. The really restricting part of our assumption is
that  in {\it classical field theory} $j^{\mu}(g)$ and ${\cal L}_1^{\nu}(g)$
are related by a certain local current conservation (see (\ref{dj(g)})
below) which is a consequence of the field equations given by $S_{\rm total}(g)$.
We show that this implies PGI-tree, i.e.~the equation (\ref{PGI}) with $T^N$ replaced 
by the contribution $T^{N\,{\rm tree}}$ of its tree diagrams (on both sides of (\ref{PGI})).

We use the formalism of \cite{DF3} and \cite{DF2}, see Appendix A. 
With that a perturbative classical field (\ref{A^ret}) 
agrees exactly with the contribution of the tree diagrams
$A^\mathrm{tree}_{S(g)}(x)$ to the corresponding field $A_{S(g)}(x)$ (\ref{intfield:QFT})
of perturbative QFT: due to $A\sim\hbar^0,\> S(g)\sim\hbar^{-1}$ and
(\ref{R-tree}) it holds \cite{DF1}
%%%%%%%%%%%%%%%
\beq
A_{S(g)}(x)=A^\mathrm{tree}_{S(g)}(x)+{\cal O}(\hbar)\quad{\rm and}\quad
A^\mathrm{tree}_{S(g)}(x)\sim\hbar^0\ ,
\eeq
%%%%%%%%%%%%%%%%%
and hence
%%%%%%%%%
\beq
A^\mathrm{tree}_{S(g)}(x)\vert_{{\cal C}_{S_0}}\equiv
R_{S_0}^\mathrm{tree}\Bigl(e_\otimes^{S(g)},A(x)\Bigr)=
R_{S_0}^\mathrm{class}\Bigl(e_\otimes^{S(g)},A(x)\Bigr)\ .\label{tree=class}
\eeq
%%%%%%%%%%%%%%%%%%
Due to this identity the classical factorization of composite fields holds also 
for  $A^\mathrm{tree}_{S(g)}(x)$ \cite{DF2},
%%%%%%%%%%%%%%%
\beq
(AB)^\mathrm{tree}_{S(g)}(x)=A^\mathrm{tree}_{S(g)}(x)\cdot B^\mathrm{tree}_{S(g)}(x)\ ,
\quad A,B\in{\cal P}\ ,\label{factorization}
\eeq
%%%%%%%%%%%%%%%%%%
and the fields $\varphi^\mathrm{tree}_{S(g)}(x)\vert_{{\cal C}_{S_0}}$
(where $\varphi$ runs through the basic fields) satisfy the classical field equations,
which form a {\it closed} system of partial differential equations. The product on the right
side of (\ref{factorization}) is the classical product (see Appendix A), not the 
$\star$-product (\ref{*-product}). The latter is related to
the Poisson bracket (of classical field theory) by 
%%%%%%%%%%%%%
\beq
\{F,G\}=\lim_{\hbar\to 0}\frac{i}{\hbar}(F\star G-G\star F)\ ,\quad F,G\in {\cal F}\ .\label{Pb}
\eeq
%%%%%%%%%%%%%%%%
%As shown e.g. in \cite{DF3} the normalization of the retarded products can be 
%chosen such that the perturbative interacting fields 
%$\varphi_{S(g)}(x)$ fulfill the field equations. With that, and due to (\ref{factorization}),
%the tree parts $\varphi^\mathrm{tree}_{S(g)}(x)$ satisfy the classical field equations,
%which form a {\it closed} system of partial differential equations.

%Since in classical field theory the local conservation of the (interacting) BRS-current is 
%derived from BRS-invariance of the Lagrangian,
%the classical field equations and the factorization (\ref{factorization}) only, this derivation 
%applies also to the tree part $j(g)^{\mu\,\mathrm{tree}}_{S(g)}(x)\vert_{{\cal C}_{S_0}}$ 
%of the BRS-current $j(g)_{S(g)}^{\mu}(x)\vert_{{\cal C}_{S_0}}$. 
%Proceeding in this way we obtain an identity for the divergence of 
%$R_{S_0}^\mathrm{tree}\Bigl(e_\otimes^{S(g)},j(g)^{\mu}(x)\Bigr)$, 
%and from this identity we will derive PGI-tree.
%However, before giving this derivation we have to do some preparations.
%by following a procedure given in \cite{DF} and \cite{DF2}.
%which we translate
%into an identity for the corresponding time ordered products $T^\mathrm{tree}(...)$. 
%From the latter identity we derive perturbative gauge invariance for tree diagrams
%by following a procedure given in \cite{DF}. 

%Since our formalism is also valid for classical field theory, the higher order terms of the 
%interaction $S(g)=\int dx\,{\cal L}(g)(x)$, the BRS-current 
%$j(g)^{\mu}(x)$ and the Q-vertex ${\cal L}_1^\nu (g)$
%are written explicitly. In the second part of this Sect. we will absorb them in an admissible
%renormalization of $T^\mathrm{tree}$ (by using the 'Main Theorem of Perturbative 
%Renormalization' \cite{DF3}) and this will yield the usual form of perturbative gauge 
%invariance \cite{S-wiley}.

Next we give some preparations concerning the BRS-structure.
In a BRS-model \cite{BRS} the basic fields are
\begin{itemize}
\item bosonic gauge fields with spin-1 ($A^\mu$) or spin-2 ($h^{\mu\nu}$),
\item for each gauge field there is precisely one pair of fermionic ghost fields $(u,\tilde u)$,
\item for each {\it massive} gauge field there is precisely one bosonic ghost field $\phi$,
\item in non-Abelian massive spin-1 gauge theories there is at least one 
physical Higgs field $H$
\item and there may be fermionic spinor fields.
\end{itemize}
The field algebras ${\cal F},\>{\cal F}^{(m)}_0\equiv\frac{\cal F}{{\cal J}^{(m)}}$ 
(classical product, see Appendix A) and ${\cal A}^{(m)}_0\equiv 
({\cal F}^{(m)}_0,\star_m)$ are
$\ZZ_2$-graded by the number of ghost fields: ${\cal F}={\cal F}_{\rm even}\oplus 
{\cal F}_{\rm odd}$. The ghost number of the action is even, and it is odd for the
BRS-current and the $Q$-vertex.

We assume that the free BRS-transformation $s_0$ acts linearly on the basic fields,
i.e.~$s_0\varphi$ ($\varphi$ a basic field) is a linear combination of partial derivatives of 
basic fields. This implies that $j^{(0)\,\mu}$ is quadratic in the (derivated) basic fields, since 
${\cal L}^{(0)}$ is quadratic in the (derivated) basic fields. 

We shall need some basic properties of the free BRS-charge $Q$ (or more precisely of
$d_Q$ (\ref{d_Q})). Formally, $Q$ is given by 
%%%%%%%%%%%
\beq
Q:= \int_{x^0={\rm const.}} d^3x\> j_{S_0}^{(0)\,0}(x^0,\vec{x})\ ,\label{Q}
\eeq
%%%%%%%%%%%%%%%%
which is a functional on ${\cal C}_{S_0}$.
The problem with this formula is that a priori $j^{(0)\,0}_{S_0}(x^0,\vec{x})$ 
can only be integrated out in $\vec{x}$ {\it and} $x^0$ and only with a test function.  
To give a rigorous definition we follow Sect. 5.1 of \cite{DF} where a 
method of Requardt \cite{R} is used. Let $k(x^0)\,h(\vec{x})\in {\cal D}(\RR^4)$,
where $\int dx^0\,k(x^0)=1$ and $h$ is a smeared characteristic function of
$\{\vec{x}\in \RR^3, |\vec{x}|\leq R\}$ for some $R>0$. We scale the test
function such that the normalization of $k$ is maintained,
%%%%%%%%%
\beq
k_\lambda (x^0)\=d\lambda\, k(\lambda x^0)\ ,\quad h_\lambda 
(\vec{x})\=d h(\lambda \vec{x})\ ,
\eeq
%%%%%%%%%%%
and want to define $Q$ as the limit
%%%%%%%%%
\beq
Q\=d\lim_{\lambda\to 0} Q_\lambda\ ,\quad 
Q_\lambda\=d \int d^4x\,k_\lambda (x^0)\,h_\lambda (\vec{x}) \,j^{(0)\,0}_{S_0}(x^0,\vec{x})\ .
\label{Q_lambda}
\eeq
%%%%%%%%%%%%%
As far as we know the existence of this limit cannot be shown generally, structural information
about the concrete model is needed. However, in this paper we are not interested in
$Q$ itself, we only study the operator $d_Q:{\cal A}_0^{(m)}
\rightarrow {\cal A}_0^{(m)}$, which is given by the graded commutator\footnote{Note
that the ghost number of $Q$ is odd.}
%%%%%%%%%%%%%%%%%
\beq
d_Q\>F_{S_0}\=d  \lim_{\lambda\to 0}\>(Q_\lambda\star F_{S_0}\mp F_{S_0}\star Q_\lambda)
=: \lim_{\lambda\to 0}\>[Q_\lambda, F_{S_0}]^\mp_\star\ ,\label{d_Q}
\eeq
%%%%%%%%%%%%%%%%%
where the minus sign appears for $F\in {\cal F}_{\rm even}$ and the plus sign for
$F\in {\cal F}_{\rm odd}$. It immediately follows that $d_Q$ is a graded 
derivation\footnote{A graded derivation $D$ of a $\ZZ_2$-graded
algebra ${\cal A}$ is defined to be a {\bf linear} map $D:{\cal A}\rightarrow {\cal A}$ with
%%%%%%%%%
\beq
D(A\cdot  B)=D(A)\cdot B +(-1)^{\epsilon (A)}\,A\cdot D(B)\ ,
\eeq
%%%%%%%%%
where $A$ is of definite degree $\epsilon(A)\in\{0,1\}$.}
with respect to the $\star$-product, provided the limit
%%%%%%%%%
\beq
d_Q\>F_{S_0}= \lim_{\lambda\to 0}\> [Q_\lambda, F_{S_0}]^\mp_\star =
\lim_{\lambda\to 0}\int dx^0\,k_\lambda (x^0)\int d^3x\,h_\lambda (\vec{x}) \,
[j^{(0)\,0}_{S_0}(x^0,\vec{x}), F_{S_0}]^\mp_\star\label{d_Q1}
\eeq
%%%%%%%%%
exists. This holds indeed true \cite{DF}: namely, because of 
$\supp [j^{(0)\,0}_{S_0}(x), F_{S_0}]^\mp_\star\subset (\supp F+(\bar V_+\cup \bar V_-))$ 
we may replace $h_\lambda (\vec{x})$ by $1$
for $\lambda >0$ sufficiently small and $R$ big compared with the support of $k$.
Note that $\int d^3x\,[j^{(0)\,0}_{S_0}(x^0,\vec{x}), F_{S_0}]^\mp_\star$
exists since the region of integration is bounded; and, due to current conservation,
it is independent of $x^0$. This yields
%%%%%%%%%
\beq
\lim_{\lambda\to 0}\> [Q_\lambda, F_{S_0}]^\mp_\star =
\int_{x^0={\rm const.}} d^3x\,[j^{(0)\,0}_{S_0}(x^0,\vec{x}), F_{S_0}]^\mp_\star\ .\label{d_Q2}
\eeq
%%%%%%%%%%%%%%
This result holds for the terms $\sim\hbar$ of $[\,\cdot\,,\,\cdot\,]^\mp_\star$ separately. Hence,
we may define
%%%%%%%%%
\beq
\{Q, F_{S_0}\}:\=d \lim_{\lambda\to 0}\> \{Q_\lambda, F_{S_0}\}
=\lim_{\hbar\to 0}\frac{i}{\hbar}\>d_Q\>F_{S_0}\label{Pb:Q}
\eeq
%%%%%%%%%
and obtain $\{Q, F_{S_0}\}=\int d^3x\,\{j^{(0)\,0}_{S_0}(x^0,\vec{x}), F_{S_0}\}$.

For the concrete models studied in Sect.~4 it holds 
%%%%%%%%
\beq
d_Q^2=0\label{nilpotent}
\eeq
%%%%%%%%%%
and
%%%%%%%%%%
\beq
\omega_0(d_Q\> F_{S_0})=0\ ,\quad\forall F\in {\cal F}\ ,
\label{Q-vaccum}
\eeq
%%%%%%%%%%%%%
as it is verified e.g. in \cite{S-wiley} (in a Krein-Fock space representation). In this paper we 
assume only the validity of (\ref{Q-vaccum}), the nil-potency will not be needed.
(Usually, the fields are represented on an inner product space such that 
$\langle F^*\Phi,\Psi\rangle =\langle \Phi,F\Psi\rangle$ (where $\langle\>\cdot\>,\>\cdot\>
\rangle$ must be indefinite) and in that representation one proves that $Q$ 
is a nilpotent and symmetric operator which annihilates the vacuum; these properties
imply (\ref{nilpotent}) and (\ref{Q-vaccum}).)

We are now going to show that $d_Q$ is a graded derivation 
also with respect to the {\bf classical} product. This follows from 
the observation that in $d_Q\, F_{S_0}$ (\ref{d_Q}) solely the terms $\sim\hbar$ of
the $\star$-product contribute. In detail:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} 
  \begin{enumerate}\item
%%%%%%%%%
\beq
d_Q\>F_{S_0}=-i\hbar\>\{Q,F_{S_0}\}\ ,\quad\forall F\in{\cal F}\ ,
\eeq
%%%%%%%%%
where $\{Q,\cdot\}$ is defined by (\ref{Pb:Q}).\\
\item
%%%%%%%%%
\beq
d_Q\>(F_{S_0}\,G_{S_0})=d_Q(F_{S_0})\>G_{S_0}\pm F_{S_0}\>d_Q(G_{S_0})\ ,
\eeq
%%%%%%%%%
where the $+$-sign holds for $F\in {\cal F}_{\rm even}$ and the $-$-sign holds 
for $F\in {\cal F}_{\rm odd}$.
 \end{enumerate}
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof} (ii) is a consequence of (i): since the graded Poisson bracket satisfies the 
graded Leibniz rule, the map $\{Q,\cdot\}$ (\ref{Pb:Q}) is a graded derivation with respect 
to the classical product.

To concentrate on the essential steps of the proof of (i) we replace $F_{S_0}$
by $\phi_1...\phi_n$ (classical product), where $\phi_j$ is (a derivative of) a basic Bose 
field $\varphi_{i_j}(x_j)$: $\phi_j=\d^{a_j}
\varphi_{i_j}(x_j)_{S_0}$. All non-vanishing terms in $[Q_\lambda,\phi_j]_\star$ have one 
contraction, hence
%%%%%%%%%
\beq
d_Q\>\phi_j=-i\hbar\>\{Q,\phi_j\}\ .
\eeq
%%%%%%%%%%%%%%%
To show that $d_Q(\phi_1...\phi_n)$ agrees with
%%%%%%%%%%%%%%%
\beq
-i\hbar\>\{Q,\phi_1...\phi_n\}=-i\hbar\sum_{k=1}^n\phi_1...\{Q,\phi_k\}...\phi_n\ ,
\eeq
%%%%%%%%%%%
we proceed by induction on the number $n$ of factors. By using 
\begin{itemize}
\item the recursion relation
%%%%%%%%%%
\beq
\phi_1...\phi_{n+1}=(\phi_1...\phi_n)\star\phi_{n+1}-\sum_{l=1}^n
(\phi_1...\hat l...\phi_n)\>\omega_0(\phi_l\star\phi_{n+1})
\eeq
%%%%%%%%%%%%%%%%%%%%
(which follows from the definition (\ref{*-product}) of the $\star$-product),
\item  our assumption that $s_0$ acts linearly on the basic fields which implies that 
$\{Q,\phi_j\}$ is a linear combination of derivated basic fields, 
\item and
%%%%%%%%%%%%%%
\beq
\omega_0(\{Q,\phi_l\}\star\phi_{n+1})+\omega_0(\phi_l\star\{Q,\phi_{n+1}\})
=\frac{i}{\hbar}\> \omega_0([Q\,,\,\phi_l\star\phi_{n+1}]_\star)=0
\eeq
%%%%%%%%%%%%%%%%
(where (\ref{Q-vaccum}) is used), 
\end{itemize}
one verifies the inductive step straightforwardly.
\end{proof} 
An alternative proof of the statement (ii) can be found in Lemma 3.1.1. of 
\cite{S-wiley}.

In this Sect.~we {\bf assume} that there exist a BRS-current $j^\mu(g)$ (\ref{j(g)}), 
an action $S_{\rm total}(g)=S_0+S(g)$  (\ref{L(g)}) and a $Q$-vertex
${\cal L}_1^{\nu}(g)$ (\ref{L_1(g)}) such that they fulfill the local current conservation
%%%%%%%%%%%%%%%%%%%%
\beq
\d_\mu\, j^\mu(g)(x)_{S_{\rm total}(g)}=(\d_\nu g)(x)\>{\cal L}_1^\nu(g)(x)_{S_{\rm total}(g)}\ ,
\label{dj(g)}
\eeq
%%%%%%%%%
in classical field theory. This identity implies that the {\it BRS-current} 
$j^\mu(g)(x)_{S_{\rm total}(g)}$ {\it is conserved for} $x\in {\cal U}$
(where ${\cal U}\subset \RR^4$ is an open set)
{\it if} $g\vert_{{\cal U}}$ {\it is constant}. We will verify the assumption (\ref{dj(g)})
for concrete models in the following Sects.. 
The current conservation (\ref{dj(g)}) holds also for the corresponding 
retarded fields (\ref{A^ret:def})
and, hence, also for the perturbative expansion (\ref{A^ret}) of the latter.
With that and due to the identity (\ref{tree=class}) we obtain a statement for the 
tree diagrams of perturbative QFT:
%%%%%%%%%%%%%%%%%%
\beq
- R^{\mathrm{tree}}_{S_0}\Bigl(e_\otimes^{S(g)},\int j(g)^{\mu}\,\d_\mu f\Bigr)=
R^{\mathrm{tree}}_{S_0}\Bigl(e_\otimes^{S(g)},\int {\cal L}_1^{\nu}(g)\,f\,\d_\nu g\Bigr)
\ .\label{curcons:R}
\eeq
%%%%%%%%%%%%%%%%%%

We are now going to absorb the higher order terms of $S(g)$, $j(g)^{\mu}$ and
${\cal L}_1^\nu (g)$ in an admissible
renormalization of the retarded product by using the 'Main Theorem of Perturbative 
Renormalization' (Sect.~4.2 of \cite{DF3}).  We assume that ${\cal L}^{(1)}$,
$j^{(0)\,\mu}$ and ${\cal L}_1^{(1)\,\nu}$ are linearly independent fields; or that
${\cal L}^{(1)}$ and $j^{(0)\,\mu}$ are linearly independent and ${\cal L}_1^{(1)\,\nu}=0$. 
(This assumption seems to hold true in all cases of interest, see Sect.~4.) 
With that we set
%%%%%%%%%%%%%%%%
\begin{gather}
\sum_{n=0}^\infty\frac{1}{n!}\,D_n\Bigl((\int g\,{\cal L}^{(1)})^{\otimes n}\Bigr)\equiv
D\Bigl(e_{\otimes}^{\int g\,{\cal L}^{(1)}}\Bigr)\=d S(g)\ ,\notag\\
D\Bigl(e_{\otimes}^{\int g\,{\cal L}^{(1)}}{\otimes} \int h_\mu\,j^{(0)\,\mu}\Bigr)\=d
\int dx\,h_\mu(x)\,j^\mu(g)(x)\ ,\notag\\
D\Bigl(e_{\otimes}^{\int g\,{\cal L}^{(1)}}{\otimes} \int h_\nu\,{\cal L}_1^{(1)\,\nu}\Bigr)\=d
\int dx\,h_\nu(x)\,{\cal L}_1^\nu(g)(x)
\end{gather}
%%%%%%%%%%%%%%%%%
and extend $D_n$ to a {\it linear} and {\it symmetrical} map
%%%%%%%%%%%%%
\beq
D_n:{\cal F}_{\rm loc}^{\otimes n}\longrightarrow {\cal F}_{\rm loc}\label{D}
\eeq
%%%%%%%%%%%%%%%%
with
\begin{itemize}
\item $D_0(1)=0$, $\quad D_1(F)=F$\ ,
\item 
%%%%%%%%%%%%%%
       \begin{equation}
       \supp\frac{\delta\,D_n(F_1\otimes\ldots\otimes F_n)}{\delta\varphi}\subset
       \bigcap_{i=1}^n \supp\frac{\delta F_{i}}{\delta\varphi}\ ,
        \end{equation}
       %%%%%%%%%%%%%%% 
\item Poincar\'e covariance and
\item $D(F^{\otimes n})^{*}=D((F^{*})^{\otimes n})$.
\end{itemize}
Part (iv) of the Main Theorem then states that\footnote{This formula has to be understood 
in the sense of formal power series in $\lambda$.} 
%%%%%%%%%%%%%%%%
   \begin{equation}
        R^N(e_{\otimes}^{\lambda G}, F) \=d 
        R(e_{\otimes}^{D(e_{\otimes}^{\lambda G})},
        D(e_{\otimes}^{\lambda G}\otimes F)) \label{ren:intf}
   \end{equation}
%%%%%%%%%%%%%%%%
defines a new retarded product $R^N$, i.e.~$R^N$ fulfills the basic properties 
(\ref{ret-prod:QFT})-(\ref{GLZ}). And, if $R$ is Poincar\'e covariant and unitary
(i.e. $R(F^{\otimes n})^*=R((F^*)^{\otimes n})$), then these properties hold also for
$R^N$. Usually $\omega_0(R_{n-1,1}(...))$ satisfies an upper bound on the scaling 
behaviour in the UV-region. The map $D$ (\ref{D}) can be chosen such that also this bound 
is maintained in the renormalization $R\longrightarrow R^N$. In terms of $R^N$
the current conservation (\ref{curcons:R}) reads
%%%%%%%%%%%%%%%%%%
\beq
- R^N_{S_0}\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}},\int j^{(0)\,\mu}\,\d_\mu f\Bigr)=
R^N_{S_0}\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}},\int {\cal L}_1^{(1)\,\nu}\,f\,\d_\nu g\Bigr)
+{\cal O}(\hbar)\ ,\label{curcons:R^N}
\eeq
%%%%%%%%%%%%%%%%%%
where we have also used (\ref{R-tree}). This identity implies
%%%%%%%%%%%%%%%%%%
\begin{gather}
- R^N_{S_0}\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}}\otimes \int g\,{\cal L}^{(1)}
,\int j^{(0)\,\mu}\,\d_\mu f\Bigr)=
R^N_{S_0}\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}},\int {\cal L}_1^{(1)\,\nu}\,f\,\d_\nu g\Bigr)\notag\\
+R^N_{S_0}\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}}\otimes \int g\,{\cal L}^{(1)}
,\int {\cal L}_1^{(1)\,\nu}\,f\,\d_\nu g\Bigr)+{\cal O}(\hbar)\ .\label{curcons:R^N1}
\end{gather}
%%%%%%%%%%%%%%%%%%

We are now going to derive PGI for $R^{N\,{\rm tree}}$.
For a given $g\in {\cal D}(\RR^4)$ let ${\cal O}$ be an open double cone such that 
$\supp g\subset {\cal O}$. Furthermore we choose $f\in {\cal D}(\RR^4)$ with
$f\equiv 1$ on a neighborhood of $\overline{\cal O}$. We decompose 
$\d_\mu f =b_\mu -a_\mu$ such that
$\supp b_\mu\cap ({\cal O}+\bar V_+)=\emptyset$ and 
$\supp a_\mu\cap ({\cal O}+\bar V_-)=\emptyset$. In the following calculation (which is mainly
taken from Sect.~5.2 of \cite{DF2}) we take into account ${\cal L}^{(1)}\sim\hbar^{-1}$,
${\cal L}_1^{(1)}\sim\hbar^{0}$ and 
%%%%%%%%%
\beq
\{Q,F_{S_0}\}=\{\int dx\, j_{S_0}^{(0)\,\mu}(x)\,b_\mu(x),F_{S_0}\}\ ,\quad\forall
F\in {\cal F}({\cal O})\ ,
\eeq
%%%%%%%%%%%%%%%%
which follows from the locality of the Poisson bracket (i.e. $\{G_{S_0},H_{S_0}\}=0$
if the supports of $G_{S_0}$ and $H_{S_0}$ are space-like separated), the current 
conservation $\d_\mu j_{S_0}^{(0)\,\mu}=0$ and the definition of $\{Q,\cdot\}$ 
(\ref{Pb:Q}). (A detailed explanation of the same conclusion 
for the commutator is given in Appendix B of \cite{DF}.) In addition we use 
Lemma 1(i), the characterization (\ref{R-tree}) of  $R^{N\,\rm tree}$, 
$R_{0,1}(F)=F$, causality (\ref{caus}) and the GLZ relation (\ref{GLZ}):
%%%%%%%%%%%%%%%%
\begin{gather}
d_Q\,R_{S_0}^{N\,{\rm tree}}\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}},\int g\,{\cal L}^{(1)}
\Bigr)%=-i\hbar\Bigl\{Q\,,\,R_{S_0}^{N\,{\rm tree}}\Bigl(...\Bigr)\Bigr\}\notag\\
=-i\hbar\Bigl\{\int j_{S_0}^{(0)\,\mu}\,b_\mu\,,\,R_{S_0}^{N\,{\rm tree}}\Bigl(...\Bigr)\Bigr\}\notag\\
=\Bigl[\int j_{S_0}^{(0)}\,b\,,\,R_{S_0}^N\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}},
\int g\,{\cal L}^{(1)}\Bigr)\Bigr]_\star\vert_{\hbar^0}\notag\\
=\Bigl[R_{S_0}^N\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}},\int j^{(0)}\,b\Bigr)\,,\,
R_{S_0}^N\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}},
\int g\,{\cal L}^{(1)}\Bigr)\Bigr]_\star\vert_{\hbar^0}\notag\\
=i\Bigl(R_{S_0}^N\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}}\otimes\int\,j^{(0)}\,b,
\int g\,{\cal L}^{(1)}\Bigr)-R_{S_0}^N\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}}\otimes
\int g\,{\cal L}^{(1)},\int j^{(0)}\,b\Bigr)\Bigr)\vert_{\hbar^0}\ ,\label{d_QR1}
\end{gather}
%%%%%%%%%%%%%%%%%
where $...\vert_{\hbar^0}$ signifies that we only mean the contribution of the terms 
$\sim\hbar^0$ (which is in all terms 
of (\ref{d_QR1})-(\ref{d_QR3}) the contribution with the lowest power of $\hbar$).
Due to the the support property of $R^N$ the last retarded product vanishes and in 
the second last we may replace $b_\mu$ by $\d_\mu f$. By using the GLZ relation 
again we obtain a form to which we can apply the classical current conservation 
(\ref{curcons:R^N}) and (\ref{curcons:R^N1}):
%%%%%%%%%%%%%%%%
\begin{gather}
=i\,R_{S_0}^N\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}}\otimes\int\,j^{(0)\,\mu}\,\d_\mu f,
\int g\,{\cal L}^{(1)}\Bigr)\vert_{\hbar^0}\notag\\
=\Bigl[R_{S_0}^N\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}},\int j^{(0)}\,\d f\Bigr)\,,\,
R_{S_0}^N\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}},
\int g\,{\cal L}^{(1)}\Bigr)\Bigr]_\star\vert_{\hbar^0}\notag\\
+i\,R_{S_0}^N\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}}\otimes
\int g\,{\cal L}^{(1)},\int j^{(0)}\,\d f\Bigr)\vert_{\hbar^0}\notag\\
=-\Bigl[R^N_{S_0}\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}},\int {\cal L}_1^{(1)}\,\d g\Bigr)
\,,\,R_{S_0}^N\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}},
\int g\,{\cal L}^{(1)}\Bigr)\Bigr]_\star\vert_{\hbar^0}\notag\\
-i\,R^N_{S_0}\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}}\otimes \int g\,{\cal L}^{(1)}
,\int {\cal L}_1^{(1)}\,\d g\Bigr)\vert_{\hbar^0}
-i\,R^N_{S_0}\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}},\int {\cal L}_1^{(1)}\,\d g\Bigr)
\vert_{\hbar^0}\ ,\label{d_QR2}
\end{gather}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $f(x)\,\d_\nu g(x) =\d_\nu g(x)$ is taken into account. By 
means of the GLZ relation
the first three retarded products can be expressed by one retarded product.
Applying (\ref{R-tree}) again we end up with
%%%%%%%%%%%%%%%%
\begin{gather}
%=-i\,R^N_{S_0}\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}}\otimes \int {\cal L}_1^{(1)}\,\d g
%,\int g\,{\cal L}^{(1)}\Bigr)\vert_{\hbar^0}
%-i\,R^N_{S_0}\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}},\int {\cal L}_1^{(1)}\,\d g\Bigr)
%\vert_{\hbar^0}\notag\\
=-i\,R^{N\,{\rm tree}}_{S_0}\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}}\otimes
\int {\cal L}_1^{(1)}\,\d g ,\int g\,{\cal L}^{(1)}\Bigr)
-i\,R^{N\,{\rm tree}}_{S_0}\Bigl(e_\otimes^{\int g\,{\cal L}^{(1)}},\int {\cal L}_1^{(1)}\,
\d g\Bigr)\ .\label{d_QR3}
\end{gather}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
(\ref{d_QR1})=(\ref{d_QR3}) is PGI-tree for $R^N$.

Next we show that PGI-tree is maintained in the transition to the corresponding 
{\it connected} time ordered product $T^N_c$. Following
Sect.~5.2 of \cite{DF1} we define recursively the {\it connected product}
%%%%%%%%%%%%%%%%%%%%
\begin{equation}
(F_1\star...\star F_n)_c\=d (F_1\star...
\star F_n)-\sum_{|P|\geq 2}\prod_{J\in P}(F_{j_1}\star...\star 
F_{j_{|J|}})_c\ ,\label{conn}
\end{equation}
%%%%%%%%%%%%%%%%
where $\{j_1,...,j_{|J|}\}=J$, $j_1<...<j_{|J|}$,  the sum runs over 
all partitions $P$ of $\{1,...,n\}$ in at least two subsets and
$\prod$ means the classical product. Identifying the vertices within each
$F_j$, $(F_1\star...\star F_n)_c$ is precisely the contribution of all connected 
diagrams to $F_1\star...\star F_n$. According to Proposition 1 of \cite{DF2} it holds
%%%%%%%%%%%
\beq
(F_1\star...\star F_n)_c={\cal O}(\hbar^{n-1})\quad{\rm if}\quad
F_1,...,F_n\sim\hbar^0\ .
\eeq
%%%%%%%%%%%%%%%%
Since solely connected diagrams contribute to the retarded products $R_{n,1}$,
we conclude from formula (E.6) in \cite{DF3} that the connected part $T^N_c$ 
of $T^N$ is obtained from $R^N$ by
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{gather}
  T^N_{n\,c}(F^{\otimes n})=\sum_{k=1}^n i^{k-n}\sum_{l_1+...+l_k=n-k}
%\frac{n!}{l_1!l_2!...l_k!}\frac{1}{(l_1+1)(l_1+l_2+2)...(l_1+l_2+...+l_k+k)}
N(n,k,l_1,...,l_k)\notag\\
\cdot\bigl(R^N_{l_1,1}(F^{\otimes l_1},F)\star...\star 
R^N_{l_k,1}(F^{\otimes l_k},F)\bigr)_c\ ,
\end{gather}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $N(n,k,l_1,...,l_k)\in\RR$ is a combinatorical factor.
We find $T^N_{n\,c}(F^{\otimes n})= {\cal O}(\hbar^{n-1})$ if $F\sim\hbar^0$.
The contribution of the tree diagrams is that part with the lowest power of $\hbar$:
%%%%%%%%%%%
\beq
T^{N\,{\rm tree}}_{n\,c}(F^{\otimes n})=\sum...\bigl(R^{N\,{\rm tree}}_{l_1,1}(...)\star...\star 
R^{N\,{\rm tree}}_{l_k,1}(...)\bigr)_c\vert_{\hbar^{n-1}}\ .
\eeq
%%%%%%%%%%%%%%%%
Because $d_Q$ is a graded derivation with respect to the classical and the $\star$-product,
it is also a graded derivation with respect to the {\it connected} product (\ref{conn}).
Hence, PGI for $R^{N\,{\rm tree}}$ implies
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{gather}
d_Q\,T^{N\,{\rm tree}}_{n\,c\>S_0}\Bigl((\int g\,{\cal L}^{(1)})^{\otimes n}\Bigr)=\sum...
\frac{d}{d\lambda}\vert_{\lambda =0}\Bigl(R^{N\,{\rm tree}}_{l_1,1\>S_0}\Bigl(
(\int g\,{\cal L}^{(1)}-i\lambda\,\int {\cal L}^{(1)}_1\,\d g)^{\otimes (l_1+1)}\Bigr)\star...\notag\\
\star R^{N\,{\rm tree}}_{l_k,1\>S_0}\Bigl(
(\int g\,{\cal L}^{(1)}-i\lambda\,\int {\cal L}^{(1)}_1\,\d g)^{\otimes (l_k+1)}\Bigr)
\Bigr)_c\vert_{\hbar^0}\notag\\
=-i\,n\>T^{N\,{\rm tree}}_{n\,c\>S_0}\Bigl(\int {\cal L}^{(1)}_1\,\d g\otimes 
(\int g\,{\cal L}^{(1)})^{\otimes (n-1)}\Bigr)\ .\label{gauinv:T_c}
\end{gather}
%%%%%%%%%%%%%%%%%%%%%%%%%%%

Finally PGI remains valid also in the step from 
$T^{N\,{\rm tree}}_{c}$ to $T^{N\,{\rm tree}}$, because each tree diagram is the 
{\it classical} product of its connected components (the latter are also tree diagrams)
and since $d_Q$ is a graded derivation with respect to the classical product.
In detail,
%%%%%%%%%%%%%%%%%%%%
\begin{equation}
T^{N\,{\rm tree}}_n(F_1\star...\star F_n)= T^{N\,{\rm tree}}_{n\,c}(F_1\star...
\star F_n)+\sum_{|P|\geq 2}\prod_{J\in P}T^{N\,{\rm tree}}_{|J|\, c}(F_{j_1}\star...\star 
F_{j_{|J|}}) ,\label{T-tree}
\end{equation}
%%%%%%%%%%%%%%%%
where $\prod$, $P$ and $J$ are as in (\ref{conn}). With that the statement is obtained 
analogously to (\ref{gauinv:T_c}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{From BRS-invariance of the Lagrangian to local 
conservation of the classical BRS-current}\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The proof in the preceding Sect.~is based on the local BRS-current conservation 
(\ref{dj(g)}) for {\bf classical field theory}. In this Sect.~we show that this 
assumption follows from {\it BRS-invariance of the Lagrangian for constant coupling}. 
The latter is verified for concrete models in Sect.~4, in particular for massless
spin-2 gauge fields.

In this procedure, solely the Lagrangian and the BRS-transformation $s$ for constant 
coupling $\kappa$ need to be given. With that we construct a BRS-transformation 
$s(g)$ for local coupling $\kappa\,g(x)$ and a corresponding local
Noether current $j^\mu(g)$. We show that the divergence of this local BRS-current
is indeed of the form  (\ref{dj(g)}) and in doing so we obtain an explicit formula
for the $Q$-vertex ${\cal L}^\mu_1(g)$. 

The Lagrangian and the BRS-transformation are assumed to be {\it formal power 
series in} $\kappa$ and we understand all equations in the sense of formal power series.
However, we point out that Sects.~3 and 4 are {\bf non-perturbative} classical field theory.
\medskip

We assume that a BRS-invariant Lagrangian
%%%%%%%%
\beq
{\cal L}_{\rm total}=\sum_{n=0}^\infty\kappa^n\,{\cal L}^{(n)}
=:{\cal L}^{(0)}+{\cal L}_{\rm int}\ ,\quad
{\cal L}^{(n)}\in {\cal P}\ ,\label{L_total}
\eeq
%%%%%%%%%%
is given, where the free part ${\cal L}^{(0)}$ is quadratic in the (derivated) basic fields
and higher than first derivatives of the basic fields do not appear in each ${\cal L}^{(n)}$. 
By BRS-invariance we mean that there exists
%%%%%%%%%%%%%
\beq 
I^\mu=\sum_{n=0}^\infty \kappa^n\, I^{(n)\,\mu}\ ,\quad
I^{(n)\,\mu}\in {\cal P}\ ,\label{I}
\eeq
%%%%%%%%%%%%%%%
such that
%%%%%%%%%%%%%
\beq
s\,{\cal L}_{\rm total}=-\d_\mu\,I^\mu\label{sL1}
\eeq
%%%%%%%%%%%%%%%
without using the field equations. We admit BRS-transformations which are formal power 
series in $\kappa$:
%%%%%%%%%%%%%
\beq 
s=\sum_{n=0}^\infty \kappa^n\, s_n\ .\label{s}
\eeq
%%%%%%%%%%%%%%%
We assume that $s$ is given on the basic fields $\varphi$ and that it is extended to a
linear map $s:{\cal P}\rightarrow {\cal P}$ (more precisely, to a formal power series of 
linear maps $s_n:{\cal P}\rightarrow {\cal P}$) by setting
%%%%%%%
\beq
s\,(\d^a\varphi)\=d\d^a(s\,\varphi)\ ,\quad a\in\NN_0^4\ ,\label{sdphi}
\eeq
%%%%%%%%%%%%%%%%
for all basic fields $\varphi$, and by requiring that $s$ is a graded derivation.
It follows 
%%%%%%%%%%
\beq
s\,(\d^\mu \,A)=\d^\mu\,(s\,A)\ ,\quad\forall A\in {\cal P}\ ,\label{[s,d]}
\eeq
%%%%%%%%%%%
and that each $s_n$ is a graded derivation which commutes with partial derivatives
(\ref{[s,d]}). In order that the BRS-symmetry (\ref{sL1}) can be used to construct the 
corresponding quantum gauge theory it is needed that $s$ is nilpotent modulo the 
field equations:
%%%%%%%%%
\beq
s^2\,(A)\>\vert_{{\cal C}_{S_{\rm total}}}=0\ ,\quad\forall A\in{\cal P}\ ,\label{s^2=0}
\eeq
%%%%%%%%%%%%%%%%
where $S_{\rm total}=\int dx\, {\cal L}_{\rm total}$. However, in the proof of 
PGI-tree given in this paper the nil-potency of $s$ is not used.
\medskip

We first recall the construction of the Noether current for {\bf constant coupling} $\kappa$
(cf.~any book on classical field theory). By using the derivation property of $s$, 
(\ref{[s,d]}) and the field equations, we get
%%%%%%%%%%%
\beq
(s\,{\cal L}_{\rm total})_{S_{\rm total}}=\Bigl(
\frac{\d{\cal L}_{\rm total}}{\d\varphi_i}\,s\varphi_i+\frac{\d{\cal L}_{\rm total}}{\d\varphi_{i,\mu}}
\,(s\varphi_i)_{,\mu}\Bigr)_{S_{\rm total}}=\d_\mu\Bigl(
\frac{\d{\cal L}_{\rm total}}{\d\varphi_{i,\mu}}\,s\varphi_i\Bigr)_{S_{\rm total}}\ ,
\label{sL2}
\eeq
%%%%%%%%%%%%%%
where it is summed over all basic fields $\varphi_i$.
The equality of (\ref{sL1}) (restricted to ${\cal C}_{S_{\rm total}}$) and (\ref{sL2}) yields that
%%%%%%%%%
\beq
j^\mu\=d -\Bigl(\frac{\d{\cal L}_{\rm total}}{\d\varphi_{i,\mu}}\,s\varphi_i+I^\mu\Bigr) =
\sum_{n=0}^\infty \kappa^n\, j^{(n)\,\mu}\ ,\label{j}
\eeq
%%%%%%%%%%%%%
is a conserved BRS-current,
%%%%%%%
\beq
\d_\mu\,j^\mu_{S_{\rm total}}=0\ .\label{dj=0}
\eeq
%%%%%%%%
For later purpose we note
%%%%%%%%%
\beq
 j^{(n)\,\mu}=-\Bigl(\sum_{k=0}^n\frac{\d{\cal L}^{(k)}}{\d\varphi_{i,\mu}}\>s_{n-k}\,\varphi_i+
I^{(n)\>\mu}\Bigr)\ .\label{j^n}
\eeq
%%%%%%%%%%
\medskip

We are now going to generalize this construction of the BRS-current to 
{\bf local couplings} $\kappa\,g(x)$, $g\in {\cal D}(\RR^4)$. 
Roughly speaking we do this by replacing $\kappa$ by $\kappa\,g(x)$ everywhere.
In detail: from ${\cal L}^{(n)}$ (\ref{L_total}) and $j^{(n)\,\mu}$ (\ref{j^n})
we construct ${\cal L}(g)$ (\ref{L(g)}) and $j^{\mu}(g)$ (\ref{j(g)}).
In the Lagrangian ${\cal L}_{\rm total}$ (\ref{L_total}) we replace the interacting part
${\cal L}_{\rm int}=\sum_{\kappa=1}^\infty \kappa^n {\cal L}^{(n)}$ by ${\cal L}(g)$ 
(\ref{L(g)}) and we use the same notations $S_0,\>S(g)$ and $S_{\rm total}(g)$
as in Sect.~2.
%%%%%%%%%%%%%%
%\beq
%S_{\rm total}(g)=\int dx\, {\cal L}_{\rm total}(g)=S_0+S(g)\ ,
%\quad S_0=\int dx\, {\cal L}^{(0)}\ ,\quad S(g)=\int dx\,{\cal L}(g)\ .
%\eeq
%%%%%%%%%%%%%
The {\it local BRS-transformation} $s(g)$ is also a formal power series in $\kappa$,
%%%%%%%%%%%%%
\beq 
s(g)=\sum_{n=0}^\infty \kappa^n\, s_n(g)\ .\label{s(g)}
\eeq
%%%%%%%%%%%%%%%
We determine $s(g)$ by requiring that 
it is a graded derivation and by its action on the basic fields $\varphi$ 
and their partial derivatives:\footnote{Motivated by Sect.~5.2 of \cite{DF2} an
alternative, more explicit definition of $s(g)$ seems to be natural: namely
(\ref{s(g)}) with
%%%%%%%%%%%%%
\beq
s_n(g):=\int dx\,(g(x))^n\,\tilde s_n(x)\ ,\quad
\tilde s_n(x):= (s_n\varphi_i)(x)\,\frac{\delta}{\delta\varphi_i(x)}\ ,\label{s(g):alternat}
\eeq
%%%%%%%%%%%%%
where it is summed over all basic fields $\varphi_i$ and
$s_n$ is given from the model with constant coupling. Obviously, the so defined $s(g)$
is a graded derivation and one easily verifies that it satisfies (\ref{s(g)phi}) and
(\ref{s(g)dphi}). Hence, this definition (\ref{s(g):alternat}) agrees with the 
definition given in the main text.}
%%%%%%%%%%%
\beq
s(g)\,\varphi(x)\=d \sum_{n=0}^\infty \kappa^n\,(g(x))^n\, s_n\,\varphi(x)\ ,\label{s(g)phi}
\eeq
%%%%%%%%%%
(where $s_n$ is given from the model with constant coupling (\ref{s})) and
%%%%%%%%%%%
\beq
s(g)\,(\d^a\varphi)\=d \d^a (s(g)\,\varphi)\ ,\quad a\in\NN_0^4\ .\label{s(g)dphi}
\eeq
%%%%%%%%%%
As in (\ref{[s,d]}) it follows $s(g)\,(\d^\mu \,A)=\d^\mu\,(s(g)\,A)\ ,\>\forall
A\in {\cal P}$. Since $s(g)$ is a graded derivation which commutes with partial derivatives,
this holds also for
%%%%%%%
\beq
s_k(g)\=d \frac{1}{k!}\>\frac{d^k}{d\kappa^k}\vert_{\kappa =0}\, s(g)\ ,\quad
k\in\NN_0\ .
\eeq
%%%%%%%%%
For a basic field $\varphi$ we obtain
%%%%%%%%%
\begin{gather}
s_k(g)\,\varphi(x)=(g(x))^k\,s_k\,\varphi(x)\ ,\notag\\
s_k(g)\,\varphi^{,\mu}=\bigl(g^k\,s_k\,\varphi\bigr)^{,\mu}=
g^k\,s_k(\varphi^{,\mu})+g^{,\mu}\,k\,g^{(k-1)}\,s_k\,\varphi\ .\label{s_1(g)}
\end{gather}
$s(g)$ is in general not nilpotent modulo the field equations. But, if $g\vert_{\cal O}=1$
for some region ${\cal O}\subset\RR^4$, we have $s(g)\vert_{{\cal F}({\cal O})}= 
s\vert_{{\cal F}({\cal O})}$ (where $s$ is the BRS-transformation of the corresponding 
model with constant coupling) and hence 
%%%%%%%
\beq
(s(g))^2\,(F)\vert_{{\cal C}_{S_{\rm total}(g)}}=0\ ,\quad\forall F\in {\cal F}({\cal O})\ .
\eeq
%%%%%%%%%%%%%%%%

As in (\ref{sL2}) the field equations for $S_{\rm total}(g)$ imply
%%%%%%%%%%%%%
\begin{gather}
\Bigl(s(g)\,{\cal L}_{\rm total}(g)\Bigr)_{S_{\rm total}(g)}=\d_\mu\Bigl(
\frac{\d{\cal L}_{\rm total}(g)}{\d\varphi_{i,\mu}}\>
s(g)\,\varphi_i\Bigr)_{S_{\rm total}(g)}\notag\\
=\d_\mu\sum_{n=0}^\infty \kappa^n\,g^n\,\sum_{k=0}^n\Bigl(
\frac{\d{\cal L}^{(k)}}{\d\varphi_{i,\mu}}\>s_{n-k}\,\varphi_i\Bigr)_{S_{\rm total}(g)}\ .
\label{s(g)L2}
\end{gather}
%%%%%%%%%%%%%%%
By using the derivation property of $s_l(g)$ and of $s_l$ we obtain
%%%%%%%%%%%%%
\begin{gather}
s_l(g)\,{\cal L}^{(k)}=\frac{\d{\cal L}^{(k)}}{\d\varphi_i}\>g^l\,s_l\,\varphi_i +
\frac{\d{\cal L}^{(k)}}{\d\varphi_{i,\mu}}\>\bigl(g^l\,s_l\,(\varphi_{i,\mu})+g_{,\mu}\,
l\,g^{(l-1)}\,s_l\,\varphi_i\bigr)\notag\\
=g^l\,s_l\,{\cal L}^{(k)}+ g_{,\mu}\,l\,g^{(l-1)}\,
\frac{\d{\cal L}^{(k)}}{\d\varphi_{i,\mu}}\>s_l\,\varphi_i\ .
\end{gather}
%%%%%%%%%%%%%%
With that we get
%%%%%%%%%%%%%
\begin{gather}
s(g)\,{\cal L}_{\rm total}(g)=\sum_{n=0}^\infty \kappa^n\,\sum_{k=0}^n\,g^k\,
s_{n-k}(g)\,{\cal L}^{(k)}\notag\\
=\sum_{n=0}^\infty \kappa^n\,g^n\,\sum_{k=0}^n s_{n-k}{\cal L}^{(k)}
+g_{,\mu}\,\sum_{n=1}^\infty \kappa^n\,g^{(n-1)}\,\sum_{k=0}^{n-1} (n-k)\>
\frac{\d{\cal L}^{(k)}}{\d\varphi_{i,\mu}}\>s_{n-k}\,\varphi_i\ .\label{s(g)L1}
\end{gather}
%%%%%%%%%%%%%%
With $s=\sum_{n=0}^\infty \kappa^n\,s_n$ and (\ref{sL1}) the first term is equal to
%%%%%%%%
\begin{gather}
\sum_{n=0}^\infty \kappa^n\,g^n\,\bigl(s\,{\cal L}_{\rm total}\bigr)^{(n)}
=-\sum_{n=0}^\infty \kappa^n\,g^n\,\d_\mu\,I^{(n)\,\mu}\notag\\
=-\d_\mu\,\sum_{n=0}^\infty \kappa^n\,g^n\,I^{(n)\,\mu}+
g_{,\mu}\,\sum_{n=1}^\infty \kappa^n\,n\,g^{(n-1)}\,I^{(n)\,\mu}\ .\label{s(g)L1a}
\end{gather}
%%%%%%%%%%%%%%%%%%%%%%
We insert (\ref{s(g)L1a}) into (\ref{s(g)L1}) and the resulting equation into the left side
of (\ref{s(g)L2}), and then we use (\ref{j^n}). This yields the local current
conservation (\ref{dj(g)}),
%%%%%%%%
%\beq
%\d_\mu\, j^\mu(g)_{S_{\rm total}(g)}(x)=g_{,\mu}(x)\>{\cal L}_1^\mu(g)_{S_{\rm total}(g)}(x)\ ,
%\label{dj(g)}
%\eeq
%%%%%%%%%
where ${\cal L}_1^\mu(g)$ is given by (\ref{L_1(g)}) and
%%%%%%%%%
\beq
{\cal L}_1^{(n)\,\mu}\=d -\Bigl(\sum_{k=0}^{n-1} (n-k)\>
\frac{\d{\cal L}^{(k)}}{\d\varphi_{i,\mu}}\>s_{n-k}\,\varphi_i
+n\>I^{(n)\,\mu}\Bigr)\ ,\quad n=1,2,...\ .\label{L_1^n}
\eeq
%%%%%%%%%%%%%%
With that the proof of PGI-tree is complete for models with a BRS-invariant 
Lagrangian (\ref{sL1}).
\medskip

$d_Q$ can be viewed as a graded derivation $d_Q:{\cal P}\vert_{{\cal C}_{S_0}}
\rightarrow {\cal P}\vert_{{\cal C}_{S_0}}$.
Interpreting $d_Q$ in this sense we would like to prove
%%%%%%%%%
\beq
d_Q\>A_{S_0}=i\>(s_0\,A)_{S_0}\ ,\quad\forall A\in {\cal P}\ .\label{d_Q=s_0}
\eeq
%%%%%%%%%%%

Since $d_Q$ and $s_0$ are both graded derivations which commute with partial derivatives
it suffices to prove (\ref{d_Q=s_0}) for $A$ running through all basic fields $\varphi_i$.
In models with solely massless fields it usually holds that $s_0\,\varphi_i$ is
a divergence, i.e.
%%%%%%%%%%%%%%%%
\beq
s_0\,\varphi_i=\d_\mu\,\phi^\mu_i\quad {\rm for\>some}\quad \phi^\mu_i\in {\cal P}\ ,
\label{s_0=div}
\eeq
%%%%%%%%%%%%%
for all basic fields $\varphi_i$. If this holds true, (\ref{d_Q=s_0}) follows from part (i) of the 
following Corollary of PGI-tree.
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{cor} Let a free Lagrangian ${\cal L}^{(0)}$ be given which is invariant with respect to a
free BRS-transformation $s_0$,
%%%%%%%%%%
\beq
s_0\,{\cal L}^{(0)}=-\d_\mu\,I^{(0)\,\mu}\quad {\it for\>some}\quad I^{(0)\,\mu}\in {\cal P}\ .
\eeq
%%%%%%%%%%%
In terms of the corresponding Noether current $j^{(0)\,\mu}$ (\ref{j^n})
we define $d_Q$ by (\ref{Q_lambda})-(\ref{d_Q}). 
 \begin{enumerate}
 	\item If $s_0\,A=\d_\mu\, B^\mu$ for some $B^\mu$, then $A$ 
satisfies the relation (\ref{d_Q=s_0}).
\item For an arbitrary $P\in {\cal P}$ it holds
%%%%%%%%
\beq
\d^\nu\,\bigl(d_Q\>P_{S_0}-i\>(s_0\,P)_{S_0}\bigr)=0\ .\label{d_Q=s_01}
\eeq
%%%%%%%%%%%
 \end{enumerate}
\end{cor}
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof} (i) To the given free Lagrangian we add the interaction
${\cal L}_{\rm int}:=\kappa\, A$ and choose $s:=s_0$. This model is BRS invariant:
%%%%%%%%%
\beq
s {\cal L}_{\rm total}=s_0\,{\cal L}^{(0)}+\kappa\,s_0\,A=-\d_\mu (I^{(0)\,\mu}-\kappa\, B^\mu)\ .
\eeq
%%%%%%%%%%%%%
So, our assumption (\ref{sL1}) is satisfied and, hence, PGI-tree holds with
%%%%%%%%%%%
\beq
{\cal L}_1^{(1)\,\mu}=-I^{(1)\,\mu}=B^\mu\ ,
\eeq
%%%%%%%%%
where (\ref{L_1^n}) is used. To first order PGI-tree reads
%%%%%%%%%%%
\beq
d_Q\>A_{S_0}=d_Q\> {\cal L}^{(1)}_{S_0}=i\>\d_\mu\,{\cal L}^{(1)\,\mu}_{1\>\>S_0}
=i\>\d_\mu\,B^\mu_{S_0}=i\>(s_0\,A)_{S_0}\ .
\eeq
%%%%%%%%%%%%%%
(ii) Since $s_0\,(\d^\nu P)=\d_\mu\,(\eta^{\mu\nu}\>s_0\,P)$, part (i)
applies to $A=\d^\nu P$.
\end{proof}

\noindent {\it Remarks: (1)} From (\ref{d_Q=s_0}) and PGI-tree it follows
%%%%%%%%%
\beq
i\,(s_0\,{\cal L}^{(1)})_{S_0}=d_Q\,{\cal L}^{(1)}_{S_0}=i\,\d_\nu {\cal L}^{(1)\,\nu}_{1\>S_0}\ .
\eeq
%%%%%%%%%
The relation
%%%%%%%%%%%
\beq
(s_0\,{\cal L}^{(1)})_{S_0}=\d_\nu {\cal L}^{(1)\,\nu}_{1\>S_0}
\eeq
%%%%%%%%%
(where ${\cal L}_1^{(1)\,\mu}$ is given by 
${\cal L}_1^{(1)\,\mu}=-\frac{\d{\cal L}^{(0)}}{\d\varphi_{i,\mu}}\>s_1\,\varphi_i
-I^{(1)\,\mu}$  (\ref{L_1^n}))
can be verified directly, i.e.~without using PGI-tree. Namely,
by using $-\d_\mu\,I^{(1)\,\mu}=s_0\,{\cal L}^{(1)}+s_1\,{\cal L}^{(0)}$ (\ref{sL1}),
the derivation property of $s_1$  and the free field equations we obtain
%%%%%%%%%
\begin{gather}
\bigl(s_0\,{\cal L}^{(1)}-\d_\mu {\cal L}_1^{(1)\,\mu}\bigr)_{S_0}=
\Bigl(-s_1\,{\cal L}^{(0)}+\d_\mu\Bigl(\frac{\d{\cal L}^{(0)}}{\d\varphi_{i,\mu}}\>
s_1\,\varphi_i\Bigr)\Bigr)_{S_0}\notag\\
=\Bigl(-\frac{\d{\cal L}^{(0)}}{\d\varphi_{i}}\>s_1\,\varphi_i-
\frac{\d{\cal L}^{(0)}}{\d\varphi_{i,\mu}}\>(s_1\,\varphi_i)_{,\mu}+
\d_\mu\Bigl(\frac{\d{\cal L}^{(0)}}{\d\varphi_{i,\mu}}\>s_1\,\varphi_i\Bigr)\Bigr)_{S_0}\notag\\
=\Bigl(\d_\mu\frac{\d{\cal L}^{(0)}}{\d\varphi_{i,\mu}}-\frac{\d{\cal L}^{(0)}}{\d\varphi_{i}}
\Bigr)_{S_0}\>(s_1\,\varphi_i)_{S_0}=0\ .
\end{gather}
%%%%%%%%%%%%%%%%%%%%%%%
\noindent {\it (2)} If higher ($\geq 2$) derivatives of the basic fields appear in 
${\cal L}_{\rm total}$ (\ref{L_total}) the construction of a conserved BRS-current is still 
possible in the case of constant coupling: the field equations
%%%%%%%%%%%
\beq
\sum_{l\in\NN_0}(-1)^l\>\d_{\mu_1}...\d_{\mu_l}\,\frac{\d\,{\cal L}_{\rm total}}
{\d\,\varphi_{i,\mu_1...\mu_l}}=0
\eeq
%%%%%%%%%%%%%
imply that the current
%%%%%%%%%%%%%
\beq
j^\mu\=d -\Bigl(\sum_{l\in\NN_0}\sum_{j=0}^l(-1)^j\>\d_{\mu_1}...\d_{\mu_j}
\Bigl(\frac{\d\,{\cal L}_{\rm total}}{\d\,\varphi_{i,\mu\mu_1...\mu_l}}\Bigr)\> 
s\varphi_{i,\mu_{j+1}...\mu_l} + I^\mu\Bigr)
\eeq
%%%%%%%%%%%%%%%
is conserved. But in case of a local coupling terms $\sim g_{,\mu\mu_1}(x),\>\sim
g_{,\mu}(x)\, g_{,\mu_1}(x),...$ appear in (\ref{s(g)L2})-(\ref{s(g)L1}) and, hence, also in the 
divergence of the local BRS-current, i.e.~on the right side of (\ref{dj(g)}).
%%%%%%%%%%%%%%%
\section{Models}\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Massless Yang-Mills theories}
%%%%%%%%%%%%%%%%%%%%%%
To point out the similarity of the BRS-symmetry for massless
spin-1 and spin-2 gauge fields we first recall the spin-1 case.
In terms of the covariant derivative
%%%%%%%%
\beq
D^\mu_{ab}\=d \delta_{ab}\d^\mu -\kappa\,f_{abc}\,A^\mu_c\ ,
\eeq
%%%%%%%%%%%%%%%%%%
(where $f_{abc}$ is totally antisymmetric and satisfies the Jacobi identity)
the BRS-transformation reads
%%%%%%%%%
\beq
s\,A_a^\mu=D^\mu_{ab}\,u_b\ ,\quad s\,u_a=-\frac{\kappa}{2}\,f_{abc}\,
u_b\,u_c\ ,\quad s\tilde u_a=-\d_\mu\,A^\mu_a\ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%%
Due to
%%%%%%%%%%%%%
\beq
F^{\mu\nu}_{a}\equiv\d^\mu A^\nu_a-\d^\nu A^\mu_a+\kappa\, g\,f_{abc}
A^\mu_b\, A_c^\nu\label{F}
\eeq
%%%%%%%%%%%%%
the Yang-Mills Lagrangian
%%%%%%%%%%%%%%%%%%%%%%
\beq
{\cal L}_{\rm YM}=-\frac{1}{4}\> F^{\mu\nu}_{a}\,F_{a\,\mu\nu}\label{L_YM}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%
is of the form ${\cal L}_{\rm YM}={\cal L}_{\rm YM}^{(0)}+\kappa\,
{\cal L}_{\rm YM}^{(1)}+\kappa^2\,{\cal L}_{\rm YM}^{(2)}$.
The gauge fixing Lagrangian is of zeroth order in $\kappa$,
%%%%%%%%%%%
\beq
{\cal L}_{\rm GF}=-\frac{1}{2}\>(\d_\nu A^\nu_a)^2\ ,
\eeq
%%%%%%%%%%
where we choose Feynman gauge.
However, the ghost Lagrangian has also a term linear in $\kappa$,
%%%%%%%%%%%
\beq
{\cal L}_{\rm ghost}=\d_\mu\tilde u_a\> s\,A^\mu_a\ .\label{L_ghost}
\eeq
%%%%%%%%%%
Note $s_k=0,\>\forall k\geq 2$, and ${\cal L}_{\rm total}^{(j)}=0,\>\forall j\geq 3$,
where ${\cal L}_{\rm total}\=d {\cal L}_{\rm YM}+{\cal L}_{\rm GF}+{\cal L}_{\rm ghost}$.
The BRS-transformation is nilpotent modulo the field equations,\footnote{If one 
introduces the Nakanishi-Lautrup fields $B_a$ \cite{LN}, the BRS transformation 
$s$ is nilpotent in ${\cal P}$ (i.e.~without using the field equations). $s$ is then
modified as follows: $s\,\tilde u_a=-B_a$ and $s\,B_a=0$.\label{fn:NL}}
%%%%%%%%%
\beq
s^2\,A_a^\mu=0\ ,\quad s^2\,u_a=0\, ,\quad
s^2\tilde u_a=\frac{\delta\, S_{\rm total}}{\delta\,\tilde u_a}\ ,
\eeq
%%%%%%%%%%
with $S_{\rm total}\=d\int dx\,{\cal L}_{\rm total}$. (Since $s$ is a graded derivation which 
commutes with partial derivatives the vanishing of $s^2$ on the basic fields implies
$s^2=0$.)

We are now going to verify the BRS-invariance of ${\cal L}_{\rm total}$.
$s\,{\cal L}_{\rm YM}$ vanishes, since the BRS-transformation of $A^\mu_a$ 
has the form of an infinitesimal gauge transformation
and since the gauge variation of ${\cal L}_{\rm YM}$
vanishes. For ${\cal L}_{\rm GF}$ and ${\cal L}_{\rm ghost}$ we obtain
%%%%%%%%%%%
\begin{gather}
s\,({\cal L}_{\rm GF}+{\cal L}_{\rm ghost})=-(\d_\nu A^\nu_a)\>\d_\mu (s\,A^\mu_a)
-(\d_\mu\d_\nu A^\nu_a)\> s\,A^\mu_a-\d_\mu\tilde u_a\>s^2A_a^\mu\notag\\
=-\d_\mu\,\bigl((\d_\nu A^\nu_a)\> D^\mu_{ab}\,u_b\bigr)=:-\d_\mu\,I^\mu\ .
\label{s(GF+ghost)}
\end{gather}
%%%%%%%%%%%%%%

Our formula (\ref{L_1^n}) yields the following explicit expressions for the $Q$-vertex,
which agree with the literature \cite{DHKS,S-wiley,BD}:
%%%%%%%%%%%%%%%%%%%%%%
\begin{gather}
{\cal L}_1^{(1)\,\nu}=f_{abc}[A_{a\,\mu}\, u_b\,(\d^\nu A^\mu_c-\d^\mu A^\nu_c)
-\frac{1}{2}u_a\,u_b\,\d^\nu\tilde u_c]\ , \\
{\cal L}_1^{(2)\,\nu}=f_{abr}f_{cdr}\,A_{a\,\mu}\, u_b\, A_c^\nu\, A_d^\mu\ , \\
{\cal L}_1^{(j)\,\nu}=0\ ,\quad\forall j\geq 3\ .
\end{gather}
%%%%%%%%%%%%%%%%%%%%%%%
For the zeroth order of the BRS-current our expression (\ref{j^n}) gives
%%%%%%%%%
\begin{gather}
j^{(0)\,\mu}=\d_\tau A^\tau_a\, \d^\mu u_a- (\d^\mu\d_\tau A^\tau_a)\, u_a\notag\\
-\d_\nu\,\bigl((A^{\mu,\nu}_a-A^{\nu,\mu}_a)\,u_a\bigr)+(\square A_a^\mu)\,u_a\ .
\end{gather}
%%%%%%%%
The terms in the second line do not contribute to $d_Q$ (\ref{d_Q1})-(\ref{d_Q2}). 
This is obvious for the last term due to $\square A_{a\,{S_0}}^\mu=0$. Turning to
the second last term  we point out that generally a term of the form
$\d_\nu(K^{\mu\nu}-K^{\nu\mu})$ does not contribute to $d_Q$ (\ref{d_Q2}): 
 %%%%%%%%%
\beq
\int_{x^0={\rm const.}} d^3x\,\d^x_l\> [(K_{S_0}^{0l}(x^0,\vec{x})
-K_{S_0}^{l0}(x^0,\vec{x})), F_{S_0}]^\mp_\star=0\ .\label{K=0}
\eeq
%%%%%%%%%%%%%%
Hence, $d_Q$ can be constructed from the free BRS-current 
$(\d A)\, \d^\mu u- (\d^\mu\d A)\, u$ as it is done in \cite{DHKS,S-wiley,BD}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Massive spin-1 fields}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
It is instructive to see how the BRS-formalism of the preceding Subsect.~is modified 
for {\it massive} fields. To simplify the notations we consider the most simple non-Abelian 
model: the $SU(2)$ Higgs-Kibble model, which describes three spin-1 fields,
$A_a^\mu,\>a=1,2,3$, and the bosonic scalar fields form a complex $SU(2)$ doublet,
%%%%%%%%%%%%%%%%%%%%
\begin{equation}
  \Phi=\frac{1}{\sqrt{2}}\left(
  \begin{array}{c}
        \phi_2+i\phi_1  \\
       v+H-i\phi_3
  \end{array}\right)\ . \label{Phi}
\end{equation}
%%%%%%%%%%%%%%%%%%
The shift $v\in\RR_+$ will be chosen such that the Higgs potential has a non-trivial 
minimum at $\phi =\frac{1}{\sqrt{2}}\left(
  \begin{array}{c}
        0  \\
       v \end{array}\right)$ (\ref{v}), as usual in the Higgs mechanism. The corresponding 
covariant derivative is
%%%%%%%%%
\beq
{\cal D}^\mu=({\bf 1}\,\d^\mu-\kappa\,\frac{i}{2}\,A^\mu_a\,\sigma_a)\ ,
\eeq
%%%%%%%%%%%%%%
where $(\sigma_a)_{a=1,2,3}$ are the Pauli matrices. Requiring renormalizability 
and $SU(2)$-invariance the Higgs Lagrangian takes the form
\beq
{\cal L}_\Phi=({\cal D}_\mu\Phi)^+({\cal D}^\mu\Phi)+\mu^2\,\Phi^+\Phi
-\lambda\,(\Phi^+\Phi)^2\ ,\label{L_Phi}
\eeq
%%%%%%%%%
Choosing $\mu^2>0$ and $\lambda>0$ one obtains
%%%%%%%%%%
\beq
v^2=\frac{\mu^2}{\lambda}\ .\label{v}
\eeq
%%%%%%%%%
The $SU(2)$-invariant Yang-Mills Lagrangian ${\cal L}_{\rm YM}$ is still given by 
(\ref{F})-(\ref{L_YM}). By the Higgs mechanism the three spin-1 fields $A^\mu_a\>
(a=1,2,3)$ get the same mass
%%%%%%%%
\beq
m=\frac{\kappa\,v}{2}>0\ .\label{m}
\eeq
%%%%%%%%%

From the quantization of spin-1 fields one knows that in the massive case the condition 
$\d_\mu A^\mu_a=0$ on observables is replaced by $(\d_\mu A^\mu_a+m\phi_a)=0$.
(For free fields this is explained in Sect.~3 of \cite{DF4}.) Therefore,
%%%%%%%%%%%
\beq
s\,\tilde u_a=-(\d_\mu A^\mu_a+m\phi_a)\ ,
\eeq
%%%%%%%%%%
and this replacement appears also in the gauge fixing term
%%%%%%%%%%%
\beq
{\cal L}_{\rm GF}=-(\d_\mu A^\mu_a+m\phi_a)^2={\cal L}_{\rm GF}^{(0)}\ ,
\eeq
%%%%%%%%%%%
where we choose again Feynman gauge. The BRS-transformation of $A^\mu_a$ and
$(\phi_a,\,H)$ has the form of an infinitesimal gauge transformation of the {\it un-shifted} fields. 
That is, $s\,A_a^\mu=D^\mu_{ab}\,u_b$ remains unchanged and
%%%%%%%%%%%%%
\beq
s\,\Phi=\frac{\kappa\,i}{2}\,\sigma_a\,u_a\>\Phi\ .
\eeq
%%%%%%%%%%
The latter reads explicitly
\begin{gather}
s\,\phi_a=m\,u_a+\frac{\kappa}{2}\,(f_{abc}\,\phi_b\,u_c+H\,u_a)\ ,\notag\\
s\,H=-\frac{\kappa}{2}\,\phi_a\,u_a\ .
\end{gather}
%%%%%%%%%%
It follows
%%%%%%%%
\beq
s\,{\cal L}_{\rm YM}=0\ ,\quad s\,{\cal L}_{\Phi}=0\ .
\eeq
%%%%%%%%%
To keep $s^2\,A^\mu_a=0=s^2\, u_a$, the 
BRS-transformation of $u_a$ is unchanged: $ s\,u_a=-\frac{\kappa}{2}\,f_{abc}\,
u_b\,u_c$. With that one easily verifies $s^2\,\phi_a=0=s^2\, H$:
%%%%%%%%%%%%
\beq
s^2\,\Phi=\frac{\kappa\,i}{2}\,\sigma_a\,(su_a)\>\Phi-
\frac{\kappa\,i}{2}\,\sigma_a\,u_a\>s\Phi=0\ ,
\eeq
%%%%%%%%%%%%%%%
due to $(\sigma_a\,u_a)\, (\sigma_b\,u_b)=i\,f_{abc}\,u_a\,u_b\,\sigma_c$.

The ghost Lagrangian is chosen such that $s\,({\cal L}_{\rm GF}+
{\cal L}_{\rm ghost})$ is a divergence: generalizing (\ref{L_ghost}) and 
(\ref{s(GF+ghost)}) one sets
%%%%%%%
\beq
{\cal L}_{\rm ghost}\=d \d_\mu\tilde u_a\>s\,A^\mu_a-m\,\tilde u_a\,s\,\phi_a
={\cal L}_{\rm ghost}^{(0)}+\kappa {\cal L}_{\rm ghost}^{(1)}\ ,
\eeq
%%%%%%%%%
which yields indeed
%%%%%%%%%%%
\beq
s\,({\cal L}_{\rm GF}+{\cal L}_{\rm ghost})=-\d_\mu\,I^\mu\ ,\quad
I^\mu\=d (\d_\nu A^\nu_a+m\phi_a)\,D^\mu_{ab}u_b=I^{(0)\,\mu}+\kappa\,\label{I:m>0}
I^{(1)\,\mu}\ .
\eeq
%%%%%%%%%%%%%%
We end up with the total Lagrangian
%%%%%%%%%
\beq
{\cal L}_{\rm total}\=d {\cal L}_{\rm YM}+{\cal L}_\Phi+
{\cal L}_{\rm GF}+{\cal L}_{\rm ghost}=-\frac{\lambda\,v^4}{4}+{\cal L}^{(0)}+
\kappa\,{\cal L}^{(1)}+\kappa^2\,{\cal L}^{(2)}\ .
\eeq
%%%%%%%%%%
The constant $-\frac{\lambda\,v^4}{4}$ is irrelevant and all terms of ${\cal L}^{(2)}$
come from ${\cal L}_{\rm YM}+{\cal L}_\Phi$. Note that ${\cal L}^{(0)}$ contains a 
divergence term, $-m\,\d_\mu(A^\mu_a\phi_a)$, which is irrelevant for the field equations 
but contributes to $j^{(n)\,\mu}$ (\ref{j^n}) and ${\cal L}^{(n)\,\mu}$ (\ref{L_1^n}).
For later purpose we give ${\cal L}^{(0)}$ explicitly:
%%%%%%%%%%%%%%
\begin{gather}
{\cal L}^{(0)}=-\frac{1}{4}(A_a^{\mu,\nu}-A_a^{\nu,\mu})(A_{a\,\mu,\nu}-A_{a\,\nu,\mu})
-\frac{1}{2}(\d A_a)^2+\frac{m^2}{2}\,A^\mu_aA_{a\,\mu}\notag\\
+\d_\mu\tilde u_a\,\d^\mu u_a -m^2\,\tilde u_au_a\notag\\
+\frac{1}{2}\,\d_\mu\phi_a\,\d^\mu\phi_a-\frac{m^2}{2}\,\phi_a^2
+\frac{1}{2}\,\d_\mu H\,\d^\mu H-\lambda\, v^2\,H^2\notag\\
-m\,\d_\mu(A^\mu_a\,\phi_a)\ .\label{L_0:m>0}
\end{gather}
As in the massless case it is only the vanishing of $s^2\tilde u_a$ which relies on the 
field equations:
%%%%%%%%%
\beq
s^2\,A_a^\mu=0\ ,\quad s^2\,u_a=0\, ,\quad
s^2\tilde u_a=\frac{\delta\, S_{\rm total}}{\delta\,\tilde u_a}\ ,\quad
s^2\,\phi_a=0\ ,\quad s^2\, H=0
\eeq
%%%%%%%%%%
and, hence, footnote \ref{fn:NL} applies also to the massive case.

Inserting (\ref{I:m>0}) and (\ref{L_0:m>0}) into our formulas 
(\ref{j^n}) (BRS-current) and (\ref{L_1^n}) ($Q$-vertex) we obtain
%%%%%%%%%
\begin{gather}
j^{(0)\,\mu}=(\d_\tau A^\tau_a+m\,\phi_a)\, \d^\mu u_a- (\d^\mu(\d_\tau A^\tau_a
+m\,\phi_a))\, u_a\notag\\
-\d_\nu\,\bigl((A^{\mu,\nu}_a-A^{\nu,\mu}_a)\,u_a\bigr)+((\square +m^2) A_a^\mu)\,u_a
\label{j:m>0}
\end{gather}
%%%%%%%%
and
%%%%%%%%%
\begin{gather}
{\cal L}_1^{(1)\,\mu}=f_{abc}[A_{a\,\nu}\, u_b\,(\d^\mu A^\nu_c-\d^\nu A^\mu_c)
-\frac{1}{2}u_a\,u_b\,\d^\mu\tilde u_c+\frac{m}{2}\,A^\mu_a\,\phi_b\,u_c
-\frac{1}{2}\,\d^\mu\phi_a\,\phi_b\,u_c]\notag\\
+\frac{1}{2}\,\d^\mu H\,u_a\,\phi_a-\frac{1}{2}\,H\,u_a\,\d^\mu\phi_a
+\frac{m}{2}\,A^\mu_a\,u_a\,H\label{L_1:m>0}
\end{gather}
%%%%%%%%%%%%%%%%%%%%%%%
and again ${\cal L}_1^{(k)\,\nu}=0\ ,\>\forall k\geq 3$. For the same reason as in 
the massless case (\ref{K=0}) the terms in the second line of (\ref{j:m>0}) do not contribute
to $d_Q$. With that our results (\ref{j:m>0}) and (\ref{L_1:m>0}) agree with the literature 
\cite{DS,BD}.
\medskip

\noindent {\it Remarks: (1)} If one omits the divergence $-m\,\d_\mu(A^\mu_a\phi_a)$ in 
${\cal L}^{(0)}$ an additional term appears in $s\,{\cal L}_{\rm total}$: $I^\mu$
is replaced by $(I^\mu-m\,s\,(A^\mu_a\phi_a))$. With that our results for $j^{(0)}$
(\ref{j:m>0}) and ${\cal L}_1^{(1)}$ (\ref{L_1:m>0}) remain unchanged.

\noindent {(2)} For massive spin-1 fields $s_0\varphi_l$ is not a divergence 
(\ref{s_0=div}) for the basic fields $\varphi_l=\tilde u,\>\phi$. However, since the free field 
equations are differential equations of {\it second} order, we obtain $i\,(s_0\varphi_l)_{S_0}
=d_Q\,\varphi_{l\,S_0}\,,\>\varphi_l=\tilde u,\,\phi$, from part (ii) of Corollary 2.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Massless spin-2 gauge fields}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this Subsect.~we complete our proof of PGI-tree for massless spin-2 gauge fields: we
show that classical gravity can be formulated by a BRS-invariant Lagrangian 
${\cal L}_{\rm total}$ (\ref{sL1}) which fits in our formalism. To satisfy the latter,
${\cal L}_{\rm total}$ must be a (formal) power series in $\kappa$ and it must be a 
polynomial only in zeroth and first derivatives of the basic fields. (It will turn out that both 
properties are non-trivial.) Such a BRS-formulation of gravity was given
by Kugo and Ojima in Sect.~2 of \cite{KO}. 
In that formalism we choose Feynman gauge $\alpha_0=1$
and eliminate the Nakanishi-Lautrup field $b_\mu$ \cite{LN} 
(by inserting the field equation for $b_\mu$). In view of
perturbation theory around the Minkowski metric $\eta^{\mu\nu}$
we introduce a field $h^{\mu\nu}$ which is the deviation from $\eta^{\mu\nu}$,
in terms Goldberg variables $\tilde g^{\mu\nu}$ (for details see e.g. \cite{S-wiley}):
%%%%%%%%%%%%%
\beq
\tilde g^{\mu\nu}(x)\=d \sqrt{-g(x)}\,g^{\mu\nu}(x)=\eta^{\mu\nu}+\kappa\,h^{\mu\nu}(x)\ .\label{h}
\eeq
%%%%%%%%%%
The inverse tensor is a kind of geometric series in $\kappa h$ (which we understand 
as formal power series in $\kappa$):
%%%%%%%%%%%%%%%%
\beq
\tilde g_{\mu\nu}\=d \frac{1}{\sqrt{-g}}\,g_{\mu\nu}=\eta_{\mu\nu}-\kappa\,h_{\mu\nu}
+\kappa^2\,h_{\mu\alpha}h^\alpha_{\>\nu}-...\ ,\label{h1}
\eeq
%%%%%%%%%%%%%%%%%
where
%%%%%%%%%%
\beq
h^\alpha_{\>\nu}=\eta_{\nu\tau}\, h^{\alpha\tau}\ ,\quad
h_{\mu\nu}=\eta_{\mu\rho}\,\eta_{\nu\tau}\,h^{\rho\tau}\ ,
\eeq
%%%%%%%%%%%%%%%%
and we set $h\=d h^\mu_\mu$. The field algebra ${\cal P}$ is the polynomial algebra 
generated by $h^{\mu\nu}$, the fermionic vector ghost fields $u^\mu,\>\tilde u_\mu$
and all partial derivatives of these fields. (The indices of $u$ and $\tilde u$ are also raised 
and lowered by means of $\eta$.) 
According to \cite{KO} the BRS-transformation is of the form $s=s_0+\kappa\, s_1$
and on  the basic fields it is given by
%%%%%%%%%
\begin{gather}
s\,h^{\mu\nu}=u^{\nu,\mu}+u^{\mu,\nu}-\eta^{\mu\nu}\,u^\rho_{,\rho}
+\kappa\,(h^{\mu\sigma}\,u^\nu_{\>,\sigma}+h^{\nu\sigma}\,u^\mu_{\>,\sigma}-
(h^{\mu\nu}\,u^\rho)_{,\rho})\ ,\notag\\
s\,u^\mu=-\kappa\,(u^\lambda\,u^\mu_{\>,\lambda})\ ,\notag\\
s\,\tilde u_\mu=-h_{\mu\rho}^{\>\>,\rho}\ .\label{s:gravity}
\end{gather}
%%%%%%%%%
(The sign of $s\,u^\mu$ is determined by the requirement $s^2\,h^{\mu\nu}=0$.) By using
%%%%%%%%%%
\beq
D^{\mu\nu}_{\>\rho}\=d (\eta^{\mu\sigma}+\kappa\,h^{\mu\sigma})\,\delta^\nu_\rho\,\d_\sigma
+(\eta^{\nu\sigma}+\kappa\,h^{\nu\sigma})\,\delta^\mu_\rho\,\d_\sigma
-\d_\rho\bigl((\eta^{\mu\nu}+\kappa\,h^{\mu\nu})\>\>\cdot\>\>\bigr)
\eeq
%%%%%%%%%%
we may also write
%%%%%%%
\beq
s\,h^{\mu\nu}=D^{\mu\nu}_{\>\rho}\,u^\rho\ .
\eeq
%%%%%%%%%%%

In terms of the Christoffel symbols 
%%%%%%%%%%
\beq
\Gamma^\alpha_{\beta\gamma}=\frac{1}{2}\,g^{\alpha\mu}\,(g_{\beta\mu,\gamma}
+g_{\mu\gamma,\beta}-g_{\beta\gamma,\mu})\label{Gamma}
\eeq
%%%%%%%%%%%%%%
the Einstein-Hilbert Lagrangian reads
%%%%%%%%
\beq
{\cal L}_{\rm E}=\frac{1}{\kappa^2}\,\sqrt{-g}\,g^{\mu\nu}\,(\Gamma^\rho_{\nu\rho,\mu}-
\Gamma^\rho_{\mu\nu,\rho}+\Gamma^\rho_{\mu\sigma}
\,\Gamma^\sigma_{\nu\rho}-\Gamma^\sigma_{\mu\nu}\,
\Gamma^\rho_{\sigma\rho})\ .
\eeq
%%%%%%%%%%
But ${\cal L}_{\rm E}$ is unsuitable for our formalism: it contains terms $\sim\kappa^{-1}$
and second derivatives of $h^{\mu\nu}$. Both shortcomings can be
removed by subtracting a divergence \cite{KO}:
%%%%%%%%%%
\begin{gather}
{\cal L}'_{\rm E}\=d {\cal L}_{\rm E}-\d_\mu {\cal D}^\mu\ ,\\
{\cal D}^\mu\=d \frac{1}{\kappa^2}\,(\tilde g^{\mu\nu}\,\Gamma^\lambda_{\nu\lambda}-
\tilde g^{\rho\sigma}\,\Gamma^\mu_{\rho\sigma})=\frac{1}{\kappa^2}\,
\bigl(\frac{1}{2}\,\tilde g^{\mu\nu}\,\tilde g_{\alpha\beta}\,\d_\nu\tilde g^{\alpha\beta}\,
+\d_\nu\tilde g^{\mu\nu}\bigr)\notag\\
=\frac{1}{\kappa}\,{\cal D}^{(-1)\,\mu}+{\cal O}(\kappa^0)\ ,\quad {\cal D}^{(-1)\,\mu}=
\frac{1}{2}\,h^{,\mu}+h^{\mu\nu}_{\> ,\nu}\ .
\end{gather}
%%%%%%%%%%
By inserting (\ref{h})-(\ref{h1}) and (\ref{Gamma}) we indeed obtain a formal power series 
in $\kappa$ for ${\cal L}'_{\rm E}$:
%%%%%%%%
\beq
{\cal L}'_{\rm E}=\sum_{n=0}^\infty\kappa^n\,{\cal L}^{\prime\,(n)}_{\rm E}\ ,\label{L'_E}
\eeq
%%%%%%%%%%
see Sect.~5.5 of \cite{S-wiley}. 
The total Lagragian ${\cal L}_{\rm total}$ of the BRS-formalism
is obtained by adding a gauge fixing term ${\cal L}_{\rm GF}$ and a ghost
term ${\cal L}_{\rm ghost}$:
%%%%%%%%%%%%%%%%%
\begin{gather}
{\cal L}_{\rm total}={\cal L}'_{\rm E}+{\cal L}_{\rm GF}+{\cal L}_{\rm ghost}=
\sum_{n=0}^\infty\kappa^n\,{\cal L}^{(n)}%={\cal L}^{(0)}+{\cal L}_{\rm int}(\kappa)
\ ,\label{L_total:gravity}\\
{\cal L}_{\rm GF}=\frac{1}{2}\,h^{\alpha\beta}_{\>,\beta}\,h_{\alpha\rho}^{\>,\rho}=
{\cal L}_{\rm GF}^{(0)}\ ,\\
{\cal L}_{\rm ghost}=-\frac{1}{2}\,(\tilde u_{\nu,\mu}+\tilde u_{\mu,\nu})\>
s\,h^{\mu\nu}={\cal L}_{\rm ghost}^{(0)}+\kappa\,{\cal L}^{(1)}_{\rm ghost}\ .
\end{gather}
%%%%%%%%%%%%%%%%%%%%
%The corresponding action is denoted by $S_{\rm total}$.
With that the BRS-transformation is nilpotent modulo the field equations:
%%%%%%%%%
\beq
s^2\,h^{\mu\nu}=0\ ,\quad s^2\,u^\mu=0\ ,\quad 
s^2\,\tilde u_\mu=-\eta_{\mu\tau}\,(D^{\tau\nu}_\rho\,u^\rho)_{,\nu}=
-\frac{\delta\,S_{\rm total}}{\delta\,\tilde u^\mu}\ .\label{s^2(tilde-u)}
\eeq
%%%%%%%%%%

We turn to the verification of the BRS-invariance of ${\cal L}_{\rm total}$.
Again, the BRS-transformation of the gauge field
$h^{\mu\nu}$ has the form of an infinitesimal gauge transformation 
(i.e.~general coordinate transformation) and, hence,
$s\,{\cal L}_{\rm E}$ is known from the gauge variation of ${\cal L}_{\rm E}$ \cite{KO}:
%%%%%%%%%%
\beq
s\,{\cal L}_{\rm E}=-\d_\mu(\kappa\,{\cal L}_{\rm E}\,u^\mu)\ .\label{sL_E}
\eeq
%%%%%%%%%%%%%
By the same calculation as in (\ref{s(GF+ghost)}) we obtain
%%%%%%%%%%
\beq
s\,({\cal L}_{\rm GF}+{\cal L}_{\rm ghost})=\d_\mu F^{\mu}\ ,\quad 
F^\mu\=d h_{\nu\rho}^{\>,\rho}\,D^{\mu\nu}_{\>\lambda}\,u^\lambda
=F^{(0)\,\mu}+\kappa\, F^{(1)\,\mu}\ .\label{s(L_GF+L_FP)}
\eeq
%%%%%%%%%%%%%
Summing up we get
%%%%%%%%%%%%%
\beq
s\,{\cal L}_{\rm total}=-\d_\mu\,I^\mu\ ,\quad 
I^\mu\=d \kappa\,{\cal L}_{\rm E}\,u^\mu+s\,{\cal D}^\mu-F^\mu\ 
+\frac{1}{\kappa}\,\d_\rho (u^{\rho,\mu}-u^{\mu,\rho})=
\sum_{n=0}^\infty \kappa^n\, I^{(n)\,\mu}\ .\label{sL:gravity}
\eeq
%%%%%%%%%%%%%%%
In $I^\mu$ we have added the conserved vector field
$\frac{1}{\kappa}\,\d_\rho (u^{\rho,\mu}-u^{\mu,\rho})$
to cancel $s_0\,{\cal D}^{(-1)\,\mu}=\square u^\mu-u^{\rho\>,\mu}_{\>,\rho}$,
because in our formalism $I^\mu$ is assumed to be a formal power 
series in $\kappa$.

By means of our formula (\ref{j^n}) we compute the zeroth order of the BRS-current.
We obtain
%%%%%%%%%%%
\beq
j_{S_0}^{(0)\,\mu}=-h^{\alpha\beta}_{S_0\>,\beta}\,\d^\mu u_{S_0\>,\alpha}+
(\d^\mu h^{\alpha\beta}_{S_0\>,\beta})\,u_{S_0\>,\alpha}
+\d_\rho(K_{S_0}^{\mu\rho}-K_{S_0}^{\rho\mu})\ ,\label{j^0-p}
\eeq
%%%%%%%%%%%%%%
where 
%%%%%%%%%%%%%%
\beq
K^{\rho\mu}\=d \frac{1}{2}\,h^{,\rho}\,u^\mu +h^{\alpha\rho,\mu}\,u_\alpha +
h^{\rho\nu}_{\>,\nu}\,u^\mu+h^{\rho\nu}\,u^\mu_{\>,\nu}\ ,
\eeq
%%%%%%%%%%%%%
for details see Appendix B. As explained in (\ref{K=0}), the term
$\d_\rho(K^{\mu\rho}-K^{\rho\mu})$ does not contribute to $d_Q$. The other terms 
%$-h^{\alpha\beta}_{S_0\>,\beta}\,\d^\mu u_{S_0\>,\alpha}+
%(\d^\mu h^{\alpha\beta}_{S_0\>,\beta})\, u_{S_0\>,\alpha}$,
are precisely the terms which are used for the construction of $d_Q$ in 
\cite{Schorn,S-wiley,G:grav}.
%%%%%%%%%%%%%%%%%%%%
\section{Outlook}\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%
What do we learn from this paper with respect to model building, i.e.~the task:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{itemize}
\item given a free BRS invariant theory $({\cal L}^{(0)},s_0)$ which satisfies (\ref{Q-vaccum}),
find a non-trivial deformation ${\cal L}^{(0)}\rightarrow {\cal L}_{\rm total}=
{\cal L}^{(0)}+{\cal L}_{\rm int}$ (\ref{L_total}) such that ${\cal L}_{\rm total}$
satisfies PGI-tree (and some obvious further conditions, e.g.~Lorentz invariance,
${\cal L}_{\rm total}^*={\cal L}_{\rm total}$ and ghost number zero)?
\end{itemize}
In the literature this problem is usually treated by making a polynomial ansatz for 
${\cal L}_{\rm int}$ and working out the condition of PGI-tree, as mentioned in point (B) 
of the introduction. Due to this paper one can proceed alternatively, namely one searches 
for deformations ${\cal L}^{(0)}\rightarrow {\cal L}_{\rm total}$ and $s_0\rightarrow s$
(with (\ref{s}), (\ref{sdphi}) and (\ref{s^2=0})) such that $s{\cal L}_{\rm total}$ is a divergence
(\ref{sL1}). This amounts to an inductive determination of the sequences 
$({\cal L}^{(n)})_n$ and $(s_n)_n$. (In Sect.~5.3 and Appendix B of \cite{DF2}
and in \cite{Hurth} analogous procedures are given to solve the local current conservation 
(\ref{dj(g)}).)  It is not yet investigated, whether this procedure yields the {\it most general}
solution of the above task, but it would be rather surprising if there
were additional solutions. 
\medskip

The method of proof given in this paper applies not only to BRS-symmetry; all 
Lagrangians ${\cal L}_{\rm total}$ and infinitesimal symmetry transformations $s$ which 
satisfy (\ref{L_total})-(\ref{sdphi}) and (\ref{Q-vaccum}) (where $d_Q$ is constructed by
(\ref{d_Q1}) from the zeroth order $j^{(0)\,\mu}$ (\ref{j^n})
of the Noether current belonging to $({\cal L}_{\rm total},s)$) are admitted. 
Since the nilpotency (\ref{s^2=0})
of the BRS-transformation is not used in our proof, the restrictions on $s$
are not strong. A trivial example are global $U(1)$-symmetries, e.g. charge 
conservation in QED: for the spinors $\psi$ and $\bar\psi$ let $s\psi=i\psi$,
$s\bar\psi=-i\bar\psi$, and the photon field $A^\mu$ is uncharged, $sA^\mu=0$.
The $s$-variation of the Lagrangian of QED, ${\cal L}^{\rm QED}_{\rm total}=
{\cal L}^{(0)}+\kappa g(x){\cal L}^{(1)}$, even vanishes:
$s {\cal L}^{\rm QED}_{\rm total}=0$. Our method yields
%%%%%%%%%%
\beq
\Bigl[Q_\psi , T^{\rm tree}_{n\>S_0}\Bigl((\int g\,{\cal L}^{(1)})^{\otimes n}\Bigr)\Bigr]=0\ ,
\eeq
%%%%%%%%%%%
where $Q_\psi\=d\int d^3x\,(\bar\psi\gamma^0\psi)_{S_0}$, cf. Appendix B of \cite{DF}.
\medskip

Can the method of this paper be applied to {\it massive} spin-2 gauge fields?
This amounts to the question whether there is a BRS-invariant Lagrangian
for such fields with the properties (\ref{L_total})-(\ref{sdphi}) and (\ref{Q-vaccum}).

In \cite{GS} the requirement of PGI-tree has been applied to {\it massive} spin-2 fields.
Starting with the most reasonable free theory (in Feynman gauge) and proceeding 
similarly to \cite{SW,S-wiley} the lowest orders 
${\cal L}^{(1)}$ and ${\cal L}^{(2)}$ of the possible interaction ${\cal L}=\sum_{n=1}^\infty
\kappa^n\,{\cal L}^{(n)}$ have been derived. The result agrees with 
%${\cal L}_{E\,\Lambda}^{\prime\,(0)}+\kappa {\cal L}_{E\,\Lambda}^{\prime\,(1)}+
%\kappa^2 {\cal L}_{E\,\Lambda}^{\prime\,(2)}$ (\ref{L'_E}),
%if one adds to ${\cal L}_E$ a term with cosmological constant $\Lambda\not= 0$:
the corresponding terms of the Einstein-Hilbert Lagrangian with cosmological 
constant $\Lambda\not= 0$:
${\cal L}_{E\,\Lambda}:={\cal L}_E-\frac{2}{\kappa^2}\,\sqrt{-g}\,\Lambda$. 
The mass $m$ 
of the free spin-2 field is related to $\Lambda$ by $m^2=2\,\Lambda$. 

${\cal L}_{E\,\Lambda}$ satisfies the BRS-invariance (\ref{sL_E}), too.
But the expansion of  ${\cal L}_{E\,\Lambda}$ in powers of $\kappa$ 
contains a term $\frac{-1}{\kappa}
\,\Lambda\,h$, which cannot be removed by adding a divergence, and the field equation to 
order $\kappa^{-1}$ is nonsense: $\Lambda=0$. (This contradiction is due to the fact that the 
Minkowski metric $\eta^{\mu\nu}$ is not a solution of the field equation to
${\cal L}_{E\,\Lambda}$.) Therefore, it seems that
the formalism of this paper cannot be applied to ${\cal L}_{E\,\Lambda}$;
omission of  the term $\frac{-1}{\kappa}\,\Lambda\,h$ destroys the BRS-invariance
of ${\cal L}_{E\,\Lambda}$.

It is doubtful whether the mass can be generated  
via Higgs mechanism (as for spin-1 fields), since we are not aware of such a 
procedure in the literature and since in \cite{GS} it has turned out that PGI-tree can be 
satisfied up to second order without introducing a physical Higgs field.
\medskip

\begin{center}
{\Large\bf Appendices}
\end{center}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{appendix}
%%%%%%%%%%%%%%%%%%%%
\section{Algebraic off-shell formalism}\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%
This Appendix is a short introduction to the formalism given in \cite{DF2} and \cite{DF3}.
Let $\varphi$ be a complex scalar field in 4 dimensions.
The classical phase space is ${\cal C}\=d \mathcal{C}^{\infty}(\RR^4,\CC)$.
The classical field $(\d^a\varphi)(x)$, $a\in\NN_0^4$, is the evaluation functional
%%%%%%%%%
\beq
(\d^a\varphi) (x)\>:\>{\cal C}\longrightarrow\CC\>,\>
(\d^a\varphi) (x)(h)=\d^ah(x)\ ,\quad \varphi^\star (x)(h)=\overline{h(x)}\ .\label{eval-func}
\eeq
%%%%%%%%%%
Let $\mathcal{F}$ be the set of all functionals\footnote{$\CC[[\hbar]]$ is the space of all
formal power series $\sum_{l=0}^\infty c_l\hbar^l$ with $c_l\in\CC$.}
$F\equiv F(\varphi):{\cal C}\longrightarrow\CC[[\hbar]]$ which have the form
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
  F(\varphi)=\sum_{n=0}^N\int dx_1\ldots dx_n\,\varphi(x_1)
  \cdots\varphi(x_n)f_n(x_1,\ldots,x_n)\ ,\quad\quad N<\infty\ ,
  \label{F(phi)} 
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $f_0\in\CC[[\hbar]]$. The higher $f_n$'s are $\CC[[\hbar]]$-valued 
distributions with compact support, which are symmetric under 
permutations of $x_1,...,x_n$ and which are 'translation invariant up to smooth
functions', i.e.
%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%
\begin{equation}
        WF(f_{n})\subset \{(x,k)\>|\>\sum_{i=1}^n k_{i}=0\}\ .
\end{equation}
%%%%%%%%%%%%
Due to (\ref{eval-func}) it holds
$F(\varphi)(h)=F(h)$. $\mathcal{F}$ is a *-algebra with
the classical product $(F_1\cdot F_2)(h):=F_1(h)\cdot F_2(h)$
and $ \langle f_{n},\varphi^{\otimes  n}\rangle ^\star=  \langle \overline{f_n},
(\varphi^\star)^{\otimes n}\rangle$. The functional
$\omega_0:\mathcal{F}\rightarrow \CC[[\hbar]],\>\omega_0(\sum_n
\langle f_{n},\varphi^{\otimes  n}\rangle)=f_0$ is the 'vacuum state'. 

The support of $F\in \mathcal{F}$ is the support of $\frac{\delta F}{\delta \varphi}$.
Let ${\cal P}$ be the algebra of all polynomials in $\varphi,\varphi^*$ and their partial 
derivatives. The vector space $\mathcal{F}_{\text{loc}}$ of {\it local} functionals
is the set of all $F\in {\cal F}$ of the form
%%%%%%%%%%%%%%%%
\begin{equation}
  F=\int dx\,\sum_{i=1}^N A_i(x)h_i(x)\ ,\quad A_i\in {\cal P}\ ,\> h_i\in {\cal D}(\RR^4)\ .
\end{equation}
%%%%%%%%%%%%%%%%

Given an action $S$, the set ${\cal C}_{S}\subset {\cal C}$ is the set of all smooth 
solutions of the Euler-Lagrange equations belonging to $S$
with compactly supported Cauchy data. We set
$F_{S}\=d F\vert_{{\cal C}_{S}}\ ,\>F\in {\cal F}$. We study actions of the form
$S_{\rm total}=S_0+S_{\rm int}$ with free part $S_0$ and compactly supported
interacting part: $S_{\rm int}\in {\cal F}_{\text{loc}}$. We always assume that 
$S_{\rm total}$ is such that the Cauchy problem has a unique solution.
Given two actions $S_{\rm total}^{(j)}=S_0+S_{\rm int}^{(j)}\> (j=1,2)$ of this kind, there 
exists a map
%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
   r_{S_{\rm total}^{(1)},S_{\rm total}^{(2)}}\>:\>
\mathcal{C}_{S_{\rm total}^{(2)}}\rightarrow
   \mathcal{C}_{S_{\rm total}^{(1)}}\ ,\ f_2\mapsto f_1\ ,
\end{equation}
%%%%%%%%%%%%%%%%%%%%%
such that $f_1$ agrees with $f_2$ outside the future of $(\supp 
\frac{\delta S_{\rm int}^{(1)}}{\delta \varphi}\cup\supp
\frac{\delta S_{\rm int}^{(2)}}{\delta \varphi})$. A {\it retarded field}
%%%%%%%%%%%
\beq
A^{\rm ret}_{S_{\rm int}}(x)\=d A(x)\circ r_{S_0+S_{\rm int},S_0}
\>:\>{\cal C}_{S_0}\longrightarrow\CC\ ,\quad A\in {\cal P}\ ,\label{A^ret:def}
\eeq
%%%%%%%%%%%%%%%%
is a functional on the free solutions which solves the field equation
for $S_{\rm total}=S_0+S_{\rm int}$ if $A=\varphi$ or $\varphi^*$. The 
perturbation expansion of a classical interacting field is the Taylor 
series of the corresponding retarded field as functional of $S_{\rm int}$,
%%%%%%%%%%%
\beq
A^{\rm ret}_{S_{\rm int}}(x)\simeq \sum_{n=0}^\infty\frac{1}{n!}
R^{\rm class}_{n,1\>S_0}(S_{\rm int}^{\otimes n},A(x))\equiv :
R^{\rm class}_{S_0}\Bigl(e_\otimes^{S_{\rm int}},A(x)\Bigr)\ ,\label{A^ret}
\eeq
%%%%%%%%%%%%%%
where $R^{\rm class}_{n,1\>S_0}: {\cal F}_{\text{loc}}^{\otimes (n+1)}
\rightarrow {\cal F}\vert_{{\cal C}_{S_0}}$
is the classical retarded product. (The lower index $S_0$ of $R^{\rm class}_{S_0}$
is redundant, however we write it in view of the conventions in QFT.)

Next we study the quantization of the free theory. 
Let $\Delta^{(m)}_+$ be the 2-point function of the free scalar field
with mass $m$. We define a $*$-product $\star\equiv\star_m$ on $\mathcal{F}$
\cite{BFFLS},
%%%%%%%%%%%%%%%%
\begin{gather}
  (F\star_m G)(\varphi)=
  \sum_{n=0}^{\infty}\frac{\hbar^n}{n!}
  \int dx_1\ldots dx_n dy_1\ldots dy_n 
  \frac{\delta^n F}{\delta\varphi(x_1)\cdots\delta\varphi(x_n)}\notag\\
  \cdot \prod_{i=1}^n \Delta^{(m)}_+(x_i-y_i) 
  \frac{\delta^n G}{\delta\varphi(y_1)\cdots\delta\varphi(y_n)}\ ,
  \label{*-product}
\end{gather}
%%%%%%%%%%%%%%%%
which is associative and non-commutative.
Let $\mathcal{J}^{(m)}\subset {\cal F}$ be the ideal (with respect to the classical product)
generated by the free field equation $(\square +m^2)\varphi =0$.
Let $\mathcal{F}^{(m)}_{0}\equiv {\cal F}/{\cal J}^{(m)}$. Due to $(\square +m^2)
\Delta^{(m)}_+=0$ the $*$-product (\ref{*-product}) induces a well defined product 
$\mathcal{F}^{(m)}_{0}\times \mathcal{F}^{(m)}_{0}\rightarrow\mathcal{F}^{(m)}_{0}$,
and we denote the corresponding algebra by ${\cal A}_0^{(m)}\equiv
(\mathcal{F}^{(m)}_{0},\star_m)$. The latter can
be faithfully represented on Fock space, the $*$-product goes over into
the operator product and, in addition, the classical product into the normally 
ordered product. $\omega_0$ induces a state on
$\mathcal{A}^{(m)}_{0}$ which corresponds to the Fock vacuum.

Finally we turn to perturbative QFT.
Let $F,S_n\in {\cal F}_{\rm loc}$ with $F,S_n\sim\hbar^0$ $(n\in {\bf N})$, and
let $S\equiv S(\kappa)=\sum_{n=1}^{\infty}\kappa^n S_n$ be a formal power series 
with $S(0)=0$.
We associate to $(F,S)$ a formal power series 
%%%%%%%%%%%
\beq
F_{S/\hbar}=\sum_{n=0}^\infty\frac{1}{n!}
R_{n,1}((S/\hbar)^{\otimes n},F)\equiv :R(e_\otimes^{S/\hbar},F)\label{intfield:QFT}
\eeq
%%%%%%%%%%%
which we interpret as the functional $F$ of the 
interacting retarded field under the influence of 
the interaction $S$ where $\kappa$ is the expansion parameter 
of the formal power series. The retarded products
%%%%%%%%%
\beq
R_{n,1}\>:\>{\cal F}_{\rm loc}^{\otimes (n+1)}\longrightarrow {\cal F}\label{ret-prod:QFT}
\eeq
%%%%%%%%%%
are {\it linear} maps which are {\it symmetric in the first $n$ factors}.
Their basic properties are
\begin{itemize}
 \item zeroth order $\quad R_{0,1}(F)=F$\ ,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Causality
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\supp R_{n,1}\subset
\{(x_1,\ldots,x_{n},x)\in\RR^{4(n+1)}\> |\> x_i\in x+\overline{V}_-,\>
\forall i=1,\ldots n\}\label{caus}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%
\item and the GLZ relation
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
i\Bigl[R\Bigl(e_\otimes^{S/\hbar},F\Bigr),R\Bigl(e_\otimes^{S/\hbar},H\Bigr)\Bigr]_\star =
R\Bigl(e_\otimes^{S/\hbar}\otimes H,F\Bigr)-(H\leftrightarrow F)\ .\label{GLZ}
\end{equation}
%%%%%%%%%%%%%%%%%
\end{itemize}
We set $R_{S_0}(...)\=d R(...)\vert_{{\cal C}_{S_0}}$. 

$R_{n,1}^{\rm tree}$ is the contribution of the tree diagrams to $R_{n,1}$. 
As shown in Sect.~5.2 of \cite{DF1}, $R_{n,1}^{\rm tree}$ is that
part of $R_{n,1}$ with the lowest power of $\hbar$, more precisely
%%%%%%%%%%%
\beq
R_{n,1}(F_1,...,F_{n+1})=R^{\rm tree}_{n,1}(F_1,...,F_{n+1})+
{\cal O}(\hbar^{n+1})\quad\mathrm{and}\quad
R^{\rm tree}_{n,1}(F_1,...,F_{n+1})\sim\hbar^{n}\label{R-tree}
\eeq
%%%%%%%%%%%%%%%%%%%%%%
if $F_1,...,F_{n+1}\sim\hbar^0$.
%%%%%%%%%%%%%%%%%%%%%%
\section{Zeroth order of the BRS-current for spin-2 gauge fields}\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From ${\cal L}^{(0)}$ (\ref{L_total:gravity}) one obtains the free field equations
%%%%%%%%%%%%%
\beq
\square\, h^{\mu\nu}_{S_0}=0\ ,\quad \square\, u^{\mu}_{S_0}=0\ ,\quad 
\square\, \tilde u_{\mu\,S_0}=0\ .\label{ffeq}
\eeq
%%%%%%%%%%%%%%%
By using (\ref{j^n}) and (\ref{sL:gravity}) we obtain
%%%%%%%%%%%%%
\beq
 j^{(0)\,\mu}=-\frac{\d{\cal L}^{(0)}}{\d\varphi_{i,\mu}}\,s_0\,\varphi_i-
\d_\nu\,{\cal D}^{(-1)\,\nu}\,u^\mu-s_0\,{\cal D}^{(0)\,\mu}-s_1\,{\cal D}^{(-1)\,\mu}
+F^{(0)\,\mu}\ ,\label{j^0}
\eeq
%%%%%%%%%%%%%%%
where we have 
taken into account ${\cal L}_{\rm E}^{(-1)}=\d_\mu\,{\cal D}^{(-1)\,\mu}$.
After restriction to ${\cal C}_{S_0}$ (\ref{ffeq}), we obtain the following non-vanishing 
contributions to $(-j^{(0)\,\mu}_{S_0})$:
%%%%%%%%%%%%%%
\begin{gather}
\Bigl(\frac{\d{\cal L}^{\prime\,(0)}_{\rm E}}{\d h^{\alpha\beta}_{\>,\mu}}
\>s_0\,h^{\alpha\beta}\Bigr)_{S_0}=\frac{1}{2}\,h^{\>,\mu}_{\alpha\beta\,S_0}\,
(u^{\alpha,\beta}_{S_0}+u^{\beta,\alpha}_{S_0})-
h^{\alpha\mu,\beta}_{S_0}\,(u_{\alpha,\beta\,S_0}+u_{\beta,\alpha\, S_0}
-\eta_{\alpha\beta}\,u^{\lambda}_{\>,\lambda\,S_0})\ ,\notag\\
\Bigl(\frac{\d{\cal L}_{\rm GF}}{\d h^{\alpha\beta}_{\>,\mu}}
\>s_0\,h^{\alpha\beta}\Bigr)_{S_0}=h^{\>,\gamma}_{\alpha\gamma\,S_0}\,
(u^{\alpha,\mu}_{S_0}+u^{\mu,\alpha}_{S_0}-
\eta^{\alpha\mu}\,u^{\lambda}_{\>,\lambda\,S_0})\ ,\notag\\
\Bigl(\frac{\d{\cal L}^{(0)}_{\rm ghost}}{\d \tilde u_{\nu,\mu}}
\>s_0\,\tilde u_{\nu}\Bigr)_{S_0}=(u^{\mu}_{\>,\tau\,S_0}+u^{\>,\mu}_{\tau\,S_0})\,
h^{\tau\rho}_{\>,\rho\,S_0}-u^{\lambda}_{\>,\lambda\,S_0}\,
h^{\mu\rho}_{\>,\rho\,S_0}\ ,\notag\\
\bigl(\d_\nu\,{\cal D}^{(-1)\,\nu}\bigr)_{S_0}\,u_{S_0}^{\mu}=
h^{\alpha\beta}_{\>,\alpha\beta\,S_0}\,u^{\mu}_{S_0}\,\notag\\
\bigl(s_0\,{\cal D}^{(0)\,\mu}+s_1\,{\cal D}^{(-1)\,\mu}\bigr)_{S_0}=
\frac{1}{2}\,h_{,\nu\,S_0}\,(u^{\mu,\nu}_{S_0}+u^{\nu,\mu}_{S_0})+\frac{1}{2}\,
h_{S_0}\,u^{\lambda,\mu}_{\>,\lambda\,S_0}-\frac{1}{2}\,(h_{S_0}\,
u^{\lambda}_{S_0})^{,\mu}_{,\lambda}\notag\\
-h^{\mu\nu}_{S_0}\,u^{\lambda}_{\>,\lambda\nu\,S_0}
+(h^{\rho\nu}_{S_0}\,u^{\mu}_{\>,\nu\,S_0})_{,\rho}
-(h^{\mu\nu}_{\>,\nu\,S_0}\,u^{\rho}_{S_0})_{,\rho}\,\notag\\
-\bigl(F^{(0)\,\mu}\bigr)_{S_0}=-h^{\tau\nu}_{\>,\nu\,S_0}\,u^{\mu}_{\>,\tau\,S_0}
-h^{\tau\nu}_{\>,\nu\,S_0}\,u^{\>,\mu}_{\tau\,S_0}+
h^{\mu\nu}_{\>,\nu\,S_0}\,u^{\lambda}_{\>,\lambda\,S_0}\ .
\end{gather}
%%%%%%%%%%%%%%%%%%%%
The sum of these terms is equal to (\ref{j^0-p}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{appendix}

\vskip0.5cm
{\bf Acknowledgments:} Discussions with Klaus Fredenhagen and G\"unter Scharf have  
been very helpful and motivating for writing this paper. 

\begin{thebibliography}{999}

\bibitem{BFFLS}Bayen,~F.,Flato,~M.,Fronsdal,~C.,Lichnerowicz,~A.,Sternheimer,~D.,
''Deformation Theory and Quantization'', {\it Annals of Physics (N.Y.)} 
{\bf 111}, 61, 111 (1978)

\bibitem{BBH} Barnich, G., Brandt, F., and Henneaux, M., ``Local BRST cohomology in
Einstein Yang-Mills theory'', {\it Nucl. Phys.} {\bf B455} (1995) 357

\bibitem{BRS} Becchi, C., Rouet, A., and Stora, R., "Renormalization of 
the Abelian Higgs-Kibble model",
{\it Commun. Math. Phys.} {\bf 42} (1975), 127\\

Becchi, C., Rouet, A., and Stora, R., "Renormalization of gauge theories", 
{\it Annals of Physics (N.Y.)} {\bf 98} (1976), 287

%\bibitem{BS}Bogoliubov, N.N., and Shirkov, D.V., {\it "Introduction to the 
%Theory of Quantized Fields"}, New York (1959)

\bibitem{BDK} Brandt, F., Dragon, N., and Kreuzer, M., ``The Gravitational Anomalies'',
{\it Nucl. Phys.} {\bf B340} (1990) 187

%\bibitem{BDF}Brunetti, R., D\"utsch, M., and Fredenhagen, K., work in progress

%\bibitem{BF}Brunetti, R., and Fredenhagen, K.,
%"Microlocal analysis and interacting quantum field 
%theories: Renormalization on physical backgrounds", {\it Commun. Math. 
%Phys.} {\bf 208} (2000) 623

\bibitem{DRM} Delbourgo, R., and Ramon Medrano, M., {\it Nucl. Phys.}
{\bf B110} (1976) 467

%\bibitem{de Witt}DeWitt, B.S.,''The spacetime approach to quantum 
%field theory'' in ``Relativity, Groups, and Topology II: Les Houches
%1983'' (B.S. DeWitt and R. Stora, eds.), part 2, North-Holland, New
%York (1984) 381-738

\bibitem{Dix} Dixson, J., Oxford thesis, unpublished

\bibitem{DF}D\"utsch, M., and Fredenhagen, K.,
"A local (perturbative) construction of observables 
in gauge theories: the example of QED", 
{\it Commun. Math. Phys.} {\bf 203} (1999) 71

\bibitem{DF1}D\"utsch, M., and Fredenhagen, K.,''Algebraic Quantum Field 
Theory, Perturbation Theory, and the Loop Expansion'', 
{\it Commun. Math. Phys.}  {\bf 219} (2001) 5

%\bibitem{DF1a}D\"utsch, M., and Fredenhagen, K.,
%``Perturbative Algebraic Field Theory, and Deformation Quantization'',
%{\it Fields Institute Communications} {\bf 30} (2001) 151-160

\bibitem{DF2}D\"utsch, M. and Fredenhagen, K.,
''The Master Ward Identity and Generalized Schwinger-Dyson Equation
in Classical Field Theory'', {\it Commun. Math. Phys.}  
{\bf 243} (2003) 275-314

\bibitem{DF3}D\"utsch, M. and Fredenhagen, K., ``Causal perturbation theory in 
terms of retarded products, and a proof of the Action Ward Identity'',
{\it Rev. Math. Phys.}  {\bf 16} (2004) 1291-1348

\bibitem{DF4}D\"utsch, M. and Fredenhagen, K., ``Perturbative renormalization and BRST'',
hep-th/0411196, to appear in 'Encyclopedia of Mathematical Physics'

\bibitem{BD}D\"utsch, M. and Boas, F.-M.,
''The Master Ward Identity'', {\it Rev. Math. Phys.} {\bf 14}
 (2002) 977-1049

\bibitem{DSchroer}D\"utsch, M., and Schroer, B.:''Massive Vector Mesons and
Gauge Theory'', {\it J. Phys. A} {\bf 33} (2000), 4317

\bibitem{DHKS} D\"utsch, M., Hurth, T., Krahe, K. and Scharf, G.,
``Causal Construction of Yang-Mills Theories I'' {\it N. Cimento A}
{\bf 106} (1993) 1029-1041

D\"utsch, M., Hurth, T., Krahe, K. 
and Scharf, G., "Causal construction of 
Yang-Mills theories II" {\it N. Cimento A} {\bf 107} (1994) 375-406

D\"utsch, M., Hurth, T. and Scharf, G., "Causal construction of 
Yang-Mills theories III" {\it N. Cimento A} {\bf 108} (1995) 679-707 

D\"utsch, M., Hurth, T. and Scharf, G., "Causal construction of 
Yang-Mills theories IV, Unitarity" {\it N. Cimento A} {\bf 108} (1995) 737-773

\bibitem{DS} D\"utsch, M. and Scharf, G.,
``Perturbative Gauge Invariance: the Electroweak Theory'',
{\it Ann. Phys. (Leipzig)}, {\bf 8} (1999) 359-387

Aste, A., D\"utsch, M. and Scharf, G.,
``Perturbative Gauge Invariance: the Electroweak Theory II'',
{\it Ann. Phys. (Leipzig)}, {\bf 8} (1999) 389-404

%\bibitem{EG}Epstein, H. and Glaser, V., "The role of locality in 
%perturbation theory", {\it Ann. Inst. H. Poincar\'e A} {\bf 19} (1973) 211

\bibitem{EG1}Epstein, H., and Glaser, V., ``Adiabatic limit in perturbation
theory'' in G.Velo and A.S.Wightman (eds.): Renormalization theory (1976)
193-254

\bibitem{G}Grigore, D. R., ``On the uniqueness of the non-abelian gauge
theories in Epstein-Glaser approach to renormalisation theory'',
hep-th/9806244, {\it Romanian J. Phys.} {\bf 44} (1999), 853

Grigore, D. R., ``The standard model and its generalisations in Epstein-Glaser
approach to renormalisation theory', 
{\it J.Phys. A} {\bf 33} (2000), 8443

Grigore, D. R., ``The standard model and its generalisations in Epstein-Glaser
approach to renormalisation theory II: the fermion sector and the axial
anomaly'', {\it J.Phys. A} {\bf 34} (2001), 5429

Grigore, D. R., ``Ward Identities and Renormalization of General Gauge Theories'',
{\it J.Phys. A} {\bf 37} (2004) 2803-2834

\bibitem{G:grav}Grigore, D. R.,''On the quantization of the linearized
gravitational field'', {\it Class.~Quantum Grav.} {\bf 17} (2000) 319-344

\bibitem{GS} Grigore, D. R., Scharf, G., ''Massive Gravity as a Quantum 
Gauge Theory'', hep-th/0404157

%\bibitem{G:scal} Grigore, D. R.,''Scale Invariance in the Causal
%  Approach to Renormalization Theory'', Ann. Phys. (Leipzig)
%{\bf 10} (2001) 473

\bibitem{Grillo} Grillo, N., ''Scalar Matter Coupled to Quantum Gravity in the Causal 
Approach'', {\it Ann. Phys.} {\bf 287} (2001) 153-190

%\bibitem{Haag} Haag, R., ``Local Quantum Physics: Fields, particles
%  and algebras'', Springer-Verlag, Berlin, 2nd ed. (1996)

%\bibitem{HW1-2} Hollands, S., Wald, R. M., ``Local Wick Polynomials
%and Time-Ordered-Products of Quantum Fields in Curved Spacetime'',
%{\it Commun. Math. Phys.}  {\bf 223} (2001) 289

%Hollands, S., Wald, R. M., ``Existence of Local Covariant
%Time-Ordered-Products of Quantum Fields in Curved Spacetime'',
%{\it Commun. Math. Phys.} {\bf 231} (2002) 309-345

%\bibitem{HW} Hollands, S., Wald, R. M., ``On the Renormalization Group
%  in Curved Spacetime'', {\it Commun. Math. Phys.} {\bf 237} (2003) 123-160

\bibitem{Hurth}
T.~Hurth and K.~Skenderis,
``The quantum Noether condition in terms of interacting fields,''
in 'New Developments in Quantum Field Theory', eds. P. Breitenlohner, 
D. Maison and J.Wess, {\it Lect.\ Notes Phys., 
Springer} {\bf 558} (2000) 86

\bibitem{KO1}Kugo, T., and Ojima, I., "Local covariant operator formalism of 
non-abelian gauge theories and quark confinement problem",
{\it Suppl. Progr. Theor. Phys.} {\bf 66} (1979), 1

\bibitem{KO} Kugo, T., Ojima, I., ``Subsidiary conditions and physical $S$-matrix
unitarity in indefinite-metric quantum gravitation theory'',
{\it Nucl. Phys.} {\bf B144} (1978) 234-252 

%\bibitem{Marolf} Marolf, D.~M.,''The Generalized Peierls Bracket'',
%{\it Ann. Phys. (N.Y.)} {\bf 236} (1994) 392

%\bibitem{Peierls}Peierls, R., ``The commutation laws of relativistic field 
%theory'', {\it Proc. Roy. Soc. (London)} {\bf A 214} (1952) 143

\bibitem{LN} Lautrup, B., {\it Kgl. Danske Videnskab. Selskab. Mat.-fys. Medd.} {\bf 35} 
No.11, 1 (1967)

Nakanishi, N., {\it Prog. Theor. Phys.} {\bf 35} 1111 (1966)

\bibitem{R} Requardt, M., "Symmetry conservation and integrals over local 
charge desities in quantum field theory", {\it Commun. Math. Phys.} {\bf 50} (1976) 259

\bibitem{Scharf} Scharf, G., 
{\it "Finite Quantum Electrodynamics. The causal approach"}, 
2nd. ed., Springer-Verlag (1995)

\bibitem{S-wiley} Scharf, G., {\it ``Quantum Gauge Theories - 
A True Ghost Story''}, John Wiley and Sons (2001)

\bibitem{SW} Scharf, G., Wellmann, M., ``Quantum Gravity from Perturbative 
Gauge Invariance'', {\it Gen. Relat. Gravit.} {\bf 33} (2001) 553

\bibitem{Schorn} Schorn, I., ``Gauge Invariance of Quantum Gravity in the Causal Approach'',
{\it Class. Quantum. Grav.}, {\bf 14} (1997) 653-669 

Schorn, I., ``Ghost couplings in causal quantum gravity'',
{\it Class. Quantum. Grav.}, {\bf 14} (1997) 671-686

%\bibitem{Ste} Steinmann, O., ``Perturbation expansions in axiomatic field 
%theory'', Lecture Notes in Physics {\bf 11}, Berlin-Heidelberg-New York:
%Springer-Verlag (1971)

%\bibitem{Ste1} Steinmann, O., ``Perturbative QED and Axiomatic Field 
%Theory'', Springer-Verlag (2000)

\bibitem{stora}Stora, R., "Local gauge groups in quantum field theory: 
perturbative gauge theories", talk given at the workshop 'Local Quantum 
Physics' at the Erwin-Schroedinger-institute, Vienna (1997)

%\bibitem{Sto}Stora, R.,
%"Differential algebras in Lagrangean field theory", 
%ETH-Z\"urich Lectures, January-February 1993;

%Popineau, G., and Stora, R.,
%"A pedagogical remark on the main theorem of perturbative 
%renormalization theory", unpublished preprint (1982)

%\bibitem{AWI}Stora, R., ``Pedagogical Experiments in Renormalized
%  Perturbation Theory'', contribution to the conference 'Theory of 
%Renormalization and Regularization', Hesselberg, Germany (2002),
%http://wwwthep.physik.uni-mainz.de/~scheck/Hessbg02.html;
%and private communication

\bibitem{TN} Townsend, P.K., and Van Nieuwenhuizen, P., {\it Nucl. Phys.}
{\bf B120} (1977) 301

\end{thebibliography}








\end{document}


hep-th0502071.extracted_bib
\begin{thebibliography}{999}

\bibitem{BFFLS}Bayen,~F.,Flato,~M.,Fronsdal,~C.,Lichnerowicz,~A.,Sternheimer,~D.,
''Deformation Theory and Quantization'', {\it Annals of Physics (N.Y.)} 
{\bf 111}, 61, 111 (1978)

\bibitem{BBH} Barnich, G., Brandt, F., and Henneaux, M., ``Local BRST cohomology in
Einstein Yang-Mills theory'', {\it Nucl. Phys.} {\bf B455} (1995) 357

\bibitem{BRS} Becchi, C., Rouet, A., and Stora, R., "Renormalization of 
the Abelian Higgs-Kibble model",
{\it Commun. Math. Phys.} {\bf 42} (1975), 127\\

Becchi, C., Rouet, A., and Stora, R., "Renormalization of gauge theories", 
{\it Annals of Physics (N.Y.)} {\bf 98} (1976), 287

%\bibitem{BS}Bogoliubov, N.N., and Shirkov, D.V., {\it "Introduction to the 
%Theory of Quantized Fields"}, New York (1959)

\bibitem{BDK} Brandt, F., Dragon, N., and Kreuzer, M., ``The Gravitational Anomalies'',
{\it Nucl. Phys.} {\bf B340} (1990) 187

%\bibitem{BDF}Brunetti, R., D\"utsch, M., and Fredenhagen, K., work in progress

%\bibitem{BF}Brunetti, R., and Fredenhagen, K.,
%"Microlocal analysis and interacting quantum field 
%theories: Renormalization on physical backgrounds", {\it Commun. Math. 
%Phys.} {\bf 208} (2000) 623

\bibitem{DRM} Delbourgo, R., and Ramon Medrano, M., {\it Nucl. Phys.}
{\bf B110} (1976) 467

%\bibitem{de Witt}DeWitt, B.S.,''The spacetime approach to quantum 
%field theory'' in ``Relativity, Groups, and Topology II: Les Houches
%1983'' (B.S. DeWitt and R. Stora, eds.), part 2, North-Holland, New
%York (1984) 381-738

\bibitem{Dix} Dixson, J., Oxford thesis, unpublished

\bibitem{DF}D\"utsch, M., and Fredenhagen, K.,
"A local (perturbative) construction of observables 
in gauge theories: the example of QED", 
{\it Commun. Math. Phys.} {\bf 203} (1999) 71

\bibitem{DF1}D\"utsch, M., and Fredenhagen, K.,''Algebraic Quantum Field 
Theory, Perturbation Theory, and the Loop Expansion'', 
{\it Commun. Math. Phys.}  {\bf 219} (2001) 5

%\bibitem{DF1a}D\"utsch, M., and Fredenhagen, K.,
%``Perturbative Algebraic Field Theory, and Deformation Quantization'',
%{\it Fields Institute Communications} {\bf 30} (2001) 151-160

\bibitem{DF2}D\"utsch, M. and Fredenhagen, K.,
''The Master Ward Identity and Generalized Schwinger-Dyson Equation
in Classical Field Theory'', {\it Commun. Math. Phys.}  
{\bf 243} (2003) 275-314

\bibitem{DF3}D\"utsch, M. and Fredenhagen, K., ``Causal perturbation theory in 
terms of retarded products, and a proof of the Action Ward Identity'',
{\it Rev. Math. Phys.}  {\bf 16} (2004) 1291-1348

\bibitem{DF4}D\"utsch, M. and Fredenhagen, K., ``Perturbative renormalization and BRST'',
hep-th/0411196, to appear in 'Encyclopedia of Mathematical Physics'

\bibitem{BD}D\"utsch, M. and Boas, F.-M.,
''The Master Ward Identity'', {\it Rev. Math. Phys.} {\bf 14}
 (2002) 977-1049

\bibitem{DSchroer}D\"utsch, M., and Schroer, B.:''Massive Vector Mesons and
Gauge Theory'', {\it J. Phys. A} {\bf 33} (2000), 4317

\bibitem{DHKS} D\"utsch, M., Hurth, T., Krahe, K. and Scharf, G.,
``Causal Construction of Yang-Mills Theories I'' {\it N. Cimento A}
{\bf 106} (1993) 1029-1041

D\"utsch, M., Hurth, T., Krahe, K. 
and Scharf, G., "Causal construction of 
Yang-Mills theories II" {\it N. Cimento A} {\bf 107} (1994) 375-406

D\"utsch, M., Hurth, T. and Scharf, G., "Causal construction of 
Yang-Mills theories III" {\it N. Cimento A} {\bf 108} (1995) 679-707 

D\"utsch, M., Hurth, T. and Scharf, G., "Causal construction of 
Yang-Mills theories IV, Unitarity" {\it N. Cimento A} {\bf 108} (1995) 737-773

\bibitem{DS} D\"utsch, M. and Scharf, G.,
``Perturbative Gauge Invariance: the Electroweak Theory'',
{\it Ann. Phys. (Leipzig)}, {\bf 8} (1999) 359-387

Aste, A., D\"utsch, M. and Scharf, G.,
``Perturbative Gauge Invariance: the Electroweak Theory II'',
{\it Ann. Phys. (Leipzig)}, {\bf 8} (1999) 389-404

%\bibitem{EG}Epstein, H. and Glaser, V., "The role of locality in 
%perturbation theory", {\it Ann. Inst. H. Poincar\'e A} {\bf 19} (1973) 211

\bibitem{EG1}Epstein, H., and Glaser, V., ``Adiabatic limit in perturbation
theory'' in G.Velo and A.S.Wightman (eds.): Renormalization theory (1976)
193-254

\bibitem{G}Grigore, D. R., ``On the uniqueness of the non-abelian gauge
theories in Epstein-Glaser approach to renormalisation theory'',
hep-th/9806244, {\it Romanian J. Phys.} {\bf 44} (1999), 853

Grigore, D. R., ``The standard model and its generalisations in Epstein-Glaser
approach to renormalisation theory', 
{\it J.Phys. A} {\bf 33} (2000), 8443

Grigore, D. R., ``The standard model and its generalisations in Epstein-Glaser
approach to renormalisation theory II: the fermion sector and the axial
anomaly'', {\it J.Phys. A} {\bf 34} (2001), 5429

Grigore, D. R., ``Ward Identities and Renormalization of General Gauge Theories'',
{\it J.Phys. A} {\bf 37} (2004) 2803-2834

\bibitem{G:grav}Grigore, D. R.,''On the quantization of the linearized
gravitational field'', {\it Class.~Quantum Grav.} {\bf 17} (2000) 319-344

\bibitem{GS} Grigore, D. R., Scharf, G., ''Massive Gravity as a Quantum 
Gauge Theory'', hep-th/0404157

%\bibitem{G:scal} Grigore, D. R.,''Scale Invariance in the Causal
%  Approach to Renormalization Theory'', Ann. Phys. (Leipzig)
%{\bf 10} (2001) 473

\bibitem{Grillo} Grillo, N., ''Scalar Matter Coupled to Quantum Gravity in the Causal 
Approach'', {\it Ann. Phys.} {\bf 287} (2001) 153-190

%\bibitem{Haag} Haag, R., ``Local Quantum Physics: Fields, particles
%  and algebras'', Springer-Verlag, Berlin, 2nd ed. (1996)

%\bibitem{HW1-2} Hollands, S., Wald, R. M., ``Local Wick Polynomials
%and Time-Ordered-Products of Quantum Fields in Curved Spacetime'',
%{\it Commun. Math. Phys.}  {\bf 223} (2001) 289

%Hollands, S., Wald, R. M., ``Existence of Local Covariant
%Time-Ordered-Products of Quantum Fields in Curved Spacetime'',
%{\it Commun. Math. Phys.} {\bf 231} (2002) 309-345

%\bibitem{HW} Hollands, S., Wald, R. M., ``On the Renormalization Group
%  in Curved Spacetime'', {\it Commun. Math. Phys.} {\bf 237} (2003) 123-160

\bibitem{Hurth}
T.~Hurth and K.~Skenderis,
``The quantum Noether condition in terms of interacting fields,''
in 'New Developments in Quantum Field Theory', eds. P. Breitenlohner, 
D. Maison and J.Wess, {\it Lect.\ Notes Phys., 
Springer} {\bf 558} (2000) 86

\bibitem{KO1}Kugo, T., and Ojima, I., "Local covariant operator formalism of 
non-abelian gauge theories and quark confinement problem",
{\it Suppl. Progr. Theor. Phys.} {\bf 66} (1979), 1

\bibitem{KO} Kugo, T., Ojima, I., ``Subsidiary conditions and physical $S$-matrix
unitarity in indefinite-metric quantum gravitation theory'',
{\it Nucl. Phys.} {\bf B144} (1978) 234-252 

%\bibitem{Marolf} Marolf, D.~M.,''The Generalized Peierls Bracket'',
%{\it Ann. Phys. (N.Y.)} {\bf 236} (1994) 392

%\bibitem{Peierls}Peierls, R., ``The commutation laws of relativistic field 
%theory'', {\it Proc. Roy. Soc. (London)} {\bf A 214} (1952) 143

\bibitem{LN} Lautrup, B., {\it Kgl. Danske Videnskab. Selskab. Mat.-fys. Medd.} {\bf 35} 
No.11, 1 (1967)

Nakanishi, N., {\it Prog. Theor. Phys.} {\bf 35} 1111 (1966)

\bibitem{R} Requardt, M., "Symmetry conservation and integrals over local 
charge desities in quantum field theory", {\it Commun. Math. Phys.} {\bf 50} (1976) 259

\bibitem{Scharf} Scharf, G., 
{\it "Finite Quantum Electrodynamics. The causal approach"}, 
2nd. ed., Springer-Verlag (1995)

\bibitem{S-wiley} Scharf, G., {\it ``Quantum Gauge Theories - 
A True Ghost Story''}, John Wiley and Sons (2001)

\bibitem{SW} Scharf, G., Wellmann, M., ``Quantum Gravity from Perturbative 
Gauge Invariance'', {\it Gen. Relat. Gravit.} {\bf 33} (2001) 553

\bibitem{Schorn} Schorn, I., ``Gauge Invariance of Quantum Gravity in the Causal Approach'',
{\it Class. Quantum. Grav.}, {\bf 14} (1997) 653-669 

Schorn, I., ``Ghost couplings in causal quantum gravity'',
{\it Class. Quantum. Grav.}, {\bf 14} (1997) 671-686

%\bibitem{Ste} Steinmann, O., ``Perturbation expansions in axiomatic field 
%theory'', Lecture Notes in Physics {\bf 11}, Berlin-Heidelberg-New York:
%Springer-Verlag (1971)

%\bibitem{Ste1} Steinmann, O., ``Perturbative QED and Axiomatic Field 
%Theory'', Springer-Verlag (2000)

\bibitem{stora}Stora, R., "Local gauge groups in quantum field theory: 
perturbative gauge theories", talk given at the workshop 'Local Quantum 
Physics' at the Erwin-Schroedinger-institute, Vienna (1997)

%\bibitem{Sto}Stora, R.,
%"Differential algebras in Lagrangean field theory", 
%ETH-Z\"urich Lectures, January-February 1993;

%Popineau, G., and Stora, R.,
%"A pedagogical remark on the main theorem of perturbative 
%renormalization theory", unpublished preprint (1982)

%\bibitem{AWI}Stora, R., ``Pedagogical Experiments in Renormalized
%  Perturbation Theory'', contribution to the conference 'Theory of 
%Renormalization and Regularization', Hesselberg, Germany (2002),
%http://wwwthep.physik.uni-mainz.de/~scheck/Hessbg02.html;
%and private communication

\bibitem{TN} Townsend, P.K., and Van Nieuwenhuizen, P., {\it Nucl. Phys.}
{\bf B120} (1977) 301

\end{thebibliography}