% A CHARACTER LIST TO TEST TRANSMISSION PROBLEMS:
% Uppercase A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
% Lowercase a b c d e f g h i j k l m n o p q r s t u v w x y z
% Digits 0 1 2 3 4 5 6 7 8 9
% Exclamation ! Double quote " Hash (number) #
% Dollar $ Percent % Ampersand &
% Acute accent ' Left paren ( Right paren )
% Asterisk * Plus + Comma ,
% Minus  Point . Solidus /
% Colon : Semicolon ; Less than <
% Equals = Greater than > Question mark ?
% At @ Left bracket [ Backslash \
% Right bracket ] Circumflex ^ Underscore _
% Grave accent ` Left brace { Vertical bar 
% Right brace } Tilde ~
% This is a Latex file. Latex twice to get references correct.
\documentstyle[12pt]{article}
\renewcommand{\baselinestretch}{1.1}
\textwidth 6truein
\textheight 8.5truein
\topmargin 12pt
\oddsidemargin 20pt
% MACROS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\dual}[1]{{}^*\hspace{1mm}#1}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\beasn}{\begin{sneqnarray}}
\newcommand{\eeasn}{\end{sneqnarray}}
\newcommand{\bref}[1]{(\ref{#1})}
% use ct in lieu of cite, just in case we want to change format:
\newcommand{\ct}[1]{\cite{#1}}
\newcommand{\eps}{\epsilon}
\newcommand{\veps}{\varepsilon}
\newcommand{\ov}[1]{\overline #1}
\newcommand{\der}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\lder}[2]{\frac{\partial_l#1}{\partial #2}}
\newcommand{\rder}[2]{\frac{\partial_r#1}{\partial #2}}
\newcommand{\gh}[1]{{\cal #1}}
\newcommand{\agh}[1]{\bar{\cal #1}}
\newcommand{\Op}[1]{\hat{#1}}
\newcommand{\bra}[1]{\langle \, #1 \, }
\newcommand{\ket}[1]{ \, #1 \, \rangle}
\newcommand{\bracket}[2]{\langle \, #1 \,  \, #1 \, \rangle}
\newcommand{\expectation}[1]{\langle \, #1 \, \rangle}
\newcommand{\invstackrel}[2]{\relstack{#2}{#1}}
\newcommand{\formula}{\vspace{20mm}}
\newcommand{\ghn}[1]{{\rm gh}\left[{#1}\right]}
\newcommand{\brst}{{\mit s}}
\newcommand{\abrst}{\bar{\mit s}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\makeatletter
% partial slash in Dirac equation:
\def\slashit#1{\slash\mkern10mu#1}
% better than \not\partial for partial slash
% math operations
\def\relstack#1#2{\mathrel{\mathop{#1}\limits_{#2}}}
\def\mathoperat{\@ifnextchar [{\@mathoperat}{\@mathoperat[rm]}}
\def\@mathoperat[#1]#2#3{\def#2{\mathop{\@nameuse{#1} #3{}}\nolimits}}
% general definitions
\def\presup#1{{}^{#1}\hspace{.15em}} %presuperscript
\def\presub#1{{}_{#1}\hspace{.12em}} %presubscript
\def\restric#1#2{{\left. #1 \right_{#2}}}
\def\dif{{\rm d}}
\def\deriv{\@ifnextchar[{\@deriv}{\@deriv[]}}
\def\@deriv[#1]#2#3{\mathchoice%
{{\dif^{#1}#2\over\dif{#3}^{#1}}}{{\dif^{#1}#2/\dif{#3}^{#1}}}%
{{\dif^{#1}#2\over\dif{#3}^{#1}}}{{\dif^{#1}#2/\dif{#3}^{#1}}}}
\def\derpar#1#2{\mathchoice%
{{\partial#1\over\partial#2}}{{\partial#1/\partial#2}}%
{{\partial#1\over\partial#2}}{{\partial#1/\partial#2}}}
\def\dderpar#1#2#3{\mathchoice%
{{\partial^2 #1\over\partial #2\,\partial #3}}%
{{\partial^2 #1/\partial #2\,\partial #3}}%
{{\partial^2 #1\over\partial #2\,\partial #3}}%
{{\partial^2 #1/\partial #2\,\partial #3}}}
\def\Ker{\mathop{\rm Ker}\nolimits}
\def\Img{\mathop{\rm Im}\nolimits}
\def\tr{\mathop{\rm tr}\nolimits}
%
% to put the section number in the equation number
\def\secteqno{\@addtoreset{equation}{section}%
\def\theequation{\thesection.\arabic{equation}}}
% to disable this:
\def\endsecteqno{\def\theequation{\@ifundefined{chapter}%
{\arabic{equation}}{\thechapter.\arabic{equation}}}}
%
% here comes samenum
\newcounter{subequation}
\def\thesubequation{\alph{subequation}}
\def\sneqnarray{\stepcounter{equation}\let\@currentlabel=\theequation
\setcounter{subequation}{1}
\def\@eqnnum{{\rm (\theequation.\thesubequation)}}
\global\@eqcnt\z@\tabskip\@centering\let\\=\@eqncr\let\@@eqncr=\@@sneqncr
$$\halign to \displaywidth\bgroup\@eqnsel\hskip\@centering
$\displaystyle\tabskip\z@{##}$&\global\@eqcnt\@ne
\hskip 2\arraycolsep \hfil${##}$\hfil
&\global\@eqcnt\tw@ \hskip 2\arraycolsep $\displaystyle\tabskip\z@{##}$\hfil
\tabskip\@centering&\llap{##}\tabskip\z@\cr}
\def\endsneqnarray{\@@sneqncr\egroup $$\global\@ignoretrue}
\def\@@sneqncr{\let\@tempa\relax
\ifcase\@eqcnt \def\@tempa{& & &}\or \def\@tempa{& &}
\else \def\@tempa{&}\fi
\@tempa \if@eqnsw\@eqnnum\stepcounter{subequation}\fi
\global\@eqnswtrue\global\@eqcnt\z@\cr}
%
\def\qed{{\unskip\nobreak\hfil\penalty50\hbox{}\nobreak\hfil
\hbox{\vrule height8pt width8pt}\parfillskip=0pt
\finalhyphendemerits=0 \par}}
% (page 106 of TeXbook), and \hbox...
%
% def. of "nobiblabels":
\def\nobiblabels{\def\@lbibitem[##1]##2{\@bibitem{##2}}}
%
% some general names
\def\Id{\mathop{\rm Id}\nolimits}
\def\pr{\mathop{\rm pr}\nolimits}
\def\Nat{{\bf N}}
\def\Zahl{{\bf Z}}
\def\Field{{\bf F}}
\def\Quot{{\bf Q}}
\def\Real{{\bf R}}
\def\Comp{{\bf C}}
\def\Hamq{{\bf H}}
\def\Torus{{\bf T}}
\def\Sph{{\bf S}}
% algebra
\def\Ker{\mathop{\rm Ker}\nolimits}
\def\Img{\mathop{\rm Im}\nolimits}
\def\tr{\mathop{\rm tr}\nolimits}
\def\rg{\mathop{\rm rg}\nolimits}
\def\Hom{\mathop{\rm Hom}\nolimits}
\def\End{\mathop{\rm End}\nolimits}
\def\Aut{\mathop{\rm Aut}\nolimits}
\def\Proj{{\bf P}}
\def\Grass{{\bf G}}
\def\Matr{{\bf M}}
\def\GLin{{\bf GL}}
\def\SLin{{\bf SL}}
\def\Aff{{\bf A}}
\def\Ort{{\bf O}}
\def\SOrt{{\bf SO}}
\def\Spin{{\bf Spin}}
\def\Unit{{\bf U}}
\def\SUnit{{\bf SU}}
\def\Symp{{\bf Sp}}
% calculus
\def\eexp{\mathop{\rm e}\nolimits}
\def\tg{\mathop{\rm tg}\nolimits}
\def\cotg{\mathop{\rm cotg}\nolimits}
\def\cosec{\mathop{\rm cosec}\nolimits}
\def\arc{\mathop{\rm arc}\nolimits}
\def\tgh{\mathop{\rm tgh}\nolimits}
\def\cotgh{\mathop{\rm cotgh}\nolimits}
%
\makeatother
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% END OF MACROS
\begin{document}
\begin{titlepage}
\begin{flushright}
CCNYHEP94/03\\
KULTF94/12\\
UBECMPF 94/15\\
UTTG1194\\
hepth/yymmnnn\\
May 1994\\
% A few revisions have occured since May 1994
\end{flushright}
\vskip 2.0mm
\begin{center}
{\LARGE\bf Antibracket, Antifields\\
\vskip 3.0mm
and GaugeTheory Quantization}\\
\vskip 6.mm
{ Joaquim Gomis$^{*1}$, Jordi Par\'{\i}s$^{\sharp 2}$
and Stuart Samuel$^{\dagger 3}$}
\vskip 0.4cm
{\small
$^*$
{\it{Theory Group, Department of Physics}}\\
{\it{The University of Texas at Austin}}\\
{\it{RLM\,5208, Austin, Texas}}\\
{\it{and}}\\
{\it{Departament d'Estructura i Constituents de la
Mat\`eria}}\\
{\it{Facultat de F\'{\i}sica, Universitat de Barcelona}}\\
{\it{Diagonal 647, E08028 Barcelona}}\\
{\it{Catalonia}}\\
[0.4cm]
$^\sharp$
{\it{Instituut voor Theoretische Fysica}}\\
{\it{Katholieke Universiteit Leuven}}\\
{\it{Celestijnenlaan 200D}}\\
{\it{B3001 Leuven, Belgium}}\\
[0.4cm]
$^\dagger$
{\it{Department of Physics}}\\
{\it{City College of New York}}\\
{\it{138th St and Convent Avenue}}\\
{\it{New York, New York 10031 U.S.A.}}\\
[1.0cm]
}
\normalsize
\end{center}
\textwidth 6.5truein
\hrule width 5.cm
{\small
\noindent $^1$
Permanent address: Dept.\ d'Estructura i
Constituents de la Mat\`{e}ria, U.\ Barcelona.\\
Email: gomis@rita.ecm.ub.es\\
\noindent $^2$
Wetenschappelijk Medewerker, I.\,I.\,K.\,W., Belgium.\\
Email: jordi=paris\%tf\%fys@cc3.kuleuven.ac.be\\
\noindent $^3$
Email: samuel@scisun.sci.ccny.cuny.edu
}
\normalsize
\vfill
\eject
\textwidth 6truein
\pagestyle{empty}
{\ }
\vskip 1.5cm
\begin{center}
{\bf Abstract}
\end{center}
\begin{quote}
%Abstract
\hspace{\parindent}
{}\ \ \ The antibracket formalism for gauge theories,
at both the classical and quantum level,
is reviewed.
Gauge transformations and the
associated gauge structure are analyzed
in detail.
The basic concepts involved
in the antibracket formalism
are elucidated.
Gaugefixing,
quantum effects,
and anomalies
within the fieldantifield formalism
are developed.
The concepts, issues and constructions
are illustrated using eight gaugetheory models.
\end{quote}
\vfill
\eject
\pagestyle{empty}
\tableofcontents
\vfill
\eject
\pagestyle{empty}
{\ }
\vskip 4.5cm
\begin{center}
{This work is dedicated to Joseph and Marie,\\ }
{to Pilar,\\ }
{and to the memory of Pere and Francesca.}
\end{center}
\end{titlepage}
\setcounter{page}{2}
\secteqno
\pagestyle{myheadings}
\markright{{\it J.\,Gomis, J.\,Par\'{\i}s and S.\,Samuel} 
Antibracket, Antifields and $\dots $}
\section{Introduction}
\label{s:i}
\hspace{\parindent}
The known fundamental interactions of nature
are all governed by gauge theories.
The presence of a gauge symmetry
indicates that a theory
is formulated in a redundant way,
in which certain local degrees of freedom
do not enter the dynamics.
Conversely,
when there are degrees of freedom,
which do not enter the lagrangian,
a theory possesses local invariances.
Although one can in principle eliminate
the gauge degrees of freedom,
there are reasons for not doing so.
These reasons include manifest covariance,
locality of interactions,
and calculational convenience.
The first example of a gauge theory
was electrodynamics.
Electric and magnetic forces are generated
via the exchange of photons.
Being particles of spin $1$,
photons involve a vector field, $A^\mu$.
However, not all four components
of the electromagnetic potential $A^\mu$
enter dynamically.
Two degrees of freedom correspond to
the two possible physical polarizations of the photon.
The longitudinal degree of freedom
plays a role in interactions
via virtual exchanges of photons.
The remaining gauge degree of freedom
does not enter the theory.
Consequently,
electromagnetism is described by a gauge theory.
When it was realized that
the weak interactions could be unified
with electromagnetism
in an $SU(2) \times U(1)$ gauge theory
\ct{glashow61a,weinberg67a,salam68a}
and that this theory is renormalizable
\ct{thooft71a,thooft71b},
the importance of nonabelian gauge theories
\ct{ym54a}
grew enormously.
The strong interactions are also governed by
an SU(3) nonabelian gauge theory.
The fourth fundamental force
is gravity.
It is based on Einstein's general theory of relativity
and uses general coordinate invariance.
When formulated in terms of a metric
or any other convenient fields,
gravity also possesses gauge symmetries.
The quantization of gauge theories
is not always straightforward.
In the abelian case,
relevant for electromagnetism,
the procedure is well understood.
In contrast,
quantization of a nonabelian theory
and its renormalization
is more complicated.
Quantization generally involves
the introduction of ghost fields.
Typically, a gaugefixing procedure
is used to render dynamical
all degrees of freedom.
Ghost fields are used
to compensate for the effects
of the gauge degrees of freedom
\ct{feynman63a},
so that unitarity is preserved.
In electrodynamics in the linear gauges,
ghosts decouple and can be ignored.
In nonabelian gauge theories,
convenient gauges
generically involve interacting ghosts.
A major step in understanding these issues
was the FaddeevPopov quantization procedure
\ct{fp67a,dewitt67a},
which relied heavily on the functionalintegral
approach to quantization
\ct{fh65a,al73a,iz80a}.
{}From this viewpoint,
the presence of ghost fields
is understood as a ``measure effect''.
In dividing out the volume of gauge transformations
in function space,
a Jacobian measure factor arises.
This factor is produced naturally
by introducing quadratic terms in the lagrangian
for ghosts and then integrating them out.
It was realized at a later stage
that the gaugefixed action
retains a nilpotent, odd, global symmetry
involving transformations of both fields and ghosts.
This BecchiRouetStoraTyutin (BRST) symmetry
\ct{brs74a,tyutin75a}
is what remains of the original gauge invariance.
In fact, for closed theories,
the transformation law for the original fields
is like a gauge transformation with gauge parameters
replaced by ghost fields.
In general,
this produces nonlinear transformation laws.
The relations among correlation functions
derived from BRST symmetry
involve the insertions of the BRST variation of fields.
These facts require the use of composite operators
and it is convenient to introduce
sources for these transformations.
The Ward identities
\ct{ward50a}
associated with the BRST invariance
treated in this way
are the SlavnovTaylor identities
\ct{slavnov72a,taylor71a}.
The SlavnovTaylor identities
and BRST symmetry
have played an important role
in quantization, renormalization, unitarity,
and other aspects
of gauge theories.
Ghosts fields have been useful
throughout the development of
covariant gaugefieldtheory quantization
\ct{ko78a,ko79a,ku82a,on82a,ab83a}.
It is desirable to have a formulation of gauge theories
that introduces them from the outset
and
that automatically incorporates BRST symmetry
\ct{baulieu85a}.
The fieldantifield formulation has these features
\ct{brs74a,zinnjustin75a,bv81a,bv83a,bv83b,bv84a}.
It relies on BRST symmetry as fundamental principle
and uses sources to deal with it
\ct{brs74a,tyutin75a,zinnjustin75a}.
It encompasses
previous ideas and developments
for quantizing gauge systems
and extends them to
more complicated situations
(open algebras, reducible systems, etc.)
\ct{fn76a,fnf76a,kallosh78a,stn78a,wh79a}.
In 1975, J. ZinnJustin,
in his study of the renormalization of
YangMills theories
\ct{zinnjustin75a},
introduced the abovementioned sources
for BRST transformations
and a symplectic structure $( \ , \ )$
(actually denoted $*$ by him)
in the space of fields and sources,
He expressed
the SlavnovTaylor identities
in the compact form $( \Gamma , \Gamma ) = 0$,
where $\Gamma$,
the generating functional
of the oneparticleirreducible diagrams,
is known as the effective action
(see also \ct{lee76a}).
These ideas were developed further
by B.\ L.\ Voronov and I.\ V.\ Tyutin in
\ct{vt82a,vt82b}
and by I.\ A.\ Batalin and G.\ A.\ Vilkovisky
in refs.\ct{bv81a,bv83a,bv83b,bv84a,bv85a}.
These authors generalized
the role of $( \ , \ )$ and of the sources
for BRST transformations
and called them
the antibracket and antifields respectively.
Due to their contributions,
this quantization procedure is
often referred to
as the BatalinVilkovisky formalism.
The antibracket formalism
gained popularity among string theorists,
when it was applied to the open bosonic string field theory
\ct{bochicchio87a,thorn87a}.
It has also proven
quite useful for the closed string field theory
and for topological field theories.
Only within the last few years
has it been applied to more general aspects
of quantum field theory.
In some sense,
the BRST approach,
which was driven, in part,
by renormalization considerations,
and the fieldantifield formalism,
which was motivated by classical considerations
such as gauge structure,
are not so different.
When sources are introduced for BRST transformations,
the BRST approach resembles the fieldantifield one.
Antifields, then, have a simple intepretation:
They are the sources
for BRST transformations.
In this sense,
the fieldantifield formalism
is a general method for dealing with gauge theories
within the context of standard field theory.
The general structure of the antibracket formalism
is as follows.
One introduces an antifield
for each field and ghost,
thereby doubling the total number of original fields.
The antibracket $( \ , \ )$ is
an odd nondegenerate symplectic form
on the space of fields and antifields.
The original classical action $S_0$
is extended to a new action $S$,
in an essentially unique way,
to arrive at a theory
with manifest BRST symmetry.
One equation,
the master equation $( S , S ) = 0$,
reproduces in a compact way
the gauge structure of the original theory
governed by $S_0$.
Although the master equation resembles
the ZinnJustin equation,
the content of the two is different
since $S$ is a functional of quantum fields and antifields
and $\Gamma$ is a functional of classical fields.
The antibracket formalism currently appears
to be the most powerful method for quantizing
a gauge theory.
Beyond tree level, order $\hbar$ terms
usually need to be added to the action,
thereby leading to a quantum action $W$.
These counterterms are expected
to render finite loop contributions,
after a suitable regularization procedure
has been introduced.
The master equation must be appropriately generalized
to the socalled quantum master equation.
It involves a potentially singular operator $\Delta$.
The regularization procedure and counterterms
should also render $\Delta$
and its action on $W$ welldefined.
Violations of the quantum master equation
are equivalent to gauge anomalies
\ct{tnp90a}.
To calculate correlation functions
and scattering amplitudes
in perturbation theory,
a gaugefixing procedure is selected.
This procedure eliminates antifields
in terms of functionals of fields.
When appropriately implemented,
propagators exist,
and the usual Feynman graph methods can be used.
In addition,
for the study of symmetry properties,
renormalization and anomalies,
a modified version of the
gaugefixing procedure is available
which keeps antifields.
In short,
the antibracket formalism
has manifest gauge invariance or BRST symmetry,
provides the extra fields needed
for covariant quantization,
permits a perturbative expansion
of the quantum theory,
and allows the study of
quantum corrections
to the symmetry structure of the theory.
The fieldantifield formalism can treat systems
that cannot be handled
by FaddeevPopov functional integration approach.
This is particularly clear for theories
in which quartic ghost interactions arise
\ct{kallosh78a,wh79a}.
FaddeevPopov quantization leads to an action
bilinear in ghost fields,
and fails
for the case of open algebras.
An open algebra occurs
when the commutator of two gauge transformations
produces a term proportional to
the equations of motion
and not just another gauge transformation
\ct{wh79a,bv84a}.
In other words,
the gauge algebra closes only onshell.
Such algebras occur in gravity
\ct{fv75b}
and supergravity
\ct{fnf76a,kallosh78a,wh79a,vannieuwenhuizen81a}
theories.
The ordinary FaddeevPopov procedure
also does not work for reducible theories.
In reducible theories, the gauge generators
are all not independent
\ct{cs74a,kr74a,ad79a,bhnw79a,%
siegel80a,townsend80a,vasiliev80a,dts81a,thierrymieg90a}.
Some modifications of the procedure
have been developed by introducing ghosts for ghosts
\ct{siegel80a,kimura81a}.
However, these modifications
\ct{siegel80a,hko81a,kimura81a,fs88a}
do not work for the general reducible theory.
Even for YangMills theories,
the FaddeevPopov procedure can fail,
if one considers exotic gaugefixing procedures
for which ``extraghosts'' appear
\ct{kallosh78a,nielsen78a,nielsen81a}.
The fieldantifield formalism is sufficiently general
to encompass previously known lagrangian
approaches to the quantization of gauge theories.
Perhaps the most attractive feature
of the fieldantifield formalism
is its imitation of a hamiltonian Poisson structure
in a covariant way.
In some instances,
the hamiltonian approach
to quantization has the advantage
of being manifestly unitary.
However, it is necessarily noncovariant
since the time variable
is treated in a manner different
from the space variables.
In addition, the gauge invariances usually must be fixed
at the outset.
In compensation for this,
one needs to impose
constraints on the Hilbert space of states.
In the fieldantifield approach,
the antibracket plays the role of the Poisson bracket.
As a consequence,
hamiltonian concepts,
such as canonical transformations,
can be formulated and used
\ct{vlt82a,vt82a,vt82b,bv84a,fh90a,tnp90a}.
At the same time,
manifest covariance and BRST invariance are maintained.
Since the antibracket formalism proceeds via
the functional integral,
the powerful techniques of functional integration
are available.
A nontrivial aspect of the fieldantifield approach
is the construction of the quantum action $W$.
When loop effects are ignored,
$W \to S$ provides the solution
to the master equation.
A straightforward but not necessarily simple
procedure is available for
obtaining $S$ given the classical action $S_0$
and its gauge invariances
for a finitereducible system.
When quantum effects are incorporated,
$W$ must satisfy
the more singular quantum master equation.
However, there is currently no known method
that guarantees the construction of $W$.
The problem is that
the fieldantifield formalism does not
automatically provide
the functional integration measure.
These issues are linked with
those associated with unitarity,
renormalization, quantum gauge invariance,
and anomalies.
Because these aspects of gauge theories
are inherently difficult,
it is not surprising
that the fieldantifield formalism
does not provide a simple solution.
Another, less serious weakness,
is that the antibracket formalism
involves quite a bit of mathematical machinery.
Sometimes, a gauge theory is expressed
in a form
which is more complicated than necessary.
This can make computations
somewhat more difficult.
The organization of this article is as follows.
Sect.\ \ref{s:ssgt}
discusses gauge structure.
Some notation is presented during the process of
introducing gauge transformations.
The distinction between irreducible
and reducible gauge theories is made.
The latter involve a redundant set of gauge invariances
so that there are relations among the gauge generators.
As a result,
there exists gauge invariances
for gauge invariances,
and ghosts for ghosts.
A theory is $L$thstage reducible
if there are gauge invariances
for the gauge invariances for the gauge invariances, etc.,
$L$fold times.
The general form of the gauge structure
for a firststage reducible case
is determined.
In Sect.\ \ref{s:egt},
specific gauge theories are presented to illustrate
the concepts
of Sect.\ \ref{s:ssgt}.
The spinless relativistic particle,
nonabelian YangMills theories,
topological YangMills theory,
the antisymmetric tensor field,
free abelian $p$form theories,
open bosonic string field theory,
the massless relativistic spinning particle,
and the firstquantized bosonic string
are treated.
The spinless relativistic particle
of Sect.\ \ref{ss:srp}
is also used to exemplify notation.
The massless relativistic spinning particle
provides an example of a simple supergravity theory,
namely a theory
with supersymmetric gauge invariances.
This system is used to illustrate
the construction of
supersymmetric and supergravity theories.
A review of the construction
of generalcoordinateinvariant theories
is given in the subsection
on the firstquantized open bosonic string.
These minireviews should be useful
to the reader who is new to these subjects.
The key concepts of
the fieldantifield formalism are elucidated
in Sect.\ \ref{s:faf}.
Antifields are introduced
and the antibracket is defined.
The latter is used to define canonical transformations.
They can be quite helpful in simplifying computations.
Next, the classical master equation
$(S,S)=0$ is presented.
When appropriate boundary conditions are imposed,
it reproduces, in a compact way,
the gauge structure
of Sect.\ \ref{s:ssgt}.
A suitable action $S$ satisfying the master equation
is called a proper solution.
Given the gaugestructure tensors
of a firststage reducible theory,
Sect.\ \ref{ss:psga}
presents the generic proper solution.
The last part
of Sect.\ \ref{s:faf}
defines and discusses the classical BRST symmetry.
Examples of proper solutions
are provided
in Sect.\ \ref{s:eps}
for the gauge field theories
presented
in Sect.\ \ref{s:egt}.
Sect.\ \ref{s:gff} begins
the passage from the classical to the quantum aspects
of the fieldantifield formalism.
The gaugefixing procedure is discussed.
The gaugefixing fermion $\Psi$ is a key concept.
It is used as a means of eliminating antifields
in terms of functions of fields.
The result
is an action that is suitable
for use in the path integral.
Only in this context
and in performing standard perturbative computations
are antifields eliminated.
It is shown that results are independent
of the choice of $\Psi$,
if the quantum action $W$ satisfies
the quantum master equation.
To implement gaugefixing,
more fields and their antifields
must be introduced.
How this works
for irreducible and firststage reducible theories
is treated first.
Then, for reference purposes,
the general $L$thstage reducible case is considered.
Deltafunction type gaugefixing is treated
in Sect.\ \ref{ss:dfgfp}.
Again, irreducible and firststage reducible cases
are presented first.
Again, for reference purposes,
the general $L$thstage reducible case is treated.
Gaugefixing by a gaussian averaging process
is discussed
in Sect.\ \ref{ss:ogfp}.
After gaugefixing,
a classical gaugefixed BRST symmetry can be defined.
See Sect.\ \ref{ss:gfcbrsts}.
The freedom to perform canonical transformations
permits one to work in any appropriate field basis.
This freedom can be quite useful.
Concepts tend to have different
interpretations in different bases.
One basis, associated with $\Psi$
and called the gaugefixed basis,
is the last topic
of Sect.\ \ref{s:gff}.
Examples of gaugefixing procedures are provided
in Sect.\ \ref{s:gfe}.
With the exception of the free $p$form theory,
the theories are the ones considered
in Sects.\ \ref{s:egt} and \ref{s:eps}.
Quantum effects and possible gauge anomalies
are analyzed in Sect.\ \ref{s:qea}.
The key concepts are quantumBRST transformations
and the quantum master equation.
Techniques for assisting in finding solutions
to the quantum master equation
are provided
in Sects.\ \ref{ss:sqme},
\ref{ss:qmevg} and \ref{ss:ctqme}.
The generating functional $\Gamma$
for oneparticleirreducible diagrams
is generalized to the fieldantifield case
in Sect.\ \ref{ss:eazje}.
This allows one to treat the quantum system
in a manner similar to the classical system.
The ZinnJustin equation is shown
to be equivalent to the quantum master equation.
When unavoidable violations of the latter occur,
the gauge theory is anomalous.
See Sect.\ \ref{ss:qmevg}.
Explicit formulas
at the oneloop level are given.
In Sect.\ \ref{s:sac},
sample anomaly calculations are presented.
It is shown that the spinless relativistic particle
does not have an anomaly.
In Sect.\ \ref{ss:acsm},
the fieldantifield treatment
of the twodimensional chiral Schwinger model
is presented.
Violations of the quantum master equation
are obtained.
This is expected since the theory is anomalous.
A similar computation is performed
for the open bosonic string.
For $D \ne 26$,
the theory is anomalous,
as expected.
Some of the details of the calculations
are relegated to Appendix C.
Section \ref{s:bdot}
briefly presents several additional topics.
The application of the fieldantifield formalism
to global symmetries is presented.
A review is given of the geometric interpretation
of E. Witten \ct{witten90a}.
The next topic is the role of locality.
This somewhat technical issue is important
for renormalizability and for cohomological aspects.
A summary of cohomological methods is given.
Next, the relation
between the hamiltonian and antibracket approaches
is discussed.
The question of unitarity is the subject
of Sect.\ \ref{ss:u}.
One place
where the fieldantifield formalism
has played an essential role
is in the $D=26$ closed bosonic string field theory.
This example is rather complicated
and not suitable for pedagogical purposes.
Nevertheless,
general aspects of the antibracket formalism
for the closed string field theory
are discussed.
Finally,
an overview is given of how
to handle anomalous systems
using an extended set of fields and antifields.
Appendix A reviews the mathematical aspects
of left and right derivatives,
integration by parts,
and chain rules for differentiation.
Appendix B discusses in more detail
the regularity condition,
which is a technical requirement
of the antibracket formalism.
At every stage of development of the formalism,
there exists some type of BRST operator.
In the space of fields and antifields
before quantization,
a classical nilpotent BRST transformation $\delta_B$
is defined by using the action $S$
and the antibracket: $\delta_B F = ( F , S )$.
{}From $\delta_B$,
a gaugefixed version $\delta_{B_\Psi}$
is obtained by imposing the conditions
on antifields
provided by the gaugefixing fermion $\Psi$.
At the quantum level,
a quantum version $\delta_{\hat B}$
of $\delta_B$ emerges.
In the context of the effective action formulation,
a transformation $\delta_{B_{cq}}$,
acting on classical fields, can be defined
by using $\Gamma$ in lieu of $S$.
Several subsections are devoted
to the BRST operator,
its properties and its utility.
The existence of a BRST symmetry
is crucial to the development.
Observables
are those functionals which are BRST invariant
and cannot be expressed as the BRST variation
of something else.
In other words,
observables correspond to the elements
of the BRST cohomology.
The nilpotency of $\delta_B$ and $\delta_{\hat B}$
are respectively
equivalent to the classical and quantum master equations.
The traditional treatment
of gauge theories using BRST invariance
is reviewed in \ct{baulieu85a}.
For this reason,
we do not discuss
BRST quantization in detail.
The antibracket formalism is rather versatile
in that one can use any set of fields (and antifields)
related to the original fields (and antifields)
by a canonical transformation.
However, under such a change,
the meaning of certain concepts change.
For example,
the gauge structure,
as determined by the master equation,
has a different interpretation
in the gaugefixed basis than in the original basis.
Most of this review
uses the second viewpoint.
The treatment in the gaugefixed basis
is handled in Sects.\ \ref{ss:gfb} and \ref{ss:eazje}.
The material in each section
strives to fulfill one of three purposes.
A key purpose
is to present computations
that lead to understanding and insight.
Sections \ref{s:ssgt}, \ref{s:faf},
\ref{ss:g}, \ref{ss:gfcbrsts}, \ref{ss:gfb},
\ref{s:qea} and \ref{s:bdot}
are mainly of this character.
The second purpose
is pedagogical.
This Introduction falls into this category
in that it gives
a quick overview of the formalism
and the important concepts.
Sections \ref{s:egt},
\ref{s:eps}, \ref{s:gfe},
and \ref{s:sac}
analyze specially chosen gauge theories
which
allow the reader
to understand the fieldantifield formalism
in a concrete manner.
Finally,
some material is included for technical completeness.
Sections \ref{ss:gfaf}\ref{ss:ogfp}
present methods
for gaugefixing
the generic gauge theory.
Parts of sections \ref{ss:irgt},
\ref{ss:gs} and \ref{ss:aoll}
are also for reference purposes.
Probably the reader should not initially
try to read these sections in detail.
Many sections serve a dual role.
A few new results on the antibracket formalism
are presented in this review.
They are included
because they provide insight for the reader.
We have tried
to have a minimum overlap
with other reviews.
In particular,
cohomological aspects
are covered in
\ct{brs74a,dixon76a,bc83a,baulieu85a,dtv85a,band86a,%
henneaux90a,ht92a,tpbook}
and moredetailed aspects of anomalies
are treated
in \ct{tpbook}.
Pedagogical treatments are given in
references \ct{ht92a,tpbook}.
In certain places,
material from
reference \ct{paris92a}
has been used.
This review
focuses on the key points and concepts
of antibracket formalism.
There is some emphasis
on applications to string theory.
Our format is to first present the material abstractly
and then to supply examples.
The reader who is new to this subject
and mainly interested in learning
may wish to reverse this order.
Exercises can be generated
by verifying the abstract results
in each of the sample gauge theories
of Sects.\ \ref{s:egt},
\ref{s:eps}, \ref{s:gfe},
and \ref{s:sac}.
Other systems,
which have been treated
by fieldantifield quantization
and may be of use to the reader,
are
the free spin $\frac52$ field
\ct{bv83b},
the spinning string
\ct{gpr90a},
the $10$dimensional
BrinkSchwarz superparticle and superstring
\ct{gglrsnv89a,gh89a,kallosh89a,lrsnv89a,%
rsnv89a,bk90a,siegel90a,bkp92a,sezgin93a}%
{\footnote{
See
\ct{sezgin93a}
for additional references.
}},
chiral gravity
\ct{jst93a},
$W_3$ gravity
\ct{hull91a,bss92a,bg93a,hull93a,dayi94a,vp94a},
general topological field theories
\ct{bms88a,lp88a,brt89a,bbrt91a,getzler92a,%
ikemori93a,lps93a,horava94a,jv94a},
the supersymmetric WessZumino model
\ct{bbow90a,hlw90a}
and
chiral gauge theories in fourdimensions
\ct{tnp90a}.
The antibracket formalism has found
various interpretations
in mathematics
\ct{gerstenhaber62a,gerstenhaber64a,getzler92a,ps92a,%
lz93a,nersesian93a,schwarz93a,schwarz93b,stasheff93a}.
Some other recent relevant work
can be found in
\ct{verlinde92a,bt93a,bt93b,bcov93a,hata93a}.
The referencing in this review
is thorough but not complete.
A restriction has been made
to only cite works directly relevant
to the issues addressed in each section.
Multiple references are done first chronologically
and then alphabetically.
The titles of references are provided
to give the reader a better indication
of the content of each work.
We work in Minkowski space throughout this article.
Functional integrals are defined
by analytic continuation using Wick rotation.
This is illustrated in the computations
of Appendix C.
We use $\eta_{\mu \nu}$ to denote the flatspace metric
with the signature convention
$ ( 1, 1, 1, \dots , 1 )$.
Flatspace indices are raised and lowered with this metric.
The epsilon tensor
$\varepsilon_{\mu_0 \mu_1 \mu_2 \dots \mu_{d1} }$
is determined by the requirement that it be
antisymmetric in all indices
and that $\varepsilon_{0 1 2 \dots {d1} } = 1$,
where $d1$ and $d$ are respectively
the dimension of space and spacetime.
We often use square brackets to indicate a functional
of fields and antifields to avoid confusion
with the antibracket, i.e.,
$S [ \Phi , \Phi^* ]$ in lieu of $S ( \Phi , \Phi^* )$.
\vfill\eject
\section{Structure of the Set of Gauge Transformations}
\label{s:ssgt}
\hspace{\parindent}
The most familiar example of a gauge structure
is the one associated
with a nonabelian YangMills theory
\ct{ym54a},
namely a Lie group.
The commutator of two Liealgebra generators
produces a Liealgebra generator.
When a basis is used,
this commutator algebra is determined
by the structure constants of the Lie group.
For example, for the Lie algebra $su(2)$
there are three generators
and the structure constant is the antisymmetric tensor
on three indices $\varepsilon^{\alpha \beta \gamma}$.
A commutator algebra,
as determined by a set of abstract structure constants,
does not necessarily lead to a Lie algebra.
The Jacobi identity,
which expresses the associativity of the algebra,
must be satisfied
\ct{vannieuwenhuizen81a}.
Sometimes, in more complicated field theories,
the transformation rules involve
fielddependent structure constants.
Such cases are sometimes
referred to as ``soft algebras''
\ct{batalin81a,sohnius83a}.
In such a situation,
the determination of the gauge algebra is more
complicated than in the YangMills case.
The Jacobi identity must be appropriately generalized
\ct{batalin81a,bv81a,dewitt84a}.
Furthermore, new structure tensors
beyond commutator structure constants may appear
and new identities need to be satisfied.
In other types of theories,
the generators of gauge transformations
are not independent.
This occurs when there is ``a gauge invariance''
for gauge transformations.
One says the system is {\it reducible}.
A simple example is a theory constructed using
a threeform $F$ which is expressed
in terms of a twoform $B$
by applying the exterior derivative $F = dB$.
The gauge invariances are
given by the transformation rule $\delta B = dA$
for any oneform $A$.
The theory is invariant under such transformations
because the lagrangian is a functional of $F$ and
$F$ is invariant: $\delta F = d \delta B = ddA = 0$.
However, the gauge invariances are not all independent
since modifying $A$ by $\delta A = d \lambda$ for some
zeroform $\lambda$ leads to no change
in the transformation for $B$.
When $A = d \lambda$, $\delta B = d A = d d \lambda = 0$.
The structure of a gauge theory is more complicated
than the YangMills case
when there are gauge invariances for gauge transformations.
Another complication occurs
when the commutator of two gauge transformations
produces a term that vanishes onshell,
i.e., when the equations of motion are used.
When equations of motion appear in the gauge algebra,
how should one proceed?
In this section we discuss the abovementioned complications
for a generic gauge theory.
The questions are
(i) what are the relevant
gaugestructure tensors and
(ii) what equations
do they need to satisfy.
The answers to these questions
lead us to the gauge structure of a theory.
This section constitutes
a somewhat technical but necessary prelude.
A reader might want to consult the examples
in Sect.\ \ref{s:egt}.
The more interesting development
of the fieldantifield formalism
begins in Sect.\ \ref{s:faf}.
\subsection{Gauge Transformations}
\label{ss:gt}
\hspace{\parindent}
This subsection introduces the notions
of a gauge theory and a gauge transformation.
It also defines notation.
The antibracket approach employs
an elaborate mathematical formalism.
Hence, one should try to become quickly
familiar with notation and conventions.
Consider a system whose dynamics is governed by
a classical action $S_0 [ \phi ]$,
which depends on $n$ different fields $\phi^i(x)$, $i=1,\cdots,n$.
The index $i$ can label spacetime indices $\mu$, $\nu$
of tensor fields,
the spinor indices of fermion fields,
and/or an index distinguishing different types of generic fields.
At the classical level, the fields are functions of spacetime.
In the quantum system, they are promoted to operators.
In this section, we treat the classical case only.
Let $\epsilon (\phi^i) = \epsilon_i$ denote
the statistical parity of $\phi^i$.
Each $\phi^i$ is either a commuting field ($\epsilon_i = 0$) or
an anticommuting field ($\epsilon_i = 1$).
One has
$\phi^i (x) \phi^j (y) =
(1)^{\epsilon_i \epsilon_j } \phi^j (y) \phi^i (x) $.
Let us assume that the action is invariant
under a set of $m_0$ ($m_0\leq n$)
nontrivial gauge transformations,
which, when written in infinitesimal form, read
\be
\delta\phi^i (x) =
\left( R^i_{\alpha} (\phi)\varepsilon^{\alpha} \right) (x) \quad ,
\quad {\rm where}
\ \alpha=1 \ {\rm or} \ 2 \ \ldots \ {\rm or} \ m_0 \quad .
\label{trans gauge}
\ee
Here, $\varepsilon^{\alpha} (x)$
are infinitesimal gauge parameters,
that is, arbitrary functions of the spacetime variable $x$,
and $R^i_{\alpha}$ are
the generators of gauge transformations.
These generators are operators
that act on the gauge parameters.
In kernel form,
$\left( R^i_{\alpha} (\phi)\varepsilon^{\alpha} \right) (x)$
can be represented as
$\int {\dif y R_{\alpha }^i\left( {x,y} \right)
\varepsilon ^{\alpha }\left( y \right)}$.
It is convenient to adopt
the following compact notation
\ct{dewitt64a,dewitt67a}.
Unless otherwise stated, the appearance of a discrete index
also indicates the presence of a spacetime variable.
We then use a generalized summation convention in which
a repeated discrete index implies not only a sum over that index
but also an integration over
the corresponding spacetime variable.
As a simple example,
consider the multiplication of two matrices $g$ and $h$,
written with explicit matrix indices.
In compact notation,
\be
{f^A}_B = {g^A}_C {h^C}_B
%\quad ,
\label{gsc1}
\ee
becomes not only a matrix product in index space
but also in function space.
Eq.\bref{gsc1} represents
\be
{f^A}_B \left( { x , y } \right) = \sum_C \int \dif z \,
{g^A}_C \left( { x , z } \right)
{h^C}_B \left( { z , y } \right)
%\quad ,
\label{gsc2}
\ee
in conventional notation.
In other words, the index $A$
in Eq.\bref{gsc1}
stands for $A$ and $x$
in Eq.\bref{gsc2}.
Likewise, $B$ and $C$
in Eq.\bref{gsc1}
represent
$\{ B, y \}$ and $\{ C, z \}$.
The generalized summation convention for $C$
in compact notation
yields a sum over the discrete index $C$
and an integration over $z$
in conventional notation
in Eq.\bref{gsc2}.
The indices $A$, $B$ and $C$
in compact notation
implicitly represent spacetime variables $x$, $y$, $z$, etc.,
and explicitly can be
field indices $i, j, k ,$ etc.,
gauge index $\alpha, \beta , \gamma ,$ etc.,
or any other discrete index
in the formalism.
With this convention, the transformation laws
\be
\delta\phi^i (x) =
\sum\limits_{\alpha }
\int {\dif y R_{\alpha }^i\left( { x,y } \right)
\varepsilon ^{\alpha }\left( y \right)}
%\quad ,
\label{transformation rule long}
\ee
can be written succinctly as
\be
\delta\phi^i=R^i_{\alpha} \varepsilon^{\alpha} \quad .
\label{transformation rule}
\ee
The index $\alpha$ in Eq.\bref{transformation rule} corresponds
to the indices $y$ and $\alpha$ in Eq.\bref{transformation rule long}.
The index $i$ in Eq.\bref{transformation rule} corresponds
to the indices $x$ and $i$ in Eq.\bref{transformation rule long}.
The compact notation is illustrated
in the example of
Sect.\ \ref{ss:srp}.
Although this notation might seem confusing at first,
it is used extensively in the antibracket formalism.
In the next few paragraphs,
we present equations in both notations.
Each gauge parameter
$\varepsilon^{\alpha}$ is either commuting,
$\epsilon (\varepsilon^{\alpha}) \equiv \epsilon_{\alpha}= 0$,
or is anticommuting, $\epsilon_{\alpha}= 1$.
The former case corresponds to an ordinary symmetry
while the latter is a supersymmetry.
The statistical parity of $R^i_{\alpha}$,
$\epsilon (R^i_{\alpha})$, is determined from Eq.\bref{trans gauge}:
$\epsilon (R^i_{\alpha}) =
\left( \epsilon_i + \epsilon_{\alpha} \right)
\ \ ({\rm mod } \ 2)$.
Let $S_{0,i} \left( { \phi , x} \right) $
denote the variation of the action
with respect to $\phi^i (x)$:
\be
S_{0,i}\left( { \phi , x} \right) \equiv
\rder{S_0 [ \phi ] }{\phi^i (x) } \quad ,
\label{def s0i}
\ee
where the subscript $r$
indicates that the derivative is
to be taken from the right (see Appendix A).
Henceforth, when a subscript index $i$, $j$, etc.,
appears after a comma
it denotes the right derivative
with respect to the corresponding field $\phi^i$, $\phi^j$, etc..
In compact notation, we write Eq.\bref{def s0i}
as $S_{0,i} = \rder{S_0}{\phi^i} $ where the index $i$ here
stands for both $x$ and $i$ in Eq.\bref{def s0i}.
The statement that the action is invariant
under the gauge transformation in Eq.\bref{trans gauge} means that
the Noether identities
\be
\int {\dif x} \sum\limits_{i=1}^n S_{0,i}
\left( { x} \right)
R^i_{\alpha}\left( { x,y } \right) = 0
%\quad ,
\label{ident noether long form}
\ee
hold, or equivalently, in compact notation
\be
S_{0,i} R^i_{\alpha} =0 \quad .
\label{ident noether}
\ee
Eq.\bref{ident noether} is derived by varying $S_0$
with respect to right variations
of the $\phi^i$ given by Eq.\bref{trans gauge}.
When using right derivatives,
the variation $\delta S_0$ of $S_0$,
or of any other object,
is given by
$
\delta S_0 = S_{0,i} \delta \phi^i
$.
If one were to use left derivatives,
the variation of $S_0$ would read
$
\delta S_0 =
\delta \phi^i { {\partial_l S_{0} } \over {\partial \phi^i} }
$.
Eq.\bref{ident noether long form}
is sometimes zero because the integrand
is a total derivative.
We assume that surface terms can be dropped in such integrals 
this is indeed the case when
Eq.\bref{ident noether long form} is applied
to gauge parameters that fall off sufficiently fast
at spatial and temporal infinity.
The Noether identities in Eq.\bref{ident noether}
are the key equations of this subsection
and can even be thought of
as the definition of when a theory is invariant
under a gauge transformation
of the form in Eq.\bref{trans gauge}.
To commence perturbation theory, one searches for solutions
to the classical equations of motion,
$S_{0,i}\left( { \phi , x} \right)=0$,
and then expands about these solutions.
We assume there exists at least one such
stationary point $\phi_0 = \{ \phi^j_0 \}$ so that
\be
\restric{S_{0,i}}{\phi_0} = 0 \quad .
\label{saddle point}
\ee
Equation \bref{saddle point}
defines a surface $\Sigma$
in function space,
which is infinite dimensional
when gauge symmetries are present.
As a consequence of the Noether identities,
the equations of motion are not independent.
Furthermore, new saddle point solutions
can be obtained by performing
gauge transformations on any particular solution.
These new solutions should not be regarded as representing
new physics however 
fields related by local gauge transformations
are considered equivalent.
The Noether identities also imply that propagators do not exist.
By differentiating the identities from the left
with respect $\phi^j$,
one obtains
$$
\frac{\partial_l}{\partial\phi^i}
\left (S_{0,j} R^j_{\alpha}\right ) =
\left(\frac{\partial_l \partial_r S_0}
{\partial\phi^i \partial\phi^j}\right) R^j_{\alpha} +
S_{0,j} \frac{\partial_l
R^j_{\alpha}}{\partial\phi^i}
(1)^{\epsilon_i \epsilon_j } = 0
\ ,
$$
\be
\Rightarrow\restric{
\left(\frac{\partial_l \partial_r S_0}
{\partial\phi^i \partial\phi^j}\right) R^j_{\alpha}}
{\phi_0} = 0
\quad ,
\label{hessian degeneracy}
\ee
i.e., the hessian
$\left(\frac{\partial_l \partial_r S_0}
{\partial\phi^i \partial\phi^j}\right)
$
of $S_0$ is degenerate
at any point on the stationary surface $\Sigma$.
The $R^i_\alpha$ are onshell null vectors of this hessian.
Since propagators involve the inverse of this hessian,
propagators do not exist for certain combinations of fields.
This means that the standard loop expansion cannot be
straightforwardly applied.
A method is required to overcome this problem.
Technically speaking,
to study the structure of the set of gauge transformations
it is necessary to assume certain {\it regularity conditions}
on the space for which the equations of motion
$S_{0,i} = 0 $ hold.
The interested reader can find these conditions in Appendix B.
A key consequence of the regularity conditions is that
if a function $F(\phi)$ of the fields $\phi$
vanishes onshell,
that is, when the equations of motion are implemented,
then $F$ must be a linear combination of the equations of motion,
i.e.,
\be
\restric{F( \phi )}{ \Sigma } = 0 \Rightarrow
F(\phi ) = S_{0,i} \lambda^i (\phi )
\quad ,
\label{consequence of rc}
\ee
where
$\restric{}{\Sigma}$
indicates the restriction to the surface
where the equations of motion hold
\ct{wh79a,bv85a,fhst89a,fisch90a,fh90a}.
Eq.\bref{consequence of rc} can be thought of
as a completeness relation
for the equations of motion.
We shall make use of Eq.\bref{consequence of rc} frequently.
Throughout Sect.\ \ref{s:ssgt},
we assume that the gauge generators are
fixed once and for all.
One could take linear combinations of the generators
to form a new set.
This would change the gaugestructure tensors
presented below.
This nonuniqueness is not essential
and is discussed
in Sect.\ \ref{ss:eu}.
To see explicit examples
of the abstract formalism that follows,
one may want to glance
from time to time
at the examples
of Sect.\ \ref{s:egt}.
\subsection{Irreducible and Reducible Gauge Theories}
\label{ss:irgt}
\hspace{\parindent}
It is important to know any dependences
among the gauge generators.
Only with this knowledge is possible
to determine the independent degrees of freedom.
The purpose of this subsection
is to analyze this issue in more detail
for the generic case.
The simplest gauge theories,
for which all gauge transformations
are independent, are called {\it irreducible}.
When dependences exist,
the theory is {\it reducible}.
In reducible gauge theories,
there is a ``kind of gauge invariance for gauge transformations''
or what one might call ``levelone'' gauge invariances.
If the levelone gauge transformations
are independent,
then the theory is called {\it firststage reducible}.
This may not happen.
Then, there are ``leveltwo'' gauge invariances, i.e.,
gauge invariances for the levelone gauge invariances
and so on.
This leads to the concept
of an {\it $L$th stage reducible theory}.
In what follows we let $m_s$ denote
the number of gauge generators at the $s$th stage
regardless of whether they are independent.
Let us define more precisely the above concepts.
Assume that all gauge invariances of a theory are known
and that the regularity condition described in Appendix B
is satisfied.
Then, the most general solution to the
Noether identities \bref{ident noether}
is a gauge transformation, up to terms proportional
to the equations of motion:
\be
S_{0,i} \lambda^i = 0 \Leftrightarrow
\lambda^i = R^i_{0\alpha_0} \lambda^{\prime \alpha_0} +
S_{0,j} T^{ji}
\quad ,
\label{completesa}
\ee
where $T^{ij}$ must satisfy the graded symmetry property
\be
T^{ij} = (1)^{\eps_i \eps_j } T^{ji}
\quad .
\label{symmetry of Tij}
\ee
The $R^i_{0\alpha_0}$ are the gauge generators
in Eq.\bref{trans gauge}.
For notational convenience,
we have appended a subscript $0$ on the gauge generator
and the gauge index $\alpha$.
This subscript indicates
the level of the gauge transformation.
The second term $S_{0,j}T^{ji}$ in Eq.\bref{completesa}
is known as a trivial gauge transformation.
Such transformations are discussed in the next subsection.
It is easily checked that the action is invariant under
such transformations due to the trivial commuting or
anticommuting properties of the $S_{0,j}$.
The first term $R^i_{0\alpha_0} \lambda^{\prime \alpha_0}$
in Eq.\bref{completesa} is similar
to a nontrivial gauge transformation
of the form of Eq.\bref{trans gauge} with
$\varepsilon^{\alpha_0} = \lambda^{\prime \alpha_0}$.
The key assumption in Eq.\bref{completesa}
is that the set of functionals $R^i_{0\alpha_0}$
exhausts onshell the relations among the equations of motion,
namely the Noether identities.
In other words, the gauge generators
are onshell a complete set.
This is essentially equivalent
to the regularity condition.
If the functionals $R^i_{0\alpha_0}$
are independent onshell
then the theory is {\it irreducible}.
In such a case,
\be
\restric{{\rm rank}\; R^i_{0\alpha_0}}{\Sigma} = m_0
\quad ,
\label{indp generadors}
\ee
where $m_0$ is the number of gauge transformations.
The rank of the hessian
\be
\mbox{\rm rank}\restric{
\left(\frac{\partial_l \partial_r S_0}
{\partial\phi^i \partial\phi^j}\right)} {\Sigma}=
n\mbox{\rm rank}\restric{R^i_\alpha}{\Sigma}
%\quad ,
\label{rank hessian}
\ee
is
$ n  m_0 $.
Define the
net number of degrees of freedom
$n_{\rm dof}$ to be the number of fields
that enter dynamically in $S_0$,
regardless of whether they propagate.%
{\footnote{
In electromagnetism,
$n_{\rm dof} =3$,
but there are only two propagating
degrees of freedom
corresponding to the two physical polarizations.}}
Then for an irreducible theory
$n_{\rm dof}$ is $n  m_0$
since there are $m_0$ gauge degrees of freedom.
Note that $n_{\rm dof}$ matches the rank of the hessian
in Eq.\bref{rank hessian}.
If, however, there are dependences among the gauge
generators, and the rank of the generators
is less than their number,
$\restric{{\rm rank}\; R^i_{0\alpha_0}}{\Sigma}^J}
\ee
Note that $\Phi_c^A$ is a functional of
$J_B$ and $\Phi_{cB}^*$.
In principle,
it is possible to invert this relation
to determine $J_A$ in terms of
$\Phi_c^B$ and $\Phi_{cB}^*$.
We indicate the solution to this inversion
by $J_{cA}$:
$
J_{cA} = J_A\left[ {\Phi_c,\Phi_c^*} \right]
$.
The effective action
is obtained by a Legendre transformation
\ct{iz80a}:
\be
\Gamma \left[ {\Phi_c,\Phi_c^*} \right] \equiv
i\hbar\ln Z_c 
J_{cA} \Phi_c^A
\quad ,
\label{def of effective action}
\ee
where $Z_c$ is $Z \left[ { J, \Phi_c^* } \right] $
evaluated at $J_A = J_{cA}$:
\be
Z_c \equiv Z\left[ {J_c,\Phi_c^*} \right]
\quad .
\label{def of Zc}
\ee
As a result, $Z_c$ is a functional
of classical fields and antifields.
A straightforward calculation of
${{\partial_r \Gamma } \over {\partial \Phi_c^A}}$ gives
$$
{{\partial_r \Gamma } \over {\partial \Phi_c^A}} =
 J_{cA}  \left( {1} \right)^{\eps_A\eps_B}
{{\partial_r J_{cB}} \over {\partial \Phi_c^A}}
\Phi_c^B 
\left. {i\hbar{{\partial_r \ln Z
\left[ {J,\Phi_c^*} \right] } \over {\partial J_B}}
{{\partial_r J_{cB} } \over {\partial \Phi_c^A}}} \right_{J=J_c}
\quad .
$$
Using Eq.\bref{def of classical field},
one finds that the last two terms cancel
so that
\be
{{\partial_r \Gamma } \over {\partial \Phi_c^A}} =
 J_{cA}
\quad .
\label{recovery of Jc}
\ee
At this stage,
we have made a transition from fields
to classical fields.
Essentially all quantities are now
functionals of $\Phi_c^A$ and $\Phi_{cA}^*$.
Given two functionals $X$ and $Y$
of $\Phi_c^A$ and $\Phi_{cA}^*$,
define the classical antibracket
$\left( { \ , \ } \right)_c $ by
\be
\left( {X,Y} \right)_c \equiv
{{\partial_r X} \over {\partial \Phi_c^A}}
{{\partial_l Y} \over {\partial \Phi_{cA}^*}} 
{{\partial_r X} \over {\partial \Phi_{cA}^*}}
{{\partial_l Y} \over {\partial \Phi_c^A}}
\quad .
\label{def of classical antibracket}
\ee
Since the classical antibracket
is defined in the same manner as the antibracket,
it satisfies the same identities
\bref{antibracket properties}  \bref{bracket derivation}.
Using Eq.\bref{recovery of Jc},
one obtains
\be
{1 \over 2}\left( {\Gamma ,\Gamma } \right)_c =
J_{cA}{{\partial_l \Gamma } \over {\partial \Phi_{cA}^*}} =
\left\langle {i\hbar\Delta W +
{1 \over 2}\left( {W,W} \right)} \right\rangle_c
\quad ,
\label{anomalous ZJ equation}
\ee
where the final equality is obtained
after some algebra, which makes use of
integration by parts.
In Eq.\bref{anomalous ZJ equation},
$ \left\langle \ \right\rangle_c $
denotes the expectation value in the presence of $J$
but expressed in terms of $\Phi_c$ and $\Phi_c^*$.
More precisely,
if $X$ is a functional of $\Phi$ and $\Phi^*$
then
\be
\left\langle X \right\rangle_c \equiv
\left. {\left\langle X \right\rangle^J} \right_{J=J_c}
\quad ,
\label{def of <>_c}
\ee
where $\left\langle \ \right\rangle^J$ is defined
in Eq.\bref{def <>^J}.
Because of the quantum master equation
\bref{qme2},
Eq.\bref{anomalous ZJ equation} becomes
\be
\left( {\Gamma ,\Gamma } \right)_c = 0
\quad ,
\label{ZJ equation}
\ee
a result known as the ZinnJustin equation
\ct{zinnjustin75a}.
Equation \bref{def of <>_c}
allows one to pass from a functional $X$
of the original fields $\Phi$ and $\Phi^*$
to a classical functional
$\left\langle X \right\rangle_c$
of classical fields
$\Phi_c$ and $\Phi_c^*$
by taking the ``classical expectation''
of $X$:
$
X\left[ {\Phi ,\Phi^*} \right] \to
\left\langle X \right\rangle_c
$.
We refer to $\left\langle X \right\rangle_c $
as the {\it classical version}
of the quantum functional $X$.
The definition is consistent with
the notation for $\Phi_c^A$ since
\be
\left\langle \Phi^A \right\rangle_c
= \Phi_c^A
\quad .
\label{consistent notation for Phi_c}
\ee
The process $X \to \left\langle X \right\rangle_c$
conforms to the idea
that a classical variable
is the expectation value
of the corresponding quantum functional.
Since $\Gamma$ satisfies the ZinnJustin equation
and plays the role of $S$ in the classical antibracket formalism,
one can construct by analogy
a ``classicalquantum'' BRST transformation $\delta_{B_{cq}}$
having the same properties as $\delta_B$.
Define
\be
\delta_{B_{cq}} X \equiv
\left( { X , \Gamma } \right)_c
\quad ,
\label{cq BRST}
\ee
where $X$ is any functional of $\Phi_c$ and $\Phi_c^*$.
The nilpotency $\delta_{B_{cq}}^2 = 0 $ of $\delta_{B_{cq}}$
follows from the ZinnJustin equation.
Define $X$ to be cqBRST invariant if $\delta_{B_{cq}} X = 0$.
According to Eq.\bref{ZJ equation},
the effective action $\Gamma$ is cqBRST invariant since
\be
\delta_{B_{cq}} \Gamma =
\left( \Gamma , \Gamma \right)_c = 0
\quad .
\label{ZJ equation2}
\ee
Several of the abovementioned classical functionals
are also cqBRST invariant.
A analysis of $ \delta_{B_{cq}}J_{cA} $ reveals that
\be
\delta_{B_{cq}} J_{cA} \equiv
\left( {J_{cA}, \Gamma } \right)_c =
{1 \over 2}\left( {1} \right)^{\eps_A+1}
{{\partial_r \left( {\Gamma ,\Gamma } \right)_c}
\over {\partial \Phi_c^A}}
= 0
\quad .
\label{cq BRST invariance of Jc}
\ee
Consider
$$
i\hbar\delta_{B_{cq}} \ln \left( {Z_c} \right) =
\left( {\Gamma +J_{cA}\Phi_c^A,\Gamma } \right)_c =
J_{cA}\left( {\Phi_c^A,\Gamma } \right)_c =
J_{cA}{{\partial_l \Gamma } \over {\partial \Phi_{cA}^*}} = 0
\quad ,
$$
where the first equality holds
because of
Eqs.\bref{ZJ equation2}
and \bref{cq BRST invariance of Jc},
and
the last equality follows
from Eqs.\bref{qme2} and \bref{anomalous ZJ equation}.
Hence,
\be
\delta_{B_{cq}}Z_c =
\left( {Z_c,\Gamma } \right)_c = 0
\quad .
\label{cq BRST invariance of Zc}
\ee
The cqBRST operator $\delta_{B_{cq}}$
is the effective classical version
of the quantumBRST operator $\delta_{\hat B} X$.
To understand this statement,
consider
$$
\delta_{B_{cq}}
\left( {Z_c\left\langle X \right\rangle_c} \right) =
\left( {Z_c\left\langle X \right\rangle_c,\Gamma } \right)_c =
{{\partial_r \left( {Z_c\left\langle X \right\rangle_c} \right)}
\over {\partial \Phi_c^A}}
{{\partial_l \Gamma } \over {\partial \Phi_{cA}^*}} 
{{\partial_r \left( {Z_c\left\langle X \right\rangle_c} \right)}
\over {\partial \Phi_{cA}^*}}
{{\partial_l \Gamma } \over {\partial \Phi_c^A}}
$$
$$
= {{\partial_r \left( {Z \left\langle X \right\rangle^J} \right)}
\over {\partial J_{cB}}}\left( {J_{cB},\Gamma } \right)_c +
\left( {1} \right)^{\eps_A} Z_c
\left\langle {\left( {{{\partial_r X}
\over {\partial \Phi_A^*}} +
{i \over {\hbar}}X{{\partial_r W}
\over {\partial \Phi_A^*}}} \right)J_A} \right\rangle_c
\quad .
$$
The last step follows because
$ \left\langle { \ } \right\rangle_c $
has dependence on
$\Phi_{c}$ and $\Phi_{c}^*$ through $J_{c}$.
After some algebra which makes use of integration by parts,
one finds that
\be
\delta_{B_{cq}}
\left( {Z_c \left\langle X \right\rangle_c} \right)
= Z_c \left\langle {\delta_{\hat B}X +
X \left( {\Delta W +
\frac{i}{2 \hbar}
\left( {W,W} \right)} \right)} \right\rangle_c +
(1)^{\eps_B} Z_c \langle X \Phi_c^B
\rangle \left( { J_{cB} , \Gamma } \right)_c
\quad .
\label{anomalous cq BRST qBRST identity}
\ee
If
$
\delta_{B_{cq}}
\left( { \left\langle X \right\rangle_c } \right)
$
is computed instead,
one obtains the connected part
of Eq.\bref{anomalous cq BRST qBRST identity}
without a $Z_c$ factor.
Since Eq.\bref{cq BRST invariance of Jc}
and the quantum master equation
\bref{qme2} are satisfied,
\be
\delta_{B_{cq}}
\left( { \left\langle X \right\rangle_c } \right) =
\left\langle { \delta_{\hat B} X} \right\rangle_c
\quad .
\label{cq BRST qBRST identity}
\ee
In other words,
the cqBRST variation of the classical version of $X$
is the classical version of the quantumBRST variation of $X$
\ct{anselmi93a}.
The effective action $\Gamma$
in the classical antibracket formalism
plays a role analogous to the proper solution $S$
in the ordinary antibracket formalism.
Properties obeyed by $\Gamma$
are the same as those obeyed by $S$.
Therefore,
one can define a BRST structure
associated with $\Gamma$
\ct{gp93c}.
The BRST structure tensors are encoded in $\Gamma$
and the relations among them
are given by $( \Gamma , \Gamma ) = 0$.
Expanding in a Taylor series in $\Phi_c^*$,
one has
\be
\Gamma \left[ { \Phi_c ,\Phi_c^* } \right] =
\Gamma_0 (\Phi_c ) + \Phi_{cA}^* \Gamma^A (\Phi_c )
+ \frac12 \Phi_{cA}^* \Phi_{cB}^* \Gamma^{BA} (\Phi_c )
+ \ldots
\quad .
\label{Gamma antifield expansion}
\ee
Recalling that there
is an undisplayed dependence
on the gaugefixing fermion $\Psi$,
the above terms have the following interpretation:
$\Gamma_0 (\Phi_c )$
is the oneparticleirreducible
generating functional
for the basic fields including all loop corrections
for the action gaugefixed using $\Psi$, i.e.,
$S_\Psi$ in Eq. \bref{S sub Psi},
and
$\Gamma^A (\Phi_c )$
is the generator of gaugefixed cqBRST transformations.
The gaugefixed cqBRST operator $\delta_{B_{cq \Psi} }$,
the analogy of $\delta_{B_{\Psi} }$,
is defined by
\be
\delta_{B_{cq \Psi}} X \equiv
\left. {\left( {X, \Gamma } \right)} \right_{\Sigma_\Psi }
\quad ,
\label{gaugefixed cqBRST}
\ee
where $X$ is a functional of the $\Phi_c^A$ only.
In the gaugefixed basis,
$\Sigma_\Psi$
in Eq.\bref{gaugefixed cqBRST}
means that classical antifields
are set to zero.
Thus, one has $\delta_{B_{cq \Psi}} \Phi_c^A = \Gamma^A (\Phi_c )$.
The tensor $\Gamma^{BA} (\Phi_c )$
is related to the onshell nilpotency
of $\delta_{B_{cq \Psi}}$.
In summary,
the quantum aspects of the classical theory described by $S$
are reproduced by an effective classical theory
governed by $\Gamma$.
\subsection{Quantum Master Equation Violations: Generalities}
\label{ss:qmevg}
\hspace{\parindent}
Suppose Eq.\bref{qme2} is not zero.
Let
\be
{\cal A} \equiv
\Delta W + { i \over {2\hbar }} \left( {W,W} \right)
%\quad ,
\label{qme anomaly}
\ee
be the violation of the quantum master equation.
A straightforward computation
using
Eqs.\bref{antibracket properties},
\bref{Delta properties} and \bref{qBRST},
but not assuming the validity of the quantum master equation,
reveals that
\be
\delta_{\hat B}{\cal A} = 0
\quad .
\label{full consistency conditions}
\ee
This equation is consistent with
Eqs.\bref{anomalous ZJ equation},
\bref{def of <>_c} and \bref{cq BRST qBRST identity}
and the Jacobi identity for the antibracket:
One has
$
\frac12 ( \Gamma,\Gamma )_c =
$
$
i \hbar \langle {\cal A} \rangle_c
$
so that
$
0 = \frac{i}{2 \hbar} ( (\Gamma,\Gamma)_c , \Gamma )_c
$
$
= ( \langle {\cal A} \rangle_c , \Gamma )_c
$
$
= \delta_{B_{cq}} \langle {\cal A} \rangle_c
$
$
= \langle \delta_{\hat B} {\cal A} \rangle_c
$.
Suppose that
\be
i {\cal A} = \delta_{\hat B}\Omega +
{{\hbar } \over 2}\left( {\Omega ,\Omega } \right) =
\left( {\Omega ,W} \right)  i\hbar \Delta \Omega +
{{\hbar } \over 2} \left( {\Omega ,\Omega } \right)
\quad ,
\label{fake anomaly}
\ee
where $\Omega$
is a local functional of the fields and antifields.
The last term may seem surprising
but it is necessary if ${\cal A}$ is to satisfy
Eq.\bref{full consistency conditions}:
When the quantum master equation is violated
as in Eq.\bref{qme anomaly},
the nilpotency of the quantum BRST operator
no longer holds,
as can be seen
from Eq.\bref{nil of qBRST}.
The last term in Eq.\bref{fake anomaly}
is required to compensate for this effect
and ensures
Eq.\bref{full consistency conditions}.
Let
\be
W^{'} = W + \hbar \Omega
\quad .
\label{remove anomaly}
\ee
Then,
using Eqs.\bref{antibracket properties} and
\bref{Delta properties},
one finds that $W^{'}$ satisfies
the quantum master equation:
$$
{1 \over 2}\left( {W^{'} , W^{'}} \right) 
i\hbar \Delta W^{'} = 0
\quad .
$$
Since $W^{'}$ has the same classical limit as $W$, namely,
its order
$ \hbar^0 $ term is $S$,
one can use $W^{'}$ in lieu of $W$ for the quantum action.
Then, since the quantum master equation is satisfied,
a quantum gauge theory can be defined.
When ${\cal A}$ cannot be expressed as
in Eq.\bref{fake anomaly}
for a local functional $\Omega$,
there is an anomaly in the quantum master equation
and an obstruction to maintaining
gauge symmetries at the quantum level.
Since anomalies involve subtleties
and singular expressions,
they are usually not too easy to compute.
The usual approach to this subject uses
a loop expansion:
\be
{\cal A} =
\sum\limits_{l=1}^\infty {}{\cal A}_l\hbar^{l1} =
{\cal A}_1 + \hbar {\cal A}_2 + \ldots
\quad .
\label{hbar expansion of anomaly}
\ee
Using Eqs.\bref{hbar expansion of W} and
\bref{hbar expansion of anomaly},
one finds the following expression
for the anomaly at the oneloop level
\be
{\cal A}_1 \equiv \Delta S + i \left( {M_1,S} \right)
\quad .
\label{oneloop anomaly}
\ee
To this order,
the condition for the absence of an anomaly
in Eq.\bref{fake anomaly}
is that ${\cal A}_1$
is a classical BRST variation, i.\ e.,
\be
{\cal A}_1 = i \left( {\Omega_1,S} \right) =
i \delta_B \Omega_1
\quad .
\label{oneloop fake anomaly}
\ee
If ${\cal A}_1$ can be expressed as
in Eq.\bref{oneloop fake anomaly}
for some local functional $\Omega_1$,
then, by setting
\be
W^{'} = S + \hbar \Omega_1
\quad ,
\label{oneloop remove anomaly}
\ee
the quantum master equation
in Eq.\bref{qme2}
is satisfied to order $\hbar$.
In words,
Eq.\bref{oneloop fake anomaly}
says that if ${\cal A}_1$ is expressible
as a local BRST variation
then effectively there is no oneloop anomaly.
Since the second term
in Eq.\bref{oneloop anomaly}
is already of this form,
the requirement becomes that $\Delta S$
should be a classicalBRST variation
of a local functional.
To order $\hbar$,
the equation $\delta_{\hat B}{\cal A} = 0$ in
Eq.\bref{full consistency conditions}
is
\be
\delta_B {\cal A}_1 =
\left( {{\cal A}_1,S} \right) = 0
\quad .
\label{oneloop consistency conditions}
\ee
In view
of Eqs.\bref{oneloop fake anomaly}
and \bref{oneloop consistency conditions},
the investigation of anomalies
is related to the local BRST cohomology
at ghost number one
\ct{brs74a,baulieu85a,hlw90a,tnp90a,tonin91a}.
Although not obvious,
it turns out that
Eq.\bref{oneloop consistency conditions}
embodies the WessZumino
anomaly consistency equations
\ct{wz71a}.
In fieldantifield formalism,
the oneloop master equation anomaly
must be classically BRST invariant.
If anomalies arise beyond the oneloop level,
Eq.\bref{full consistency conditions}
provides the full quantum consistency conditions:
The master equation anomaly must
be quantumBRST invariant.
Note that $\rm{gh} \left[ {\cal A} \right] = 1$.
Of the fields
in Eq.\bref{field set},
only
${\cal C}^{\alpha_0}_0 \equiv {\cal C}^{\alpha}$
has ghost number one.
Hence, one may write
\be
i {\cal A} =
a_\alpha \left( \phi \right){\cal C}^\alpha + \dots
\quad ,
\label{gh structure of anomaly}
\ee
where the omitted terms involve antifields.
Sometimes these terms are absent
so that
Eq.\bref{gh structure of anomaly}
gives the structure
of the quantummasterequation anomaly.
Although not obvious,
the coefficients
$a_\alpha \left( \phi \right)$
are the usual gauge anomalies
\ct{tnp90a}.
In other words,
a quantummasterequation anomaly and a gauge anomaly
are equivalent.
\subsection{{Canonical Transformations and the Quantum \hfil}
\break
Master Equation}
\label{ss:ctqme}
\hspace{\parindent}
Canonical transformations
preserve the quantum master equation
as long as $W$ is appropriately transformed
\ct{ht92a}.
Consider an infinitesimal canonical transformation
as in Eq.\bref{canonical transformation}
governed by $F$.
Normally a functional $G$ transforms as
$G \to G + \varepsilon (G,F)$.
For $W$, however, one must add an extra term
$i \hbar \varepsilon \Delta F$ to
compensate for ``measure effects''.
Hence, the transformation rule for $W$ is taken to be
\be
W \to W + \varepsilon ( W, F) +
i \hbar \varepsilon \Delta F +
O \left( \varepsilon^2 \right)
= W  \varepsilon \delta_{\hat B} F +
O \left( \varepsilon^2 \right)
\quad .
\label{w ct law}
\ee
According
to Eq.\bref{qme symmetry},
Eq.\bref{w ct law}
is a symmetry
of the quantum master equation
\ct{bv84a,henneaux90a,fisch90a,dejonghe93a,tpbook}.
If $W$ is a solution
to the quantum master equation,
changing $W$ as
in Eq.\bref{w ct law}
will not upset the solution.
The same conclusion holds for finite transformations
governed by $F_2$
in Eq.\bref{f2 can trans}.
In this case,
one transforms
from $\{ \Phi , \Phi^* \}$ variables
to $\{ \tilde \Phi , \tilde \Phi^* \}$ variables
via Eq.\bref{f2 can trans}.
The transformation rule for $W$ is
\ct{bv84a}
\be
\widetilde W [ \tilde \Phi , \tilde \Phi^*] =
W [ \Phi , \Phi^* ]  \frac{i\hbar}{2} \ln J
\quad ,
\label{w fct law}
\ee
where the jacobian factor $J$
is the berezinian governing
the change from $\{ \Phi , \Phi^* \}$ variables
to tilde variables.
Then,
$\widetilde W$ satisfies
the quantum master equation exactly in the tilde variables
if $W$ satisfies it in the $\{ \Phi , \Phi^* \}$ variables.
A detailed proof of this result
and a formula for $J$
can be found
in ref.\ct{tnp90a}.
Here, we provide a few key steps.
Define $\widetilde \Delta$
as the analog of $\Delta$
in the transformed variables, i.e.,
$$
\widetilde \Delta \equiv
\left( {1} \right)^{\epsilon_A+1}
{{\partial_r} \over {\partial \tilde \Phi^A}}
{{\partial_r} \over {\partial \tilde \Phi_A^*}}
\quad .
$$
If $G$ is an arbitrary functions of
$\tilde \Phi$ and $\tilde \Phi^*$,
then
\be
\widetilde \Delta G \left[ \tilde \Phi , \tilde \Phi^* \right] =
\Delta G
\left[ \tilde \Phi \left[ \Phi , \Phi^* \right] ,
\tilde \Phi^* \left[ \Phi , \Phi^* \right] \right]
 \frac12 \left( G, \ln J \right)
\quad .
\label{fct of Delta}
\ee
When $\Delta$ acts on $G$ on the righthand side
of Eq.\bref{fct of Delta},
the tilde fields should be regarded as functionals
of $\Phi$ and $\Phi^*$, as indicated.
The chain rule for derivatives is then used.
This produces $\widetilde \Delta G$ plus an extra term,
which is equal to $\frac12 ( G, \ln J )$ and
needs to be subtracted to obtain the identity
in Eq.\bref{fct of Delta}.
Using Eqs.\bref{w fct law} and \bref{fct of Delta},
one finds,
$$
i \hbar \widetilde \Delta \widetilde W 
\frac12 \left( \widetilde W , \widetilde W \right) =
i \hbar \left\{ \Delta \widetilde W 
\frac12 \left( \widetilde W, \ln J \right) \right\} 
\frac12 \left( \widetilde W , \widetilde W \right)
$$
$$
= i \hbar \left\{ {
\Delta W  \frac{i\hbar}{2} \Delta \ln J 
\frac12 \left( W  \frac{i\hbar}{2} \ln J , \ln J \right)
} \right\}
 \frac12 \left( { W  \frac{i\hbar}{2} \ln J ,
W  \frac{i\hbar}{2} \ln J
} \right)
$$
$$
= i \hbar \Delta W  \frac12 \left( W , W \right) +
\frac{\hbar^2}{2}
\left\{ { \Delta \ln J 
\frac14 \left( \ln J , \ln J \right)
} \right\}
\quad .
$$
It can be shown
\ct{bv84a,tnp90a}
that
$
\Delta \ln J = \frac14 \left( \ln J , \ln J \right)
$,
so that the last term is zero.
Hence, if
$W$ satisfies the quantum master equation,
then $\widetilde W$ satisfies the tilde version
of the quantum master equation.
For infinitesimal transformations,
$$
\tilde \Phi^A = \Phi^A 
\varepsilon \left( \Phi^A , F \right) +
O \left( \varepsilon^2 \right)
\quad ,
$$
$$
\tilde \Phi_A^* = \Phi_A^* 
\varepsilon \left( \Phi_A^* , F \right) +
O \left( \varepsilon^2 \right)
\quad ,
$$
\be
J = 1  2 \varepsilon \Delta F +
O \left( \varepsilon^2 \right)
\quad .
\label{going to infinitesimal}
\ee
Then,
$$
\widetilde W \left[ \Phi , \Phi^* \right] =
W \left[ \Phi , \Phi^* \right] +
\varepsilon \left( W , F \right) +
i \hbar \Delta F + O \left( \varepsilon^2 \right)
\quad ,
$$
so that
$$
W \left[ \Phi , \Phi^* \right] \to
\widetilde W \left[ \Phi , \Phi^* \right] =
W  \varepsilon \delta_{\hat B} F +
O \left( \varepsilon^2 \right)
\quad ,
$$
and one recovers
Eq.\bref{w ct law}.
The identity
$
\Delta \ln J  \frac14 \left( \ln J , \ln J \right) \sim
O \left( \varepsilon^2 \right)
$
follows
from Eq.\bref{going to infinitesimal} and
$\Delta^2 = 0$.
One can take advantage of canonical transformations
in analyzing potential anomalies
by going to a basis
for which the computation is simpler.
\subsection{The Anomaly at the OneLoop Level}
\label{ss:aoll}
\hspace{\parindent}
The quantities that appear in the violation
of the quantum master equation in
Sect.\ \ref{ss:sqme}
involve both fields and antifields.
As a consequence,
one must use the action
before any elimination of antifields.
On the other hand,
since propagators are needed
to perform perturbative computations,
a gaugefixing procedure is required.
Both these requirements can be satisfied
by working
in the gaugefixed basis described
in Sect.\ \ref{ss:gfb}.
It is achieved by performing
a canonical transformation with
the gaugefixing fermion $\Psi$
so that
$
\Phi_A^* \to \Phi_A^* +
{{\partial \Psi } \over {\partial \Phi^A}}
$.
Throughout the rest of this section,
we assume that an admissible $\Psi$
has been selected and
that the shift to the gaugefixed basis
has been performed.
According to the result
in Sect.\ \ref{ss:ctqme},
if the quantum master equation
is satisfied and a canonical transformation is performed
to a new basis
then, by appropriately adjusting the action,
the quantum master equation is satisfied
in the new basis.
Hence, the existence or nonexistence
of an anomaly is independent
of the choice of basis,
although the form of the anomaly may depend
on this choice.
There are different ways
of obtaining the anomaly.
We mostly follow the approach
of ref.\ct{tnp90a}
and briefly mention other methods
at the end of this subsection.
Reference \ct{tnp90a} obtained general formulas
for the antifieldindependent part
of the oneloop anomaly using a PauliVillars
regularization scheme.
Since the derivation is somewhat technical,
we present only the final results.
For more details,
see refs.\ct{dtnp89a,tnp90a,vanproeyen91a,%
dstvnp92a,dejonghe93a,tp93a,vp94a,tpbook}.
In particular,
refs.\ct{dejonghe93a,tpbook}
have an extensive discussion
of PauliVillars regularization
in the antibracket formalism
to which we refer the reader.
The goal of the next few paragraphs
is to obtain a regularized expression
for $\Delta S$, denoted by
$ \left( { \Delta } S \right) _{\rm reg} $.
The anomaly ${\cal A}_1 $ is essentially
$ \left( { \Delta } S \right) _{\rm reg}$
since the $M_1$ term
in Eq.\bref{oneloop anomaly}
is eliminated as a possible violation
of the quantum master equation
via Eq.\bref{oneloop remove anomaly}
with $\Omega_1 =  M_1$,
i.e., the counterterm
$ \hbar M_1$ is added to the action.
Define
\be
{K^A}_{B} \equiv
\lder{}{\Phi^*_A}\rder{}{\Phi^B}
S \left[ \Phi, \Phi^* \right]
\quad ,
\label{def of K}
\ee
\be
Q_{AB} \equiv \frac{\partial_l} {\partial\Phi^A}
\frac{\partial_r}{\partial\Phi^B}
S \left[ \Phi, \Phi^* \right]
\quad .
\label{def of Q}
\ee
Note that $Q$ involves derivatives with respect to fields
and not antifields.
If expanded about a stationary point,
$Q$ becomes the quadratic form for the fields.
In such an expansion,
the inverse of $Q$ is the propagator.
The properness condition in
the gaugefixed basis guarantees
that propagators exist.
An operator ${\cal O}$
used to regulate $\Delta S$
is related to $Q$ by
\be
{{\cal O}^A}_B \equiv
\left( { T^{1} } \right)^{AC} Q_{CB}
\quad ,
\label{def of O}
\ee
where $T_{AC}$ is an arbitrary invertible matrix
satisfying
$T_{BA}=(1)^{\eps_A + \eps_B + \eps_A \eps_B} T_{AB}$.
The inverse of $T$ obeys
$(T^{1})^{BA}=(1)^{\eps_A \eps_B} (T^{1})^{AB}$.
The Grassmann statistics of
$T_{AB}$ and $(T^{1})^{AB}$
are $\eps_A + \eps_B$ (mod 2).
Eq.\bref{def of O} implies
$Q_{AB} = T_{AC} {{\cal O}^C}_B$.
In the regularization scheme
of ref.\ct{tnp90a},
the matrix $T$ appears in the mass term
for the regulating PauliVillars fields.
In that approach,
the violation of the quantum master equation
is shifted from the $\Delta$ term
to the $( S, S )$ term.
The regulated expression for
$\left( \Delta S \right) _{\rm reg}$
is \ct{tnp90a}
\be
\left( \Delta S \right) _{\rm reg} =
{\left[ { {F^A}_B
{ \left( { \frac{1}{1  { {\cal O} \over M } } }
\right)^B }_{A}
} \right]}_0
\quad ,
\label{Delta2 S reg}
\ee
where
\be
{F^A}_C \equiv
{K^A}_{C} +
\frac12 (T^{1})^{AD}
\left( \delta_B T \right)_{DC}(1)^{\epsilon_C}
\quad ,
\label{def of F}
\ee
and where $\delta_B T$ denotes
the classical BRST transform of $T$:
$\delta_B T = \left( { T , S } \right)$.
In Eq.\bref{Delta2 S reg},
the sum over $A$ and $B$, leading to the trace,
involves the quadratic form of fields only
and not that of antifields.
The subscript $0$
on the square brackets
in Eq.\bref{Delta2 S reg}
indicates that the term
independent of $M$ is to be extracted.
Here, $M$ denotes a regulator mass.
The oneloop nature
of Eq.\bref{Delta2 S reg}
is evident by the presence
of the propagator factor
$
 i / \left( { M  {\cal O}} \right)
\to
{ \left( i M \right) }^{1}
/ \left( { 1  {\cal O} / M } \right)
$
and the sum over the index $A$
indicating a trace.
When ${\cal O}$ is quadratic in spacetime derivatives,
one lets
\be
{\cal R} = {\cal O}
\quad , \quad \quad
{\cal M}^2 = M
\quad ,
\label{def2 of R}
\ee
where ${\cal R}$ denotes
the quadratic regulator operator
and ${\cal M}$ denotes the regulating mass or cutoff.
When ${\cal O}$ is linear in spacetime derivatives,
it is convenient to multiply
on the right in the trace
in Eq.\bref{Delta2 S reg}
by
$
1 / \left( 1 + {\cal O} / M \right)
\ \left( 1 + {\cal O} / M \right)
$
and carry out the multiplication of
$ 1 / \left( 1  {\cal O} / M \right) $
with $ 1 / \left( 1 + {\cal O} / M \right) $.
Eq.\bref{Delta2 S reg}
can then be manipulated into the form
\ct{tnp90a}
\be
\left( \Delta S \right)_{\rm reg} =
{\left[ { {{F^\prime}^A}_B
{ \left( { \frac{1}{ 1  { {\cal R} \over {\cal M}^2 } }
} \right)^B }_{A}
} \right]}_0
\label{Delta S reg}
\quad ,
\ee
where
\be
{\cal R} = {\cal O}^2
\quad , \quad \quad
{\cal M} = M
\quad ,
\label{def1 of R}
\ee
and
\be
{ F^{\prime A} }_C \equiv
{F^A}_C  \frac{1}{2 M}
{\left( { \delta_B {\cal O} } \right)^A}_C
(1)^{\epsilon_C}
%\quad ,
\label{def of F prime}
\ee
with
$\delta_B {\cal O} = \left( {\cal O} , S \right)$.
Summarizing,
in the quadratic momenta case,
$\left( \Delta S \right) _{\rm reg}$
is given by
Eqs.\bref{Delta2 S reg} and \bref{def of F}.
This is the same as
using Eq.\bref{Delta S reg}
with ${\cal R}$ and ${\cal M}$ given
in Eq.\bref{def2 of R}
and with $F$
in Eq.\bref{def of F}
replacing $F^\prime$.
In the linear momenta case,
Eqs.\bref{Delta S reg}\bref{def of F prime} are used.
For the situation in which
${\cal O}$ is quadratic in momenta
or in the case where $\delta_B {\cal O}$
does contribute
in Eq.\bref{Delta S reg},
one can replace
$1 / \left( { 1  {\cal R} / {\cal M}^2 } \right)$ by
${\rm exp} \left( { {\cal R} / {\cal M}^2 } \right) $.
This follows by writing
$
1 / \left( { 1  {\cal R} / {\cal M}^2 } \right) =
\int^\infty_0 \exp\left[ {  \lambda
\left( 1  {\cal R} / {\cal M}^2 \right) } \right]
\dif \lambda
$,
and inserting
in Eq.\bref{Delta2 S reg}
or Eq.\bref{Delta S reg}:
$$
\int_0^\infty {\dif \lambda } \exp \left( {\lambda } \right)
\left[ { {F^A}_B
{\left( { \exp
\left( { {\lambda {\cal R} } \over { {\cal M}^2} } \right)
} \right)^B}_A
} \right]_0 =
$$
$$
\int_0^\infty { \dif \lambda }\exp \left( {\lambda } \right)
\sum\limits_{n=p}^\infty f_n
\left[ {
\left( {{ {\lambda } \over {{\cal M}^2}}}
\right)^n} \right]_0 = f_0 =
$$
$$
\sum\limits_{n=p}^\infty
f_n \left[ {
\left( {{ {1} \over {{\cal M}^2}}} \right)^n}
\right]_0 =
\left[ { {F^A}_B
{ \left( { \exp
\left( { {\cal R} \over {{\cal M}^2} } \right)
} \right)^B}_A
} \right]_0
\quad ,
$$
where a Laurent expansion
in ${1 \over {{\cal M}^2} }$
has been performed.
The resulting expression,
\be
(\Delta S)_{\rm reg} =
{ \left[ {F^A}_{B}
{ {\left( {
\exp \left( { {\cal R} \over {{\cal M}^2} } \right)
} \right)}^B }_{A} \right]
}_0
\quad ,
\label{Delta S Fujikawa}
\ee
corresponds to the Fujikawa form
\ct{fujikawa80a}
of the regularization
\ct{tnp90a} .
In the limit
${\cal M} \to \infty$,
terms of order $1 / {\cal M}^n$ for $n > 0$ vanish,
whereas terms with $n < 0$ blow up.
The regularization scheme consists
of dropping the terms that blow up.
In the PauliVillars regularization,
this is achieved by adding fields
with appropriate statistics, couplings and masses
to cancel all $n < 0$ terms.
As a consequence,
only the $n=0$ term remains.
In the quadratic momentum case,
Eq.\bref{Delta2 S reg}
can be manipulated into the following supertrace form
\ct{gp93c}
\be
(\Delta S)_{\rm reg}
= {\left[\frac12 {({\cal R}^{1}\delta {\cal R})^A}_B
{\left( { \frac{1}{1  { {\cal R} \over {{\cal M}^2} } }
} \right)^B}_A
(1)^{\epsilon_A} \right]}_0
\quad ,
\label{Delta3 S reg}
\ee
which only involves
the quadratic regulator ${\cal R}$.
Eq.\bref{Delta3 S reg} shows that
if $\delta\gh R = [\gh R, G]$ for some $G$
then the anomaly vanishes,
as a consequence of the cyclicity of the trace
\ct{gp93c}.
Formally,
Eq.\bref{Delta3 S reg} is
\be
(\Delta S)_{\rm reg} =
\delta_B {\left[ {
\frac12 { \left( {
\ln \left( \frac{ {\cal R} }{ {\cal M}^2  {\cal R} }
\right)
} \right)^A }_A (1)^{\epsilon_A}
} \right] }_0
\quad ,
\label{Delta4 S reg}
\ee
which is tantamount to demonstrating
that $(\Delta S)_{\rm reg}$ satisfies
the oneloop anomaly consistency condition
$\delta_B (\Delta S)_{\rm reg} = 0$
in Eq.\bref{oneloop consistency conditions}
since $(\Delta S)_{\rm reg}$ is a classical BRST variation
and $\delta_B$ is nilpotent.
If the nonBRSTinvariant part of the quantity
corresponding to
${ \left[ { \dots } \right] }_0 $
in Eq.\bref{Delta4 S reg}
is local%
{\footnote{
The BRSTinvariant part,
which may be nonlocal,
gives zero contribution to the anomaly
since $\delta_B$ is applied to it.}},
then there is no oneloop anomaly
according
to Eqs.\bref{oneloop fake anomaly}
and \bref{oneloop remove anomaly}.
The expression in square brackets
in Eq.\bref{Delta4 S reg}
turns out to be
the oneloop contribution to the effective action.
Eq.\bref{Delta4 S reg}
says that the oneloop anomaly
is the BRST variation
of this oneloop contribution.
In the approach of
ref.\ct{tnp90a},
Eq.\bref{Delta2 S reg}, \bref{Delta S reg},
or \bref{Delta S Fujikawa}
is evaluated
using standard perturbation theory
about a stationary point.
Antifields are finally set to zero
and ${\cal R}$
is evaluated at $\Sigma$, i.e, onshell.
Another approach to anomalies,
which retains antifields, is developed
in refs.\ct{vanproeyen91a,gp93c,tp93a,vp94a}.
At one loop,
results agree with the above.
It has the advantage of making it easier
to compute antifielddependent terms
in the anomaly, if present.
Such terms might arise
if there is an anomalous nonclosure
of BRST transformations
or some other difficulty with a BRSTstructure equation.
Antifields may be retained or eliminated
at any stage of a computation.
A third approach
is to use the effective action $\Gamma$
in Eq.\bref{anomalous ZJ equation}
\ct{baulieu85a,hlw90a,tonin91a}.
In this method,
antifields must be retained.
Information about anomalies
can be obtained using cohomolgical arguments
based on the WessZumino consistency conditions
\ct{wz71a}.
One must compute the coefficients of candidate terms
using perturbative methods
\ct{dixon76a,bc83a,dtv85a,band86a}.
Eq.\bref{Delta3 S reg} demonstrates
that fields
for which $\delta {\cal R}$ is zero,
do not contribute to the anomaly.
Nonpropagating degrees of freedom,
such as gaugefixed fields and
deltafunctiongenerating Lagrange multipliers $\pi$,
are not expected
to contribute
\ct{tpbook}
because anomalies arise
from loop effects.
In practice,
the evaluation of anomalies is performed perturbatively.
Consequently,
one expands around a stationary point.
In the gaugefixed basis,
this involves expanding about $\Sigma$,
that is,
around $\Phi^*_A =0$,
if the method retains antifields,%
{\footnote{
In methods for which antifields are eliminated,
one does not expand about $\Phi^*_A =0$
but simply sets $\Phi^*_A =0$.}}
and about $\Phi^A = \Phi_0^A$,
where the $\Phi_0^A$ satisfy
the gaugefixed equations of motion.
One must be careful to compute
${ F^{\prime A} }_C$
in Eq.\bref{def of F prime}
before expanding about the perturbative saddle point.
In a deltafunction implementation
of gaugefixing,
it is advantageous,
at the beginning of a computation,
to perform a canonical transformation
that shifts the Lagrange multipliers $\pi$
by solutions to equations of motion.
These equations
are generated by the fields
which are being gaugefixed.
Such a canonical transformation ensures
that gaugefixed fields and the $\pi$
do not mix onshell
with the other fields of the system.
This is illustrated
in the first and third sample computations
of Sect.\ \ref{s:sac}.
The choice of $T$
in Eq.\bref{def of O},
which determines the regulator ${\cal O}$,
is at one's disposal.
The requirements on $T$
are that it be invertible
and that it lead to a quadratic regulator
${\cal R}$ that is negative definite
after a Wick rotation to Euclidean space.
Modifying $T$
changes the form of the anomaly.
In particular,
when more than one gauge symmetry is present,
varying $T$
changes the coefficients $a_\alpha$
in Eq.\bref{gh structure of anomaly}.
If some nonzero $a_\alpha$ are made zero
and viceversa,
the anomaly is shifted
from being associated with one type of gauge symmetry
to another.
This is analogous
to the well known situation
for anomalous chiral gauge theories
in four dimensions:
The anomaly can be moved from the axial vector sector
to the vector sector, if so desired.
See, for example, Sect.\ 4.1 of ref.\ct{tnp90a}.
Although we do not present any examples
of this effect,
it is well illustrated in ref.\ct{tnp90a}.
In performing anomaly calculations,
it is useful to choose $T$ to render
a computation as simple as possible.
For a similar reason,
it is also useful to perform
certain canonical transformations
before commencing a calculation.
\vfill\eject
% This is part3, the last part, and must be placed after part2
% FieldAntifield Review
\section{Sample Anomaly Calculations}
\label{s:sac}
\hspace{\parindent}
In this section,
we present computations of
$\left( {\Delta S} \right)_{\rm reg}$
to see whether the quantum master equation is violated
at the oneloop level.
In general,
the analysis is complicated and lengthy.
For this reason,
we treat only three cases:
the spinless relativistic particle,
the chiral Schwinger model,
and the firstquantized bosonic string.
We use the method of
ref.\ct{tnp90a},
which we have outlined
in Sect.\ \ref{ss:aoll}.
The first step in the procedure is
to transform to the gaugefixed basis.
One then has the option
of performing additional canonical transformations.
They can be used to partially diagonalize the system,
so that potential contributions to the anomaly
can be calculated separately from various sectors.
The second step
is to compute the matrices
${K^A}_{B}$ and $Q_{AB}$
in Eqs.\bref{def of K} and \bref{def of Q}.
The third step is to select a $T_{AB}$ matrix
so that the operator ${{\cal O}^A}_B$
in Eq.\bref{def of O}
can be obtained.
A judicious choice of $T_{AB}$ can simplify
a computation.
One then obtains ${F^A}_{B}$
from Eq.\bref{def of F}
and ${\cal R}$
from Eq.\bref{def2 of R} or Eq.\bref{def1 of R}.
The final step is to use the anomaly formula
in Eq.\bref{Delta S Fujikawa}.
Standard perturbation theory is performed,
in which one expands about a stationary point
and sets antifields to zero.
For sample computations
using a method that retain antifields
throughout the computation,
see ref.\ct{gp93c,vp94a}.
Other useful results for anomaly calculations
can be found in \ct{tnp90a,dstvnp92a,%
dejonghe93a,gp93c,hull93a,ms93a,%
bh94a,brandt94a,mpsx94a,vp94a}
and references therein.
\subsection{Computation for the Spinless Relativistic Particle}
\label{ss:csrp}
\hspace{\parindent}
In this subsection,
we show that the spinless relativistic particle
of Sect.\ \ref{ss:srp2}
possesses no anomaly.
This example is useful
for illustrating the formalism
of Sect.\ \ref{ss:aoll}
because the computation is relatively simple.
We use
\be
\Psi =\int {\dif \tau }
\bar {\cal C} \left( {e  \rho} \right)
%\quad ,
\label{rsp psi rho}
\ee
for the gaugefixing fermion,
where $\rho$ is an arbitrary function of $\tau$.
This allows us to judge
potential dependence on the gaugefixing procedure
by varying $\rho$.
Next,
a canonical transformation is performed
to the gaugefixed basis
of Sect.\ \ref{ss:gfb}
using $\Psi$
in Eq.\bref{rsp psi rho}.
One obtains
\be
S \to \int {\dif \tau }
\left\{ {
{{1 \over 2}
\left( {{{\dot x^2} \over e}  m^2 e } \right) +
x_\mu^*\dot x^\mu {{ {\cal C} } \over e} +
\left( {e^* + \bar {\cal C} } \right)
\dot {\cal C} +
\bar \pi \left( {
\bar {\cal C}^* + e  \rho } \right)}
} \right\}
\quad .
\label{rsp gfb S}
\ee
It is advantageous to perform a canonical transformation
that shifts $\bar \pi$
by the solution
of the equation of motion generated by $e$.
According to the result
in Sect.\ \ref{ss:ctqme},
canonical transformations
do not affect the existence or nonexistence
of violations of the quantum master equation.
The variation of $S_\Psi$
with respect to $e$
yields
\be
\bar \pi  {1 \over 2} {{\dot x^2} \over {e^2}}
 \frac12 m^2 = 0
\quad .
\label{eos for e}
\ee
The relevant canonical transformation is
$$
\bar \pi \to \bar \pi +
{1 \over 2}{{\dot x^2} \over {e^2}} + \frac12 m^2
\quad ,
$$
\be
e^*\to e^* +
\bar \pi^*{{\dot x^2} \over {e^3}} \ ,
\quad \quad x_\mu^*\to x_\mu^* +
{d \over {d\tau }}
\left( {{{\bar \pi^*\dot x_\mu } \over {e^2}}} \right)
\quad ,
\label{rsp pi ct}
\ee
with the other fields and antifields left unchanged.
The action becomes
$$
S \to \int { \dif\tau }
\left\{
{\left( {{1 \over e} 
{\rho \over {2e^2}}} \right)\dot x^2 
{1 \over 2}\rho m^2 + \bar{\cal C} \dot{\cal C} +
\bar \pi \left( {e  \rho } \right) +
} \right.
$$
\be
\left. {
x_\mu^* \dot x^\mu {{{\cal C}} \over e} +
e^* \dot{\cal C} +
\bar{\cal C}^*{1 \over 2}
\left( {{{\dot x^2} \over {e^2}} + m^2} \right) 
{{ \bar \pi^* {\cal C} \dot x_\mu } \over {e^2}}
{d \over {d\tau }}
\left( {{{\dot x^\mu } \over e}} \right)
} \right\}
\quad .
\label{rsp gfb shifted S}
\ee
Let us first determine the overall structure
of the computation.
{}From Eq.\bref{rsp gfb shifted S},
one finds that the nonzero entries
of the matrix ${K^A}_{B}$ are
\be
K = \bordermatrix{
& x^\mu & e & \bar \pi & \bar {\cal C} & {\cal C} \cr
x_\mu^* & * & * & 0 & 0 & * \cr
e^* & 0 & 0 & 0 & 0 & * \cr
\bar \pi^* & * & * & 0 & 0 & * \cr
\bar {\cal C}^* & * & * & 0 & 0 & 0 \cr
{\cal C}^* & 0 & 0 & 0 & 0 & 0 \cr
}
\quad ,
\label{rsp non zero K entries}
\ee
where the columns and rows
are labelled by the corresponding
fields and antifields.
We select $T_{AB}$ to be proportional to the identity matrix,
except in the ghost sector for which
\be
T_{\bar{\cal C}{\cal C}} =
\bordermatrix{
& \bar{\cal C} & {\cal C} \cr
\bar{\cal C} & 0 & 1 \cr
{\cal C} & 1 & 0 \cr}
\quad ,
\label{rsp T_ghost}
\ee
and in the $x$ sector for which
\be
\left( {T_x} \right)_{\mu \nu } = \eta_{\mu \nu }
\quad .
\label{rsp T_x}
\ee
In perturbation theory,
the regulator ${\cal R}$
is evaluated at the stationary point
of the gaugefixed action.
For $e$,
this corresponds to
\be
\left. e \right_\Sigma = \rho
\quad ,
\label{e stationary point}
\ee
where $\Sigma$ indicates
the stationarypoint surface in field space.
Using Eq.\bref{rsp gfb shifted S},
a straightforward calculation reveals
that the nonzero entries of
$\left. { {\cal R} } \right_\Sigma$ are
\be
\left. { {\cal R} } \right_\Sigma =
\bordermatrix{
& x^\mu & e & \bar \pi & \bar {\cal C} & {\cal C} \cr
x^\mu & * & 0 & 0 & 0 & 0 \cr
e & 0 & * & * & 0 & 0 \cr
\bar \pi & 0 & * & 0 & 0 & 0 \cr
\bar {\cal C} & 0 & 0 & 0 & * & 0 \cr
{\cal C} & 0 & 0 & 0 & 0 & * \cr
}
\quad .
\label{rsp non zero regulator entries}
\ee
Eq.\bref{rsp non zero regulator entries}
shows that propagation is diagonal within three sectors:
the $x^\mu$ sector, the $e$$\bar \pi$ sector
and the ghost sector.
As expected,
the canonical transformation
in Eq.\bref{rsp pi ct}
decouples
$e$ from $x^\mu$:
For the shifted action
in Eq.\bref{rsp gfb shifted S},
one has
\be
\left. {
{{\partial _l\partial _rS}
\over {\partial e\partial x^\mu }}
} \right_\Sigma =
\left. {
\left( {2{d \over {d\tau }}
{1 \over {e^2}}\dot x^\mu +
2{d \over {d\tau }}{\rho \over {e^3}}\dot x^\mu } \right)
} \right_\Sigma = 0
\quad .
\label{x e decoupling}
\ee
Because a constant $T_{AB}$ matrix has been selected,
the nonzero entries of ${F^A}_B$
in Eq.\bref{def of F} are
the same as
in Eq.\bref{rsp non zero K entries}.
{}From the structure of
Eqs.\bref{rsp non zero K entries}
and \bref{rsp non zero regulator entries},
one sees that the anomaly computation
separates
into contributions from
the $x^\mu$ sector, the $e$$\bar \pi$ sector
and the ghost sector.
The propagating fields are
$x^\mu$, $\bar {\cal C}$ and ${\cal C}$.
The field $\bar \pi$ serves as a Lagrange multiplier
for setting $e$ equal to $\rho$.
Hence, $e$ and $\bar \pi$ are nonpropagating
and should not contribute to the anomaly
according to the analysis
in Sect.\ \ref{ss:aoll}
\ct{tpbook}.
For this particular system,
the contribution is zero
because ${F^A}_B$ in the $e$$\bar \pi$ sector
is offdiagonal.
It is also clear that
${\cal C}$ and $\bar {\cal C}$
do not contribute to the anomaly
since ${F^A}_B$ is zero for all ghost entries.
One only needs to consider the $x^\mu$ sector.
In what follows
we use a subscript $x$ for quantities
associated with $x^\mu$.
Applying
Eqs.\bref{def of K} and \bref{def of Q}
to Eq.\bref{rsp gfb shifted S},
one arrives at
\be
{\left( {K_x} \right)^\mu}_\nu =
{\delta^\mu}_\nu e^{1}
{\cal C} {d \over {d\tau }}
\quad ,
\label{rsp K_x}
\ee
and
\be
\left( {Q_x} \right)_{\mu \nu } =
\eta_{\mu \nu } Q_x
\quad ,
\label{rsp Q_x}
\ee
where $Q_x$ without $\mu \nu$ subscripts is defined by
\be
Q_x \equiv 2 {d \over {d\tau }}e^{1}{d \over {d\tau }} +
{d \over {d\tau }}\rho e^{2}{d \over {d\tau }}
\quad .
\label{rsp Q_x noindex}
\ee
Here and below, the derivative
${ {d} \over {d\tau} }$
acts on everything to the right
including $e$, $\rho$ and
the function to which $Q_x$ is applied.
Some contributions
to Eq.\bref{rsp Q_x noindex}
come from the shifts
in Eq.\bref{rsp pi ct}.
We have also dropped terms
proportional to
$\bar \pi^{*}$ and $\bar {\cal C}^*$
because they will not contribute,
when expanding about the stationary point.
Since the ${T_x}$ matrix is
$
\left( {T_x} \right)_{\mu \nu } = \eta_{\mu \nu }
$,
the regulator matrix
in Eq.\bref{def2 of R}
is
\be
{\left( {{\cal R}_x} \right)^\mu}_\nu = {\delta^\mu}_\nu Q_x
\quad .
\label{rsp R_x}
\ee
Because $\delta_B {T_x} = 0$,
\be
{\left( {F_x} \right)^\mu}_\nu = {\left( {K_x} \right)^\mu}_\nu
\quad ,
\label{rsp F_x}
\ee
where ${\left( {K_x} \right)^\mu}_\nu$ is given
in Eq.\bref{rsp K_x}.
All the relevant matrices
of Sect.\ \ref{ss:aoll}
for the computation of the anomaly
have been obtained.
At this stage,
one expands about the stationary point
of the gaugefixed action.
The field $e$ is set equal to the function $\rho$
according to Eq.\bref{e stationary point}.
The regulator matrix becomes
\be
\left. { {\left( {{\cal R}_x} \right)^\mu}_\nu }
\right_\Sigma =
 {\delta^\mu}_\nu {d \over {d\tau }}
\rho^{1}{d \over {d\tau }}
\ ,
\label{R_x at Sigma}
\ee
so that onshell
$$
\left( {\Delta S} \right)_{\rm reg} =
$$
\be
D \int {\dif \tau }
\left[ {\int\limits_{\infty }^\infty
{{{\dif k} \over {2\pi }}}\ \exp \left( {ik\tau } \right)
\ \rho^{1} {\cal C} {d \over {d\tau }}
\ \exp \left( {{{{d \over {d\tau }}
\rho^{1}{d \over {d\tau }}}
\over { {\cal M}^2}}} \right)
\ \exp \left( {ik\tau } \right)} \right]_0
\quad ,
\label{rsp Delta S reg}
\ee
where we have used the form of
$\left( {\Delta S} \right)_{\rm reg}$
in Eq.\bref{Delta S Fujikawa}.
The trace over field indices $A$
produces a factor of $\int {\dif \tau }$
and a factor of $D$
(because the number of $x^\mu$ fields is $D$ and
each contributes equally).
The operator trace is evaluated using
a complete set of momentumspace functions,
thereby generating
the factors $\exp \left( {\pm ik\tau } \right)$.
The calculation
in Eq.\bref{rsp Delta S reg}
is performed in Appendix C,
where it is shown that
the integrand is an odd function of $k$.
Consequently,
\be
\left( {\Delta S} \right)_{\rm reg} = 0
\quad .
\label{rsp Delta S reg result}
\ee
The calculation
of the $x^\mu$ contribution
is even simpler
using Eq.\bref{Delta3 S reg}.
To compute $\delta_B {\cal R}$,
note that
\be
\delta_B e = \left( {e,S} \right) =
\dot {\cal C}
\quad .
\label{rsp BRST var of e}
\ee
Using this equation,
Eq.\bref{rsp Q_x noindex} and Eq.\bref{rsp R_x},
one finds
\be
{\left( {\delta_B {\cal R}_x} \right)^\mu}_\nu =
{\delta^\mu}_\nu \left( {2{d \over {d\tau }}
e^{2} \dot {\cal C} {d \over {d\tau }} 
2{d \over {d\tau }}\rho e^{3}
\dot {\cal C} {d \over {d\tau }}} \right)
\quad ,
\label{rsp BRST var of R_x}
\ee
so that
\be
\left. { {\left( {\delta_B {\cal R}_x}
\right)^\mu}_\nu } \right_\Sigma = 0
\quad .
\label{rsp BRST var of R_x at Sigma}
\ee
When $ \delta_B {\cal R}_x $ is substituted
into Eq.\bref{Delta3 S reg},
one gets
$$
\left( {\Delta S} \right)_{\rm reg} = 0
\quad ,
$$
in agreement
with Eq.\bref{rsp Delta S reg result}.
The absence of a violation
of the quantum master equation
means that the spinless relativistic particle theory
is gaugeinvariant
even at the quantum level.
\subsection{The Abelian Chiral Schwinger Model}
\label{ss:acsm}
\def\Dslash{\slash\mkern12mu D}
\def\Aslash{\slash\mkern12mu A}
\hspace{\parindent}
In this subsection,
we analyze the abelian chiral Schwinger model
in twodimensions.
It is an anomalous gauge theory
and
a particularly simple example
that illustrates the formalism
of Sect.\ \ref{ss:aoll}.
The model contains an abelian gauge field,
i.\ e., ``a photon'' $A_\mu$,
and a charged lefthanded fermion.
It is governed by the following classical action
\be
S_0
\left[ { A_\mu , \psi , \bar\psi } \right] =
\int \dif^2 x
\left[ {
\frac{1}{4e^2}
F^{\mu\nu} F_{\mu\nu} +
\bar\psi i \Dslash \psi
} \right]
\quad ,
\label{acsm1}
\ee
where $e$ is the electromagnetic coupling constant.
Although we take $\psi$ to be a Dirac fermion,
we use a covariant derivative
that couples the photon only
to the the rightmoving component:
\be
i\Dslash =
i \slashit \partial + \Aslash P_{}
\quad .
\label{acsm2}
\ee
In other words,
$P_{} \psi$ is charged but $P_{+} \psi$ is neutral.
Here $P_{\pm}$ are the chiral projectors
\be
P_{} = \frac12 \left( { 1  \gamma_5 } \right)
\ , \quad \quad
P_{+} = \frac12 \left( { 1 + \gamma_5 } \right)
\quad .
\label{acsm3}
\ee
In twodimensions,
they project onto right and leftmoving states.
Hence, the leftmoving fermion $P_{+} \psi$
is a free particle and decouples.
A slash through a vector $V_{\mu}$ represents
$\gamma^{\mu} V_{\mu}$:
$\slashit{V} = \gamma^{\mu} V_{\mu}$.
In twodimensions the $\gamma^{\mu}$
are $2 \times 2$ matrices
satisfying
$
\gamma^\mu \gamma^\nu +
\gamma^\nu \gamma^\mu =
2\eta^{\mu \nu }
$,
and $\gamma_5 $ is defined as
$\gamma_5 \equiv \gamma^0\gamma^1$.
The action in Eq.\bref{acsm1} is invariant
under the finite gauge transformations
\be
A'_{\mu} = A_{\mu} + \partial_\mu\veps
\ , \quad \quad
\psi' = \exp\{iP_{}\veps\}\psi
\ , \quad \quad
\bar\psi' = \bar\psi\exp\{iP_{+}\veps\}
\quad .
\label{acsm4}
\ee
Although this model
has not been discussed
in previous sections,
it is straightforward to apply
the fieldantifield formalism \ct{bm91a,bm94a}.
The proper solution for the gauge sector
corresponds to the solution
for the YangMills example
given in Sect.\ \ref{sss:ymt} for $d=2$
and for a $U(1)$ group.
In addition to the antifield $A^*_\mu$ of the photon,
one has commuting antifields
$\psi^*$ and $\bar\psi^*$
for the fermions.
The proper solution of the master equation is
\be
S = S_0 + \int\dif^2 x
\left[ {
A^*_\mu\partial^\mu \gh C +
i (\psi^*)^t P_{}\psi\,\gh C
i \bar\psi^* P_{+}^t \bar\psi^t \gh C
} \right]
\quad ,
\label{acsm5}
\ee
where $\gh C$ is the ghost field
associated with the gauge parameter $\veps$.
The superscript $t$ stands for transpose:
$
\psi =\left( {
\matrix{
\psi^1 \cr
\psi^2 \cr }
} \right)
$
and
$
\bar \psi =\left( { \bar\psi^1 , \bar\psi^2 } \right)
$,
so that
$
\bar\psi^t =
\left( { \matrix{
\bar\psi^1 \cr
\bar\psi^2 \cr } } \right)
$
and
$
(\psi^*)^t =
\left( { (\psi^1)^* , (\psi^2)^* } \right)
$.
A formal computation
using the expression for $\Delta$
in Eq.\bref{def Delta}
reveals that only the fermion sector
contributes to $\Delta S$.
A more detailed analysis using a regularization procedure
confirms this.%
{\footnote{
The computation made in ref.\ct{dstvnp92a}
for a pure YangMills theories in four dimensions
supports the idea
that gauge fields and ghosts
produce a BRST trivial contribution
to $(\Delta S)_{\rm reg}$.
}}
Therefore,
we focus on the contribution to $\Delta S$
{}from $\psi$ and $\bar\psi$.
Gaugefixing is not needed because
propagators for the fermions already exist.
For these reasons,
it is not necessary to consider
gaugefixing auxiliary fields,
nor a nonminimal proper solution.
Using Eqs.\bref{def of K} and \bref{acsm5},
one finds that the $K$ matrix is given by
\be
K = i {\cal C}
\left( { \matrix{
P_ & 0_2 \cr
0_2 & P_+^t \cr}
} \right)
\quad ,
\label{acsm6}
\ee
where all entries are $2 \times 2$ matrices,
e.\ g.,
$
0_2 =
\left( { \matrix{
0 & 0 \cr
0 & 0 \cr } } \right)
$,
so that $K$ is a $4 \times 4$ matrix.
In Eq.\bref{acsm6} and throughout this subsection,
we label the rows and columns of matrices
in the order
$\psi^1 , \psi^2 , \bar\psi^1 , \bar\psi^2$.
{}From Eqs.\bref{def of Q} and \bref{acsm1},
the matrix $Q$ for the fermion sector is
\be
Q =
\left( {
\matrix{ 0_2 & i \tilde{\Dslash} \cr
i \Dslash & 0_2 \cr }
} \right)
\quad ,
\label{acsm7}
\ee
where $\tilde{\Dslash}$ is defined by
$
\int {\dif^2x} \bar \psi i \Dslash \psi =
\int {\dif^2x} \psi^t i \tilde{\Dslash} \bar \psi^t
$.
More precisely,
\be
i \tilde{\Dslash} =
i \slashit{\partial}^t  \Aslash^t P_+^t
\quad ,
\label{acsm8}
\ee
where, here, the superscript $t$ indicates
the transpose of a matrix in Diracindex space.
For the matrix $T$, we choose
\be
T = \left( {
\matrix{ 0_2 & 1_2 \cr
1_2 & 0_2 \cr }
} \right)
\quad ,
\label{acsm9}
\ee
where
$
1_2 =
\left( \matrix{1 & 0 \cr
0 & 1 \cr} \right)
$.
Because $\delta_B T = 0$,
the matrix $F$
in Eq.\bref{def of F}
is equal to the matrix $K$
in Eq.\bref{acsm6}.
Using Eq.\bref{def of O}, one finds
\be
{\cal O} = \left( { \matrix{
i \Dslash & 0_2 \cr
0_2 & i\tilde{\Dslash} \cr}
} \right)
\quad .
\label{acsm10}
\ee
Then from Eq.\bref{def1 of R},
one obtains
\be
R = \left( {
\matrix{
 \Dslash \Dslash & 0_2 \cr
0_2 &  \tilde{\Dslash} \tilde{\Dslash} \cr}
} \right)
\quad ,
\label{acsm11}
\ee
for the regulator matrix.
Let us use the following representation
of the gamma matrices:
\be
\gamma^0 = \sigma^1 =
\left( \matrix{ 0 & 1 \cr
1 & 0 \cr} \right)
\ , \quad \quad
\gamma^1 = i\sigma^2 =
\left( \matrix{ 0 & 1 \cr
1 & 0 \cr} \right)
\quad ,
\label{acsm12}
\ee
so that
\be
\gamma^5 =
\gamma^0\gamma^1 = \sigma^3 =
\left( \matrix{1 & 0 \cr
0 & 1 \cr} \right)
\quad ,
\label{acsm13}
\ee
and
\be
P_ =
\left( \matrix{ 1 & 0 \cr
0 & 0 \cr} \right)
\ , \quad \quad
P_+ =
\left( \matrix{ 0 & 0 \cr
0 & 1 \cr} \right)
\quad .
\label{acsm14}
\ee
With this representation,
\be
i \Dslash =
\left( { \matrix{
0 & i\partial_+ \cr
i\partial_ + A_ & 0 \cr} } \right)
\ , \quad \quad
i \tilde{\Dslash} =
\left( { \matrix{
0 & i\partial_  A_ \cr
i\partial_+ & 0 \cr} } \right)
\quad ,
\label{acsm15}
\ee
where
\be
\partial_\pm = \partial_0 \pm \partial_1
\ , \quad \quad
A_ = A_0  A_1
\quad .
\label{acsm16}
\ee
The entries in $R$
of Eq.\bref{acsm11}
are
$$
 \Dslash \Dslash =
\left( { \matrix{
i\partial_+ \left( { i\partial_ + A_ } \right) & 0 \cr
0 & \left( { i\partial_ + A_ } \right) i\partial_+ \cr
} } \right)
\quad ,
$$
\be
 \tilde{\Dslash} \tilde{\Dslash} =
\left( { \matrix{
\left( { i\partial_  A_ } \right) i\partial_+ & 0 \cr
0 & i \partial_+ \left( { i\partial_  A_ } \right) \cr
} } \right)
\quad .
\label{acsm17}
\ee
The derivative $\partial_+$ acts to the right,
so that
it differentiates $A_$
as well as any function
to which $R$ is applied.
Note that $R$ is diagonal.
In general,
the matrix $F'$
in Eq.\bref{def of F prime}
has two contributions.
However, the term proportial to $\delta_B {\cal O}$
does not contribute, upon taking the trace,
because ${\cal O}$
is offdiagonal and $R$ is diagonal
when the above gamma matrices are used.
Hence, one may take
\be
F' = F = K
\quad ,
\label{acsm18}
\ee
where $K$ is given in Eq.\bref{acsm6}.
Summarizing, for the computation of $\Delta S$
in Eq.\bref{Delta S Fujikawa},
one uses $F=K$
in Eq.\bref{acsm6}
and $R$ given
by Eqs.\bref{acsm11} and \bref{acsm17}.
Incorporating the projectors $P_{\pm}$ in $K$,
one obtains
\be
\Delta S =
 i \left[{
Tr \, {\cal C}
\left( {
\exp{ \left( {
{ { R_+ } \over {{\cal M}^2} }
} \right) } 
\exp{ \left( {
{ { R_ } \over {{\cal M}^2} }
} \right) }
} \right)
} \right]_0
\ ,
\label{acsm19}
\ee
where there is no trace in Diracindex space,
only in function space.
In Eq.\bref{acsm19},
\be
R_\pm =
\partial^\mu \partial_\mu
\pm i (\partial_+ A_) \pm i A_ \partial_+
\quad .
\label{acsm20}
\ee
where
we use a parenthesis around
$(\partial_+ A_)$
to indicate that $\partial_+$
acts only on $A_$.
In Appendix C,
the computation
of Eq.\bref{acsm20}
is performed.
One finds
\be
{\cal A}_1 = \Delta S =
\frac{i}{4\pi}
\int \dif^2 x \, {\cal C}
\left( {
\epsilon^{\mu \nu} \partial_\mu A_\nu
 \partial_\mu A^\mu
} \right)
\quad ,
\label{acsm21}
\ee
where $\epsilon^{10} = 1 = \epsilon^{01} $.
Note that the anomaly is consistent since
\be
\delta_B {\cal A}_1 =
\frac{i}{4\pi}
\int \dif^2 x \, {\cal C}
\left( {
\epsilon^{\mu \nu} \partial_\mu \partial_\nu {\cal C}
 \partial_\mu \partial^\mu {\cal C}
} \right)
= 0
\quad .
\label{acsm22}
\ee
The first term vanishes by the antisymmetry property
of $\epsilon^{\mu \nu}$,
while the second term vanishes by integration by parts
and the anticommuting nature of ${\cal C}$.
A local counterterm $\Omega_1$
cannot be added to the action
to eliminate ${\cal A}_1$
via Eq.\bref{oneloop remove anomaly}.
If one takes
$\Omega_1 = \frac{1}{8\pi} \int \dif^2 x A^\mu A_\mu$,
then the second term
in Eq.\bref{acsm21}
is eliminated,
but the term
$
\frac{i}{8\pi} \int \dif^2 x \, {\cal C}
\epsilon^{\mu \nu} F_{\mu\nu}
$
in ${\cal A}_1$ remains.
\subsection{Anomaly in the Open Bosonic String}
\label{ss:aobs}
\hspace{\parindent}
In this subsection,
we investigate the violation
of the quantum master equation
for the firstquantized open bosonic string
when the dimension of spacetime is not $26$.
The bosonic contribution was explicitly computed
in refs.\ct{tnp90a,gp93c}.
We gaugefix the action
using the fermion $\Psi$
in Eq.\bref{fqbs psi}.
It depends on the conformal factor $\rho$.
It turns out that
when $D \ne 26$,
there is an anomaly.
As a consequence,
the theory depends on $\rho$.
Although the theory is classically gaugeinvariant,
one of the four gauge symmetries
is violated by quantum effects.
This anomalous gauge symmetry
cannot be fixed for $D \ne 26$.
Let us apply the oneloop anomaly analysis
given in Sect.\ \ref{ss:aoll}
to the bosonic string.
First, we perform a canonical transformation
with $\Psi$ so that
$
\Phi_A^* \to \Phi_A^* +
{{\partial \Psi } \over {\partial \Phi^A}}
$.
This leads to the following shifts in antifield fields
$$
{e^{*\tau}}_\tau \to {e^{*\tau}}_\tau +
\bar {\cal C}+\bar {\cal C}_\sigma
\ ,\quad \quad
{e^{*\sigma}}_\sigma
\to {e^{*\sigma}}_\sigma +
\bar {\cal C}\bar {\cal C}_\sigma \ ,
$$
$$
{e^{*\tau}}_\sigma\to {e^{*\tau}}_\sigma +
\bar {\cal C}_{\tau \sigma }+\bar {\cal C}_\tau \ ,
\quad \quad {e^{*\sigma}}_\tau
\to {e^{*\sigma}}_\tau + \bar {\cal C}_{\tau \sigma } 
\bar {\cal C}_\tau \ ,
$$
$$
\bar {\cal C}^{*\tau } \to \bar {\cal C}^{*\tau } +
{e_\tau}^{\sigma}  {e_\sigma}^{\tau} \ ,
\quad \quad \bar {\cal C}^{*\sigma } \to
\bar {\cal C}^{*\sigma }+{e_\tau}^{\tau} {e_\sigma}^{\sigma} \ ,
$$
\be
\bar {\cal C}^*\to \bar {\cal C}^* +
{e_\tau}^{\tau} +{e_\sigma}^{\sigma} 2\rho^{1/2} \ ,
\quad \quad \bar {\cal C}^{*\tau \sigma } \to
\bar {\cal C}^{*\tau \sigma } +
{e_\tau}^{\sigma} +{e_\sigma}^{\tau}
\label{fgbs ct with psi}
\quad .
\ee
Additional terms are produced in the total action
$S_{\rm total} = S + S_{\rm aux}$
of Eqs.\bref{fqbs S} and \bref{aux S for fqbs}
given by
$$
S \to S +
\int {\dif\tau }\int\limits_0^\pi {\dif\sigma }
\left\{ {
\left( {\bar {\cal C} + \bar {\cal C}_\sigma } \right)
\left( {{\cal C}^n\partial_n{e_\tau}^{\tau} 
{e_\tau}^n\partial_n{\cal C}^\tau } \right) +
\left( {\bar {\cal C}\bar {\cal C}_\sigma } \right)
\left( {{\cal C}^n\partial_n{e_\sigma}^{\sigma} 
{e_\sigma}^n\partial_n{\cal C}^\sigma } \right)
} \right.
$$
$$
+ \left( {\bar {\cal C}_{\tau \sigma } +
\bar {\cal C}_\tau } \right)
\left( {{\cal C}^n\partial_n{e_\tau}^{\sigma} 
{e_\tau}^n\partial_n{\cal C}^\sigma } \right) +
\left( {\bar {\cal C}_{\tau \sigma } 
\bar {\cal C}_\tau } \right)
\left( {{\cal C}^n\partial_n{e_\sigma}^{\tau} 
{e_\sigma}^n\partial_n{\cal C}^\tau } \right)
$$
$$
+\left( {\left( {\bar {\cal C} + \bar {\cal C}_\sigma } \right)
{e_\tau}^{\tau} + \left( {\bar {\cal C} 
\bar {\cal C}_\sigma } \right)
{e_\sigma}^{\sigma} +\left( {\bar {\cal C}_{\tau \sigma } +
\bar {\cal C}_\tau } \right){e_\tau}^{\sigma} +
\left( {\bar {\cal C}_{\tau \sigma } 
\bar {\cal C}_\tau } \right){e_\sigma}^{\tau} } \right){\cal C}
$$
\be
\left. {\left( {\left( {\bar {\cal C}+\bar {\cal C}_\sigma } \right)
{e_\sigma}^{\tau} +\left( {\bar {\cal C}\bar {\cal C}_\sigma } \right)
{e_\tau}^{\sigma} +\left( {\bar {\cal C}_{\tau \sigma } +
\bar {\cal C}_\tau } \right){e_\sigma}^{\sigma} +
\left( {\bar {\cal C}_{\tau \sigma } 
\bar {\cal C}_\tau } \right){e_\tau}^{\tau} } \right)
{\cal C}^{\tau \sigma }} \right\}
\ ,
\label{S with psi ct}
\ee
and
$$
S_{\rm aux} \to \int {\dif\tau }\int\limits_0^\pi {\dif\sigma }
\left\{ {\bar \pi \left( {\bar {\cal C}^* +
{e_\tau}^{\tau} + {e_\sigma}^{\sigma} 2\rho^{1/2}} \right) +
} \right.
$$
\be
\left.{
\bar \pi_\tau \left( {\bar {\cal C}^{*\tau }+{e_\tau}^{\sigma} 
{e_\sigma}^{\tau} } \right) + \bar \pi_\sigma
\left( {\bar {\cal C}^{*\sigma }+{e_\tau}^{\tau} 
{e_\sigma}^{\sigma} } \right)+\bar \pi_{\tau \sigma }
\left( {\bar {\cal C}^{*\tau \sigma }+{e_\tau}^{\sigma} +
{e_\sigma}^{\tau} } \right)} \right\}
\ .
\label{S aux with psi ct}
\ee
At this stage,
it is desirable to shift fields
by the solutions to the equations of motion
of the ${e_a}^m$,
by using a canonical transformation.
Such a shift guarantees that
the quadratic form $Q_{AB}$
is onshell diagonal
in the ${e_a}^m$ sector.
This avoids mixing of the $X^\mu$ and ghost sectors
with the $\bar \pi$${e_a}^m$ sector.
Variations of $S_{\rm total}$ in the gaugefixed basis
with respect to the ${e_a}^m$
produce the equations for the four $\bar \pi$.
Using a subscript $0$ to denote the solutions,
and ignoring terms proportional
to antifields,
one finds that
$$
2\left( {\bar \pi } \right)_0 \equiv
\left( {{{\partial S} \over {\partial {e_\tau}^{\tau} }} +
{{\partial S} \over {\partial {e_\sigma}^{\sigma} }}
 2\bar \pi } \right) =
$$
$$
{ \left( { {e_\tau}^{\tau} + {e_\sigma}^{\sigma} } \right) }
e {{\cal L}_X} 
{e} \left( {\partial_\tau X^\mu
D_\tau X_\mu \partial_\sigma X^\mu D_\sigma X_\mu } \right) +
$$
$$
2\partial_n\left( {\bar {\cal C}{\cal C}^n} \right) +
\bar {\cal C}
\left({ \partial_n{\cal C}^n  2{\cal C} } \right)
+ \bar {\cal C}_{\tau \sigma }
\left( {\partial_\tau {\cal C}^\sigma +
\partial_\sigma {\cal C}^\tau +
2 {\cal C}^{\tau \sigma } } \right) +
$$
$$
\bar {\cal C}_\tau \left( {\partial_\tau {\cal C}^\sigma 
\partial_\sigma {\cal C}^\tau } \right) +
\bar {\cal C}_\sigma \left( {\partial_\tau {\cal C}^\tau 
\partial_\sigma {\cal C}^\sigma } \right)
\quad ,
$$
$$
2\left( {\bar \pi_{\tau \sigma }} \right)_0 \equiv 
\left( {{{\partial S} \over {\partial {e_\tau}^{\sigma} }} +
{{\partial S} \over {\partial {e_\sigma}^{\tau} }} 
2\bar \pi_{\tau \sigma }} \right) =
$$
$$
{ \left( { {e_\tau}^{\sigma} + {e_\sigma}^{\tau} } \right) }
e {{\cal L}_X} 
{e} \left( {\partial_\sigma X^\mu D_\tau X_\mu 
\partial_\tau X^\mu D_\sigma X_\mu } \right) +
$$
$$
2\partial_n
\left( {\bar {\cal C}_{\tau \sigma }{\cal C}^n} \right) +
\bar {\cal C}
\left( {\partial_\tau {\cal C}^\sigma +
\partial_\sigma {\cal C}^\tau } +
2 {\cal C}^{\tau \sigma } \right) +
\bar {\cal C}_{\tau \sigma }
\left( { \partial_n{\cal C}^n  2 {\cal C} } \right)
$$
$$
\bar {\cal C}_\tau \left( {\partial_\tau {\cal C}^\tau 
\partial_\sigma {\cal C}^\sigma } \right) 
\bar {\cal C}_\sigma \left( {\partial_\tau {\cal C}^\sigma 
\partial_\sigma {\cal C}^\tau } \right)
\quad ,
$$
$$
2\left( {\bar \pi_\tau } \right)_0 \equiv
\left( {{{\partial S} \over {\partial {e_\tau}^{\sigma} }} 
{{\partial S} \over {\partial {e_\sigma}^{\tau} }} 
2\bar \pi_{\tau }} \right) =
$$
$$
{ \left( {{e_\tau}^{\sigma} {e_\sigma}^{\tau} } \right) }
e {{\cal L}_X} 
{e} \left( {\partial_\sigma X^\mu D_\tau X_\mu +
\partial_\tau X^\mu D_\sigma X_\mu } \right) +
$$
$$
2\partial_n\left( {\bar {\cal C}_\tau {\cal C}^n} \right) 
\bar {\cal C}\left( {\partial_\tau {\cal C}^\sigma 
\partial_\sigma {\cal C}^\tau } \right) 
\bar {\cal C}_{\tau \sigma }
\left( {\partial_\tau {\cal C}^\tau 
\partial_\sigma {\cal C}^\sigma } \right)
$$
$$
+ \bar {\cal C}_\tau
\left( {\partial_n{\cal C}^n2{\cal C}} \right) +
\bar {\cal C}_\sigma
\left( {\partial_\tau {\cal C}^\sigma +
\partial_\sigma {\cal C}^\tau 
2{\cal C}^{\tau \sigma }} \right)
\quad ,
$$
$$
2\left( {\bar \pi_\sigma } \right)_0 \equiv
\left( {{{\partial S} \over {\partial {e_\tau}^{\tau} }} 
{{\partial S} \over {\partial {e_\sigma}^{\sigma} }} 
2\bar \pi_\sigma } \right) =
$$
$$
{ \left( { {e_\sigma}^{\sigma}  {e_\tau}^{\tau} } \right) }
e {{\cal L}_X} 
e \left( {\partial_\tau X^\mu D_\tau X_\mu +
\partial_\sigma X^\mu D_\sigma X_\mu } \right) +
$$
$$
2\partial_n
\left( {\bar {\cal C}_\sigma {\cal C}^n} \right) +
\bar {\cal C}\left( {\partial_\tau {\cal C}^\tau 
\partial_\sigma {\cal C}^\sigma } \right) +
\bar {\cal C}_{\tau \sigma }
\left( {\partial_\tau {\cal C}^\sigma 
\partial_\sigma {\cal C}^\tau } \right)
$$
\be
+ \bar {\cal C}_\tau \left( {\partial_\tau {\cal C}^\sigma +
\partial_\sigma {\cal C}^\tau 
2{\cal C}^{\tau \sigma }} \right) +
\bar {\cal C}_\sigma
\left( {\partial_n{\cal C}^n2{\cal C}} \right)
\quad ,
\label{pi_0 defs}
\ee
where the $X^\mu$lagrangian density ${\cal L}_X$ is defined to be
\be
{\cal L}_X \equiv
{e \over {2}}\left( {D_\tau X^\mu D_\tau X_\mu 
D_\sigma X^\mu D_\sigma X_\mu } \right)
\quad .
\label{L_X def}
\ee
The canonical transformation of interest is given by
$$
\bar \pi \to \bar \pi +
\left( {\bar \pi } \right)_0 \ ,
\quad \quad \bar \pi_n\to \bar \pi_n +
\left( {\bar \pi_n} \right)_0\ ,
\quad \quad \bar \pi_{\tau \sigma }
\to \bar \pi_{\tau \sigma } +
\left( {\bar \pi_{\tau \sigma }} \right)_0
\quad ,
$$
\be
\Phi_A^* \to \Phi_A^*  \bar \pi^*{{\partial_r
\left( {\bar \pi } \right)_0} \over {\partial \Phi^A}} 
\bar \pi^{*\tau }{{\partial_r
\left( {\bar \pi_\tau } \right)_0} \over {\partial \Phi^A}} 
\bar \pi^{*\sigma }{{\partial_r
\left( {\bar \pi_\sigma } \right)_0}
\over {\partial \Phi^A}} 
\bar \pi^{*\tau \sigma }{{\partial_r
\left( {\bar \pi_{\tau \sigma }} \right)_0}
\over {\partial \Phi^A}}
\quad ,
\label{pi e ct}
\ee
where in the first equation,
$n$ stands for $\tau$ or $\sigma$.
The goal of the next few paragraphs
is to obtain
${F^A}_B$ and ${{\cal R}^A}_B$
so that the anomaly in Eq.\bref{Delta S reg}
can be computed.
One must first calculate
$ {K^A}_{B} $ and $ Q_{AB} $
and specify $ T_{AB} $.
Simplifications occur for the following reasons.
In the final gaugefixed form of the action
in Eq.\bref{fqbs gf action},
the propagating fields are
$X^\mu$, ${\bar {\cal C}}_n$ and ${\cal C}^n$
where $n$ represents
$\tau$ and $\sigma$.\footnote{
This is made clear by setting
$
{\cal C} = \hat {\cal C} +
\frac{1}{2} \left( { \partial_\tau {\cal C}^\tau +
\partial_\sigma {\cal C}^\sigma } \right) 
2 \rho^{1/2} {\cal C}^n \partial_n \rho^{1/2}
$
and
$
{\cal C}^{\tau\sigma} = \hat {\cal C}^{\tau\sigma} 
\frac{1}{2} \left( { \partial_\tau {\cal C}^\sigma +
\partial_\sigma {\cal C}^\tau } \right)
$.
Throughout this subsection,
one should think of
${\cal C}$ and ${\cal C}^{\tau\sigma}$
as standing for these combinations of fields.
In terms of
$\hat {\cal C}$ and $\hat {\cal C}^{\tau\sigma}$,
the gaugefixed action
in Eq.\bref{fqbs gf action}
is block diagonal
and $\bar {\cal C}$, $\bar {\cal C}^{\tau\sigma}$,
$\hat {\cal C}$ and $\hat {\cal C}^{\tau\sigma}$
are nonpropagating fields.
The calculations in this subsection
should be performed in terms of
$\hat {\cal C}$ and $\hat {\cal C}^{\tau\sigma}$.
At the end of the computation,
one sets
$
\hat {\cal C} = {\cal C} 
\frac{1}{2} \left( { \partial_\tau {\cal C}^\sigma 
\partial_\sigma {\cal C}^\tau } \right) +
2 \rho^{1/2} {\cal C}^n \partial_n \rho^{1/2}
$
and
$
\hat {\cal C}^{\tau\sigma} = {\cal C}^{\tau\sigma} +
\frac{1}{2} \left( { \partial_\tau {\cal C}^\sigma 
\partial_\sigma {\cal C}^\tau } \right)
$,
thereby returning to the original fields.}
By the argument
in Sect.\ \ref{ss:aoll},
only these fields contribute;
the ${F^A}_B$ and ${{\cal R}^A}_B$
associated with nonpropagating fields
do not enter the calculation
\ct{tpbook}.
Furthermore,
by a judicial choice of $T_{AB}$,
${{\cal R}^A}_B$ can be made diagonal.
Hence, only the diagonal components of
$ {K^A}_{B} $, ${\delta_B {\cal O}^A}_B$ and ${F^A}_B$
for $X^\mu$, ${\bar {\cal C}}_n$ and ${\cal C}^n$
need to be computed.
In what follows,
we use a subscript $X$,
${\bar {\cal C}}$ and ${\cal C}$
to denote respectively
a matrix restricted to
the subspace corresponding to
$X^\mu$, ${\bar {\cal C}}_n$ and ${\cal C}^n$.
The subscript ``ghost'' is used
to denote the combined
${\bar {\cal C}}_n$ and ${\cal C}^n$ subspace.
After the above two canonical transformations
have been performed,
the computation is straightforward.
{}From Eq.\bref{def of K},
one finds that the ${K^A}_{B}$ in the three sectors are
$$
{\left( {K_X} \right)^\mu}_\nu =
{\delta^\mu}_\nu {\cal C}^n \partial_n
\quad ,
$$
$$
{\left( {K_{\bar {\cal C}}} \right)^\tau}_\tau =
{\left( {K_{\bar {\cal C}}} \right)^\sigma}_\sigma =
 {\cal C}^n\partial_n
{3 \over 2}\left( {\partial_n{\cal C}^n} \right)
+ {\cal C}
\quad ,
$$
$$
{\left( {K_{{\cal C}}} \right)^\tau}_\tau =
 {\cal C}^n\partial_n +
{1 \over 2}\left( {\partial_n{\cal C}^n} \right) +
{1 \over 2}\left( {\partial_\tau C^\tau 
\partial_\sigma C^\sigma } \right)
\quad ,
$$
\be
{\left( {K_{{\cal C}}} \right)^\sigma}_\sigma =
 {\cal C}^n\partial_n +
{1 \over 2} \left( {\partial_n{\cal C}^n} \right) 
{1 \over 2}\left( {\partial_\tau C^\tau 
\partial_\sigma C^\sigma } \right)
\quad ,
\label{fqbs K comp}
\ee
where the presence of a parenthesis
around a derivative indicates
that it acts only on fields
within the parenthesis.
Throughout this subsection,
the absence of a parenthesis
means that the derivative
acts on everything to the right.
The terms for ${K_X}$ and ${K_{{\cal C}}}$ arise
from differentiating
Eq.\bref{fqbs S},
whereas those for ${K_{\bar {\cal C}}}$ come
from $S_{\rm aux}$ in
Eq.\bref{S aux with psi ct}
after the $\bar \pi_n$ shifts
in Eq.\bref{pi e ct}
have been performed.
The quadratic forms $Q_{AB}$
are
\be
\left( {Q_X} \right)_{\mu \nu } =
\eta_{\mu \nu } Q_X
\quad ,
\label{Q_X mu nu }
\ee
where
$$
Q_X = D_\tau^{\dagger} {e}
\left( { 1  e \rho^{1/2}
{ \left( { {e_\tau}^{\tau} + {e_\sigma}^{\sigma} } \right) }
} \right) D_\tau 
D_\sigma^{\dagger} {e}\left( { 1  e \rho^{1/2}
{ \left( {{e_\tau}^{\tau} + {e_\sigma}^{\sigma} } \right) }
} \right) D_\sigma
$$
\be
+ D_\tau^{\dagger} {{ e \rho^{1/2}} }\partial_\tau 
\partial_\tau {{ e \rho^{1/2}} } D_\tau 
D_\sigma^{\dagger} {{e \rho^{1/2}} e}\partial_\sigma +
\partial_\sigma {{e\rho^{1/2}} } D_\sigma
%\quad ,
\label{Q_X}
\ee
for the $X^\mu$ sector,
where the covariant derivatives
$D_\tau$ and $D_\sigma$
are given in Eq.\bref{gc cov der},
and where
$
D_\tau^{\dagger} =  \partial_\tau {e_\tau}^{\tau}
 \partial_\sigma {e_\tau}^{\sigma}
$
and
$
D_\sigma^{\dagger} =  \partial_\tau {e_\sigma}^{\tau}
 \partial_\sigma {e_\sigma}^{\sigma}
$.
Since derivatives act to the right,
$\partial_\tau$ and $\partial_\sigma$ in
$D_\tau^{\dagger}$ in first the term in Eq.\bref{Q_X}
act on
$
{e}\left( { 1 e \rho^{1/2}
{ \left( { {e_\tau}^{\tau} + {e_\sigma}^{\sigma} } \right) }
} \right)
$,
on the vielbeins in $D_\tau^{\dagger}$ and $D_\tau$,
and on any function
to which the operator $Q_X$ is applied.
Likewise, for the other derivatives.
The terms in Eq.\bref{Q_X}
come from the original action $S_0$
in Eq.\bref{bosonic string action},
as well as from the $\bar \pi$ shifts
of Eq.\bref{pi e ct}
in $S_{\rm aux}$
of Eq.\bref{S aux with psi ct}.
For the ghost sector,
let
\be
V = \left( \matrix{
\bar {\cal C}_\tau \hfill\cr
\bar {\cal C}_\sigma \hfill\cr
C^\tau \hfill\cr
C^\sigma \hfill\cr }
\right)
\quad .
\label{fqbt def of V}
\ee
Then the
ghost part of the action is
\be
S_{\rm ghost} =
{1 \over 2}\int {\dif\tau }\int\limits_0^\pi {\dif\sigma }
\ V^t Q_{\rm ghost}V
\quad ,
\label{fqbt S ghost}
\ee
where the superscript $t$ on $V^t$ stands for transpose.
The ghost quadratic form is
\be
Q_{\rm ghost} =
\left( \matrix{
0\hfill &0\hfill &{\rho^{1/2}\partial_\sigma }\hfill
&{{\rho^{1/2}\partial_\tau }}\hfill \cr
0\hfill &0\hfill &{\rho^{1/2}\partial_\tau }\hfill
&{\rho^{1/2}\partial_\sigma }\hfill \cr
{{\partial_\sigma }\rho^{1/2}}\hfill
&{{\partial_\tau \rho^{1/2}}}\hfill &{0}\hfill &0\hfill \cr
{\partial_\tau \rho^{1/2}}\hfill
&{{\partial_\sigma }\rho^{1/2}}\hfill
&0\hfill &0\hfill \cr} \right)
\quad .
\label{Q ghost}
\ee
The terms in $Q_{\rm ghost}$ originate from
Eq.\bref{S with psi ct} and
from the $\bar \pi$ shifts
of Eq.\bref{pi e ct}
in $S_{\rm aux}$
of Eq.\bref{S aux with psi ct}.
The dependence on ${e_a}^m$ cancels
between the two contributions
leaving only a dependence on $\rho$.
For the matrix $T_{AB}$
of Sect.\ \ref{ss:aoll},
we choose
\be
\left( {T_X} \right)_{\mu \nu } = e \eta_{\mu \nu }
\quad ,
\label{T_X}
\ee
and
\be
T_{\rm ghost} =  i
\left( \matrix{
0\hfill &{e^{2}}\hfill &0\hfill &0\hfill \cr
{e^{2}}\hfill &0\hfill &0\hfill &0\hfill \cr
{0}\hfill &0\hfill &{0}\hfill &{e^{2}}\hfill \cr
0\hfill &0\hfill &{e^{2}}\hfill &0\hfill \cr
} \right)
\quad ,
\label{T_ghost}
\ee
where $T_{\rm ghost}$ acts in the space of antighosts and ghosts
given in Eq.\bref{fqbt def of V}.
In the $X^\mu$ sector
the operator ${\cal O}$,
defined in Eq.\bref{def of O},
is $e^{1} {\delta^\mu}_\nu {Q_X} $
and is equal to
${\left( {{\cal R}_X} \right)^\mu}_\nu$,
where ${Q_X}$ is given
in Eq.\bref{Q_X}.
In the ghost sector,
${\cal O}$ is
\be
{\cal O}_{\rm ghost} =
i\left( \matrix{ 0_2 \hfill &{e^{2}}\hfill
\rho^{1/2}\slashit \partial \cr
{{e^{2} \slashit \partial }
\rho^{1/2}}\hfill & 0_2 \hfill \cr} \right)
\quad ,
\label{O_ghost}
\ee
where each entry is a two by two matrix.
It is somewhat accidental that the Dirac operator
in twodimensional Minkowski space
\be
\slashit \partial \equiv \gamma^n \partial_n =
\gamma^\tau \partial_\tau +
\gamma^\sigma \partial_\sigma
\quad ,
\label{def of partial slash}
\ee
enters in Eq.\bref{O_ghost},
if the gamma matrices
\be
\gamma^\tau = \sigma^3 =
\left( \matrix{1\hfill &0\hfill \cr
0\hfill &{1}\hfill \cr} \right)\ ,
\quad \quad
\gamma^\sigma = i\sigma^2 =
\left( \matrix{0\hfill &{1}\hfill \cr
1\hfill &0\hfill \cr} \right)
%\quad ,
\label{def of gamma matrices}
\ee
are used.
They satisfy
\be
\gamma^n \gamma^m + \gamma^m \gamma^n = 2 \eta^{nm}
\quad .
\label{gamma matrix com rels}
\ee
The ghostsector regulator matrix
${\cal R}_{\rm ghost}$ is
\be
{\cal R}_{\rm ghost}={\cal O}^2 =
\left( \matrix{
{{\cal R}_{\bar {\cal C}}}\hfill &0_2 \hfill \cr
0_2 \hfill &{{\cal R}_{{\cal C}}}\hfill \cr} \right)
\quad ,
\label{R ghost}
\ee
where
\be
{\cal R}_{\bar {\cal C}} = e^{2}\rho^{1/2}
\slashit \partial e^{2}
\slashit \partial \rho^{1/2} \ ,
\quad \quad
{\cal R}_{{\cal C}} =
e^{2}\slashit \partial e^{2}\rho^{1} \slashit \partial
\quad .
\label{R C bar and R C}
\ee
To compute ${F^A}_C$
in Eq.\bref{def of F},
one needs the BRST variation of $T_{AB}$.
Using
\be
\delta_B e =
{\cal C}^n\partial_ne + e\partial_n{\cal C}^n 
2e{\cal C}
%\quad ,
\label{delta B of e}
\ee
and Eqs.\bref{T_X} and \bref{T_ghost},
it is straightforward to calculate
$
{ \left( T^{1} \right) }^{AD} \left( \delta_B T \right)_{DC}
$.
The result is combined with ${K^A}_{C}$
in Eq.\bref{fqbs K comp}
to arrive at
$$
{ \left( F_X \right)^\mu}_\nu = { \delta^\mu }_\nu F_X
\quad ,
$$
where
\be
F_X = {\cal C}^n\partial_n +
{1 \over 2}\left( {\partial_n{\cal C}^n} \right) +
{1 \over 2 }e^{1}{\cal C}^n
\left( {\partial_ne} \right)  {\cal C}
\quad ,
\label{fqbt F_X}
\ee
for the $X^\mu$ sector.
For the ghost sector
$$
{\left( {F_{\bar {\cal C}}} \right)^\tau}_\tau =
{\left( {F_{\bar {\cal C}}} \right)^\sigma}_\sigma =
{\cal C}^n\partial_n{1 \over 2}
\left( {\partial_n{\cal C}^n} \right) +
e^{1}{\cal C}^n\left( {\partial_ne} \right){\cal C}
\quad ,
$$
$$
{\left( {F_{{\cal C}}} \right)^\tau}_\tau =
{\cal C}^n\partial_n{1 \over 2}
\left( {\partial_n{\cal C}^n} \right) 
e^{1}{\cal C}^n\left( {\partial_ne} \right) +
2{\cal C}+{1 \over 2}\left( {\partial_\tau {\cal C}^\tau 
\partial_\sigma {\cal C}^\sigma } \right)
\quad ,
$$
\be
{\left( {F_{{\cal C}}} \right)^\sigma}_\sigma =
{\cal C}^n\partial_n{1 \over 2}
\left( {\partial_n{\cal C}^n} \right) 
e^{1}{\cal C}^n\left( {\partial_ne} \right) +
2{\cal C}{1 \over 2}\left( {\partial_\tau {\cal C}^\tau 
\partial_\sigma {\cal C}^\sigma } \right)
\quad .
\label{fqbt F_ghost}
\ee
Although $\delta_B {\cal O}_{\rm ghost}$ is nonzero,
it does not contribute because
the regulator matrix ${\cal R}$ is block diagonal
in the $\bar {\cal C}_n$ and ${\cal C}^n$ sectors.
It turns out that,
even in each sector,
only diagonal terms contribute
because of the nature of ${\cal R}$.
Hence,
in Eq.\bref{fqbt F_ghost}
we display only the diagonal part of ${F^A}_{C}$.
Finally,
since
the contribution from ${\cal C}_\tau$
turns out to be equal to
the contribution from ${\cal C}_\sigma$,
the two
$
\partial_\tau {\cal C}^\tau 
\partial_\sigma {\cal C}^\sigma
$
terms in Eq.\bref{fqbt F_ghost} for
$ F_{\cal C} $ cancel.
For the rest of this section,
we drop these terms.
The final step is the computation of
$(\Delta S)_{\rm reg}$.
Since $\delta_B {\cal O}$ does not contribute,
we may
use Eq.\bref{Delta S Fujikawa}.
At this stage,
we expand about the classical saddle point,
denoted by $\Sigma$,
corresponding to the solution to the equations of motion.
We set
the $\bar \pi$ equal to zero
and evaluate the ${e_a}^m$
as in Eq.\bref{fqbs gf conditions}.
To determine the regulators and ${F^A}_B$
at $\Sigma$,
note that
\be
\left. e \right_\Sigma = \rho
\quad .
\label{e at Sigma}
\ee
It is useful to express the regulators
in terms of symmetric operators $\tilde {\cal R}$.
For $X^\mu$,
\be
\left. {\left( {{\cal R}_X} \right)} \right_\Sigma =
\rho^{1}\left( {\partial_\tau \partial_\tau +
\partial_\sigma \partial_\sigma } \right) =
\rho^{1/2}\tilde {\cal R}_X\rho^{1/2}
\quad ,
\label{R_X at Sigma}
\ee
where
\be
\tilde {\cal R}_X =
\rho^{1/2}\partial^n\partial_n\rho^{1/2}
\quad .
\label{def of R_X tilde}
\ee
For ghosts,
\be
\left. {{\cal R}_{\bar {\cal C}}} \right_\Sigma =
\rho \tilde {\cal R}_{\bar {\cal C}}\rho^{1}\ ,
\quad \quad
\left. {{\cal R}_{{\cal C}}} \right_\Sigma =
\rho^{1}\tilde {\cal R}_{{\cal C}}\rho
\quad ,
\label{R_ghost at Sigma}
\ee
where
\be
\tilde {\cal R}_{\bar {\cal C}} =
\rho^{1/2} \slashit \partial \rho^{2}
\slashit \partial \rho^{1/2} \ ,
\quad \quad
\tilde {\cal R}_{{\cal C}} =
\rho^{1}\slashit \partial \rho \slashit \partial \rho^{1}
\quad .
\label{def of R_ghost tilde}
\ee
The conjugating factors
in Eqs.\bref{R_X at Sigma} and \bref{R_ghost at Sigma}
can be commuted past the operators ${F^A}_B$
to arrive at the following equivalent form
for $\left( {\Delta S} \right)_{\rm reg}$
$$
\left( {\Delta S} \right)_{\rm reg} =
\left[ {
{\left( {\tilde F_X} \right)^\mu}_\nu
{\left( {\exp
\left( {{{\tilde {\cal R}_X} \over {{\cal M}^2}}} \right)}
\right)^\nu}_\mu +
} \right.
$$
\be
\left. {
{\left( {\tilde F_{\bar {\cal C}}} \right)^n}_n
{\left( {\exp
\left( {{{\tilde {\cal R}_{\bar {\cal C}}} \over
{{\cal M}^2}}} \right)}
\right)^n}_n +
{\left( {\tilde F_{{\cal C}}} \right)^n}_n
{\left( {\exp \left( {{{\tilde {\cal R}_{{\cal C}}}
\over {{\cal M}^2}}} \right)}
\right)^n}_n
} \right]_0
\quad ,
\label{fqbs Delta S reg}
\ee
where
$$
{\left( {\tilde F_X} \right)^\mu}_\nu =
{\delta^\mu}_\nu \left( {{\cal C}^n\partial_n +
{1 \over 2}
\left( {\partial_n{\cal C}^n} \right){\cal C}} \right)
\quad ,
$$
$$
{\left( {\tilde F_{\bar {\cal C}}} \right)^\tau}_\tau =
{\left( {\tilde F_{\bar {\cal C}}} \right)^\sigma}_\sigma =
{\cal C}^n\partial_n 
{1 \over 2}\left( {\partial_n{\cal C}^n} \right) 
{\cal C}
\quad ,
$$
\be
{\left( {\tilde F_{{\cal C}}} \right)^\tau}_\tau =
{\left( {\tilde F_{{\cal C}}} \right)^\sigma}_\sigma =
{\cal C}^n\partial_n 
{1 \over 2}\left( {\partial_n{\cal C}^n} \right) +
2{\cal C}
\quad .
\label{def of F tilde}
\ee
The $T_{AB}$ matrix has been judiciously chosen
so that the violation in the quantum master equation
is proportional to ${\cal C}$.
The coefficient of ${\cal C}^n$
in Eq.\bref{fqbs Delta S reg} vanishes.
To see this,
let $\psi_r$ and $ E_r^2$
be the eigenfunctions and eigenvalues
of any of the $\tilde {\cal R}$ operators:
\be
\tilde {\cal R} \psi_r =  E_r^2 \psi_r
\quad .
\label{psi_r}
\ee
The $\psi_r$ can be chosen to be real.
The terms
in \bref{def of F tilde}
involving ${\cal C}^n$
enter in the combination
$
{{\cal C}^n \partial_n +
{1 \over 2}\left( {\partial_n{\cal C}^n} \right)}
$.
Such a combination gives a zero contribution to
$ \left( {\Delta S} \right)_{\rm reg}$ since
$$
\left( {{\cal C}^n \partial_n +
{1 \over 2}\left( {\partial_n{\cal C}^n} \right)} \right)
\left( {\exp
\left( {{{\tilde {\cal R}} \over {{\cal M}^2}}} \right)} \right) =
$$
$$
\int {\dif\tau }\int\limits_0^\pi {\dif\sigma }\sum\limits_r
\psi_r\left( {\tau ,\sigma } \right)
\left( {{\cal C}^n\partial_n +
{1 \over 2}
\left( {\partial_n{\cal C}^n} \right)} \right)\psi_r
\left( {\tau ,\sigma } \right)
\exp \left( {{{E_r^2} \over {{\cal M}^2}}} \right) =
$$
$$
{1 \over 2}
\int {\dif\tau }\int\limits_0^\pi {\dif\sigma }\sum\limits_r
\partial_n\left( {\psi_r^2\left( {\tau ,\sigma } \right)
{\cal C}^n\exp \left( {{{E_r^2} \over
{{\cal M}^2}}} \right)} \right)
\to 0
\quad ,
$$
where, in the last step,
we assume that quantities fall off sufficiently fast
at large $\tau$
and obey appropriate boundary conditions at
$\sigma = 0$ and $\sigma = \pi$.
The reader can also verify
the absence of a ${\cal C}^n$ anomaly directly
by using the methods in Appendix C.
To evaluate the terms in
$\left( {\Delta S} \right)_{\rm reg}$ proportional
to ${\cal C}$,
let
\be
H_r \equiv \rho^{\left( {r+1} \right)/2}
\slashit \partial \rho^r
\slashit \partial \rho^{\left( {r+1} \right)/2}
\quad .
\label{def of H_n}
\ee
The tilde regulators
in Eqs.\bref{def of R_X tilde} and
\bref{def of R_ghost tilde}
are
\be
\tilde {\cal R}_X = H_0 \ ,
\quad \quad \tilde {\cal R}_{\bar {\cal C}}=H_{2} \ ,
\quad \quad \tilde {\cal R}_{{\cal C}}=H_1
\quad ,
\label{R tilde in terms on H_n}
\ee
except that, for $X^\mu$,
$\tilde {\cal R}_X$ acts in a $D$dimensional space,
whereas
$H_0$ acts in a twodimensional space,
since
$
 \slashit \partial \slashit \partial =
I_2 \partial^n \partial_n =
I_2 \left( { \partial_\tau \partial_\tau +
\partial_\sigma \partial_\sigma } \right)
$,
where $I_2$ denotes the twodimensional unit matrix.
The nonzero terms in
$\left( {\Delta S} \right)_{\rm reg}$
in Eq.\bref{fqbs Delta S reg}
all involve
\be
{{Tr} \over 2}
\left[ {\exp \left( {{{H_r} \over
{{\cal M}^2}}} \right)} \right]_0
\equiv
\int {\dif\tau }\int\limits_0^\pi {\dif\sigma } \ \kappa_r
\quad .
\label{def of kappa_n}
\ee
The trace ${Tr}$ is over both $2$ by $2$ gamma space
and function space.
The coefficients $\kappa_r$,
which are computed in Appendix C, are
\be
\kappa_r = {1 \over {24\pi i}}
\left( {3r+1} \right)\partial^n\partial_n\ln
\left( \rho \right)
\quad .
\label{kappa_n results}
\ee
The contributions to
the violation of the quantum master equation
from the
$X^\mu$, $\bar {\cal C}_n$, and ${\cal C}_n$ sectors
are respectively $ (1) D \kappa_{0} $,
$ (1) 2 \kappa_{2} $, and
$ (+2) 2 \kappa_{1} $,
where the factors in parentheses
are the coefficients of ${\cal C}$
in the $\tilde F$
in Eq.\bref{def of F tilde}.
The total anomaly ${\cal A}$ onshell is
\ct{polyakov81a,fujikawa82a}
$$
\left( {\Delta S} \right)_{\rm reg} =
i{{\left( {D26} \right)} \over {24\pi }}
\int {\dif\tau }\int\limits_0^\pi {\dif\sigma }
\ {\cal C}\partial^n\partial_n\ln \left( \rho \right)
$$
\be
= i{{\left( {D26} \right)} \over {24\pi }}
\int {\dif\tau }\int\limits_0^\pi {\dif\sigma }
\ {\cal C}
\left.
\partial^n\partial_n\ln \left( e \right)
\right_\Sigma
\quad .
\label{fqbs anomaly}
\ee
It is absent when $D = 26$.
For $D \ne 26 $
no local counter term $\Omega_1$ can be added
to cancel the violation of the quantum master equation
via Eq.\bref{oneloop fake anomaly}
and the theory is anomalous.
\vfill\eject
\section{Brief Discussion of Other Topics}
\label{s:bdot}
\hspace{\parindent}
The following are discussed in this section:
applications to global symmetries,
a geometric interpretation of the fieldantifield formalism,
locality, cohomology,
the equivalence
of lagrangian and hamiltonian approaches,
unitarity,
the antibracket formalism in a general coordinate system,
the $D=26$ closed bosonic string field theory, and
the extended formalism for anomalous gauge theories.
One topic not addressed
is the antiBRST symmetry
\ct{cf76a,ojima80a,ab83a,baulieu85a,%
blt90a,gprr90a,gr90a,hull90a,%
bcg92a,henneaux92a,ikemori93a}.
\subsection{Applications to Global Symmetries}
\label{ss:ags}
\hspace{\parindent}
Certain models
with continuous rigid symmetries
share some of the characteristics of
gauge theories,
such as the closure only onshell
of the commutator algebra
and the presence of
fielddependent structure constants.
Global supersymmetric theories
without auxiliary fields
and
models employing nonlinear realizations
of rigid symmetries
are often
examples of algebras that do not close offshell.
The antibracket formalism
can be used to assist
in the analysis of such theories
\ct{bbow90a,hlw90a}.
Even though, in the rigidsymmetry case,
the parameters
$\varepsilon^{\alpha}$
in the transformation law
in Eq.\bref{trans gauge}
are not functions of the spacetime variable $x$,
there is still the notion of a symmetry structure.
In other words,
the analogs of the structure equations
in Sect.\ \ref{s:ssgt},
such as the Noether identity,
the Jacobi identity, etc.,
still exist.
There are two differences for the globally symmetric case:
(a) everywhere $\varepsilon^{\alpha}$ appears,
it is a constant
and (b)
the compact notation
for Greek indices,
associated with gauge transformations,
involves a discrete sum
but not an integral over spacetime.
For Latin indices,
associated with the $\phi^i$,
repeated indices still indicate a spacetime integral.
Taking into account the above two differences,
the equations
in Sect.\ \ref{s:ssgt}
hold for the globally symmetric case.
The development of an antibracketlike
formalism proceeds as
in Sect.\ \ref{s:faf}.
Since
the gauge parameters
$\varepsilon^{\alpha}$
are not functions of the spacetime variable $x$,
one introduces
{\it constant} ghosts ${\cal C}^\alpha$.
The antifields $\phi_i^*$ for the original fields
$\phi^i$ are spacetime functions,
but the antifields for ghosts are constants.
Grassmann statistics and ghost numbers
are assigned as in the gaugetheory case.
In the antibracket and elsewhere,
functional derivatives with respect to ghosts
and antifields of ghosts
are replaced by ordinary partial derivatives.
The proper solution $S$
in Eq.\bref{fsr proper solution}
is a generating functional for the structure tensors.
The structure equations are encoded in
the classical master equation $ ( S , S ) = 0$.
Of course,
since global symmetries
do not affect the rank of the hessian of $S$
at a stationary point,
the concept of properness has little meaning:
If one wants to treat all global symmetries
via an antibracketlike formalism,
one should proceed by mimicking the gauge case.
Since global symmetries
do not upset the development of perturbation theory,
no gaugefixing procedure
via a fermion $\Psi$ is implemented.
In the quantum theory,
the ghosts are only technical tools.
They should not be considered as quantum fields.
Antifields are still interpreted
as sources for rigid symmetries.
To perform standard perturbation theory,
antifields can be set to zero.
Alternatively,
one can differentiate with respect to antifields
before setting them to zero
to obtain global Ward identities.
A third approach is
to introduce sources $J$ for fields
via Eq.\bref{Z of J},
retain antifields,
and construct the effective action $\Gamma$
as described
in Sect.\ \ref{ss:eazje}.
Anomalous violations of global symmetries
can be analyzed by searching for violations
of the ZinnJustin equation $( \Gamma , \Gamma )_c = 0$
(see Eq.\bref{anomalous ZJ equation}).
Examples of anomalous global symmetries
are the axial vector currents
of massless fourdimensional QCD.
An application to the $D=4$ supersymmetric
WessZumino model is given
in ref.\ct{hlw90a}.
\subsection{A Geometric Interpretation}
\label{ss:agi}
\hspace{\parindent}
This subsection
discusses a geometric interpretation
of the fieldantifield formalism,
as presented by E.\ Witten
\ct{witten90a}.
See also
refs.\ct{khudaverdian91a,henneaux92a,bt93a,schwarz93a,kn93a}.
The geometric intepretation is made clearer
if we first assume
that no fermionic fields are present.
Let ${\cal M}$ denote the manifold
of infinitedimensional function space.
The classical fields $\Phi^A$ of the theory
are local coordinates for ${\cal M}$.
Then,
${ {\partial} \over {\partial \Phi^A } }$
is a local basis
for the tangent space ${\cal TM}$
of vector fields.
Likewise, $d \Phi^A$ is a basis
for the cotangent space ${\cal T}^{*}{\cal M}$
consisting of differential forms.
There exists a natural quadratic form
on ${\cal TM} \oplus {\cal T}^{*}{\cal M}$
given by
$
\langle d \Phi^A ,
{ {\partial} \over {\partial \Phi^B } } \rangle
= \delta_B^A
$,
$\langle d \Phi^A , d \Phi^B \rangle = 0$,
$
\langle { {\partial} \over {\partial \Phi^A } } ,
{ {\partial} \over {\partial \Phi^B } } \rangle
= 0
$.
Introduce, in an ad hoc manner,
two quantities $z^A$ and $w_A$ and
associate $z^A$ with $d \Phi^A$
and $w_A$ with ${ {\partial} \over {\partial \Phi^A } }$, i.e.,
\be
z^A \leftrightarrow d \Phi^A
\ , \quad \quad
w_A \leftrightarrow { {\partial} \over {\partial \Phi^A } }
\quad .
\label{witten association}
\ee
Consider the Clifford algebra
for $z^A$ and $w_A$
determined by the quadratic form
$\langle \ , \ \rangle $, namely
\be
\{ z^A , w_B \} = \delta_B^A
\ , \quad \quad
\{ z^A , z^B \} =0
\ , \quad \quad
\{ w_A , w_B \} =0
\quad ,
\label{clifford algebra}
\ee
where $\{ \ , \ \} $ denotes
the anticommutator:
$ \{ x , y \} = x y + y x $.
A possible representation
of the Clifford algebra
regards the $z^A$ as creation operators
and the $w_A$ as destruction operators.
Then, the most general state
at a point $\Phi$ on the manifold
is created by
$
\Omega \left( { \Phi , z } \right) =
\Omega_0 \left( { \Phi } \right) +
\Omega_A \left( { \Phi } \right) z^A +
\frac12 \Omega_{AB} \left( { \Phi } \right) z^B z^A +
\dots
$
acting on a Fockspace vacuum $  0 \rangle $,
defined by
$w_A  0 \rangle = 0$ for all $A$,
i.\ e.,
it is annihilated by all the $w_A$.
Representing the $w_A$
as ${ {\partial} \over {\partial z^A } }$,
$  0 \rangle $ can be taken to be $1$
when considered as a function of the $z^A$.
With the association
$z^A \leftrightarrow d \Phi^A $,
one sees that $\Omega$ is equivalent
to an element of the exterior algebra
of differential forms on ${\cal M}$,
in which differential forms are multiplied by
using the wedge product $\wedge$.
When supplemented with the exterior derivative $d$,
this structure becomes the de Rham complex
\ct{flanders63a,egh80a}.
In summary,
one has an irreducible representation
of the Clifford algebra
in Eq.\bref{clifford algebra}
at each point of the manifold.
The Clifford algebra
in \bref{clifford algebra}
is symmetrical in its treatment
of the elements $z$ and $w$.
Hence,
one can reverse the above viewpoint
and regard the $w_A$ as creation operators
and the $z^A$ as annihilation operators.
In this picture,
let us identify the antifields $\Phi_A^*$
of the antibracket formalism
with the vectorfieldlike objects $w_A$.
The most general state
at a point $\Phi$
is created by
\be
F \left[ \Phi, \Phi^* \right] =
F_0 ( \Phi ) + F^A ( \Phi ) \Phi_A^* +
\frac12 F^{AB} ( \Phi ) \Phi_B^* \Phi_A^* + \dots
%\quad ,
\label{dual clifford element}
\ee
acting on a state
$  0 \rangle^{\prime} $
that is annihilated by all $z^A$.
In this picture, denoted by $R^{\prime}$ by E.\ Witten,
$z^A = { {\partial}_r \over {\partial \Phi_A^* } } $,
$w_A = \Phi_A^*$,
and $  0 \rangle^{\prime} $
can be taken to be $1$
when regarded as a function of the $\Phi_A^*$.
Two elements $F$ and $G$ in the form
of Eq.\bref{dual clifford element}
are multiplied using
$
\{ \Phi_A^* , \Phi_B^* \} = 0
$.
Exploiting the association
$
d \Phi^A \to z^A
= { {\partial}_r \over {\partial \Phi_A^* } }
$
in the $R^{\prime}$ picture,
the exterior derivative
$
d \equiv
{ {\partial}_r \over { \partial \Phi^A } } d \Phi^A
$
becomes
$
{ {\partial}_r \over { \partial \Phi^A } }
{ {\partial}_r \over { \partial \Phi_A^* } } F
=  \Delta F
$
when acting on a general functional $F$
of the type
in Eq.\bref{dual clifford element}.
Here, we have used the definition
of $\Delta$ given
in Eq.\bref{def Delta}.
In computing $\Delta F$,
one treats
$\Phi^A$ and $\Phi_B^*$ as independent variables.
Because the exterior derivative is nilpotent,
$\Delta$ satisfies $\Delta^2 = 0$.
In short,
one arrives at a dual picture of the de Rham complex.
It is isomorphic to the standard de Rham complex
but not in a natural way
because there is no preferred manner of associating
the above two Fockspace vacuums
$  0 \rangle $ and
$  0 \rangle^{\prime} $.
The state
$  0 \rangle^{\prime} $
of the $R^{\prime}$ picture
is represented as
$ \left( f_{12 \dots } \right) \prod_A z^A$
in the first picture,
where $f_{12 \dots }$ is arbitrary.
A natural choice {\it does} exist if ${\cal M}$
is endowed with a measure
$
d \mu =
\left( {\mu_{12 \dots} } \right)
d \Phi^1 \wedge d \Phi^2 \wedge \dots
$.
Then, one can take $f=\mu$.
If fermionic fields are present,
${\cal M}$ is a supermanifold.
Then, $\{ \ , \ \} $ appears as
a graded commutator:
$
\{ x , y \} =
x y  (1)^{\epsilon_x \epsilon_y} y x
$.
Note that
$
\epsilon \left( { z^A } \right) =
\epsilon_A + 1 =
\epsilon \left( { w_A } \right)
$,
so that $z^A$ and $w_A$ have
the opposite statistics of $\Phi^A$.
For the bosonic case,
$ \epsilon_A = 0$ for all $A$,
and $\{ \ , \ \} $
becomes the usual anticommutator.
In the $R^{\prime}$ picture,
the Clifford algebra
in Eq.\bref{clifford algebra}
is satisfied when
\be
w_A = \Phi_A^*
\ , \quad \quad
z^A =
(1)^{\epsilon_A}
{ {\partial}_r \over {\partial \Phi_A^* } }
\quad .
\label{witten identification}
\ee
Then, $d$ becomes,
when acting on $F$
of Eq.\bref{dual clifford element},
\be
(1)^{\epsilon_A + 1}
{ {\partial}_r \over { \partial \Phi^A } }
{ {\partial}_r \over { \partial \Phi_A^* } } F
= \Delta F
\quad .
\label{dual DeRhamlike d}
\ee
The nilpotent operator $\Delta$
of the fieldantifield formalism
is identified with minus
the exterior derivative.
For elements $F$ and $G$
in Eq.\bref{dual clifford element},
the antibracket $( \ , \ )$ is defined by
$
\left( { F \left[ \Phi, \Phi^* \right] ,
G \left[ \Phi, \Phi^* \right] } \right)
$
$
\equiv
{{ \partial_r F }
\over {\partial \Phi^A }}
{{ \partial_l G }
\over {\partial \Phi_A^* }}

{{ \partial_r F }
\over {\partial \Phi_A^* }}
{{ \partial_l G }
\over {\partial \Phi^A }}
$.
Using $\Delta$,
the antibracket can be expressed as
\be
\Delta ( FG ) 
F \Delta ( G )  (1)^{\epsilon_G} \Delta ( F ) G =
(1)^{\epsilon_G}
\left( { F [ \Phi, \Phi^* ] , G [ \Phi, \Phi^* ] } \right)
\quad .
\label{DeRhamlike antibracket2}
\ee
Eq.\bref{DeRhamlike antibracket2}
shows that the antibracket is the obstruction
of $\Delta$ to be a derivation from the right.
Once $\Delta$ has been defined,
one can take the lefthand side of
Eq.\bref{DeRhamlike antibracket2}
to be the definition of the antibracket
after multiplying by $ (1)^{\epsilon_G}$.
Summarizing, one has the following analogy.
If one thinks of function space as a supermanifold,
then antifields are the basis vectors
in the $R^{\prime}$ picture of the de Rham complex.
The operator $\Delta$ is
the analog of the exterior derivative.
The antibracket is the obstruction
to $\Delta$ for it to be a derivation.
Interestingly,
the quantum master equation
\be
\frac12 ( W , W )  i \hbar \Delta W = 0
%\quad ,
\label{qme in introduction}
\ee
has the same form as the equation of motion
for a ChernSimons theory
\ct{witten90a}.
The analog of a gauge transformation
in ChernSimons theory
is a quantum BRST transformation
\ct{lt85a}
in the antibracket formalism.
It is not always too easy
to obtain solutions $W$
to the quantum master equation.
In some sense,
finding an appropriate $W$
is equivalent to obtaining the correct measure,
i.e., specifying $\mu$.
\subsection{Locality }
\label{ss:l}
\hspace{\parindent}
An important but technical aspect
of quantum field theories is locality.
Here,
we study this issue
in the antibracket formalism
\ct{henneaux91a,ht92a,paris92a,gp93b,bbh94a,vp94a}.
In going from the classical action $S_0$
to the proper solution $S$
and to the quantum action $W$,
lagrangian terms are added.
In a theory defined by a local classical action,
the question is whether these terms
are also local.
Local interactions involve fields and
derivatives, up to a finite order, of fields
multiplied at the same spacetime point.
Nonlocal terms are likely to lead to difficulties
such as nonrenormalizability, nonunitarity
or violations of causality.
The discussion of the gauge structure algebra
in Sect.\ \ref{s:ssgt}
used extensively the consequences of
the regularity condition
in Eq.\bref{consequence of rc}.
An examination of the proof
of Eq.\bref{completesa}
reveals that certain operators need to be inverted
so that nonlocal effects are possible.
Indeed,
it is easy to find a $\lambda^i$
so that the solution to $S_{0,i} \lambda^i = 0$
in Eq.\bref{completesa}
involves a nonlocal operator $T^{ji}$ or a function
$\lambda^{\prime \alpha_0}$ that does not fall off fast
at large spacetime distances.
Nonlocality often occurs when the quantity of interest
vanishes because it is an integral of a total derivative.
As an example,
consider $n$ free quantum mechanical particles
governed by the action
$S_0 = \frac12 \int \dif t \ \dot q^i \dot q_i $.
Note that $S_{0,i} =  \ddot q_i $.
Let $\lambda^i = \dot q^i$.
Then,
$
S_{0,i}\lambda^i =  \frac12 \int \dif t
{d \over {dt}} \left( { \dot q^i \dot q_i } \right) \to 0
$.
The solution
in Eq.\bref{completesa}
is $\lambda^{\prime \alpha_0}=0$
and
$$
T^{ji} \left( {t',t} \right) =
\left\{ \matrix{ \ \ {{\delta ^{ji}} \over 2} \ ,
\quad \quad {\rm for \ } \ t>t' \quad , \hfill\cr
 {{\delta ^{ji}} \over 2} \ ,
\quad \quad \hbox{\rm for \ } t 0$
reflects the consequences of the regularity condition
used in Sect.\ \ref{s:ssgt}.%
{\footnote{
If one does not use a proper solution for $S$,
$H_{k} \left( { \delta_{kt} } \right)$
can be nonempty for $k > 0$.}}
Although the above discussion has been formal,
a more rigorous analysis can be given.
See refs.\ct{fh90a,henneaux90a}.
Insight into the physical significance
of the KoszulTate cohomology is gained
by computing
$H_{0} \left( { \delta_{kt} } \right)$.
A general closed element
of the zero ghostnumber sector
is a functional $\alpha_0$ of the fields $\phi$
since
$
\delta_{kt}\alpha_0\left( \phi \right)
$
is automatically zero
due to $\delta_{kt} \phi^i = 0$.
Let $\beta_{1}$ be a general element
of the $1$ ghost sector, i.e,
$
\beta_{1} =  \phi_i^*\lambda^i
$,
where we include a minus sign for convenience,
and where the Grassmann nature of $\lambda^i$ is
$
\epsilon \left( {\lambda^i} \right) =
\epsilon_i
$.
Apply the KoszulTate differential
to $\beta_{1}$ to obtain
$
\delta_{kt}\beta_{1} =
S_{0,i}\lambda^i
$.
One concludes that
if
$
\alpha_0 = S_{0,i}\lambda^i
$
then $\alpha_0$ is exact.
Therefore $\alpha_0$ and $\alpha_0{'}$
are not related
under the equivalence relation $\sim \atop 0$
if $\alpha_0$ and $\alpha_0{'}$ differ
on the stationary surface $\Sigma$
where the equations of motion $S_{0,i} = 0$ hold.
Consequently,
$
H_0\left( {\delta_{kt}} \right)
$
corresponds to the set of distinct functions on $\Sigma$.
More precisely,
\be
H_0\left( {\delta_{kt}} \right)
= \left\{ {
\alpha_0 \left( \phi \right) \mid
\alpha_0 \sim \alpha_0^{'} \; ,\;
\mbox{if}\; \alpha_0 
\alpha_0^{'} =
S_{0,i} \lambda^i\; \mbox{\rm for some}\; \lambda^i
} \right\}
\quad .
\label{H_0 of delta_kt}
\ee
Suppose that a theory has no gauge invariances.
Then the classical observables correspond to
functionals taking on distinct values on $\Sigma$.
This space is
$ H_0\left( {\delta_{kt}} \right) $.
If a theory has gauge invariances,
then the observables should
be the gaugeinvariant elements of
$ H_0\left( {\delta_{kt}} \right) $.
To facilitate the issue of gauge invariance,
one introduces the vertical differential $\delta_{g}$
\ct{fhst89a,henneaux90a}.
An alternative name for $\delta_{g}$
is the ``exterior derivative
along the gauge orbit''.
It is defined as
\be
\delta_g X =
\left. { \left( { \delta_B X } \right) } \right_{G_{}} =
\left. { \left( {X,S} \right) } \right_{G_{}}
\quad ,
\label{def of delta_g}
\ee
where $G_{}$ corresponds to the condition
of setting antifields to zero
and going onshell with respect to the original fields, i.e.,
\be
G_{} =
\left\{ { {\Phi_A^* = 0} , \rder{S}{\phi^i} = 0 } \right\}
\quad .
\label{def of G}
\ee
Because $\delta_g$ is defined in terms of $\delta_B$,
it is a derivation from the right:
\be
\delta_g \left( {XY} \right) =
X \delta_g Y +
\left( {1} \right)^{\eps_y}\left( {\delta_g X} \right) Y
\quad .
\label{derivation property of delta_g}
\ee
Antifields can be ignored
in evaluating the vertical differential
because they are either set to zero
or transformed to zero since
\be
\delta_g \Phi_A^* =
\left. {\left( {{{\partial_l S} \over {\partial \Phi^A}}}
\right)} \right_{G_{}} = 0
\quad ,
\label{delta_g on antifields}
\ee
which follows from ghost number considerations
or Eq.\bref{def of G}.
On fields, one has
\be
\delta_g \Phi^A =
\left. { \left( {{{\partial_l S} \over {\partial \Phi_A^*}}
} \right)} \right_{G_{}}
\quad .
\label{delta_g on fields}
\ee
To check the nilpotency of $\delta_g$,
one only needs to check that
$ \delta_g \delta_g \Phi^A = 0 $.
A straightforward calculation gives
$$
\delta_g^2 \Phi^A =
\left. {\left( {
{{\partial_r \partial_l S} \over
{\partial \Phi^B\partial \Phi_A^*}}
{{\partial_l S} \over {\partial \Phi_B^*}}
} \right)} \right_{G_{}}
$$
$$
= \left. {\left( {{{\partial_l} \over {\partial \Phi_A^*}}
\left( {{{\partial_r S} \over {\partial \Phi^B}}
{{\partial_l S} \over {\partial \Phi_B^*}}} \right) 
\left( {1} \right)^{\left( {\eps_A +1} \right)\eps_B}
{{\partial_r S} \over {\partial \Phi^B}}
{{\partial_l\partial_l S} \over {\partial \Phi_A^*\partial \Phi_B^*}}
} \right)} \right_{G_{}}
$$
$$
= {1 \over 2}\left. {\left( {
{{\partial_l \left( {S,S} \right)} \over {\partial \Phi_A^*}}
} \right)} \right_{G_{}}=0
\quad ,
$$
where Eqs.\bref{master equation}
and \bref{delta_g on antifields}
have been used.
As a consequence of nilpotency,
a cohomology with respect to $\delta_g$ can be defined.
The physical relevance of $\delta_g$
can be seen by computing $ H_0 \left( {\delta_g} \right)$.
The action of $\delta_g$ on the original fields $\phi^i$ is
\be
\delta_g \phi^i = R_\alpha^i{\cal C}^\alpha
\quad .
\label{delta_g on phi^i}
\ee
Without loss of generality,
a functional $\alpha_0$ with ghost number $0$
can be taken to be a functional of the $\phi^i$ only.
Such a functional is closed if
$\delta_g \alpha_0 \left( \phi \right) = 0$.
Using Eq.\bref{delta_g on phi^i},
one finds
$$
\delta_g \alpha_0 =
\alpha_{0,i} R_\alpha^i {\cal C}^\alpha =
0 \ \Rightarrow \alpha_0 {\rm \ is \ gauge \ invariant \ }
\quad .
$$
Since any functional $\beta_{1}$ with ghost number $1$
is annihilated by $\delta_g$,
a closed $\alpha_0$ cannot be exact:
$
\alpha_0 \ne \delta_g \beta_{1}
$.
The conclusion is that
\be
H_0\left( {\delta_g} \right) =
\mbox{\rm the set of gaugeinvariant functionals}
\quad .
\label{H_0 of delta_g}
\ee
With the above insights,
one realizes that observables should
roughly correspond to
$
H_0 \left( {\delta_{kt}} \right)
\cap H_0\left( {\delta_g} \right)
$.
However, a difficulty arises.
The intersection
$
H_0 \left( {\delta_{kt}} \right)
\cap H_0\left( {\delta_g} \right)
$
does not make sense
unless
$
\delta_{kt} \delta_g + \delta_g \delta_{kt}
= 0
$.
The situation
is analogous to the one in quantum mechanics
where one seeks a state that is simultaneously
the eigenvector of two different operators.
Such a state is possible if the two operators commute.
Because of the graded nature
of $\delta_{kt}$ and $\delta_g$,
the analog condition is that
$\delta_{kt}$ and $\delta_g$ anticommute.
The difficulty can be posed as a question:
Should one take the gaugeinvariant elements of
$H_0 \left( {\delta_{kt}} \right) $
or should one take the elements of
$H_0\left( {\delta_g} \right)$
modulo the equivalence relation
of Eq.\bref{H_0 of delta_kt}?
If
$
\delta_{kt} \delta_g + \delta_g \delta_{kt}
= 0
$,
then the above two procedures yield the same result.
In such a case,
one can define a nilpotent BRST operator
$\delta_B \equiv \delta_{kt} + \delta_g$,
and the observables correspond to the BRST cohomology.
Unfortunately,
$
\delta_{kt} \delta_g + \delta_g \delta_{kt}
\ne 0
$
in general.
An inspection of $\delta_B$,
$\delta_{kt}$ and $\delta_g$ reveals that
$\delta_B = \delta_{kt} + \delta_g + {\it extra \ terms}$.
The {\it extra terms} render $\delta_B$ nilpotent,
by compensating
for the failure of the anticommutivity of
$\delta_{kt}$ and $ \delta_g$.
The BRST operator is the natural extension of
$\delta_{kt} + \delta_g$.
The elements of the cohomology of $\delta_B$
are the classical observables
\ct{fh90a,henneaux90a}.
They are the definition
of what one means
by the ``gaugeinvariant functionals on $\Sigma$''.
When quantum effects are incorporated,
the quantum BRST transformation $\delta_{\hat B}$
is relevant.
As discussed
in Sect.\ \ref{ss:qBRSTt},
the quantum observables
correspond to the elements
of the cohomology of $\delta_{\hat B}$.
Because canonical transformations
preserve the antibracket,
the cohomology of $\delta_{B}$
is independent of the basis,
as can be seen as follows.
Given a proper solution $S [ \Phi , \Phi^* ]$
in one (untilde) basis,
a solution $\tilde S [ \tilde \Phi , \tilde \Phi^* ]$
in another (tilde) basis
is given by
$
\widetilde S [ \tilde \Phi , \tilde \Phi^* ]
\equiv S [ \Phi , \Phi^* ]
$.
Likewise,
given any functional $X [ \Phi , \Phi^* ]$,
one can define a functional $\widetilde X$
of tilde fields using
$
\widetilde X [ \tilde \Phi , \tilde \Phi^* ]
\equiv X [ \Phi , \Phi^* ]
$.
The tilde antibracket of $ \widetilde X$ and $\widetilde S$,
as a function of tilde fields and antifields,
equals
$(X,S)$ as a function of untilde fields.
Hence, $\widetilde X$ is closed if and only if $X$ is,
and $\widetilde X$ is exact if and only if $X$ is.
Consequently, there is an exact isomorphism
of the cohomologies.
Since the gaugefixed BRST transformation
$\delta_{B_\Psi}$ is not nilpotent,
one cannot directly define a cohomology
associated with $\delta_{B_\Psi}$.
However, according
to Eq.\bref{nilpot cal of gf BRST},
$\delta_{B_\Psi}^2$ is proportional to the equations
of motion
of the gaugefixed action $S_{\Psi }$.
Define an equivalence relation,
denoted by $\approx$,
that equates two quantities
if they differ by terms proportional to
the equations of motion for $S_{\Psi }$.
Then, one has $\delta_{B_\Psi}^2 \approx 0$
and a gaugefixed BRST cohomology
can be defined.
What is the relation between
$H_n \left( { \delta_{B_\Psi} } \right)$
and $H_n \left( { \delta_{B} } \right)$?
The connection is best seen
by going to the gaugefixed basis
for $\delta_{B}$.
Let
\be
Y ( \tilde \Phi , \tilde \Phi^*) =
y ( \tilde \Phi ) +
y^A ( \tilde \Phi ) \tilde \Phi_A^* + \dots
%\quad ,
\label{yseries}
\ee
be the antifield expansion of a functional $Y$
in the gaugefixed basis
$\tilde \Phi^A$ and $\tilde \Phi_A^*$
of Sect.\ \ref{ss:gfb}.
Eqs.\bref{gaugefixed BRST} and
\bref{gf BRST trans for antifields}
imply
\be
\delta_B Y =
\delta_{B_\Psi} y 
y^A \lder{S_\Psi}{\tilde \Phi^A}
+ O ( \tilde \Phi^* )
\quad ,
\label{delta relations1}
\ee
so that
\be
\left.{ (\delta_B Y ) }
\right_{ \{ { \tilde \Phi^* = 0 } \} } \approx
\delta_{B_\Psi} y
\quad ,
\label{delta relations2}
\ee
since the last term
in Eq.\bref{delta relations1}
is proportional to gaugefixed equations of motion.
Eq.\bref{delta relations2} implies
that if $Y$ is $\delta_B$closed
then $y$ is $\delta_{B_\Psi}$closed,
and that if $Y$ is $\delta_B$exact
then $y$ is $\delta_{B_\Psi}$exact.
This is not enough to establish any relation
between
$H_n \left( { \delta_{B_\Psi} } \right)$
and $H_n \left( { \delta_{B} } \right)$.
Given a element of $y$
of $H_n \left( { \delta_{B_\Psi} } \right)$,
one must uniquely construct
an element $Y$
of $H_n \left( { \delta_{B_\Psi} } \right)$,
under the condition that
$ \left.{ Y } \right_{ \{ { \tilde \Phi^* = 0 } \} } = y $.
In other words,
one must find the higherorder terms
in Eq.\bref{yseries}.
References
\ct{henneaux89a,fh90a,fisch90a}
succeeded in doing this.
For additional discussion,
see refs.\ct{ht92a,tpbook}.
The cohomologies governed by
$\delta_{B_\Psi}$ and $\delta_{B}$
are equivalent.
\subsection{Equivalence with the Hamiltonian BFV Formalism}
\label{ss:ewhBFVf}
\hspace{\parindent}
Gauge theories can also be analyzed
using a hamiltonian formalism.
For the generic theory,
the BatalinFradkinVilkovisky (BFV) approach
\ct{fv75a,bv77a,ff78a,bf83a,bf86a}
is quite useful.
For the simplest theories,
such as particle models, YangMills theory, and gravitation,
it is not difficult to show that it yields
results equivalent to the fieldantifield formulation.
Demonstrating the equivalence in general,
at the classical level
or formally at the quantum level without regularization,
has been the subject of the work
in refs.\ct{bgpr89a,fh89a,siegel89a,siegel89b,%
dfgh91a,ggt91a,ggt92a,nr92a,paris92a,%
dejonghe93a,dejonghe93b}.
A review of the BFV hamiltonian formalism
is given
in ref.\ct{henneaux85a}.
Here, we present only the key ideas.
Let $S_0$ be
the classical action
as determined from a lagrangian $L$
by $S_0 = \int \dif t \ L$.
The hamiltonian $H_{S_0}$
associated with ${S_0}$
is constructed
in the standard manner using%
{\footnote
{In this subsection, we use the convention
that a field index also
represents a spatial position.
An index
appearing twice
represents a sum not only that index
but also an integration over space.
This is the hamiltonian analog of the compact
notation described
in Sect.\ \ref{ss:gt}.
The difference, here, is that time is not included
as part of the integration. }
}
\be
H_{S_0} [ \phi, \pi ] \equiv \dot \phi^i \pi_i  L
\quad ,
\label{H construction}
\ee
where a dot over a field indicates a time derivative,
where the conjugate momentum of $\phi^i$
is $\pi_i \equiv \lder{S_0}{\dot \phi^i}, i=1, \dots , n$,
and where
$H_{S_0}$ is obtained as a function of the $\phi$
and $\pi$
by solving for $\dot \phi$ in terms of the $\pi$ and
$\phi$.
For some systems,
this velocitymomentum inversion process is not possible
due to the presence of primary constraints.
Even in this case, a hamiltonian
$H_{S_0}$ can be uniquely constructed
on the surface of these primary
constraints.
We symbolically represent the procedure
of obtaining a hamiltonian from an action
diagrammatically as
$$
\matrix{
S_0 \cr
\Big\downarrow \cr
H_{S_0} \cr
}
$$
For a wide class of gauge theories,
$H_{S_0}$ is of the form
\be
H_{S_0} = H_0 \left[ { \varphi , \pi } \right] +
\lambda^\alpha T_\alpha \left[ { \varphi, \pi } \right]
\quad ,
\label{H_S_0}
\ee
where
the original $n$ variables $\phi^i$
are split into dynamical degrees of freedom
$\varphi^a, a =1, \dots , m \le n$
and Lagrange multipliers $\lambda^\alpha$
for the (secondary) constraints $T_\alpha$.
In Eq.\bref{H_S_0},
$H_0$ and $T_\alpha$ are functions of the $\varphi$
and their momenta only.
The velocities $\dot\lambda^\alpha$ are
usually assumed not to appear
in $S_0$.
This means that
the momenta of the $\lambda^\alpha$,
namely $\pi_\alpha$,
are primary constraints
and do not enter in $H_{S_0}$.
For example,
in a YangMills theory,
the hamiltonian density ${\cal H}_0$ is
${\cal H}_0 = \frac12 E_a^i E^a_i + \frac14 F^a_{ij} F^{ij}_a$,
where $ E_a^{i} =  F_a^{0i} = F_{a0i}$ are the canonical momenta
for the potentials $A^a_i$,
the constraints $T_\alpha$
correspond to Gauss's law:
$
T_a =  {D_{ia}}^{b} E_{b}^i
$,
and the Lagrange multipliers $\lambda^\alpha$
are $A_0^a$.
For simplicity, assume that the constraints $T_\alpha$
and the hamiltonian $H_0$
are first class, i.e,
\be
\{ T_\alpha, T_\beta \}_{PB} =
T_{\alpha\beta}^\gamma T_\gamma
\ ,\quad \quad
\{ H_0, T_\alpha \}_{PB} = V_\alpha^{\beta} T_\beta
\quad ,
\label{first class}
\ee
where $\{ \ , \ \}_{PB}$ denotes the graded Poisson bracket
defined by
\be
\{ F , G \}_{PB} =
(1)^{\eps_i} \rder{F}{\phi^i} \lder{G}{\pi_i} 
\rder{F}{\pi_i} \lder{G}{\phi^i}
\quad .
\label{Poisson bracket}
\ee
Here, the sum over $i$ is such that all fields
and momenta are included.
If the constraints are second class,
Dirac brackets
\ct{dirac64a}
must be used.
Note that $\{ \pi_i , \phi^j \}_{PB} =  \delta_i^j $.
The BFV program is based on BRST invariance.
One introduces ghosts and their conjugate momenta.
The ghosts needed
correspond to the minimal set,
introduced
in Eq.\bref{field set}.
For the irreducible case,
they are the ${\cal C}^\alpha$.
We use $\agh P_a$
to denote the momentum associated with a ghost
${\cal C}^a$,
where $a$ is a label that enumerate all ghosts.
The Poisson bracket
in Eq.\bref{Poisson bracket}
is then extended to include a sum over ghosts.
With these conventions,
$ \{ \agh P_b , {\cal C}^a \}_{PB} =  \delta_b^a $.
The ghost numbers and statistics of the BFV ghosts
are the same as
in Sect.\ \ref{ss:fa}.
For momenta,
$
{\rm gh} \left[ { \agh P_a } \right] =
 {\rm gh} \left[ { {\cal C}^a } \right]
$
and
$
\eps \left( \agh P_a \right) =
\eps \left( { {\cal C}^a } \right)
$.
A canonical generator of the BRST transformations
$Q_B$ and an extended hamiltonian $H$
are constructed,
using the requirement that $Q_B$ be nilpotent
and that $H$ be BRST invariant:
\be
\{ Q_B , Q_B \}_{PB} = 0
\ , \quad \quad
\{ H , Q_B \}_{PB} = 0
\quad .
\label{Q_B H algebra}
\ee
They can be expanded as a power series in ghost fields,
for which the first few terms are
\be
H = H_0 +
{\cal C}^\alpha V_\alpha^\beta \agh P_\beta
+ \dots
\ , \quad \quad
Q_B = {\cal C}^\alpha T_\alpha 
\frac12 (1)^{\eps_\beta} {\cal C}^\beta {\cal C}^\gamma
T_{\gamma\beta}^\alpha \agh P_\alpha + \dots
\ \ .
\label{hamiltonian ghost expansion}
\ee
It turns out that the requirement of $\{ Q_B , Q_B\}_{PB} = 0$
reproduces the relations defining the structure of the gauge algebra
at hamiltonian level.
In other words, $Q_B$ plays a role analogous to
the proper solution $S$ of the antibracket formalism.
In this approach,
${\cal O}$ is an observable if it is BRSTinvariant, i.e.,
$ \{ {\cal O} , Q_B \}_{PB} = 0$.
Thus, the hamiltonian is an observable.
Two observables ${\cal O}_1$ and ${\cal O}_2$
are considered equivalent if
${\cal O}_2 = {\cal O}_1 + \{ {\cal O}^{'} , Q_B \}_{PB}$,
for some ${\cal O}^{'}$.
A state $\ket\psi$ is called physical
if $Q_B \ket \psi = 0$.
Two states $\ket {\psi_1}$ and $\ket {\psi_2}$ are considered equivalent if
$\ket {\psi_2} = \ket {\psi_1} + Q_B \ket{ \psi'}$,
for some $\ket{\psi'}$.
Given a suitable hamiltonian $H$,
a lagrangian can be constructed via
\be
\exp \left( { \frac{i}{\hbar}
S_H [ \Phi , \dot \Phi ] } \right)
= \int \frac{\left[ \dif \Pi \right] }{2 \pi i \hbar}
\exp \left[ { { \frac{i}{\hbar}
\int \dif t \ ( \dot \Phi^i
\Pi_i  H [ \Phi , \Pi ] ) } } \right]
\quad ,
\label{action construction}
\ee
where $\Phi$ denotes all degrees of freedom and $\Pi$
denotes the corresponding momenta.
We indicate the process of constructing an action $S_H$
from a hamiltonian $H$
by the following diagram
$$
\matrix{
S_H \cr
\Big\uparrow \cr
H \cr
}
$$
In the BFV formalism,
to obtain a hamiltonian $H_\Psi$,
which is appropriate for insertion in the functional integral,
a fermion $\Psi$ with ghost number minus one
is used.
As in the antibracket formalism,
BRST trivial pairs exist.
Given two fields $\Lambda$ and $\Sigma$,
and their conjugate momenta,
$\agh P_\Lambda$ and $\agh P_\Sigma$,
a term $ \int \dif^{d1} x \ \Sigma\, \agh P_{\Lambda}$
can be added to $Q_B$ without ruining nilpotency.
The next step in the BFV program
is to introduce
additional fields and their momenta
and add them as trivial pairs to $Q_B$.
These fields are
the analogs of the auxiliary gaugefixing fields
of Sect.\ \ref{ss:gfaf}.
They include antighosts, extraghosts,
and the Lagrangemultiplier fields
of Eq.\bref{auxiliary fields}.
The fermion $\Psi$
in the hamiltonian formulation
must satisfy conditions similar to those
in Sect.\ \ref{ss:dfgfp}
for the $\Psi$ in the antibracket formalism.
We denote the BRST charge extended by
the inclusion of the additional trivial terms
by $Q^{\rm nm}_B$.
The hamiltonian $H_\Psi$ is given by
\be
H_\Psi = H  \{ \Psi , Q^{\rm nm}_B \}_{PB}
\quad .
\label{def of H_Psi}
\ee
Let
$
Z_\Psi = \int [ \dif \Phi ]
\exp \left( { \frac{i}{\hbar} S_{H_\Psi} } \right)
$,
where $ S_{H_\Psi}$ is constructed from $H_\Psi$
via Eq.\bref{action construction}.
The FradkinVilkovisky theorem
\ct{fv75a}
states that
$ Z_\Psi$ is independent of $\Psi$.
The equivalence
of the BFV hamiltonian and antibracket methods
is established
if the remaining leg of the following diagram
$$
\matrix{
& S_0 & \longrightarrow & S_\Psi & \cr
& \Big\updownarrow &&& \cr
& H_{S_0} & \longrightarrow & H_\Psi & \cr
}
$$
can be completed.
In other words,
is $S_{H_\Psi}$,
as constructed from ${H_\Psi}$
via Eq.\bref{action construction},
equivalent to $S_\Psi$
as obtained from the antibracket formalism?
Likewise,
is $H_{S_\Psi}$, as constructed from ${S_\Psi}$
via Eq.\bref{H construction},
equivalent to the BFV hamiltonian ${H_\Psi}$?
Another question
is whether the gaugefixed BRST charge $Q_{\rm Noether}$,
as constructed from $S_\Psi$ using Noether's theorem,
coincides with the BRST charge
$Q^{\rm nm}_B$ for the BFV formalism.
The affirmative answer to the above questions,
obtained
in refs.\ct{fh89a,dfgh91a},
implies that construction processes in
$$
\matrix{
& S_0 & \longrightarrow & S_\Psi & \cr
& \Big\updownarrow & & \Big\updownarrow & \cr
& H_{S_0} & \longrightarrow & H_\Psi & \cr
}
$$
commute to give equivalent results.
One can also ask whether
an equivalence occurs before the introduction
of gaugefixing and $\Psi$:
$$
\matrix{
S \ \cr
\Big\downarrow ? \cr
H \ \cr
}
$$
Clearly, a straightforward correspondence
cannot exist because $S$ contains antifields.
However, at least for closed irreducible theories,
if certain antifields are set to zero and others
are identified with ghost momenta,
then an equivalence of $H_S$,
as constructed from $S$
via Eq.\bref{H construction},
and the BFV $H$ is achieved
\ct{bgpr89a}.
Similar results have been obtained
in refs.\ct{siegel89a,dfgh91a}.
If sources for the
BRST transformations are included
at the hamiltonian level,
the above correspondence can be made clearer.
Then, the sources in the hamiltonian formulation
can be identified with antifields
in the antibracket formalism.
This method was used
in refs.\ct{ggt91a,dejonghe93a,dejonghe93b}
to establish
the equivalence in the gaugefixed basis.
An open problem is to extend all of the above analysis
to the quantum case in a rigorous manner.
That situation is more difficult
due to operator ordering problems
and the singular character of field theories.
\subsection{Unitarity}
\label{ss:u}
\hspace{\parindent}
The difficulty in proving unitarity
in covariant approaches
to quantizing gauge theories
is due to the presence of ghosts
and
of unphysical degrees of freedom with negative norms.
One often deals
with indefinitemetric Hilbert spaces.
Unitarity can be spoiled
in theories with
kinetic energy terms of the wrong sign
and/or
nonhermitian interaction terms.
Wrong sign kinetic energy terms
almost always arise in gauge theories
with particles of spin one or higher.
Due to the sign of the metric component $\eta_{00}$,
there are potential difficulties with
the temporal components,
such as $A_0$ in electromagnetism,
$A_0^a$ in YangMills theories,
and $g_{0i}$ in gravity.
FaddeevPopov and other gaugefixing ghosts
enter in loops with the wrong sign,
and would lead to a violation of unitarity,
if their contributions were considered in isolation.
Let us summarize how unitarity is established
in certain covariant quantization procedures.
First of all,
one needs to assume that
there are not any nonhermitian
interactions in the original theory and that
the spatial components of tensors have the correct sign
in kinetic energy terms.
In other words,
the theory should be ``naively'' unitary.
The first approach is as follows.
In some theories,
there exists a unitary gauge,
in which it is evident that
the unphysical excitations are not present.
If one can establish the gauge invariance
of the $S$matrix,
then unitarity can be proven by
going from a covariant gauge to an unitary one
\ct{fs80a}.
Unfortunately, this method is only well developed
for irreducible theories with closed algebras.
For reducible systems,
this approach often encounter difficulties,
although for some specific examples
it has been successfully implemented
\ct{fs88a}.
Another method for checking unitarity
is in perturbation theory
via Feynman diagrams
\ct{thooft71a,tv73a}.
Using the WardTakahashi
\ct{ward50a,takahashi57a} or SlavnovTaylor identities
\ct{taylor71a,slavnov72a},
as well as the LandauCutkosky rules
\ct{elop66a},
one tries to show
that contributions from the unphysical polarizations
of the classical fields
are cancelled by contributions from ghost fields
or from other sources.
A third approach proceeds via canonical quantization.
The ``physical sector''
is selected out
by imposing some subsidiary conditions
that
remove negative norm states.
The physical sector
should be stable under time evolution
and
should involve a nonnegative metric.
A wellknown example of this approach
is the GuptaBleuler procedure
\ct{bleuler50a,gupta50a}
for quantizing QED.
All components of the electromagnetic field $A^\mu$
are used;
however only states $\ket \varphi_{\rm phys}$
satisfying
\be
(\partial_\mu A^\mu)^{+}\ket\varphi_{\rm phys} = 0
%\quad ,
\label{gupta}
\ee
are considered,
where $(\partial_\mu A^\mu)^{+}$ denotes
the positive frequency components of $\partial_\mu A^\mu$.
This condition determines
the physical sector ${\cal H}_{\rm phys}$
in the GuptaBleuler procedure.
Unfortunately,
when applied to nonabelian YangMills theories,
this method fails to
preserve ${\cal H}_{\rm phys}$ under time evolution.
To quantize covariantly nonabelian gauge theories,
refs.\ct{cf76b,ko78a,ko79a}
proposed
\be
Q_B \ket\varphi_{\rm phys} = 0
\quad ,
\label{subs cond}
\ee
where $Q_B$ is the hermitian nilpotent BRST operator.
Eq.\bref{subs cond}
is the basis for BRST quantization.
We use $V_{\rm phys}$ to denote the space of states
annihilated by $Q_B$.
In the BRST approach,
the hamiltonian is automatically hermitian
so that the $S$matrix is a unitarity operator in $V_{\rm phys}$.
However, there is a possible difficulty with $V_{\rm phys}$.
Despite the fact that $Q_B$ commutes with the hamiltonian,
the positive semidefiniteness
of the norm of
$V_{\rm phys}$
is not ensured.
The question of unitarity in
BRST quantization
becomes that of proving
the positive semidefiniteness of $V_{\rm phys}$,
and must be analyzed model by model.
However,
T.\ Kugo and I.\ Ojima \ct{ko78a,ko79a}
(see also \ct{no90a}),
obtained criteria under which unitarity does hold.
They established a connection
with the metric structure of $V_{\rm phys}$
and the multiplets
of the algebra generated by the conserved BRST charge
$Q_B$ and the conserved ghost number charge $Q_C$.%
{\footnote{ We assume there are no anomalies
associated with $Q_B$ and $Q_C$.
For the firstquantized string,
this is actually not the case for $Q_C$,
but BRST quantization is still possible
\ct{hwang83a,ko83a,fgz86a}.
For similar analyses in other models
see refs.\ct{bmp92a,bbrt93a,bln93a,bmp93a}.
}}
With our conventions, $Q_C$ is antihermitian:
$Q_C^{\dagger}=Q_C$.
These generators satisfy
\be
[ Q_C , Q_B ] = Q_B
\ , \quad\quad
[ Q_C , Q_C ] = 0
\ , \quad\quad
\frac12 \{ Q_B , Q_B \} = Q_B^2 = 0
\quad .
\nonumber
\ee
Three types of multiplets are possible:
\medskip
(a) ``True physical states'': BRST singlets with zero ghost number.
\medskip
(b) Doublets: pairs of BRST singlets related by ghost conjugation.
\medskip
(c) Quartets: pairs of BRST doublets related by ghost conjugation.
\medskip
Roughly speaking, ghost conjugation is the operation
that interchanges
ghosts and antighosts.
Under this operation,
the sign of the ghost number of a state is flipped.
In the next three paragraphs,
we explain the classification of the multiplets.
One can choose states to be eigenfunctions of $Q_c$.
Let $\ket g$ be a state with a nonzero ghost number $g$.
Then $\ket g$ has zero norm
since $\bra g Q_c \ket g = g \bracket{g}{g} = g \bracket{g}{g}$,
the first equality arising when $Q_c$ acts to the right,
and the second equality arising when $Q_c$ acts to the left.
Nonzero matrix elements occur only when bra and ket states
have opposite ghost numbers.
Under application of $Q_B$,
the ghost number of a state is increased by one.
Such states $\ket s = Q_B\ket{s'}$ also have null norms
since $\bracket{s}{s} = \bra{s'}Q_B Q_B\ket{s'} = 0$.
Due to the nilpotency of $Q_B$,
the representations are either BRST singlets or BRST doublets.
A BRST singlet
$\ket s$ satisfies $Q_B\ket{s} = 0$
and $\ket{s} \ne Q_B\ket{s'}$ for any $\ket{s'}$.
If $Q_B\ket{s'} = \ket{s} \ne 0$,
then $\ket{s'}$ is a member of the BRST doublet
consisting of $\ket{s'}$ and $\ket{s}$.
The upper member of a doublet $\ket{s}$
is annihilated by $Q_B$,
since $Q_B \ket{s} = Q_B Q_B\ket{s'} = 0$,
and it carries one unit of ghost number more than $\ket{s'}$:
$Q_c \ket{s} = Q_c \ket{s'} + 1$.
If $\ket{s}$ is a BRST singlet and carries
ghost number zero, then it is of type (a).
If $\ket{s}$ is a BRST singlet and carries
nonzero ghost number $g$, then it is of type (b).
Under ghost conjugation, another BRST single
with ghost number $g$ is created,
thus forming the pair.
If $\ket{s}$ and $\ket{s'}$ constitute a BRST doublet,
then ghost conjugation produces another BRST doublet
and a type (c) multiplet is obtained.
For an irreducible gauge theory,
T.\ Kugo and I.\ Ojima in \ct{ko78a,ko79a}
proved that
(i) {\it if type} (a) {\it states have positive definite norm}
and
(ii) {\it if type} (b) {\it states are absent},
then quartets only appear in
$V_{\rm phys}$
through zero norm combinations.
Consequently,
when (i) and (ii) are satisfied,
$V_{\rm phys}$ has a positive
semidefinite norm.
To obtain a unitary theory,
one mods out the nullnorm states:
Two states are identified if they differ by a nullnorm vector.
Clearly, nullnorm states are identified
with the null state.
The moddingout procedure
automatically restricts states
to the zeroghost number sector,
since states with nonzeroghost number have zero norms.
Furthermore, because BRSTtrivial states
$ Q_B\ket{s'}$ are nullnorm vectors,
all that remains after modding out
are the nontrivial elements
of the $g=0$ BRST cohomology, i.e.,
states with ghost number zero that are annihilated by $Q_B$
and that cannot be expressed as
$ Q_B\ket{s'}$ for any state $\ket{s'}$.
This sector is preserved under time evolution
because $Q_B$ and $Q_C$ commute with the hamiltonian.
In the $g=0$ sector,
it makes sense to identify nullnorm states
with the null vector
because they decouple
from matrix elements involving observables,
such as the hamiltonian.
Observables ${\cal O}$ are BRSTinvariant operators:
$[ {\cal O} , Q_B ] = 0$.
If $\ket t$ is a BRSTtrivial state,
so that $\ket t = Q_B\ket{s'}$,
and if $\ket s$ is any element
of $V_{\rm phys}$, so that $Q_B \ket s = 0$,
then
$
\bra s {\cal O} \ket t =
\bra s {\cal O} Q_B\ket{s'} =
\bra s Q_B {\cal O} \ket{s'} = 0
$.
For reducible systems,
ghosts for ghosts and extraghosts arise,
some of which have zero ghost number.
Hence
a third condition arises for reducible theories:
(iii)
{\it a state of $V_{\rm phys}$
involving ghosts in the $g=0$ sector
must be a member of a quartet multiplet}.
This guarantees that they are null vectors
and do not ruin the
positive semidefiniteness of $V_{\rm phys}$.
The above conditions provide
criteria for establishing the positivity of the norm
and hence unitarity in a covariant formulation.
Reference
\ct{cf76b,ko78a,ko79a}
established unitarity for YangMills theories
by proving (i) and (ii) for this case.
In perturbation theory
and in a Fock space representation,
A.\ Slavnov
in \ct{slavnov89a}
used (i)(iii)
to obtain simpler criteria.
The most important requirements,
apart from the positivity of the norm of type (a) states,
were that $Q_B$ be nilpotent
and that it have nontrivial action
on all ghost fields or their conjugate momenta.
Under these conditions,
$V_{\rm phys}$ has a positive semidefinite norm.
Then,
S.\ A.\ Frolov and A.\ Slavnov
\ct{fs89a}
using the hamiltonian BFVBRST formalism
for lagrangians $L$ of the form
in Eqs.\bref{H construction} and \bref{H_S_0},
verified the abovementioned conditions perturbatively.
The analysis was simplified because
one could use the free BRST charge $Q^{(0)}_B$.
The criteria became that $Q^{(0)}_B Q^{(0)}_B = 0$
and that $Q^{(0)}_B$
have nontrivial action on all ghosts.
Given the validity of perturbation theory,
their result on the unitarity of a gauge theory
holds for the finite reducible case.
S.\ A.\ Frolov and A.\ Slavnov
in ref.\ct{fs90a,slavnov90a},
were able to translate
the above program into a lagrangian approach,
by using an effective action $A_{eff}$.
The action and BRST charge
were perturbatively expanded in a series:
$A_{eff} = A_{eff}^{(0)} + \dots$ and
$Q_B = Q^{(0)}_B + \dots$.
The term $A_{eff}^{(0)}$
was the leading order part
of the general gaussian gaugefixed action $S_{\Psi}$
of the fieldantifield formalism
presented
in Sect.\ \ref{ss:ogfp}.
Requiring nilpotency and BRST invariance of the action
lead to
a series of recursion relations
for the higher order terms
in $A_{eff}$ and $Q_B$.
The action $A_{eff}$, thus obtained,
is constructed using unitarity requirements.
Finally, when certain conditions on the
rank of the
gauge generators are imposed,
the free BRST charge
is seen to act nontrivially on ghosts fields
and unphysical polarizations of the classical fields,
thereby yielding a unitary theory
if the classical gaugeinvariant degrees of freedom
have a positive norm.
The problem of unitarity
in the fieldantifield formalism was addressed in
\ct{gp92a,paris92a,oppt93a,opt93a}.
A perturbative solution of the proper solution $S$
was obtained in
\ct{gp92a,paris92a,gp93b}
(see also ref.\ct{bh93a}).
Then,
a general gaussian gaugefixing procedure was performed,
using a fermion $\Psi$
of the type given
in Sects.\ \ref{ss:dfgfp} and \ref{ss:ogfp}.
It was shown that
BRST invariance of the gaugefixed action
and nilpotency of the gaugefixed BRST transformation
lead to the same recursion relations
obtained in \ct{fs90a,slavnov90a},
and that
the leading two terms of $S_\Psi$ agree
with $A_{eff}$.
The conclusion
is that the fieldantifield formalism
produces an action $S_\Psi$ that coincides
with $A_{eff}$
of ref.\ct{fs90a,slavnov90a}
obtained by unitarity considerations.
The above approaches to unitarity
are formal
in that the difficulties
with fieldtheoretic infinities are not addressed.
The renormalizability or nonrenormalizability
is not used.
To proceed rigorously,
one needs to regulate the theory
with a cutoff,
verify unitarity,
and then make sure that unitarity remains
as the cutoff is removed.
The issue of locality also enters here.
For example,
it may happen that $A_{eff}$ or $S$
contains nonlocal terms.
This does not necessarily ruin unitarity,
but might signal that the theory is nonrenormalizable
or illdefined.
Studies of unitarity without using perturbation theory
for general systems with finite degrees of freedom,
such as in quantum mechanics,
have been carried out
in ref.\ct{marnelius93a}.
\subsection{The Antibracket Formalism in General Coordinates}
\label{ss:afgc}
\hspace{\parindent}
The antibracket formalism in a general coordinate system
has been developed
in refs.\ct{witten90a,khudaverdian91a,%
bt93a,kn93a,schwarz93a,schwarz93b,hz94a}.
A brief overview is given
in \ct{ht92a}.
This approach sometimes goes by the name
{\it covariant formulation of the fieldantifield formalism}.
Possible applications are in mathematics
\ct{ps92a,av93a,lz93a,nersesian93a,horava94a} and
in string field theory
\ct{ps92a,witten92a,lz93a,hz94a,horava94a,sz94a,sz94b}.
Consider a
supermanifold ${\cal M}$ of type $(N,N)$,
meaning that there are
$N$ bosonic and $N$ fermionic coordinates.
Collectively denote these as $z^a$, $a=1,\ldots, 2N$.
In this coordinate system,
a local basis for the cotangent space ${\cal{T^*M}}$
consists of the $1$forms $d z^a$, $a=1,\ldots, 2N$.
The Grassmann parity
of a differential is the same as that
of the corresponding coordinate:
$\eps(d z^a)=\eps(z^a) = \eps_a$.
Introduce an odd twoform $\zeta$,
$\eps(\zeta)=1$,
which is nondegenerate and closed, i.e.,
$d \zeta = 0$.
In the local basis,
$\zeta$ is expressed as
\be
\zeta =  \frac12 \zeta_{ab}(z) d z^b \wedge d z^a =
\frac12 d z^b \wedge \zeta_{ba}(z) d z^a
\ , \quad \quad
\zeta_{ab}=(1)^{\eps_a \eps_b + 1} \zeta_{ba}
\quad ,
\label{zeta}
\ee
where $\eps ( \zeta_{ab} ) = \eps_a + \eps_b + 1 $ (mod 2).
Let $\zeta^{ab}$ be the inverse
of the matrix $\zeta_{ab}$.
It obeys
$
\zeta^{ab} =
(1)^{\eps_a + \eps_b + \eps_a \eps_b } \zeta^{ba}
$.
One then defines the antibracket
via Eq.\bref{sym form of antibracket}
but using $\zeta^{ab} (z)$.
Alternatively,
let $X$ be a function on ${\cal M}$.
Then one can define a vector field
$
{\mathop V \limits^\leftarrow }_X =
{{\mathop \partial \limits^\leftarrow }
\over {\partial z^a }}
\zeta^{ab}\lder{X}{z^b}
$
that acts
from right to left on functions $Y$ of ${\cal M}$:
$
[Y] {\mathop V \limits^\leftarrow }_X =
\rder{Y}{z^a}\zeta^{ab}\lder{X}{z^b}
$.
Then the antibracket
can be written as
\be
(X,Y) = \zeta[ {\mathop V \limits^\leftarrow }_X ,
{\mathop V \limits^\leftarrow }_Y ] =
[X] {\mathop V \limits^\leftarrow }_Y
\quad ,
\label{gc def of antibracket}
\ee
where the first equality follows from
$$
dz^b \wedge dz^a [\rder{}{z^c} , \rder{}{z^d} ] =
\delta_c^a \delta_b^d 
(1)^{\eps_a \eps_b} \delta_c^b \delta_a^d
\quad .
$$
In this way,
$\{ {\cal M}, \zeta \}$
becomes an odd symplectic structure.
The antibracket, defined as above,
obeys the properties
in Eqs.\bref{antibracket properties}
and \bref{bracket derivation}.
It turns out that $d \zeta = 0$
is necessary for the Jacobi identity
in Eq.\bref{antibracket properties}.
For ordinary symplectic manifolds,
there exists a natural volume element $d \mu$
obtained by wedging $\zeta$ with itself $N$ times.
Unfortunately, for an old sympletic manifold,
$\zeta \wedge \zeta = 0$.
Hence, a measure must be introduced by hand:
\be
d \mu(z) = \rho (z) \prod_{a=1}^{2N} d z^a
\quad ,
\label{volume element}
\ee
where $\rho(z)$ is a density.
The divergence of a generic vector field
$
{\mathop V \limits^\leftarrow } =
{{\mathop \partial \limits^\leftarrow }
\over {\partial z^a }} V^a
$
is defined in the usual way by
\be
{\rm div}_\rho {\mathop V \limits^\leftarrow } \equiv
\frac1{\rho}(1)^{\eps_a} \lder{(\rho V^a )}{z^a}
\quad .
\label{gc divergence}
\ee
Then,
the laplacian $\Delta_\rho$
acting on a function $X$ is defined
by taking the divergence
of the corresponding vector field via
\be
\Delta_\rho X\equiv (1)^{\eps_X}
\frac12{\rm div}_\rho
{\mathop V \limits^\leftarrow }_X =
(1)^{\eps_X +\eps_a }
\frac1{2 \rho} \lder{}{z^a}
\left(\rho\zeta^{ab}\lder{X}{z^b}\right)
\label{gc delta}
\quad .
\ee
Since one would like to use $\Delta_\rho$
as the general coordinate version
of $\Delta$
of Sect.\ \ref{ss:g},
one wants it to be nilpotent.
However, this is not necessarily the case since
$$
\Delta_\rho \Delta_\rho =
\frac12 \left[ {
\Delta_\rho \left(\frac1{\rho}
(1)^{\eps_a + \eps_b}
\lder{}{z^a} \left(\rho\zeta^{ab}\right)\right)
} \right]
\lder{}{z^b}
\quad .
$$
Hence, one requires $\rho$ to satisfy
\be
\Delta_\rho \left(\frac1{\rho}
(1)^{\eps_a + \eps_b}
\lder{}{z^a} \left(\rho\zeta^{ab}\right)\right)
= 0
\quad .
\label{density requirement}
\ee
When Eq.\bref{density requirement}
holds,
$\Delta_\rho$ is formally nilpotent and
a graded derivation of both functional multiplication
and the antibracket, i.e.,
it satisfies
Eq.\bref{Delta properties}
with $\Delta \to \Delta_\rho$.
It turns out that
Eq.\bref{density requirement}
is the necessary and sufficient condition
for the existence of Darboux coordinates locally.
For such coordinates,
$\rho = 1$ and $\zeta^{ab}$
takes the form
in Eq.\bref{sym form of antibracket}.
Then $z^a$ for $a=1,\ldots, N$ can
be identified with fields
and $z^a$ for $a=N+1,\ldots, 2N$ can
be identified with antifields.
Hence,
we employed the Darboux coordinate system
for the antibracket formalism
in Sects.\ \ref{s:faf}  \ref{s:qea}.
Darboux coordinates suffice
as long as global issues are not important.
Quantization in a general coordinate system proceeds
as in the Darboux case.
Everywhere $\Delta$ appears
in Sects.\ \ref{s:gff}  \ref{s:qea},
one replaces it by $\Delta_\rho$.
The functionalintegral measure
also must be modified.
Integration is restricted
to an $N$dimensional submanifold ${\cal N}$.
Since little distinction is made
between fermionic and bosonic coordinates
in the covariant formulation,
${\cal N}$ can be an arbitrary $(k, Nk)$ submanifold
as long as $\zeta[ V, V' ] = 0$ on ${\cal{N}}$
for any two tangent vectors $V,\, V'\in {\cal{T N}}$.
One considers a basis
$\{e_1,\ldots, e_N; h^1, \ldots, h^N\}$
for ${\cal{T M}}$,
such that
$\{e_1,\ldots, e_N\}$ is a basis for
${\cal{T N}}$
and $\zeta [ e_i, h^j ] = \delta_i^j$.
Then the volume element on ${\cal N}$ is
\be
d\mu_{\cal N}(e_1,\ldots, e_N) =
\left[ d\mu(e_1,\ldots, e_N ; h^1,\ldots,h^N) \right]^{1/2}
\label{gc measure}
\quad .
\ee
Since integration in function space is restricted
to ${\cal N}$,
the above procedure corresponds to a gaugefixing procedure.
The submanifold ${\cal N}$
can be defined using $N$ linearly independent
constraints $\Psi_A(z)=0$ satisfying
\be
( \Psi_A, \Psi_B ) = T^C_{AB}(z) \Psi_C
\quad .
\label{involution}
\ee
The vectors ${\mathop V \limits^\leftarrow }_{\Psi_A}$
are a basis for the tangent space ${\cal{T N}}$ of ${\cal N}$.
Furthermore,
as a consequence
of Eq.\bref{involution},
$
\zeta[ {\mathop V \limits^\leftarrow }_{\Psi_A} ,
{\mathop V \limits^\leftarrow }_{\Psi_B} ] = 0
$
on ${\cal N}$,
which is a consistency check.
To make contact with
the gaugefixing procedure
of Sect.\ \ref{s:gff},
one goes to the Darboux coordinate system
and chooses
$
\Psi_A = \Phi^*_A  \der{\Psi}{\Phi^A}
$.
One disadvantage of the general coordinate approach,
is that the concept of ghost number becomes obscure.
\subsection{The D=26 Closed Bosonic String Field Theory}
\label{ss:d26cbsft}
\hspace{\parindent}
A review of the current formulation of the closed bosonic string
has been given
in ref.\ct{zwiebach93a}.
Here, we present some of the salient points.
At the firstquantized level,
closed strings possess holomorphic factorization.
This means that, with the possible exception of zero modes,
the integrands of closedstring amplitudes
factorize into two openstringlike integrands,
one for leftmoving degrees of freedom
and one for rightmoving degrees of freedom.
At the secondquantized level,
there is a similar splitting.
Hence, a closed string field $A$ is a tensor product
of a left string field $A_L$
with a right string field $A_R$,
so that one can write $A$ as
\be
A = \left( { A_L; A_R } \right)
\quad ,
\label{cs field factorization}
\ee
or as a sum of terms of the form
in Eq.\bref{cs field factorization}.
The field $A_L$ (respectively, $A_R$)
is precisely of the form
of the open string case,
except a subscript $L$ (respectively, $R$)
is appended to all quantities.
One exception is the zero modes
of $X^\mu (\sigma)$, namely
the position and momentum operators.
They are the same for both left and right sectors,
so that
$x_L^\mu = x_R^\mu \equiv x^\mu$
and
$p_L^\mu = p_R^\mu \equiv p^\mu$.
The total string ghost number is the sum
of the left and right string ghost numbers, i.e.,
$
g\left( A \right) =
g_L \left( {A_L} \right) + g_R \left( {A_R} \right)
$.
One can attempt to construct closedstring field theory
along the lines
of the open string case described
in Sect.\ \ref{ss:obsft}.
It is easy to see that
not all the open string axioms can
be extended.
When the axioms hold,
PatonChan factors
\ct{pc69a}
can be appended to the string field
leading to a nonabelian YangMills gauge group.
However, closed string theories
cannot possess such a nonabelian gauge structure
\ct{schwarz82a}.
Define closedstring integration
$\int\limits_{\rm closed}$
as the product
of left and right open string integrals via
$
\int\limits_{\rm closed} {} \equiv \int_L {} \int_R {}
$,
i.e.,
for fields in the form
in Eq.\bref{cs field factorization},
one has
\be
\int\limits_{\rm closed} A = \int_L {A_L} \int_R {A_R}
\quad .
\label{cs integration}
\ee
To generate a nonzero integral
in Eq.\bref{cs integration},
$A$ must have a leftghost number of $3$
and a rightghost number of $3$ and
consequently a total ghost number of $6$:
$$
\int\limits_{\rm closed} A = 0
\ , \quad
{\rm if} \ g \left( A \right) \ne 6
\quad .
$$
Let $\sigma$ be the firstquantized variable
parametrizing the string.
It varies between $0$ and $2 \pi$
and is periodic.
To define
the closedstring star operation $\circ$,
pick two antipodal points,
e.g. $\sigma =0$ and $\sigma=\pi$.
This divides a string into two halves.
Then, $\circ$ is defined in analogy
to the open string case.
One half of one string overlaps
with one half of the other string
and what remains is the product string.
The firstquantized BRST charge $Q$
is the sum of the right and left parts:
\be
Q = Q_L + Q_R
\quad ,
\label{cs Q}
\ee
and it carries ghost number one:
$
g \left( Q \right) = 1
$.
Eq.\bref{cs Q} is equivalent to
\be
Q A =
\left( {Q_L A_L; A_R} \right) +
\left( {A_L; Q_R A_R} \right)
\quad .
\label{cs Q2}
\ee
Even though $\int$, $\circ$ and $Q$
have been defined,
there is a difficulty in obtaining a free action.
Let $C$ denote the closedstring field.
The naive term
$
\int {C \circ Q C}
$
vanishes because of ghost number considerations.
The total ghost number of the integrand,
which is
$
2g \left( C \right) + 1
$,
must be equal to $6$.
This constraint cannot be satisfied because
$g \left( C \right)$ is an integer.
To correct the problem,
various schemes can be used
\ct{bfmpp86a,lr86a,kaku88a}.
Let
\be
\bar c_0^ \equiv \bar c_0^L  \bar c_0^R
\ , \quad \quad
c_0^ \equiv \frac12 \left( { c_0^L  c_0^R } \right)
\quad .
\label{c minus}
\ee
Impose the following two constraints
on $C$
\be
\bar c_0^ C = 0
\ , \quad \quad
\left( { L_0^L  L_0^R } \right) C = 0
\quad .
\label{c bar minus constraint}
\ee
Anticipating that the freetheory equation
is $Q C = 0$,
one sees that these constraints are consistent since
$
\left\{ {Q, \bar c_0^} \right\} = L_0^L  L_0^R
$,
$
\left[ {Q , L_0^L  L_0^R } \right] = 0
$ and
$
\left[ {\bar c_0^ , L_0^L  L_0^R } \right] = 0
$.
The condition
$\left( { L_0^L  L_0^R } \right) C = 0$
is quite natural.
The operator
$L_0^L  L_0^R$ is the generator for rigid rotation
of the firstquantized parameter $\sigma$,
which labels the points along the string.
Since, in the closedstring case,
$\sigma$ is periodic,
there is no preferred choice of an origin.
The condition
$\left( { L_0^L  L_0^R } \right) C = 0$
does not lead to an equation of motion for $C$
since the $ \frac12 \partial_\mu \partial^\mu $
terms in $L_0^L$ and $L_0^R$ cancel.
{}From Eq.\bref{L 0},
one sees that it implies
that the mass ${\cal M}_L$ of the leftsector must equal
the mass ${\cal M}_R$ of the rightsector,
a wellknown constraint of firstquantized
closedstring states.
However, the operation $\circ$
does not preserve the constraint.
One can modify $\circ$ to $\hat \circ$ by averaging
over a rigid rotation that rotates the product string
over angles ranging from $ 0 $ to $2 \pi$.
The new $\hat \circ$ no longer is associative,
as can be checked by drawing some pictures.
The ghost number problem
can be fixed by inserting a factor of $c_0^$.
Define the quadratic form
\be
\left\langle {A,B} \right\rangle =
\int\limits_{\rm closed} {A \hat \circ c_0^ B}
\quad .
\label{cs two form}
\ee
Let the free action be
\be
S_0^{\left( 2 \right)} =
{1 \over 2}\left\langle { C, QC } \right\rangle
\quad .
\label{cs free action}
\ee
It is invariant under the gauge transformations
$
\delta C = Q \Lambda
$,
where $g \left( { \Lambda } \right) =1$
since $g \left( { C } \right) =2$.
There are many ways
of resolving
the difficulty with ghost number
of the free action of closedstring field theory
but they are equivalent to the above.
The treelevel threepoint interaction is
\be
S_0^{\left( 3 \right)} =
{{2g} \over 3} \int\limits_{\rm closed}
C \hat \circ C \hat \circ C
\quad .
\label{cs 3point interaction}
\ee
For onshell external states,
this interaction correctly produces threepoint interactions.
Treelevel gauge invariance
is violated for the theory described by
$ S_0^{\left( 2 \right)} + S_0^{\left( 3 \right)}$
due to the violation of the associativity axiom.
However, by adding higherorder terms
gauge invariance can be restored
\ct{kaku88a,kaku88b,kl88a,kks89a,sz89a,%
ks90a,zwiebach90a}.
The new interactions can be defined by relatively
simple geometrical constraints
\ct{zwiebach90a,zwiebach93a}.
This leads to a treelevel
nonpolynomial closedstring field theory.
Unfortunately, the classical theory needs
further modification at the quantum level.
Oneloop and higherloop amplitudes are not produced
by using only treelevel vertices.
It is at this stage where the antibracket formalism
has been of great utility.
Interaction terms proportional to powers of $\hbar$
need to be added
in a manner similar
to Eq.\bref{hbar expansion of W}.
To ensure that the theory
is quantummechanically gauge invariant,
the work
in refs.\ct{zwiebach90a,sz93a,zwiebach93a,zwiebach93b,%
hz94a,sz94a}
has relied on the antibracket formalism.
The guiding principle is that
the quantum closedstring field theory
must satisfy
the quantum master equation.
The antibracket is defined
using the quadratic form
in Eq.\bref{cs two form}
\ct{sz93a,zwiebach93a,zwiebach93b}.
As in the open string field theory,
the system is infinitely reducible
so that there are ghosts for ghosts ad infinitum.
The fields can be collected into one object
$\Psi_c$ in a manner similar
to the open string case in Eq.\bref{psi sum}
\be
\Psi_c \equiv \ldots +
\stackrel{{\textstyle{s+4}}}{{}^*{\cal C}_s^*}
+ \ldots +
\stackrel{{\textstyle{4}}}{{}^*{\cal C}_0^*}
+ \stackrel{{\textstyle{3}}}{{}^*C^*}
+ \stackrel{{\textstyle{2}}}{C}
+ \stackrel{{\textstyle{1}}}{{\cal C}_0}
+ \ldots +
\stackrel{{\textstyle{s+1}}}{{\cal C}_s}
+ \ldots
\quad ,
\label{cs psi sum}
\ee
where the ghost number is indicated above the field.
The closestring Hodge operation,
denoted by a superscript $*$ in front of a field,
is defined using the quadratic form
in Eq.\bref{cs two form}.
It takes a $p$form into a $5p$,
where the order of string form
is the same as the ghost number.
Let
$ \varphi_s $
be a complete set of normalized firstquantized states
for $g \left( {\varphi_s} \right) \le 2$.
Let ${}^* \varphi^s $ denote the corresponding state
transformed by the Hodge star operation.
The ${}^* \varphi^s $
are a normalized complete set of states for ghost numbers
greater than $2$.
With these definitions,
$\Psi_c$
in Eq.\bref{cs psi sum}
can be written
in terms of ordinary particle fields $\psi^s$
via
$
\Psi_c =
\sum_s \left(
{ \varphi_s \psi^s +
{}^* \varphi^s \psi_s^*
} \right)$
where $\psi_s^*$ are the antifields of $\psi^s
$.
The quantum master equation is
then the same as the particle case,
namely
$
{1 \over 2} \left( { W( \Psi_c ), W( \Psi_c ) } \right) =
i\hbar \Delta W( \Psi_c )
$,
where the antibracket is
$
\left( {
X \left( { \Psi_c } \right) ,
Y \left( { \Psi_c } \right)
} \right)
= \frac{\partial_r X} {\partial\psi^s}
\frac{\partial_l Y} {\partial \psi^*_s}
 \frac{\partial_r X} {\partial \psi^*_s}
\frac{\partial_l Y} {\partial\psi^s}
$
and
$
\Delta =
\left( {1} \right)^{\epsilon_s+1}
{{\partial_r } \over {\partial \psi^s}}
{{\partial_r } \over {\partial \psi_s^*}}
$.
The solution of the quantum master equation
for the closedstring field theory
is presented
in refs.\ct{zwiebach93a,zwiebach93b}.
This tourdeforce work goes beyond
the goals of our review.
The reader interested in this topic
can consult the above references
for more discussion.
The current formulation of string fields theories
is developed around a particular
spacetime background.
Any background is permitted,
as long as it leads
to a nilpotent firstquantized BRST charge.
Such BRST charges
correspond to twodimensional conformal field theories
with the total central charge
of the Virasoro algebra equal to zero.
Usually, the flat spacetime background
in $26$ dimensions is used.
Since string theories contain gravity,
it should be possible to pass
from one background to another.
It is an interesting question
of whether there
is background independence of string field theory
\ct{sen90a,ss92a,witten92a,sz93a,witten93a,sz94a}.
A proof for bosonic string field theories
has been obtained
in refs.\ct{sz93a,sz94a},
for backgrounds infinitesimally close.
The antibracket formalism has played
an important role in the analysis.
The basic idea is that string field theory,
formulated about a particular background ${\cal B}$,
corresponds to a particular solution $S_{\cal B}$
of the classical master equation.
Reference \ct{sz93a}
demonstrated that,
for two nearby backgrounds
${{\cal B}_1}$ and ${{\cal B}_2}$,
$S_{{\cal B}_1}$ and $S_{{\cal B}_2}$
are related by a canonical transformation
of the antibracket.
The conclusion is that string field theory
is background independent,
although not manifestly.
Barring difficulties with singular expressions,
ref.\ct{sz94a}
has extended the result to the quantum case.
For the quantum system, a particular background ${\cal B}$
corresponds to a solution
$W_{\cal B}$ of the quantum master equation.
\subsection{{Extended Antibracket Formalism for \hfil}
\break
Anomalous Gauge Theories}
\label{ss:eafat}
\hspace{\parindent}
In certain cases,
it is possible
to quantize an anomalous gauge theory.
An example is
the firstquantized bosonic string
for $D \ne 26$
discussed in Sect.\ \ref{ss:aobs}.
Polyakov \ct{polyakov81a}
quantized this system
in the presence of a conformal anomaly.
A new degree of freedom,
the Liouville mode, emerged.
Another example is the chiral Schwinger model
in two dimensions.
Despite its anomalous behavior,
it is consistent and unitary
\ct{jr85a}.
In refs.\ct{faddeev84a,fs84a},
additional degrees of freedom were introduced
into fourdimensional anomalous YangMills theories
to cancel the anomalies in the path integral.
This cancellation can be obtained by
adding a WessZumino term
for the extra degrees of freedom.
A careful treatment of
the integration measure in
FaddeevPopov pathintegral quantization
shows how such a WessZumino term can arise naturally
\ct{paris93a}.
For earlier approaches to this subject in the case of
the Schwinger model,
see \ct{bsv86a,ht87a}.
A treatment of anomalous chiral QCD in two dimensions
within the fieldantifield formalism was obtained
in refs.\ct{bm91a,bm94a}.
Methods of quantizing anomalous gauge theories
using the antibracket formalism
were developed
in \ct{bm91a,bm93a,gp93a,gp93c,gp93d,jst93a,bm94a}.
Let us describe in general terms
the extended antibracket method
of refs.\ct{gp93a,gp93c,gp93d}.
For simplicity we consider the closed irreducible case.
At the classical level,
the number of dynamical local degrees of freedom
$n_{\rm dof}$ is
the total number of fields $n$
minus the number of gauge invariances $m_0$, i.e.,
$n_{\rm dof} = n  m_0$.
Suppose there are
$r$ anomalous gauge invariances.
Then, due to quantum effects,
$r$ of the $m_0$ gauge degrees of freedom
enter the theory dynamically.
Hence, the true net number of degrees of freedom at quantum level is
$n  m_0 + r$.
Following the ideas
of refs.\ct{wz71a,polyakov81a,zumino83a,faddeev84a,fs84a}
for treating anomalous gauge theories,
one wants to have $r$ extra degrees of freedom.
The proposal is to augment the original set of fields,
$\phi^i$, $i = 1, 2, \dots , n$ with $r$ new fields
$\widehat \phi^{\hat i}$, ${\hat i} = 1, 2, \dots , r$,
in such a way that
the original gauge structure continues to be maintained
at classical level.
Roughly speaking, the $\widehat \phi$ are fields
parametrizing the anomalous part of the gauge group.
In what follows, a ``hat'' on a quantity indicates
that the quantity is associated
with the extra degrees of freedom
or that the quantity has been generalized
to the extended system.
The key step is to extend the antibracket formalism
to include $\widehat \phi$ variables.
To implement this idea,
one demands the extra fields to transform under the
action of the gauge group:
\be
\delta \widehat \phi^{\hat i} =
\widehat R^{\hat i}_{\alpha}
\left[ \phi , \widehat \phi \right]
\varepsilon^{\alpha}
\quad .
\label{extended trans gauge}
\ee
Let $\widehat \phi_{\hat i}^*$
be the antifield of $\widehat \phi^{\hat i}$.
Given Eq.\bref{fsr proper solution},
the classical gauge structure of
the extended theory
should be governed by an action
$
\widetilde S=S+ \widehat \phi_{\hat i}^*
\widehat R^{\hat i}_{\alpha} \gh C^{\alpha}
$.
Since field indices range over
$a = 1, 2, \dots , n, n+1, \dots, n+r$, the compact notation
$
\phi^a \equiv
[ { \phi^i , \widehat\phi^{\hat i} } ]
$
and
$
R^a_\alpha \equiv
[ { R^i_\alpha , \widehat R^{\hat i}_\alpha } ]
$
is useful.
To maintain the gauge structure,
the extended generators $R^a_\alpha$
must satisfy the original gauge algebra.
In other words,
when the field content is extended,
Eq.\bref{algebra oberta} must still hold,
where the tensors $T$ and $E$ are
functions of the $\phi^i$ only
and have the same values as in the unextended theory.
This requirement leads
to a set of equations for the generators
$ \widehat R^{\hat i}_{\alpha}$
in Eq.\bref{extended trans gauge}.
Formulas for $\widehat R^{\hat i}_{\alpha}$
in the closed case for which $E=0$
are given in refs.\ct{gp93a,gp93c}.
In the closed case,
it is possible to
solve the antifield independent part of
the original quantum master
equation at oneloop
in a quasilocal way
in the extended theory.
In particular, the WessZumino term
$ M_1 [ \phi,\widehat \phi ]$
can be written as
\be
M_1 \left[ { \phi,\widehat \phi } \right] =
i\int_0^1 ds \gh A_{\hat i}
\left( { F \left[ { \phi,\widehat \phi s } \right]
} \right) {\widehat \phi}^{\hat i}
\quad ,
\label{wz}
\ee
where the $\gh A_{\hat i}$
are the antifieldindependent part of the anomalies
and $F^i$ is the finite gauge transformation
of the classical fields $\phi^i$ under the anomalous
part of the group.
The BRST variation of $M_1$ gives the original anomaly:
$( M_1 , \widetilde S) = i {\cal A}_\alpha {\cal C}^\alpha$.
However, in the extended antibracket formalism
the action $\widetilde S$ is not proper
\ct{jst93a}.
To have a well defined perturbation expansion,
it is necessary to modify $\widetilde S$
to a new extended action $\widehat S$
that satisfies the classical master equation
\be
\left( { \widehat S , \widehat S } \right) = 0
\quad ,
\label{extended cme}
\ee
and is proper, i.e.,
\be
{\rm rank \ }
\restric{
\frac{\partial_l \partial_r \widehat S}
{\partial z^a \partial z^b} }
{ \rm {onshell} }
= n + m_0 + r
\quad ,
\label{extented def of proper solution}
\ee
where the $z^a$ include
$n$ fields, $r$ extended fields, $m_0$ ghosts,
and their antifields.
One proposal for $\widehat S$ would be
$\widetilde S+\hbar M_1$.
A difficulty is the presence of an order $\hbar$ term.
However, in certain cases
a canonical transformation
which scales $\widehat\phi^{\hat i}$ by $1/\hbar^{\frac12}$
\ct{jst93a}
can be performed
to overcome the problem.
The general structure for $\widehat S$
for anomalous gauge theories
with an anomalous abelian subgroup is
\be
\widehat S =
S 
\frac{i}2 \widehat\phi^{\hat i}
\widehat{D}_{ {\hat i} {\hat j} } \widehat\phi^{\hat j}
+ \widehat\phi^*_{\hat i} \widehat{T}^{\hat i}_{{\beta} {\hat j} }
\widehat\phi^{\hat j} \gh C^{\beta}
\quad ,
\label{extended S}
\ee
where $\widehat{D}$ and $\widehat{T}$ are tensors
which can be found
in ref.\ct{gp93c}.
In particular,
$\widehat{D}_{ {\hat i} {\hat j} }$ is related
to the BRST variation of the anomalies $\gh A_{\hat i}$.
Since $\widehat S$ satisfies the master equation,
a classical BRST symmetry can be defined using
$\delta_B X = \left( { X , \widehat S } \right)$.
The final stage is to find a solution $\widehat W$
to the quantum master equation in the extended space.
Because of the abovementioned scaling of $\widehat \phi$,
$\widehat W$ is a series in $\hbar^{\frac12}$
\ct{jst93a,vp94a}
rather than $\hbar$:
$
\widehat W =
\widehat S + \hbar^{\frac12} \widehat M_{\frac12} + \dots
$.
After regulating the extended theory,
$\left( { \Delta \widehat S } \right)_{\rm reg} $
is computed.
By appropriately adjusting $\widehat S$ and $\widehat M_{\frac12}$,
the antifieldindependent part
of the complete oneloop obstruction $\widehat{\cal A}_1$
to the quantum master equation in the extended space
can be cancelled.
In terms of the unscaled extra variables,
this adjustment corresponds to a finite renormalization
of the original expression of the WessZumino term
in Eq.\bref{wz}, $M_1 \rightarrow \widehat M_1$.
Note that
the locality of $\widehat M_{\frac12}$
cannot be guaranteed.
Likewise,
locality of the renormalized WessZumino term
is not assured due to the integral over the variable $s$
in expressions like \bref{wz}.
In some cases,
such as the firstquantized bosonic string
\ct{gp93a,gp93c} or the abelian Schwinger model
\ct{paris93a},
only local action terms are generated.
Then, the anomalous theory makes sense at the quantum level.
However,
for chiral QCD in two dimensions
\ct{gp93a},
the integral over $s$ remains.
Even in these cases,
the violation of locality is in some sense not severe:
The equations of motion are local,
a situation referred to as quasilocal.
When the quantum extended theory is welldefined,
the final stage,
namely gaugefixing,
proceeds in a manner similar to the nonanomalous case
\ct{gp93a}.
\medskip
\medskip
\noindent
{\large \bf Acknowledgements}
\hspace{\parindent}
We thank
G.\ Barnich, C.\ Batlle, F.\ De Jonghe,
M.\ Henneaux, A.\ K.\ Kosteleck\'y,
J.\ M.\ Pons, J.\ Roca, R.\ Siebelink,
A.\ Slavnov, P.\ Townsend, W.\ Troost,
S.\ Vandoren, A.\ Van Proeyen and F.\ Zamora
for discussions.
This work is supported in part
by the
Comisi\'on Interministerial para la Ciencia
y la Technolog\'ia
(project number AEN0695),
by a Human Capital and Mobility Grant
(ERB4050PL930544),
by the National Science Foundation
(grant number PHY9009850),
by a NATO Collaborative Research Grant
(0763/87),
by the Robert A.\ Welch Foundation,
and
by the United States Department of Energy
(grant number DEFG0292ER40698).
\vfill\eject
\appendix
\section{Appendix: Right and Left Derivatives}
\label{s:appendixa}
\hspace{\parindent}
In this appendix, we provide more details about
left and right derivatives
\ct{berezin66a,bl75a,leites80a,dewitt84b,berezin87a}.
For any function or functional $X$ of $\phi$,
they are defined as
\be
{{\partial_l X} \over {\partial \phi }} \equiv
{{\mathop \partial \limits^\rightarrow }
\over {\partial \phi }} X
\ , \quad \quad
{{\partial_r X} \over {\partial \phi }} \equiv
X {{\mathop \partial \limits^\leftarrow }
\over {\partial \phi }}
\quad .
\label{def lr der}
\ee
Left derivatives are the ones usually used.
Right derivatives act from right to left.
The differential $d X(\phi)$ of $X$ is
\be
d X(\phi) =
d \phi {{\partial_l X} \over {\partial \phi }} =
{{\partial_r X} \over {\partial \phi }} d \phi
\quad .
\label{dif of X}
\ee
The formula for the variation $\delta X$ of $X$
with respect to $\phi$
resembles Eq.\bref{dif of X}:
\be
\delta X(\phi) =
\delta \phi {{\partial_l X} \over {\partial \phi }} =
{{\partial_r X} \over {\partial \phi }} \delta \phi
\quad .
\label{variations of X}
\ee
What is the relation between left and right derivatives?
If $\phi$ is commuting
($\epsilon \left( \phi \right) = 0$),
then
$
{{\partial_l X} \over {\partial \phi }} =
{{\partial_r X} \over {\partial \phi }}
$,
so that one only needs to be careful when
$\phi$ is anticommuting
($\epsilon \left( \phi \right) = 1$).
Assume
$\phi$ is anticommuting.
Then $\phi \phi = 0$.
Without loss of generality we may assume that
$X = \phi Y + Z $
where $Y$ and $Z$ have no $\phi$ dependence.
The left and right derivatives of $X$ are then
$
{{\partial_l X} \over {\partial \phi }} = Y
$
and
$
{{\partial_r X} \over {\partial \phi }} =
\left( {1} \right)^{ \epsilon_Y } Y =
\left( {1} \right)^{\epsilon \left( \phi \right)
\left( \epsilon_X + 1 \right) } Y
$.
For all cases,
\be
{{\partial_l X} \over {\partial \phi }} =
\left( {1} \right)^{\epsilon \left( \phi \right)
\left( \epsilon_X + 1 \right) }
{{\partial_r X} \over {\partial \phi }}
\quad .
\label{l and r der relation}
\ee
As a pedagogical exercise,
let us derive the graded antisymmetry property
of the bracket in Eq.\bref{antibracket properties}.
Start with the definition of
$\left( {Y,X} \right)$
in Eq.\bref{antibracket def}
and interchange the order of derivatives to obtain
$$
\left( {Y,X} \right) =
{{\partial_r Y} \over {\partial \Phi^A}}
{{\partial_l X} \over {\partial \Phi_A^*}} 
{{\partial_r Y} \over {\partial \Phi_A^*}}
{{\partial_l X} \over {\partial \Phi^A}}=
$$
$$
\left( {1} \right)^{\left( {\epsilon_Y +
\epsilon_A } \right)
\left( {\epsilon_X +
\epsilon_A + 1} \right)}
{{\partial_l X} \over {\partial \Phi_A^*}}
{{\partial_r Y} \over {\partial \Phi^A}} 
\left( {1} \right)^{\left( {\epsilon_X +
\epsilon_A } \right)
\left( {\epsilon_Y +
\epsilon_A + 1} \right)}
{{\partial_l X} \over {\partial \Phi^A}}
{{\partial_r Y} \over {\partial \Phi_A^*}}
\quad .
$$
Then using Eq.\bref{l and r der relation},
one finds that the above is equal to
$$
\left( {1} \right)^{\epsilon_X \epsilon_Y +
\epsilon_X + \epsilon_Y + 1}
{{\partial_r X} \over {\partial \Phi_A^*}}
{{\partial_l Y} \over {\partial \Phi^A}} 
\left( {1} \right)^{\epsilon_X \epsilon_Y +
\epsilon_X + \epsilon_Y + 1}
{{\partial_r X} \over {\partial \Phi^A}}
{{\partial_l Y} \over {\partial \Phi_A^*}}
$$
$$
= \left( {1} \right)^{
\left( {\epsilon_X + 1} \right)
\left( {\epsilon_Y + 1} \right)}
\left( {X,Y} \right)
\quad .
$$
This is the desired result.
As another exercise,
let us verify the formulas for
$\left( {F,F} \right)$
and
$\left( {B,B} \right)$
in Eq.\bref{() properties}
When $Y=X$,
the second term in Eq.\bref{antibracket def}
can be written as
$$
{{\partial_r X} \over {\partial \Phi_A^*}}
{{\partial_l X} \over {\partial \Phi^A}} =
\left( {1} \right)^{\left( {
\epsilon_X + 1} \right)
\left( {\epsilon_X + 1} \right)}
{{\partial_r X} \over {\partial \Phi^A}}
{{\partial_l X} \over {\partial \Phi_A^*}}
\quad ,
$$
using the same manipulations as in the previous paragraph.
When $X=F$ is anticommuting,
the sign factor is plus and the second term
in Eq.\bref{antibracket def}
cancels the first.
When $X=B$ is commuting,
the sign factor is minus
and the two add.
Another useful result concerns
integration by parts.
When $\phi$ is commuting,
one has the standard formula
$
\int {\dif \phi }{{\partial_r X} \over
{\partial \phi }} Y =
\int {\dif \phi } X
{{\partial_l Y} \over {\partial \phi }}
$.
In such a case,
$
{{\partial_r } \over {\partial \phi }} =
{{\partial_l } \over {\partial \phi }}
$,
so that the left and right subscripts
on $\partial$ are inconsequential.
Suppose $\epsilon \left( \phi \right) = 1$.
Then,
$
\int {\dif \phi }{{\partial_r X} \over {\partial \phi }}Y
$
and
$
\int {\dif \phi }X{{\partial_l Y} \over {\partial \phi }}
$
are both zero
unless both $X$ and $Y$ are linear in $\phi$.
Without loss of generality,
we may assume that
$X = x \phi $
and
$Y = \phi y$
where $x$ and $y$ are independent of $\phi$.
Then,
$
\int {\dif \phi }{{\partial_r X} \over {\partial \phi }}Y =
\int {\dif \phi }x\left( {\phi y} \right) =
\left( {1} \right)^{\epsilon \left( x \right) }xy
$
and
$
\int {\dif \phi }X{{\partial_lY} \over {\partial \phi }} =
\int {\dif \phi }\left( {x\phi } \right)y =
\left( {1} \right)^{\epsilon \left( x \right) }xy
$
lead to the same result.
Summarizing, all cases are contained in the formula
\be
\int {\dif \phi }{{\partial_r X} \over {\partial \phi }}Y =
\left( {1} \right)^{\epsilon\left( \phi \right)+1}
\int {\dif \phi }X{{\partial_l Y} \over {\partial \phi }}
\quad .
\label{int by parts}
\ee
For second derivatives of $X$, one has
$$
{{\partial_r \partial_l X} \over
{\partial \phi \partial \phi '}} =
{{\partial_l \partial_r X} \over
{\partial \phi '\partial \phi }}
\quad ,
$$
$$
{{\partial_l \partial_l X} \over
{\partial \phi \partial \phi '}} =
\left( {1} \right)^{\epsilon \left( \phi \right)
\epsilon \left( {\phi '} \right)}
{{\partial_l \partial_l X} \over
{\partial \phi '\partial \phi }}
\quad ,
$$
\be
{{\partial_r \partial_r X} \over
{\partial \phi \partial \phi '}} =
\left( {1} \right)^{\epsilon \left( \phi \right)
\epsilon \left( {\phi '} \right)}
{{\partial_r \partial_r X} \over
{\partial \phi '\partial \phi }}
\quad .
\label{second der relations}
\ee
In the first equation,
derivatives act from different directions
and hence commute.
In the second and third equations,
one must be careful of the order.
If $X$ is a functional of $Y$
which is a function of $\phi$,
one has the chain rules
$$
{{\partial_r X \left( {Y\left( \phi \right)} \right)}
\over {\partial \phi }}={{\partial_r X}
\over {\partial Y}}
{{\partial_r Y} \over {\partial \phi }}
\quad ,
$$
\be
{{\partial_l X \left( {Y\left( \phi \right)} \right)}
\over {\partial \phi }}=
{{\partial_l Y} \over {\partial \phi }}
{{\partial_l X} \over {\partial Y}}
\quad .
\label{chain rules}
\ee
\vfill\eject
\section{Appendix: The Regularity Condition}
\label{s:appendixb}
\hspace{\parindent}
If a theory is invariant under the gauge transformations
in Eq.\bref{trans gauge}
then the lagrangian does not depend on all degrees of freedom.
In other words,
$
S_0 \left[ \phi^{\prime} \right]
= S_0 \left[ \phi \right]
$
where $\phi^{\prime}$ stands for finitely transformed fields
produced by any of the infinitesimal variations
in Eq.\bref{trans gauge}.
When this relation is expanded to first order
in the gauge parameters $\varepsilon^\alpha$,
the Noether relations in Eq.\bref{ident noether} are obtained.
Let $\phi_0$ be the stationary point
about which one would like
to perform the perturbative expansion.
Then, in a neighborhood of $\phi_0$
there are other stationary points
given by performing finite gauge transformations on $\phi_0$.
Let $\Sigma$ locally be the surface around $\phi_0$
in $\phi$ configuration space
where the equations of motion vanish.
The regularity condition assumes that
the dimension of $\Sigma$
is maximal and that the quadratic form
generated by expanding the lagrangian
to second order in fields
has a rank $n_{\rm dof}$
on this surface
\ct{bv85a}.
Hence, the number of fields that enter
dynamically in $S_0$ is $n_{\rm dof}$.
The regularity assumption is
important for implementing perturbation theory
since the propagator 
which is the inverse of this quadratic form 
then exists.
Summarizing, the regularity condition is
\be
{\rm rank \ }
\restric{{\partial _l S_{0,i}} \over {\partial \phi ^j}}{ \Sigma } =
\restric{\frac{\partial _l \partial _r S_0}
{\partial \phi ^i \partial \phi ^j}
}{ \Sigma }
= n_{\rm dof}
\quad ,
\label{rc1}
\ee
where $\Sigma$ is the stationary surface defined implicitly by
\be
\restric{S_{0,i}}{ \Sigma } = 0
\quad .
\label{def of sigma}
\ee
In other words,
the onshell degeneracy
of the hessian
in Eq.\bref{rc1}
is completely due to
the $n  n_{\rm dof}$ independent
null vectors $R^i_\alpha$
associated with gauge transformations
and does not come from some other source
\ct{bv84a,bv85a}:
\be
\restric{\frac{\partial _l \partial _r S_0}
{\partial \phi ^i \partial \phi ^j}
R^i_\alpha}{ \Sigma }=0
\quad .
\label{rc1b}
\ee
An example of a lagrangian that does not satisfy
the regularity condition is ${\cal L} = \phi^{4}$
with no kinetic energy term for $\phi$.
The stationary point $\phi_0 = 0$
has a vanishing quadratic
form even though there is no gauge invariance.
In such a case one can proceed by arbitrarily
adding and subtracting some kinetic energy term
and treating $\phi^4$ minus this kinetic energy term
as a perturbation.
However, throughout this review we assume
that such singular cases do not arise.
In principle, one can separate the degrees of freedom
into propagating degrees of freedom
$\varphi^s$, $s= \ 1,\ 2, \dots , \ n_{\rm dof}$
and gauge degrees of freedom
$\chi^a$, $a= \ 1,\ 2, \dots , \ n  n_{\rm dof}$.
An efficient separation is often difficult
and the $\varphi^s$ and $\chi^a$
are usually quite complicated and nonlocal
functionals of the $\phi^i$.
The regularity conditions are then given by
$$
S_{0,a} = 0 \ , \quad {\rm identically \quad ,}
$$
\be
{\rm rank }
\restric{{\partial _l S_{0,s}}
\over {\partial \phi ^i}}{ \phi = \phi_0 }
= n_{\rm dof}
\quad .
\label{rc2}
\ee
The regularity condition assumes that
the equations of motion $S_{0,i}$
constitute a regular representation
of the stationary surface $\Sigma$.
This means that the functions $S_{0,i}$
can be locally split into independent,
$G_s$, and dependent ones, $G_a$,
in such a way that
\begin{enumerate}
\item $G_a=0$ are a direct consequence of $G_s=0$, and
\item The rank of the matrix of the gradients $dG_s$
is maximal on $\Sigma$.
\end{enumerate}
In other words,
the regularity condition ensures
that locally the changes of variables
$\phi^i \rightarrow [\varphi^s , \chi^a ]$ or
$\phi^i \rightarrow [G_s , G_a]$
makes sense
\ct{bv84a,bv85a,fh90a}.
When the regularity condition is fulfilled,
it can be shown that any smooth function
that vanishes on the stationary surface $\Sigma$
can be written as
a combination of the equations of motion
\ct{bv85a,fisch90a,fh90a,fhst89a},
i.e.,
\be
\restric{F(\phi)}{\Sigma} = 0 \Rightarrow
F(\phi) = \lambda^j S_{0,j}
\quad ,
\label{consequence of rc2}
\ee
where the $\lambda^j$ may be functions of the $\phi^i$.
No restrictions are made on the
$\lambda^j ( \phi )$.
Putting restrictions can lead to
violations of \bref{consequence of rc2}.
An example is
presented in
\ct{vp94a}.
By considering only
local functionals,
ref.\ct{vp94a} found cases
for which
Eq.\bref{consequence of rc2}
could not be satisfied
as a local combination of the
equations of motion.
For more details on regularity conditions
as well as derivations of the above results
consult references
\ct{wh79a,bv84a,bv85a,fhst89a,%
fisch90a,fh90a,henneaux90a}.
\vfill\eject
\section{Appendix: Anomaly Trace Computations}
\label{s:appendixc}
\hspace{\parindent}
In this appendix,
we perform the functional trace calculations
of Sect.\ \ref{s:sac}.
One key idea is to use the Dysonlike expansion
\ct{fh65a,iz80a}
$$
\exp \left[ {{\cal R}_0 + V} \right] =
\exp \left[ {{\cal R}_0} \right] +
\int\limits_0^1 {\dif s}
\exp \left[ {\left( {1s} \right){\cal R}_0} \right]
V \exp \left[ {s{\cal R}_0} \right] +
$$
\be
\int\limits_0^1 {\dif u}\int\limits_0^u {\dif s}
\exp \left[ {\left( {1u} \right){\cal R}_0} \right]
V \exp \left[ {\left( {us} \right){\cal R}_0} \right]
V \exp \left[ {s{\cal R}_0} \right] + \ldots
\quad .
\label{dyson expansion}
\ee
Typically,
${\cal R}_0$ is independent of the cutoff ${\cal M}$,
and $V$ goes like inverse powers of ${\cal M}$
so that only a few terms
in Eq.\bref{dyson expansion}
need to be kept.
The anomaly equation
Eq.\bref{Delta S reg}
involves a functional trace
\ct{fujikawa80a}.
If one uses momentumspace eigenfunctions
to saturate the sum,
then expressions such as
$$
\exp \left( { i k \cdot x} \right)
\left( { O \left( {\partial_\mu } \right) } \right)
\exp \left( {ik\cdot x} \right)
$$
arise
where $O \left( {\partial_\mu } \right) $
is an arbitrary operator, or a product of operators,
involving the derivative $\partial_\mu$.
By commuting $\exp \left( {ik\cdot x} \right)$
through the expression,
one arrives at
\be
\exp \left( { i k \cdot x} \right)
\left( { O \left( {\partial_\mu } \right) } \right)
\exp \left( {ik\cdot x} \right) =
\left( { O \left( {\partial_\mu +ik_\mu }
\right) } \right) 1
\quad .
\label{momentum conjugation}
\ee
When derivatives in
$O \left( {\partial_\mu +ik_\mu } \right)$
act on the function $1$, they produce zero.
\medskip
For the spinless relativistic particle system,
we begin by taking
Eq.\bref{rsp Delta S reg}
and commuting
$\exp \left( {ik\tau } \right)$
through the expression,
using Eq.\bref{momentum conjugation},
to obtain
$$
\left( {\Delta S} \right)_{\rm reg} =
$$
$$
D\int {\dif \tau } \left[ {
\int\limits_{\infty }^\infty
{{ {\dif k} \over {2\pi } }} \rho^{1}
{\cal C}
\left( { {d \over {d\tau } } + ik } \right)
\ \exp \left( {{ {\left( { {d \over {d\tau } } + ik } \right)
\rho^{1} \left( { {d \over {d\tau }} + ik } \right)} \over
{\cal M}^2} } \right) 1
} \right]_0
\ .
$$
Rescaling $k$ by ${\cal M}$ produces
$$
\left( {\Delta S} \right)_{\rm reg} =
D\int {\dif \tau }\int\limits_{\infty }^\infty
{{{\dif k} \over { 2 \pi }}}
\left[ {{\cal M} \rho^{1} {\cal C}
\left( {{d \over {d\tau }} +
i {\cal M} k} \right) \ \times } \right.
$$
\be
\left. {
\exp \left( {
{ {k^2} \over \rho } 
i{ k \over {\cal M} }
\left( {\rho^{1} {d \over {d\tau }} +
{d \over {d\tau } } \rho^{1} } \right) 
{1 \over {\cal M}^2} {d \over {d\tau }}
\rho^{1} {d \over {d\tau } }
} \right) 1
} \right]_0
\quad .
\label{rsp delta S cal 1}
\ee
Eq.\bref{rsp delta S cal 1}
is in the form
of Eq.\bref{dyson expansion}
where $ {\cal R}_0 = { {k^2} / \rho }$.
We use the Dysonlike expansion
in Eq.\bref{dyson expansion}
and pick out the ${\cal M}$independent term
to arrive at
$$
\left( {\Delta S} \right)_{\rm reg} =
$$
$$
iD\int {\dif \tau }\int\limits_{\infty }^\infty
{{{\dif k} \over {2\pi }}}
\ k \rho^{1} {\cal C}
\left\{ {
{d \over {d\tau }}
\int\limits_0^1 {\dif s}
\exp \left( {\left( {1s} \right){{k^2} \over \rho }} \right)
\left( {\rho^{1}{d \over {d\tau }} +
{d \over {d\tau }}\rho^{1}} \right)
\exp \left( {s{{k^2} \over \rho }} \right)
} \right.
$$
$$
+ \int\limits_0^1 {\dif s}
\exp \left( {\left( {1s} \right){{k^2} \over \rho }}
\right)\left( {{d \over {d\tau }}
\rho^{1}{d \over {d\tau }}} \right)
\exp \left( {s{{k^2} \over \rho }} \right)
$$
$$
+ k^2 \int\limits_0^1
{\dif u}\int\limits_0^u {\dif s}
\exp \left( {\left( {1u} \right){{k^2} \over \rho }} \right)
\left( {\rho^{1}{d \over {d\tau }} +
{d \over {d\tau }}\rho^{1}} \right) \ \times
$$
\be
\left. {
\exp \left( {\left( {us} \right)
{{k^2} \over \rho }} \right)
\left( {\rho^{1}{d \over {d\tau }} +
{d \over {d\tau }}
\rho^{1}} \right)
\exp \left( {s {{k^2} \over \rho }} \right)
} \right\}
\quad .
\label{rsp delta S cal 2}
\ee
The first term
in Eq.\bref{rsp delta S cal 2}
is zero because it is a total derivative.
To calculate the other two terms
rotate to Euclidean space using
$k \to ik_E$.
Then the expression
${{k^2} / \rho } \to {{k_E^{2} } / \rho } $
in the exponents yields
gaussian damping factors,
so that the integrals are convergent.
Even before evaluating the derivatives
${d \over {d\tau }}$,
it is clear that the integrand
is an odd function of $k$
and hence produces a zero integral.
\medskip
For the chiral Schwinger model,
one starts with Eq.\bref{acsm19}.
Using momentumspace wave functions,
the functional trace is
$$
\Delta S =
 i \int \dif^2 x \, {\cal C} \times
$$
\be
\left[{
\int \frac{\dif^2 k}{ \left( {2\pi} \right)^2}
\exp{ \left( ik \cdot x \right) }
\left( {
\exp{ \left( {
{ { R_+ } \over {{\cal M}^2} }
} \right) } 
\exp{ \left( {
{ { R_ } \over {{\cal M}^2} }
} \right) }
} \right)
\exp{ \left( ik \cdot x \right) }
} \right]_0
\ ,
\label{acsma0}
\ee
Equation \bref{momentum conjugation}
is used to eliminate the
$\exp { \left( {\pm i k \cdot x} \right) } $
factors.
Then, one scales the momentum $k$ by ${\cal M}$.
The expression for $\Delta S$ becomes
\be
\Delta S =
i\int {\dif^2 x} \, {\cal C}
\int {{{\dif^2 k} \over {\left( {2\pi } \right)^2}}
\left[ {{\cal M}^2
\left( {\exp \left( {\tilde R_+} \right) 
\exp \left( {\tilde R_} \right)} \right)1} \right]_0}
\quad ,
\label{acsma1}
\ee
where
\be
\tilde R_\pm =
k^\mu k_\mu 
{{2ik^\mu \partial_\mu \pm A_k_+} \over {{\cal M}}} +
{{\partial^\mu \partial_\mu
\pm i\left( {\partial_+ A_} \right)
\pm iA_\partial_+} \over {{\cal M}^2}}
\quad .
\label{acsma2}
\ee
The parenthesis around
$(\partial_+ A_)$
indicates that $\partial_+$
acts only on $A_$.
The notation $\left[ \right]_0$
selects the term in Eq.\bref{acsma1}
independent of ${\cal M}$.
Hence,
one expands
in the factors in the exponentials
proportional to ${\cal M}^{1}$ and ${\cal M}^{2}$.
This results in two terms,
$\Delta S_1$ from the leading Taylor series term
in ${\cal M}^{2}$,
and $\Delta S_2$ from the second order term
in ${\cal M}^{1}$:
$
\Delta S = \Delta S_1 + \Delta S_2
$,
where
$$
\Delta S_1 =
2\int {\dif^2 x} \, {\cal C}\left( {\partial_+ A_} \right)
\int {{{\dif^2 k} \over {\left( {2\pi } \right)^2}}}
\exp \left( {k^\mu k_\mu } \right)
\quad ,
$$
\be
\Delta S_2 =
i\int {\dif^2 x} \, {\cal C}
\int {{{\dif^2 k} \over {\left( {2\pi } \right)^2}}}
\exp \left( {k^\mu k_\mu } \right)
2ik^\mu \left( { \partial_\mu A_ } \right) k_+
\quad .
\label{acsma3}
\ee
Integrals are defined by analytic continuation
using Wick rotation, that is,
the $k_0$ integral is performed
by going to Euclidean space.
The line integral contained in
$\int\limits_{\infty }^\infty {\dif k_0 }$
can be rotated counterclockwise by $90^{\rm o}$.
Then, one sets
$
k_0 = ik_0^E
$:
\be
\int\limits_{\infty }^\infty {\dif k_0 } =
\int\limits_{i\infty }^{i\infty } {\dif k_0 } =
i\int\limits_\infty^{\infty } {\dif k_0^E=}
i\int\limits_{\infty }^\infty {\dif k_0^E}
\quad .
\label{wick rotation}
\ee
Integrals are then convergent
since
\be
\exp \left[ { {k^2} } \right] =
\exp \left[ {+ k_0^2  k_1^2 } \right] \to
\exp \left[ { {k_0^{E2}}  k_1^2 } \right]
\quad ,
\label{acsma4}
\ee
provides a gaussian damping factor.
The following integration table is obtained
\be
\int {{{\dif^2 k} \over {\left( {2\pi } \right)^2}}}
\exp \left( {k^\mu k_\mu } \right)
\left\{ \matrix{1\hfill\cr
k_0^2\hfill\cr
k_1^2\hfill\cr
k_0k_1\hfill\cr} \right\} =
{i \over {8\pi }}
\left\{ \matrix{2\hfill\cr
1\hfill\cr
1\hfill\cr
0\hfill\cr} \right\}
\quad .
\label{acsma5}
\ee
Hence,
$$
\Delta S_1 =
{{2i} \over {4\pi }}\int {\dif^2 x}
\, {\cal C}\left( {\partial_+ A_} \right)
\quad ,
$$
\be
\Delta S_2 =
{i \over {4\pi }}\int {\dif^2 x}
\, {\cal C}\left( {\partial_+ A_} \right)
\quad .
\label{acsma6}
\ee
Adding the two contributions,
\be
\Delta S =
{i \over {4\pi }}\int {\dif^2 x}
\, {\cal C}\left( {\partial_+ A_} \right)
\quad .
\label{acsma7}
\ee
The factor $\partial_+ A_$
can be written in Lorentz covariant form
using
\be
\partial_+ A_ =
\partial_0 A_0\partial_1 A_1 
\partial_0 A_1+\partial_1 A_0 =
\partial_\mu A^\mu +
\varepsilon^{\mu \nu }\partial_\mu A_\nu
\quad .
\label{acsma8}
\ee
Substituting Eq.\bref{acsma8} into Eq.\bref{acsma7}
produces the result in Eq.\bref{acsm21}.
\medskip
For the firstquantized bosonic string theory,
we need to compute $\kappa_r$
of Eq.\bref{def of kappa_n}.
Using momentumspace eigenfunctions,
it can be expressed as
\ct{fujikawa82a,fis90a}
\be
\kappa_r = {{tr} \over 2}
\left[ {\int {{{\dif^2k} \over {\left( {2\pi } \right)^2}}}
\exp \left( {ik\cdot x} \right)
\exp \left( {{{H_r} \over {{\cal M}^2}}} \right)
\exp \left( {ik\cdot x} \right)} \right]_0
\quad ,
\label{kappa_n trace}
\ee
where $tr$ is a trace
over twobytwo gammamatrix space.
In Eq.\bref{kappa_n trace},
$
k \cdot x =  k_\tau \tau + k_\sigma \sigma
$.
Using the definition of $H_r$
in Eq.\bref{def of H_n}
and commuting
$\exp \left( {ik\cdot x} \right)$
through the expression
via Eq.\bref{momentum conjugation},
Eq.\bref{kappa_n trace} becomes
$$
\kappa_r = {{tr} \over 2}
\left[ {\int {{{\dif^2k} \over {\left( {2\pi } \right)^2}}}
\exp \left( {{{\rho^{\left( {r+1} \right)/2}
\left( {\slashit \partial +i\slashit k} \right)
\rho^r\left( {\slashit \partial +i\slashit k} \right)
\rho^{\left( {r+1} \right)/2}} \over {{\cal M}^2}}} \right)
1} \right]_0
$$
$$
= {{tr} \over 2}
\left[ {{\cal M}^2\int {{{\dif^2k} \over {\left( {2\pi } \right)^2}}}
\exp \left( {{{k^2} \over \rho }  i{A \over {{\cal M}}} +
{B \over {{\cal M}^2}}} \right)1} \right]_0
\quad ,
$$
where we have performed the rescaling
$
k \to {\cal M} k
$,
and where
\be
A \equiv \rho^{\left( {r+1} \right)/2}
\left( {\slashit \partial \rho^r \slashit k +
\slashit k\rho^r\slashit \partial } \right)
\rho^{\left( {r+1} \right)/2}
\quad ,
\label{def of A operator}
\ee
\be
B \equiv \rho^{\left( {r+1} \right)/2}
\slashit \partial \rho^r\slashit \partial
\rho^{\left( {r+1} \right)/2}
\quad .
\label{def of B operator}
\ee
For the next step,
we use the Dysonlike expansion
in Eq.\bref{dyson expansion}.
Recalling that
$[ \ ]_0$
indicates that the ${\cal M}$independent term
is to be selected,
$\kappa_r$ becomes a sum of two terms
$$
\kappa_r = B {\mbox{term}} + AA {\mbox{term}}
\quad ,
$$
where
$$
AA {\mbox{term}} =
 {{tr} \over 2}\int {{{\dif^2k} \over {\left( {2\pi } \right)^2}}}
\int\limits_0^1 {\dif u}\int\limits_0^u {\dif s} \ \times
$$
$$
\exp \left[ {\left( {1u} \right){{k^2} \over \rho }} \right]
A \exp \left[ {\left( {us} \right){{k^2} \over \rho }} \right]
A \exp \left[ {s{{k^2} \over \rho }} \right]1
\quad ,
$$
and
$$
B {\mbox{term}} = {{tr} \over 2}
\int {{{\dif^2k} \over {\left( {2\pi } \right)^2}}}
\int\limits_0^1 {\dif s}
\exp \left[ {\left( {1s} \right){{k^2} \over \rho }} \right]
B \exp \left[ {s{{k^2} \over \rho }} \right]1
\quad .
$$
In $AA {\mbox{term}}$,
carry out the differentiations $\slashit \partial$
in both $A$ operators
using the definition of $A$
in Eq.\bref{def of A operator}
to obtain
$$
AA {\mbox{term}} =
\int {{{\dif^2k} \over {\left( {2\pi } \right)^2}}}
\int\limits_0^1 {\dif u}\int\limits_0^u {\dif s} \ k^2
\exp \left[ { {k^2} } \right]
\left\{ {{{\partial^n\partial_n\rho } \over \rho }
\left( { 2 s k^21} \right) + } \right.
$$
$$
\left. {{{\partial^n\rho \partial_n\rho } \over {\rho^2}}
\left( {{1 \over 2}\left( {r^25} \right) 
7 s k^2  u k^2 + 2 s u k^4} \right)} \right\}
\quad ,
$$
where another rescaling
$
k \to k \rho^{1/2}
$
has been performed.
Next, rotate the $k$ integration from Minkowski space
to Euclidean space.
The line integral contained in
$\int\limits_{\infty }^\infty {\dif k_\tau }$
can be rotated counterclockwise by $90^{\rm o}$.
One sets
$
k_\tau = ik_\tau^E
$
and uses Eq.\bref{wick rotation}
for $k_0 = k_\tau$.
The exponential factor
$
\exp \left[ { {k^2} } \right] =
\exp \left[ {+ k_\tau^2  k_\sigma^2 } \right]
$
in $AA {\mbox{term}}$
becomes
$
\exp \left[ { {k_\tau^{E2}}  k_\sigma^2 } \right]
$.
One can then do the $s$, $u$ and $k$ integrations
since the latter are now convergent.
The result is
\be
AA {\mbox{term}} = {1 \over {4\pi i}}
\left( {{{\partial^n\partial_n\rho } \over \rho }{1 \over 6} 
{{\partial^n\rho \partial_n\rho } \over {\rho^2}}
\left( {{{r^2+1} \over 4}} \right)} \right)
\quad .
\label{result for AAterm}
\ee
The $B {\mbox{term}}$ is treated similarly.
One carries out the differentiations $\slashit \partial$
in the $B$ operator and
rescales $k$ by
$ \rho^{1/2}$
to arrive at
$$
B {\mbox{term}} = {{tr} \over 2}
\int {{{\dif^2k} \over {\left( {2\pi } \right)^2}}}
\int\limits_0^1 {\dif s}\exp \left[ {k^2} \right]
\left\{ {{{\partial^n\partial_n\rho } \over \rho }
\left( {sk^2  {{r+1} \over 2}} \right) + } \right.
$$
$$
\left. {{{\partial^n\rho \partial_n\rho } \over {\rho^2}}
\left( {{{\left( {3r} \right)
\left( {r+1} \right)} \over 4}  3 s k^2
+ s^2 k^4} \right)} \right\}
\quad .
$$
After rotating to Euclidean space,
all integrations can be formed.
The $B {\mbox{term}}$ is
\be
B {\mbox{term}} = {1 \over {4\pi i}}
\left( {{{\partial^n\partial_n\rho } \over \rho }
{r \over 2} +
{{\partial^n\rho \partial_n\rho } \over {\rho^2}}
\left( {{{r^2} \over 4} 
{r \over 2} + {1 \over {12}}} \right)} \right)
\quad .
\label{result for Bterm}
\ee
The sum of the $AA {\mbox{}}$ and $B {\mbox{terms}}$
in Eqs.\bref{result for AAterm}
and \bref{result for Bterm}
is the quoted result for $\kappa_r$
in Eq.\bref{kappa_n results}.
\vfill\eject
\begin{thebibliography}{999}
\label{references}
\addcontentsline{toc}{section}{References}
\def\CMP#1#2#3{ {Comm.\,Math.\,Phys.\,}
{\bf {#1}} ({#3}) {#2}}
\def\IJMPA#1#2#3{ {Int.\,J.\,Mod.\,Phys.\,}
{\bf A{#1}} ({#3}) {#2}}
\def\JMP#1#2#3{ {J.\,Math.\,Phys.\,}
{\bf {#1}} ({#3}) {#2}}
\def\MPLA#1#2#3{ {Mod.\,Phys.\,Lett.\,}
{\bf A{#1}} ({#3}) {#2}}
\def\NC#1#2#3{ {Nuovo\,Cim.\,}
{\bf {#1}} ({#3}) {#2}}
\def\NPB#1#2#3{ {Nucl.\,Phys.\,}
{\bf B{#1}} ({#3}) {#2}}
\def\PL#1#2#3{ {Phys.\,Lett.\,}
{\bf {#1}} ({#3}) {#2}}
% change in title occurs at volume 170
\def\PLB#1#2#3{ {Phys.\,Lett.\,}
{\bf B{#1}} ({#3}) {#2}}
\def\PR#1#2#3{ {Phys.\,Rep.\,}
{\bf {#1}} ({#3}) {#2}}
\def\PRL#1#2#3{ {Phys.\,Rev.\,Lett.\,}
{\bf {#1}} ({#3}) {#2}}
\def\PRD#1#2#3{ {Phys.\,Rev.\,}
{\bf D{#1}} ({#3}) {#2}}
\def\OPR#1#2#3{ {Phys.\,Rev.\,}
{\bf {#1}} ({#3}) {#2}}
\def\TMP#1#2#3{ {Theor.\,Math.\,Phys.\,}
{\bf {#1}} ({#3}) {#2}}
\bibitem{al73a}
E.\,S.\,Abers and B.\,W.\,Lee,
{\it Gauge Theories},
\PR{C9}{1}{1973}.
\bibitem{ad93a}
J.\,Alfaro and P.\,H.\,Damgaard,
{\it Origin of Antifields
in the BatalinVilkovisky Lagrangian Formalism},
\NPB{404}{751}{1993};
{\it Generalized Lagrangian Master Equations},
\PLB{334}{369}{1994}.
\bibitem{ab83a}
L.\,AlvarezGaum\'e and L.\,Baulieu,
{\it The Two Quantum Symmetries Associated
with a Classical Symmetry},
\NPB{212}{255}{1983}.
\bibitem{anselmi93a}
D.\,Anselmi,
{\it Predictivity and Nonrenormalizability},
SISSA preprint SISSA14793EP,
(Sept., 1993),
hepth/9309085.
\bibitem{av93a}
S.\,Aoyama and S.\,Vandoren,
{\it The BatalinVilkovisky Formalism
on Fermionic K\"ahler Manifolds},
\MPLA{8}{3773}{1993}.
\bibitem{ad79a}
C.\,Aragone and S.\,Deser,
{\it Consistency Problems in Hypergravity},
\PL{86B}{161}{1979}.
\bibitem{bsv86a}
O.\,Babelon, F.\,A.\,Schaposnik and C.\,M.\,Viallet,
{\it Quantization of Gauge Theories with Weyl Fermions},
\PL{177B}{385}{1986}.
\bibitem{band86a}
G.\,Bandelloni,
{\it YangMills Cohomology in Four Dimensions},
\JMP{27}{2551}{1986};
{\it Nonpolynomial YangMills Cohomology},
\JMP{28}{2775}{1987}.
\bibitem{bfmpp86a}
T.\,Banks, D.\,Friedan, E.\,Martinec, M.\,E.\,Peskin and
C.\,R.\,Preitschopf,
{\it All Free String Theories Are Theories of Forms},
\NPB{274}{71}{1986}.
\bibitem{bcl76a}
A.\,Barducci, R.\,Casalbouni and L.\,Lusanna,
{\it Supersymmetries and the Pseudoclassical
Relativistic Electron},
\NC{35A}{377}{1976}.
\bibitem{bbh94a}
G.\,Barnich, F.\,Brandt and M.\,Henneaux,
{\it Local BRST Cohomology in the Antifield Formalism:
I.\,General Theorems},
preprint ULBth94/06, NIKHEFH 9413,
hepth/9405109.
\bibitem{bcg92a}
G.\,Barnich, R.\,Constantinescu and P.\,Gr\'egoire,
{\it BRSTAntiBRST Antifield Formalism.
The Example of the FreedmanTownsend Model},
\PLB{293}{353}{1992}.
\bibitem{bh93a}
G.\,Barnich and M.\,Henneaux,
{\it Consistent Couplings Between Fields
with a Gauge Freedom and
Deformations of the Master Equation},
\PLB{311}{123}{1993}.
\bibitem{bh94a}
G.\,Barnich and M.\,Henneaux,
{\it Renormalization of Gauge Invariant Operators
and Anomalies in YangMills Theory},
\PRL{72}{1588}{1994}.
\bibitem{}
F.\,Bastianelli,
{\it Ward Identities for Quantum Field Theories
with External Fields},
\PLB{263}{411}{1991}.
\bibitem{}
F.\,Bastianelli,
{\it A Locally Supersymmetric Action for the Bosonic String},
\PLB{322}{340}{1994}.
\bibitem{batalin81a}
I.\,A.\,Batalin,
{\it Quasigroup Construction and First Class Constraints},
\JMP{22}{1837}{1981}.
\bibitem{bf83a}
I.\,A.\,Batalin and E.\,S.\,Fradkin,
{\it A Generalized Canonical Formalism and
Quantization of Reducible Gauge Theories},
\PL{122B}{157}{1983}.
\bibitem{bf86a}
I.\,A.\,Batalin and E.\,S.\,Fradkin,
{\it Operator Quantization and
Abelization of Dynamical Systems
Subject to FirstClass Constraints},
Riv.\,Nuov.\,Cim.\,{\bf 9} (1986) 1.
\bibitem{blt90a}
I.\,A.\,Batalin, P.\,M.\,Lavrov and I.\,V.\,Tyutin,
{\it Covariant Quantization of Gauge Symmetries
in the Framework of Extended BRST Symmetry},
\JMP{31}{1487}{1990};
{\it An $Sp(2)$Covariant Quantization
of Gauge Theories with Linearly Dependent Generators},
\JMP{32}{532}{1991}.
\bibitem{bt93a}
I.\,A.\,Batalin and I.\,V.\,Tyutin,
{\it On the Multilevel Generalization
of the FieldAntifield Formalism},
\MPLA{8}{3673}{1993}.
\bibitem{bt93b}
I.\,A.\,Batalin and I.\,V.\,Tyutin,
{\it On Possible Generalizations of FieldAntifield Formalism},
\IJMPA{8}{2333}{1993}.
\bibitem{bv77a}
I.\,A.\,Batalin and G.\,A.\,Vilkovisky,
{\it Relativistic $S$Matrix of Dynamical Systems
with Boson and Fermion Constraints},
\PL{69B}{309}{1977}.
\bibitem{bv81a}
I.\,A.\,Batalin and G.\,A.\,Vilkovisky,
{\it Gauge Algebra and Quantization},
\PL{102B}{27}{1981}.
\bibitem{bv83a}
I.\,A.\,Batalin and G.\,A.\,Vilkovisky,
{\it Feynman Rules for Reducible Gauge Theories},
\PL{120B}{166}{1983}.
\bibitem{bv83b}
I.\,A.\,Batalin and G.\,A.\,Vilkovisky,
{\it Quantization of Gauge Theories
with Linearly Dependent Generators},
\PRD{28}{2567}{1983};
Errata: {\bf D30} (1984) 508.
\bibitem{bv84a}
I.\,A.\,Batalin and G.\,A.\,Vilkovisky,
{\it Closure of the Gauge Algebra,
Generalized Lie Algebra Equations and Feynman Rules},
\NPB{234}{106}{1984}.
\bibitem{bv85a}
I.\,A.\,Batalin and G.\,A.\,Vilkovisky,
{\it Existence Theorem for Gauge Algebra},
\JMP{26}{172}{1985}.
\bibitem{batlle88a}
C.\,Batlle,
{\it Teories de Camps de Cordes},
Thesis, Universitat de Barcelona (1988).
\bibitem{bg88a}
C.\,Batlle and J.\,Gomis,
{\it Lagrangian and Hamiltonian BRST Structures
of the Antisymmetric Tensor Gauge Theory},
\PRD{38}{1169}{1988}.
\bibitem{bgpr89a}
C.\,Batlle, J.\,Gomis, J.\,Par\'{\i}s and J.\,Roca,
{\it Lagrangian and Hamiltonian BRST Formalisms},
\PLB{224}{288}{1989};
{\it FieldAntifield Formalism and Hamiltonian BRST Approach},
\NPB{329}{139}{1990}.
\bibitem{baulieu85a}
L.\,Baulieu,
{\it Perturbative Gauge Theories},
\PR{129}{1}{1985}.
\bibitem{bbow90a}
L.\,Baulieu, M.\,Bellon, S.\,Ouvry and J.C.\,Wallet,
{\it BatalinVilkovisky Analysis of Supersymmetric Systems},
\PLB{252}{387}{1990}.
\bibitem{bbs88a}
L.\,Baulieu, E.\,Bergshoeff and E.\,Sezgin,
{\it Open BRST Algebras, Ghost Unification
and String Field Theory},
\NPB{307}{348}{1988}.
\bibitem{bs88a}
L.\,Baulieu and I.M.\,Singer,
{\it Topological YangMills Symmetry},
Nucl.Phys.\ {\bf B} (Proc.\,Suppl) {\bf 5B} (1988) 12.
\bibitem{brs74a}
C.\,Becchi, A.\,Rouet and R.\,Stora,
{\it The Abelian Higgs Kibble Model,
Unitarity of the $S$Operator},
\PL{52B}{344}{1974};
{\it Renormalization of the Abelian HiggsKibble Model},
\CMP{42}{127}{1975};
{\it Renormalization of Gauge Theories},
Ann.\,Phys.\,{\bf 98} (1976) 287.
\bibitem{bhnw79a}
F.\,A.\,Berends, J.\,W.\,van Holten, P.\,van Nieuwenhuizen and
B.\,de Wit,
{\it On Spin $5/2$ Gauge Fields},
\PL{83B}{188}{1979}.
\bibitem{berezin66a}
F.\,A.\,Berezin,
{\it The Method of Second Quantization},
(Academic Press, New York, 1966).
\bibitem{berezin87a}
F.\,A.\,Berezin,
{\it Introduction to Superanalysis},
Mathematical Physics and Applied Mathematics, n$^{\rm o}$ 9,
(Reidel, Dordrecht, 1987).
\bibitem{bl75a}
F.\,A.\,Berezin and D.\,A.\,Leites,
{\it Supermanifolds},
Sov.\,Math.\,Dokl.\,{\bf 16} (1975) 1218.
\bibitem{bm77a}
F.\,A.\,Berezin and M.\,S.\,Marinov,
{\it Particle Spin Dynamics as the Grassmann
Variant of Classical Mechanics},
Ann.\,Phys.\,{\bf 104} (1977) 336.
\bibitem{bbrt93a}
E.\,Bergshoeff, J.\,de Boer, M.\,de Roo and T.\,Tjin,
{\it On the Cohomology of the NonCritical $W$ String},
\NPB{420}{379}{1994}.
\bibitem{bk90a}
E.\,Bergshoeff and R.\,Kallosh,
{\it Unconstrained BRST for Superparticles},
\PLB{240}{105}{1990}.
\bibitem{bkp92a}
E.\,Bergshoeff, R.\,Kallosh and A.\,Van Proeyen,
{\it Superparticle Actions and Gauge Fixings},
Clas.\,Quantum Grav.\,{\bf 9} (1992) 321.
\bibitem{bss92a}
E.\,Bergshoeff, A.\,Sevrin and X.\,Shen,
{\it A Derivation of the BRST Operator
for NonCritical $W$Strings},
\PLB{296}{95}{1992}.
\bibitem{}
N.\,Berkovits, M.\,T.\,Hatsuda and W.\,Siegel,
{\it The Big Picture},
\NPB{371}{434}{1992}.
\bibitem{bcov93a}
M.\,Bershadsky, S.\,Cecotti, H.\,Ooguri and C.\,Vafa,
{\it KodairaSpencer Theory of Gravity
and Exact Results for Quantum String Amplitudes},
Harvard Univ.\ preprint HUTP93A025,
(Sept, 1993),
hepth/9309140.
\bibitem{bln93a}
%do not change bibitem (warner left out originally)
M.\,Bershadsky, W.\,Lerche, D.\,Nemeschansky
and N.\,P.\,Warner,
{\it A BRST Operator for NonCritical Strings},
\PLB{292}{35}{1993};
{\it Extended $N=2$ Superconformal Structure
of Gravity and $W$Gravity Coupled to Matter},
\NPB{401}{304}{1993}.
\bibitem{bbrt91a}
D.\,Birmingham, M.\,Blau, M.\,Rakowski and G.\,Thompson,
{\it Topological Field Theory},
\PR{209}{129}{1991}.
\bibitem{brt89a}
D.\,Birmingham, M.\,Rakowski and G.\,Thompson,
{\it BRST Quantization of Topological Field Theories},
\NPB{315}{577}{1989}.
\bibitem{}
M.\,Blagojevi\'c and B.\,Sazdovi\'c,
{\it OffShell BRST Quantization
of Reducible Gauge Theories},
\PLB{223}{325}{1989}.
\bibitem{}
A.\,Blasi and G.\,Bandelloni,
{\it Antisymmetric Tensor Gauge Theory},
Clas.\,Quantum Grav.\,{\bf 10}, (1993) 1249.
\bibitem{bleuler50a}
K.\,Bleuler,
{\it A New Method for the Treatment
of Longitudinal and Scalar Photons},
Helv.\,Phys.\,Acta {\bf 23} (1950) 567.
\bibitem{bs89a}
R.\,Bluhm and S.\,Samuel,
{\it The OffShell KobaNielsen Formula},
\NPB{323}{337}{1989}.
\bibitem{bs89b}
R.\,Bluhm and S.\,Samuel,
{\it OffShell Conformal Field Theory
at the OneLoop Level},
\NPB{325}{275}{1989}.
\bibitem{bochicchio87a}
M.\,Bochicchio,
{\it String Field Theory in the Siegel Gauge},
\PL{188B}{330}{1987};
{\it Gauge Fixing for the Field Theory
of the Bosonic String},
\PL{193B}{31}{1987}.
\bibitem{bc83a}
L.\,Bonora and P.\,CottaRamusino,
{\it Some Remarks on BRST Transformations,
Anomalies and the Cohomology of the Lie Algebra
of the Group of Gauge Transformations},
\CMP{87}{589}{1983}.
\bibitem{borel53a}
A.\,Borel,
{\it Sur la Cohomologie des Espaces Fibres Principaux
et Espaces Homogenes de Groupes de Lie Compacts},
Ann.\,Math.\,{\bf 57} (1953) 115.
\bibitem{bmp92a}
P.\,Bouwknegt, J.\,McCarthy and K.\,Pilch,
{\it BRST Analysis of Physical States
for $2$$d$ Gravity Coupled to $c \le 1$ Matter},
\CMP{145}{541}{1992}.
\bibitem{bmp93a}
P.\,Bouwknegt, J.\,McCarthy and K.\,Pilch,
{\it Semiinfinite Cohomology of $W$ Algebras},
Lett.\,Math.\,Phys.\,{\bf 29} (1993) 91.
\bibitem{bm91a}
N.\,R.\,F.\,Braga and H.\,Montani,
{\it BatalinVilkovisky Lagrangian Quantization
of the Chiral Schwinger Model},
\PLB{264}{125}{1991}.
\bibitem{bm93a}
N.\,R.\,F.\,Braga and H.\,Montani,
{\it The WessZumino Term in the FieldAntifield Formalism},
\IJMPA{8}{2569}{1993}.
\bibitem{bm94a}
N.\,R.\,F.\,Braga and H.\,Montani,
{\it BRST Quantization of the Chiral Schwinger Model
in the Extended FieldAntifield Space},
\PRD{49}{1077}{1994}.
\bibitem{brandt94a}
F.\,Brandt,
{\it Antifield Dependence of Anomalies},
\PLB{320}{57}{1994}.
\bibitem{bszvh77a}
L.\,Brink, S.\,Deser, B.\,Zumino,
P.\,di Vecchia and P.\,Howe,
{\it Local Supersymmetry for Spinning Particles},
\PL{64B}{435}{1977}.
\bibitem{bms88a}
R.\,Brooks, D.\,Montano and J.\,Sonnenschein,
{\it Gauge Fixing and Renormalization
in Topological Quantum Field Theory},
\PLB{214}{91}{1988}.
\bibitem{cs74a}
E.\,Cremmer and J.\,Scherk,
{\it Spontaneous Dynamical Breaking
of Gauge Symmetry in Dual Models},
\NPB{72}{117}{1974}.
\bibitem{cst86a}
E.\,Cremmer, A.\,Schwimmer and C.\,Thorn,
{\it The Vertex Function in Witten's Formulation
of String Field Theory},
\PLB{179}{57}{1986}.
\bibitem{cf76a}
G.\,Curci and R.\,Ferrari,
{\it Slavnov Transformations and Supersymmetry},
\PL{63B}{91}{1976}.
\bibitem{cf76b}
G.\,Curci and R.\,Ferrari,
{\it An Alternative Approach
to the Proof of Unitarity for Gauge Theories},
Nuovo Cim.\, {\bf 35A}(1976) 273.
\bibitem{}
O.\,F.\,Dayi,
{\it Odd Time Formulation of the BatalinVilkovisky
Method of Quantization},
\MPLA{4}{361}{1989}.
\bibitem{dayi94a}
O.\,F.\,Dayi,
{\it BV and BFV Formulation of a Gauge Theory
of Quadratic Lie Algebras in $2$$D$ and
a Construction of $W_3$ Topological Gravity},
Tubitak Research Institute preprint,
(Jan., 1994),
hepth/9401148.
\bibitem{agm88a}
S.\,P.\,de Alwis, M.\,T.\,Grisaru and L.\,Mezincescu,
{\it Quantization and Unitarity
in Antisymmetric Tensor Gauge Theories},
\NPB{303}{57}{1988}.
\bibitem{bg93a}
J.\,de Boer and J.\,Goeree,
{\it The Effective Action of $W_3$ to All Orders},
\NPB{401}{348}{1993}.
\bibitem{dejonghe93a}
F.\,De Jonghe,
{\it The BatalinVilkovisky Lagrangian Quantization Scheme:
With Applications to the Study of Anomalies
in Gauge Theories},
Leuven Univ.\ preprint,
(Dec., 1993),
hepth/9403143.
\bibitem{dejonghe93b}
F.\,De Jonghe,
{\it SchwingerDyson BRST Symmetry
and the Equivalence of Hamiltonian
and Lagrangian Quantisation},
\PLB{316}{503}{1993}.
\bibitem{jst93a}
F.\,De Jonghe, R.\,Siebelink and W.\,Troost,
{\it Hiding Anomalies},
\PLB{306}{295}{1993}.
\bibitem{dstvnp92a}
F.\,De Jonghe, R.\,Siebelink, W.\,Troost,
S.\,Vandoren, P.\,van Nieuwenhuizen and A.\,Van Proeyen,
{\it The Regularized BRST Jacobian
of Pure YangMills Theory},
\PLB{289}{354}{1992}.
\bibitem{jv94a}
F.\,De Jonghe and S.\,Vandoren,
{\it Construction of Topological Field Theories
Using the BatalinVilkovisky Quantization Scheme},
\PLB{324}{328}{1994}.
\bibitem{dts81a}
S.\,Deser, P.\,K.\,Townsend and W.\,Siegel,
{\it Higher Rank Representations of Lower Spin},
\NPB{184}{333}{1981}.
\bibitem{wh79a}
B.\,de Wit and J.\,W.\,van Holten,
{\it Covariant Quantization of Gauge Theories with Open Algebra},
\PLB{79}{389}{1979}.
\bibitem{dewitt64a}
B.\,DeWitt,
{\it Dynamical Theory of Groups and Fields},
in Relativity, Groups and Topology,
Les Houches Summer School, Session XIII,
edited by C.\,DeWitt and B.\,DeWitt,
(Gordon Breach, 1964).
\bibitem{dewitt67a}
B.\,DeWitt,
{\it Quantum Theory of Gravity.\,I.\,
The Canonical Theory},
\OPR{160}{1113}{1967};
{\it Quantum Theory of Gravity.\,II.\,
The Manifestly Covariant Theory},
\OPR{162}{1195}{1967};
{\it Quantum Theory of Gravity.\,III.\,
Applications of the Covariant Theory},
\OPR{162}{1239}{1967}.
\bibitem{dewitt84a}
B.\,S.\,DeWitt,
{\it The Spacetime Approach to Quantum Field Theory},
in Relativity, Groups and Topology, II,
Les Houches Summer School, session XL,
edited by B.\,S.\,DeWitt and R.\,Stora,
(NorthHolland, Amsterdam, 1984).
\bibitem{dewitt84b}
B.\,S.\,DeWitt,
{\it Supermanifolds},
(Cambridge Univ.\ Press, Cambridge, 1984).
\bibitem{diaz88a}
A.\,H.\,Diaz,
{\it The Nonabelian Antisymmetric Tensor Field Revisited},
\PLB{203}{408}{1988}.
\bibitem{dtnp89a}
A.\,Diaz, W.\,Troost,
P.\,van Nieuwenhuizen and A.\,Van Proeyen,
{\it Understanding Fujikawa Regulators
from PauliVillars Regularization of Ghost Loops},
\IJMPA{4}{3959}{1989}.
\bibitem{dirac64a}
P.\,A.\,M.\,Dirac,
{\it Lectures on Quantum Mechanics},
(Yeshiva University, New York, 1964).
%(QC174.1.D55).
\bibitem{dixon76a}
J.\,A.\,Dixon,
{\it Cohomology and Renormalization of Gauge Theories I,II,III,IV},
unpublished (19761979);
\CMP{139}{495}{1991}.
\bibitem{donalson83a}
S.\,Donaldson,
{\it An Application of Gauge Theory
to the Topology of Four Manifolds},
J.\,Diff.\,Geom.\, {\bf 18} (1983) 279;
{\it Polynomial Invariants of Smooth
FourManifolds},
Topology {\bf 29} (1990) 257.
\bibitem{dfgh91a}
A.\,Dresse, J.\,M.\,L.\,Fisch,
P.\,Gr\'egoire and M.\,Henneaux,
{\it Equivalence of the Hamiltonian
and Lagrangian Path Integrals
for Gauge Theories},
\NPB{354}{191}{1991}.
\bibitem{dhtv91a}
M.\,DuboisViolette, M.\,Henneaux, M.\,Talon
and C.\,M.\,Viallet,
{\it Some Results on Local Cohomologies in Field Theory},
\PLB{267}{81}{1991}.
\bibitem{dtv85a}
M.\,DuboisViolette, M.\,Talon and C.\,M.\,Viallet,
{\it B.\,R.\,S.\, Algebras. Analysis
of the Consistency Equations in Gauge Theory},
\CMP{102}{105}{1985}.
\bibitem{dyson49a}
F.\,J.\,Dyson,
{\it The $S$Matrix in Quantum Electrodynamics},
Phys.\,Rev.\,{\bf 75} (1949) 1736.
\bibitem{elop66a}
R.\,J.\,Eden, P.\,V.\,Landshoff,
D.\,I.\,Olive and J.\,C.\,Polkinghorne,
{\it The Analytic $S$Matrix},
(Cambridge University Press, Cambridge, UK, 1966).
%(QC174.5.E3).
\bibitem{egh80a}
T.\,Eguchi, P.\,B.\,Gilkey and A.\,J.\,Hanson,
{\it Gravitation, Gauge Theories
and Differential Geometry},
\PR{66}{213}{1980}.
\bibitem{faddeev84a}
L.\,D.\,Faddeev,
{\it Operator Anomaly for the Gauss Law},
\PL{145B}{81}{1984}.
\bibitem{fp67a}
L.\,D.\,Faddeev and V.\,N.\,Popov,
{\it Feynman Diagrams for the YangMills Field},
\PL{25B}{29}{1967}.
\bibitem{fs84a}
L.\,D.\,Faddeev and S.\,L.\,Shatashvili,
{\it Algebraic and Hamiltonian Methods
in the Theory of NonAbelian Anomalies},
\TMP{60}{770}{1984};
{\it Realization of the Schwinger Term in the Gauss Law
and the Possibility of Correct Quantization of a Theory
with Anomalies},
\PL{167B}{225}{1986}.
\bibitem{fs80a}
L.\,D.\,Faddeev and A.\,A.\,Slavnov,
{\it Gauge Fields: Introduction to Quantum Theory},
(BenjaminCummings, Reading, MA, 1980).
%(QC793.3.F5 S5213).
\bibitem{feynman63a}
R.\,P.\,Feynman,
{\it Quantum Theory of Gravitation},
Acta Phys.\,Pol.\,{\bf 24} (1963) 697.
\bibitem{fh65a}
R.\,P.\,Feynman and A.\,R.\,Hibbs,
{\it Quantum Mechanics and Path Integrals},
(McGrawHill, New York, 1965).
\bibitem{fisch90a}
J.\,M.\,L.\,Fisch,
{\it On the BatalinVilkovisky
AntibracketAntifield BRST Formalism
and Its Applications},
Univ.\ Libre de Bruxelles preprint ULBTH29001,
(Jan., 1990).
\bibitem{fh89a}
J.\,M.\,L.\,Fisch and M.\,Henneaux,
{\it The AntifieldAntibracket Formalism
for Constrained Hamiltonian Systems},
\PLB{226}{80}{1989}.
\bibitem{fh90a}
J.\,M.\,L.\,Fisch and M.\,Henneaux,
{\it Homological Perturbation Theory
and the Algebraic Structure
of the AntifieldAntibracket Formalism
for Gauge Theories},
\CMP{128}{627}{1990}.
\bibitem{fhst89a}
J.\,M.\,L.\,Fisch, M.\,Henneaux, J.\,Stasheff
and C.\,Teitelboim,
{\it Existence, Uniqueness and Cohomology
of the Classical BRST Charge
with Ghosts for Ghosts},
\CMP{120}{379}{1989}.
\bibitem{flanders63a}
H.\,Flanders,
{\it Differential Forms},
(Academic Press, 1963).
%(QA381.F56).
\bibitem{ff78a}
E.\,S.\,Fradkin and T.\,E.\,Fradkina,
{\it Quantization of Relativistic Systems with Boson
and Fermion First and SecondClass Constraints},
\PL{72B}{343}{1978}.
\bibitem{fv75a}
E.\,S.\,Fradkin and G.\,A.\,Vilkovisky,
{\it Quantization of Relativistic Systems with Constraints},
\PL{55B}{224}{1975}.
\bibitem{fv75b}
E.\,S.\,Fradkin and G.\,A.\,Vilkovisky,
{\it Unitarity of $S$Matrix in Gravidynamics
and General Covariance in Quantum Domain},
Nuovo Cim.\,Lett.\,{\bf 13} (1975) 187.
\bibitem{fgst88a}
D.\,Z.\,Freedman, S.\,Giddings, J.\,Shapiro
and C.\,B.\,Thorn,
{\it The Nonplanar OneLoop Amplitude
in Witten's String Field Theory},
\NPB{298}{253}{1988}.
\bibitem{ft81a}
D.\,Z.\,Freedman and P.\,K.\,Townsend,
{\it Antisymmetric Tensor Gauge Theories
and NonLinear $\sigma$Models},
\NPB{177}{282}{1981}.
\bibitem{fn76a}
D.\,Z.\,Freedman and P.\,van Nieuwenhuizen,
{\it Properties of Supergravity Theory},
\PRD{14}{912}{1976}.
\bibitem{fnf76a}
D.\,Z.\,Freedman, P.\,van\,Nieuwenhuizen and S.\,Ferrara,
{\it Progress Toward a Theory of Supergravity},
\PRD{13}{3214}{1976}.
\bibitem{fgz86a}
I.\,B.\, Frenkel, H.\,Garland and G.\,J.\,Zuckerman,
{\it SemiInfinite Cohomology and String Theory},
Proc.\,Natl.\,Acad.\,Sci.\,{\bf 83} (1986) 8442.
\bibitem{fs88a}
S.\,A.\,Frolov and A.\,A.\,Slavnov,
{\it Quantization of NonAbelian
Antisymmetric Tensor Field},
Theor.\,Math.\,Phys.\,{\bf 75} (1988) 470.
\bibitem{fs89a}
S.\,A.\,Frolov and A.\,A.\,Slavnov,
{\it Physical Subspace Norm
in Hamiltonian BRSTQuantization},
\PLB{218}{461}{1989}.
\bibitem{fs90a}
S.\,A.\,Frolov and A.\,A.\,Slavnov,
{\it Construction of the Effective Action
for General Gauge Theories Via Unitarity},
\NPB{347}{333}{1990}.
\bibitem{fs90b}
S.\,Frolov and A.\,Slavnov,
{\it Lagrangian BRST Quantization and Unitarity},
\TMP{85}{1237}{1990}.
\bibitem{fujikawa80a}
%don't change bibitem date
K.\,Fujikawa,
{\it PathIntegral Measure for GaugeInvariant
Fermion Theories},
\PRL{42}{1195}{1979};
{\it Path Integral for Gauge Theories with Fermions},
\PRD{21}{2848}{1980}.
\bibitem{fujikawa82a}
K.\,Fujikawa,
{\it Path Integral of Relativistic Strings},
\PRD{25}{2584}{1982}.
\bibitem{fis90a}
K.\,Fujikawa, T.\,Inagaki and H.\,Suzuki,
{\it BRS Current and Related Anomalies
in Two Dimensional Gravity and String Theories},
\NPB{332}{499}{1990}.
\bibitem{gglrsnv89a}
S.\,J.\,Gates Jr., M.\,T.\,Grisaru,
U.\,Lindstr\"om, M.\,Ro\v{c}ek,
W.\,Siegel, P.\,van Nieuwenhuizen and A.\,E.\,van de Ven,
{\it LorentzCovariant Quantization
of the Heterotic Superstring},
\PLB{225}{44}{1989}.
\bibitem{ggrs83a}
S.\,J.\,Gates Jr., M.\,T.\,Grisaru,
M.\,Ro\v{c}ek and W.\,Siegel,
{\it Superspace or One Thousand
and One Lessons in Supersymmetry},
(Benjamin/Cummings, Reading, MA, 1983).
\bibitem{gerstenhaber62a}
M.\,Gerstenhaber,
{\it The Cohomology Structure of an Associative Ring},
Ann.\,Math.\,{\bf 78} ({1962}) {267}.
\bibitem{gerstenhaber64a}
M.\,Gerstenhaber,
{\it On the Deformation of Rings and Algebras},
Ann.\,Math.\,{\bf 79} ({1964}) {59}.
\bibitem{getzler92a}
E.\,Getzler,
{\it BatalinVilkovisky Algebras and
TwoDimensional Topological Field Theories},
\CMP{159}{265}{1994}.
\bibitem{giddings86a}
S.\,Giddings,
{\it The Veneziano Amplitude
from Interacting String Field Theory},
\NPB{278}{242}{1986}
\bibitem{glashow61a}
S.\,L.\,Glashow,
{\it Partial Symmetries of Weak Interactions},
Nucl.\,Phys.\, {\bf 22} (1961) {579}.
\bibitem{gp92a}
J.\,Gomis and J.\,Par\'{\i}s,
{\it Unitarity and the FieldAntifield Formalism},
\NPB{368}{311}{1992}.
\bibitem{gp93a}
J.\,Gomis and J.\,Par\'{\i}s,
{\it FieldAntifield Formalism for Anomalous Gauge Theories},
\NPB{395}{288}{1993}.
\bibitem{gp93b}
J.\,Gomis and J.\,Par\'{\i}s,
{\it Perturbation Theory and Locality
in the FieldAntifield Formalism},
\JMP{34}{2132}{1993}.
\bibitem{gp93c}
J.\,Gomis and J.\,Par\'{\i}s,
{\it Anomalies and WessZumino Terms
in an Extended Regularized FieldAntifield Formalism},
\NPB{431}{378}{1994},
hepth/9401161.
\bibitem{gp93d}
J.\,Gomis and J.\,Par\'{\i}s,
{\it Anomalous Gauge Theories Within the FieldAntifield Formalism},
in Proc.\,of the
International EPS Conference on High Energy Physics,
HEP93, Marseille, (July, 1993), p.\ 101102.
\bibitem{gprr90a}
J.\,Gomis, J.\,Par\'{\i}s, K.\,Rafanelli and J.\,Roca,
{\it BRST and AntiBRST Symmetries
for the Spinning Particle},
\PLB{246}{435}{1990}.
\bibitem{gpr90a}
J.\,Gomis, J.\,Par\'{\i}s and J.\,Roca,
{\it BRST Structures of the Spinning String},
\NPB{339}{711}{1990}.
\bibitem{gr90a}
J.\,Gomis and J.\,Roca,
{\it The AntiBRST Symmetry
in the FieldAntifield Formalism},
\NPB{343}{152}{1990}.
\bibitem{goto71a}
T.\,Goto,
{\it Relativistic Quantum Mechanics
of One Dimensional Mechanical Continuum
as Subsidiary Condition of Dual Resonance Model},
Prog.\,Theor.\,Phys.\,{\bf 46} (1971) 1560.
\bibitem{gh89a}
M.\,B.\,Green and C.\,M.\,Hull,
{\it Covariant Quantum Mechanics of the Superstring},
\PLB{225}{57}{1989}.
\bibitem{ggt91a}
G.\,V.\,Grigorian, R.\,P.\,Grigorian and I.\,V.\,Tyutin,
{\it Equivalence of Lagrangian
and Hamiltonian BRST Quantizations:
Systems with First Class Constraints},
Sov.\,J.\,Nucl.\,Phys.\,{\bf 50} (1991) 1058.
\bibitem{ggt92a}
G.\,V.\,Grigorian, R.\,P.\,Grigorian and I.\,V.\,Tyutin,
{\it Equivalence of Lagrangian
and Hamiltonian BRST Quantizations:
the General Case},
\NPB{379}{304}{1992}.
\bibitem{gj87a}
D.\,Gross and A.\,Jevicki,
{\it Operator Formulation
of Interacting String Field Theory (I)},
\NPB{283}{1}{1987}.
\bibitem{gj87b}
D.\,Gross and A.\,Jevicki,
{\it Operator Formulation
of Interacting String Field Theory (II)},
\NPB{287}{225}{1987}.
\bibitem{gupta50a}
S.\,N.\,Gupta,
{\it Theory of Longitudinal Photons
in Quantum Electrodynamics},
Proc.\,Phys.\,Soc.\,{\bf A63} (1950) 681.
\bibitem{ht87a}
K.\,Harada and I.\,Tsutsui,
{\it On the PathIntegral Quantization
of Anomalous Gauge Theories},
\PLB{183}{311}{1987}.
\bibitem{}
H.\,Hata,
{\it Construction of the Quantum Action
for Path Integral Quantization
of String Field Theory},
\NPB{339}{663}{1990}.
\bibitem{hata93a}
H.\,Hata,
{\it ``Theory of Theories''
Approach to String Theory},
\PRD{50}{4079}{1994}.
\bibitem{hko81a}
H.\,Hata, T.\,Kugo and N.\,Ohta,
{\it SkewSymmetric Tensor Gauge Field Theory
Dynamically Realized in the QCD $U(1)$ Channel},
\NPB{178}{527}{1981}.
\bibitem{hz94a}
H.\,Hata and B.\,Zwiebach,
{\it Developing the Covariant BatalinVilkovisky Approach
to String Field Theory},
Ann.\,Phys.\,{\bf 229} (1994) 177.
\bibitem{henneaux85a}
M.\,Henneaux,
{\it Hamiltonian Form of the Path Integral
for Theories with a Gauge Freedom},
\PR{126}{1}{1985}.
\bibitem{henneaux89a}
M.\,Henneaux,
{\it On the Algebraic Structure of the BRST Symmetry},
Lectures given at Banff Summer School
in Theoretical Physics
on Physics, Geometry and Topology,
in Banff NATO ASI 1989, p.81104.
%(QC20.N22 1989).
\bibitem{henneaux90a}
M.\,Henneaux,
{\it Lectures on
the AntifieldBRST Formalism
for Gauge Theories},
Nucl.\,Phys.\,{\bf B} (Proc.\,Suppl.) {\bf {18A}} (1990) 47.
\bibitem{}
M.\,Henneaux,
{\it Elimination of the Auxiliary Fields
in the Antifield Formalism},
\PLB{238}{299}{1990}.
\bibitem{henneaux91a}
M.\,Henneaux,
{\it Spacetime Locality of the BRST Formalism},
\CMP{140}{1}{1991}.
\bibitem{henneaux92a}
M.\,Henneaux,
{\it Geometric Interpretation
of the Quantum Master Equation
in the BRST AntiBRST Formalism},
\PLB{282}{372}{1992}.
\bibitem{henneaux93a}
M.\,Henneaux,
{\it Remarks on the Renormalization
of Gauge Invariant Operators in YangMills Theory},
\PLB{313}{35}{1993}, (Erratum \PLB{316}{633}{1993}).
\bibitem{ht92a}
M.\,Henneaux and C.\,Teitelboim,
{\it Quantization of Gauge Systems},
(Princeton University Press, Princeton, 1992).
\bibitem{horava94a}
P.\,Horava,
{\it Space Time Diffeomorphism
and Topological $W_\infty$
in Two Dimensional Topological String Theory},
\NPB{414}{485}{1994}.
\bibitem{hlw90a}
P.\,S.\,Howe, U.\,Lindstr\"om and P.\,White,
{\it Anomalies and Renormalization
in the BRSTBV Framework},
\PLB{246}{430}{1990}.
\bibitem{hull90a}
C.\,M.\,Hull,
{\it The BRST and AntiBRST Invariant Quantization
of General Gauge Theories},
\MPLA{5}{1871}{1990}.
\bibitem{hull91a}
C.\,M.\,Hull,
{\it MatterDependent $W$Gravity Anomalies
of NonLinearly Realised Symmetries},
\PLB{265}{347}{1991}.
\bibitem{hull93a}
C.\,M.\,Hull,
{\it $W$ Gravity Anomalies with Ghost Loops
and Background Charges},
\IJMPA{8}{2419}{1993}.
\bibitem{hwang83a}
S.\,Hwang,
{\it Covariant Quantization of the String
in Dimensions $D \le 26$ Using a
BecchiRouetStora Formulation},
\PRD{28}{2614}{1983}.
\bibitem{ikemori93a}
H.\,Ikemori,
{\it Extended Form Method of Antifield BRST Formalism
for Topological Quantum Field Theories},
Clas.\,Quantum Grav.\,{\bf 10}, (1993) 233.
\bibitem{iz80a}
C.\,Itzykson and J.\,B.\,Zuber,
{\it Quantum Field Theory},
(McGrawHill, New York, 1980).
%(QC174.45.I77).
\bibitem{jr85a}
R.\,Jackiw and R.\,Rajaraman,
{\it VectorMeson Mass Generation by Chiral Anomalies},
\PRL{54}{1219}{1985}.
\bibitem{jl76a}
S.\,D.\,Joglekar and B.\,W.\,Lee,
{\it Gauge Theory of Renormalization
of Gauge Invariant Operators},
Ann.\,Phys.\,{\bf 97} (1976) 160.
\bibitem{kaku88a}
M.\,Kaku,
{\it Introduction to Superstrings},
(SpringerVerlag, New York, 1988).
%(QC794.6.S85 K35).
\bibitem{kaku88b}
M.\,Kaku,
{\it Geometrical Derivation
of String Field Theory from First Principles:
Closed Strings and Modular Invariance},
\PRD{38}{3052}{1988};
{\it Nonpolynomial ClosedString Field Theory},
\PRD{41}{3734}{1990}.
\bibitem{kl88a}
M.\,Kaku and J.\,Lykken,
{\it Modular Invariant Closed String Field Theory},
\PRD{38}{3067}{1988}.
\bibitem{kr74a}
M.\,Kalb and P.\,Ramond,
{\it Classical Direct Interstring Action},
\PRD{9}{2273}{1974}.
\bibitem{kallosh78a}
R.\,E.\,Kallosh,
{\it Modified Rules in Supergravity},
\NPB{141}{141}{1978}.
\bibitem{kallosh89a}
R.\,Kallosh,
{\it SuperPoincar\'e Covariant Quantized GS Heterotic
String as a Free Conformal Theory},
\PLB{224}{273}{1989}.
\bibitem{ko83a}
M.\,Kato and K.\,Ogawa,
{\it Covariant Quantization of String Based
on BRS Invariance},
\NPB{212}{443}{1983}.
\bibitem{khudaverdian91a}
O.\,M.\,Khudaverdian,
{\it Geometry of Superspace with Even and Odd Brackets},
\JMP{32}{1934}{1991}.
\bibitem{kn93a}
O.\,M.\,Khudaverdian and A.\,P.\,Nersesian,
{\it On the Geometry of the BatalinVilkovisky Formalism},
\MPLA{8}{2377}{1993}.
\bibitem{kibble61a}
T.\,W.\,Kibble,
{\it Lorentz Invariance and the Gravitational Field},
\JMP{2}{212}{1961}.
\bibitem{kimura81a}
T.\,Kimura,
{\it Antisymmetric Tensor Gauge Field in General Covariant Gauges},
Prog.\,Theo.\,Phys.\,{\bf 64} (1980) 357;
{\it Quantum Theory of Antisymmetric Higher Rank
Tensor Gauge Field
in Higher Rank Dimensional SpaceTime},
Prog.\,Theo.\,Phys.\,{\bf 65} (1981) 338.
\bibitem{koszul50a}
J.\,L.\,Koszul,
{\it Sur un Type d'Alg\'ebres Differ\'entielles
en Raport avec la Transgression},
Bull.\,Soc.\,Math.\,France {\bf 78} (1950) 5.
\bibitem{kks89a}
T.\,Kugo, H.\,Kunitomo and K.\,Suehiro,
{\it NonPolynomial Closed String Field Theory},
\PLB{226}{48}{1989}.
\bibitem{ko78a}
T.\,Kugo and I.\,Ojima,
{\it Manifestly Covariant Canonical Formalism
of YangMills Theories, Physical State Subsidiary
Conditions and Physical $S$Matrix Unitarity},
\PLB{73}{459}{1978}.
\bibitem{ko79a}
T.\,Kugo and I.\,Ojima,
{\it Local Covariant Operator Formalism
of NonAbelian Gauge Theories
and Quark Confinement Problem},
Prog.\,Theor.\,Phys.\,Suppl.\,{\bf 66} (1979) 1.
\bibitem{ks90a}
T.\,Kugo, and K.\,Suehiro,
{\it Nonpolynomial Closed String Field Theory:
Action and Gauge Invariance},
\NPB{337}{343}{1990}.
\bibitem{ku82a}
T.\,Kugo and S.\,Uehara,
{\it General Procedure of Gauge Fixing
Based on BRS Invariance Principle},
\NPB{197}{378}{1982}.
\bibitem{lp88a}
J.\,M.\,F.\,Labastida and M.\,Pernici,
{\it A Gauge Invariant Action
in Topological Quantum Field Theory},
\PLB{212}{56}{1988}.
\bibitem{lt85a}
P.\,M.\,Lavrov and I.\,V.\,Tyutin,
{\it Effective Action in Gauge Theories
of a General Form},
Sov.\,J.\,Nucl.\,Phys.\,{\bf 41} (1985) 1049.
\bibitem{lee76a}
B.\,W.\,Lee, {\it Gauge Theories},
in Methods in Field Theory, Les Houches Summer School 1975,
edited by R.\,Balian and J.\,ZinnJustin
(NorthHolland, Amsterdam, 1976).
\bibitem{leites80a}
D.\,A.\,Leites,
{\it Introduction to the Theory of Supermanifolds},
Russ.\,Math.\,Surv.\,{\bf 35} (1980) 1.
\bibitem{lz93a}
B.\,H.\,Lian and G.\,J.\,Zuckerman,
{\it New Perspectives
on the BRSTAlgebraic Structure of String Theory},
\CMP{154}{613}{1993}.
\bibitem{lrsnv89a}
U.\,Lindstr\"om, M.\,Ro\v{c}ek, W.\,Siegel,
P.\,van Nieuwenhuizen and A.\,E.\,van de Ven,
{\it LorentzCovariant Quantization
of the Superparticle},
\PLB{224}{285}{1989};
{\it GaugeFixing Redundant Symmetries
in the Superparticle},
\PLB{228}{53}{1989}.
\bibitem{lps93a}
C.\,Lucchesi, O.\,Piguet and S.\,P.\,Sorella,
{\it Renormalization
and Finiteness of Topological BF Theories},
\NPB{395}{325}{1993}.
\bibitem{lr86a}
J.\,Lykken and S.\,Raby,
{\it NonCommutative Geometry
and the Closed Bosonic String},
\NPB{278}{256}{1986}.
\bibitem{ms93a}
N.\,Maggiore and S.\,P.\,Sorella,
{\it Perturbation Theory for Antisymmetric Tensor Fields
in FourDimensions},
\IJMPA{8}{929}{1993}.
\bibitem{marnelius93a}
R.\,Marnelius,
{\it General State Spaces for BRST Quantizations},
\NPB{391}{621}{1993}.
\bibitem{mpsx94a}
R.\,Mohayee, C.\,N.\,Pope, K.\,S.\,Stelle and K.\,W.\,Xu,
{\it Canonical BRST Quantization of World Sheet Gravities},
Texas A\&M preprint CTPTAMU1894,
(April, 1994),
hepth/9404170.
\bibitem{no90a}
N.\,Nakanishi and I.\,Ojima,
{\it Covariant Operator Formalism of Gauge
Theories and Quantum Gravity},
World Scientific Lecture Notes, Vol.\, {\bf 27} (1990).
\bibitem{nambu70a}
Y.\,Nambu,
{\it Duality and Hadrodynamics},
Lectures at the Copenhagen Summer Symposium, (1970),
published in
Strings, Lattice Gauge Theory, High Energy Phenomenology,
Proc.\ of the Winter School Panchgani,
edited by V.\,Singh and S.\,R.\,Wadia,
(World Scientific, 1987).
\bibitem{nersesian93a}
A.\,P.\,Nersesian,
{\it On Geometry of Supermanifolds
with Odd and Even Kahlerian Structures},
\TMP{96}{866}{1993}.
\bibitem{nielsen78a}
N.\,K.\,Nielsen,
{\it Ghost Counting in Supergravity},
\NPB{140}{494}{1978}.
\bibitem{nielsen81a}
N.\,K.\,Nielsen,
{\it BRS Invariance of Supergravity in a Gauge
Involving an Extra Ghost},
\PL{103B}{197}{1981}.
\bibitem{nr92a}
Kh.\,S.\,Nirov and A.\,V.\,Razumov,
{\it FieldAntifield and BFV Formalisms
for Quadratic Systems
with Open Gauge Algebras},
\IJMPA{7}{5719}{1992}.
\bibitem{ojima80a}
I.\,Ojima,
{\it Another BRS Transformation},
Prog.\,Theo.\,Phys.\,{\bf 64} (1981) 625.
\bibitem{oppt93a}
C.\,Ord\'o\~nez, J.\,Par\'{\i}s,
J.\,M.\,Pons and R.\,Toldr\`a,
{\it BV Analysis for Covariant and Noncovariant Actions},
\PRD{48}{3818}{1993}.
\bibitem{opt93a}
C.\,Ord\'o\~nez, J.\,M.\,Pons and R.\,Toldr\`a,
{\it Unitarity and BatalinVilkovisky
Path Integral Quantization for Gauge Theories},
\PLB{302}{423}{1993}.
\bibitem{on82a}
F.\,R.\,Ore, Jr.\, and P.\,van Nieuwenhuizen,
{\it Local BRST Symmetry and the
Geometry of GaugeFixing},
\NPB{204}{317}{1982}.
\bibitem{paris92a}
J.\,Par\'{\i}s,
{\it Anomalies i Unitarietat
en el Formalisme de Camps i Anticamps},
Thesis,
Universitat de Barcelona (1992).
\bibitem{paris93a}
J.\,Par\'{\i}s,
{\it FaddeevPopov Method for Anomalous Quasigroups},
\PLB{300}{104}{1993}.
\bibitem{pc69a}
J.\,E.\,Paton and H.M.\,Chan,
{\it Generalized Veneziano Model with Isospin},
\NPB{10}{516}{1969}.
\bibitem{ps92a}
M.\,Penkava and A.\,Schwarz,
{\it On Some Algebraic Structure
Arising in String Theory},
UC Davis preprint PRINT930001,
(Dec., 1992),
hepth/9212072.
\bibitem{polyakov81a}
A.\,M.\,Polyakov,
{\it Quantum Geometry of Bosonic Strings},
\PL{103B}{207}{1981}.
\bibitem{rsnv89a}
M.\,Ro\v{c}ek, W.\,Siegel, P.\,van Nieuwenhuizen and
A.\,E.\,van de Ven,
{\it Covariant Superparticle Quantization
in a Super Maxwell Background},
\PLB{227}{87}{1989}.
\bibitem{sz89a}
M.\,Saadi and B.\,Zwiebach,
{\it Closed String Theory from Polyhedra},
Ann.\,Phys.\,{\bf 192} (1989) 213.
\bibitem{salam68a}
A.\,Salam,
{\it Weak and Electromagnetic Interactions},
in Elementary Particle Theory,
edited by N.\,Svartholm,
(Almquist and Forlag, Stockholm, 1968),
p.367377.
%(QC721.N57 1968).
\bibitem{samuel86a}
S.\,Samuel,
{\it The Physical and Ghost Vertices
in Witten's String Field Theory},
\PLB{181}{255}{1986}.
\bibitem{samuel87a}
S.\,Samuel,
{\it Introduction to String Field Theory},
in Strings and Superstrings,
edited by J.\,P.\,Mittelbrunn, M.\,Ram\'onMedrano
and G.\,S.\,Rodero,
El Escorial, Spain, June, 1987,
(World Scientific, Singapore, 1988);
Also see Appendices AC of
{\it Mathematical Formulation of Witten's
Superstring Field Theory},
\NPB{296}{187}{1988}.
\bibitem{samuel88a}
S.\,Samuel,
{\it Covariant OffShell String Amplitudes},
\NPB{308}{285}{1988};
{\it OffShell String Physics from
Conformal Field Theory},
\NPB{308}{317}{1988}.
\bibitem{samuel90a}
S.\,Samuel,
{\it Solving the Open Bosonic String
in Perturbation Theory},
\NPB{341}{513}{1990}.
\bibitem{schwarz93a}
A.\,Schwarz,
{\it Geometry of BatalinVilkovisky Quantization},
\CMP{155}{249}{1993}.
\bibitem{schwarz93b}
A.\,Schwarz,
{\it Semiclassical Approximation
in BatalinVilkovisky Formalism},
\CMP{158}{373}{1993}.
\bibitem{schwarz82a}
J.\,H.\,Schwarz,
{\it Superstring Theory},
\PR{C89}{223}{1982}.
\bibitem{schwinger51a}
J.\,Schwinger,
{\it On the Green's Function of Quantized Fields.\ I.},
Proc.\,Nat.\,Acad.\,Sci.\,{\bf 37} (1951) 452;
{\it On the Green's Function of Quantized Fields.\ II.},
Proc.\,Nat.\,Acad.\,Sci.\,{\bf 37} (1951) 455.
\bibitem{ss92a}
N.\,Seiberg and S.\,Shenker,
{\it A Note on Background (In)dependence},
\PRD{45}{4581}{1992}.
\bibitem{sen90a}
A.\,Sen,
{\it On the Background Independence of String Field Theory},
\NPB{345}{551}{1990};
{\it On the Background Independence of String Field Theory II.\,
Analysis of Onshell $S$Matrix Elements},
\NPB{347}{270}{1990};
{\it On the Background Independence of String Field Theory III.\,
Explicit Field Redefinitions},
\NPB{391}{550}{1993}.
\bibitem{sz93a}
A.\,Sen and B.\,Zwiebach,
{\it A Proof of Local Background Independence
of Classical Closed String Field Theory},
\NPB{414}{649}{1994}.
\bibitem{sz94a}
A.\,Sen and B.\,Zwiebach,
{\it Quantum Background Independence
of Closed String Field Theory},
\NPB{414}{485}{1994}.
\bibitem{sz94b}
A.\,Sen and B.\,Zwiebach,
{\it A Note on Gauge Transformations
in BatalinVilkovisky Theory},
\PLB{320}{29}{1994}.
\bibitem{sezgin93a}
E.\,Sezgin,
{\it Aspects of Kappa Symmetry},
Texas A\&M preprint CTPTAMU2893,
(Oct., 1993),
hepth/9310126.
\bibitem{siegel80a}
W.\,Siegel,
{\it Hidden Ghosts},
\PL{93B}{170}{1980}.
\bibitem{siegel84a}
W.\,Siegel,
{\it Covariantly SecondQuantized String},
\PL{142B}{276}{1984};
{\it Covariantly SecondQuantized String II},
\PL{151B}{391}{1985};
{\it Covariantly SecondQuantized String III},
\PL{151B}{396}{1985}.
\bibitem{siegel89a}
W.\,Siegel,
{\it BatalinVilkovisky from Hamiltonian BRST},
\IJMPA{4}{3951}{1989}.
\bibitem{siegel89b}
W.\,Siegel,
{\it Relation Between BatalinVilkovisky
and First Quantized Style BRST},
\IJMPA{4}{3705}{1989}.
\bibitem{siegel90a}
W.\,Siegel,
{\it Lorentz Covariant Gauges for GreenSchwarz Superstrings},
in Strings 89,
eds.\,R.\,Arnowitt et al.
(World Scientific, Singapore, 1990).
\bibitem{slavnov72a}
A.\,A.\,Slavnov,
{\it Ward Identities in Gauge Theories},
\TMP{10}{99}{1972}.
\bibitem{slavnov89a}
A.\,A.\,Slavnov,
{\it Physical Unitarity in the BRST Approach},
\PLB{217}{91}{1989}.
\bibitem{slavnov90a}
A.\,A.\,Slavnov,
{\it BRSTQuantization and Unitarity},
Nucl.\,Phys.\,{\bf B} (Proc.\,Suppl.) {\bf B15} (1990) 107.
\bibitem{sohnius83a}
M.\,F.\,Sohnius,
{\it Soft Gauge Algebras},
Z.\,Phys.\,{\bf C18} (1983) 229.
\bibitem{stasheff93a}
J.\,Stasheff,
{\it Closed String Field Theory,
Strong Homotopy Lie Algebras
and the Operad Actions of Moduli Space},
Univ.\ of North Carolina preprint UNCMATH931,
(April, 1993),
hepth/9304061.
\bibitem{stn78a}
G.\,Sterman, P.\,K.\,Townsend and P.\,van Nieuwenhuizen,
{\it Unitarity, Ward Identities,
and New Quantization Rules of Supergravity},
\PRD{17}{1501}{1978}.
\bibitem{takahashi57a}
Y.\,Takahashi,
{\it On the Generalized Ward Identity},
\NC{6}{371}{1957}.
\bibitem{tate57a}
J.\,Tate,
{\it Homology of Noetherian Rings and Local Rings},
Illinois J.\,Math.\, {\bf 1} (1957) 14.
\bibitem{taylor71a}
J.\,C.\,Taylor,
{\it Ward Identities and Charge Renormalization
of the YangMills Field},
\NPB{33}{436}{1971}.
\bibitem{thierrymieg90a}
J.\,ThierryMieg,
{\it BRS Structure of
the Antisymmetric Tensor Gauge Theories},
\NPB{335}{334}{1990}.
\bibitem{thooft71a}
G.\,t'Hooft,
{\it Renormalization of Massless YangMills Fields},
\NPB{33}{173}{1971}.
\bibitem{thooft71b}
G.\,t'Hooft,
{\it Renormalizable Lagrangians for Massive YangMills Fields},
\NPB{35}{167}{1971}.
\bibitem{tv73a}
G.\,t'Hooft and M.\,Veltman,
{\it Diagrammar},
in Louvain 1973, Particle Interactions
at Very High energies, Part B,
(New York, 1973), p.177322.
\bibitem{thorn87a}
C.\,B.\,Thorn,
{\it Perturbation Theory for Quantized String Fields},
\NPB{287}{61}{1987}.
\bibitem{thorn89a}
C.\,B.\,Thorn,
{\it String Field Theory},
\PR{175}{1}{1989}.
\bibitem{tonin91a}
M.\,Tonin,
{\it Covariant Quantization and
Anomalies of the GS Heterotic $\sigma$Model},
\IJMPA{6}{315}{1991}.
\bibitem{tonin92a}
M.\,Tonin,
{\it Dimensional Regularization and Anomalies
in Chiral Gauge Theories},
talk presented at
Topical Workshop on Nonperturbative Aspect
of Chiral Gauge Theories,
Rome, Italy, (March, 1992),
in Rome Chiral Gauge, p.137151.
%(QCD161.T685 1992)
\bibitem{townsend80a}
P.\,K.\,Townsend,
{\it Gauge Invariance for Spin $1/2$},
\PL{90B}{275}{1980}.
\bibitem{tnp90a}
W.\,Troost, P.\,van Nieuwenhuizen and A.\,Van Proeyen,
{\it Anomalies and
the BatalinVilkovisky Lagrangian Formalism},
\NPB{333}{727}{1990}.
\bibitem{tp93a}
W.\,Troost and A.\,Van Proeyen,
{\it Regularization and the BV Formalism},
in Proc.\,of Strings 93,
Berkeley, (May, 1993).
\bibitem{tpbook}
W.\,Troost and A.\,Van Proeyen,
{\it An Introduction to
BatalinVilkovisky Lagrangian Quantisation},
Leuven Notes in Math.\,Theor.\,Phys., in preparation.
\bibitem{tyutin75a}
I.\,V.\,Tyutin,
{\it Gauge Invariance in Field Theory
and Statistical Mechanics},
Lebedev preprint n$^{\rm o}$ 39 (1975), unpublished.
\bibitem{ts86a}
I.\,V.\,Tyutin and Sh.\,Shvartsman,
{\it BRST Operator and Open Algebra},
\PLB{169}{225}{1986}.
\bibitem{utiyama54a}
R.\,Utiyama,
{\it Invariant Theoretical Interpretation of Interaction},
{Phys.\,Rev.\,}{\bf 101} (1956) 1597.
\bibitem{vp94a}
S.\,Vandoren and A.\,Van Proeyen,
{\it Simplifications in Lagrangian BV Quantization
Exemplified by the Anomalies of Chiral $W_3$ Gravity},
\NPB{411}{257}{1994}.
\bibitem{vannieuwenhuizen81a}
P.\,van Nieuwenhuizen,
{\it Supergravity},
\PR{68}{189}{1981}.
\bibitem{vanproeyen91a}
A.\,Van Proeyen,
{\it BatalinVilkovisky Lagrangian Quantization},
talk presented in Proc.\ of
Strings and Symmetries Stony Brook,
Strings: Stony Brook, 1991, p.388406.
%(QCD161.S711).
\bibitem{vasiliev80a}
% do not change bibitem name
M.\,A.\,Vasil'ev,
{\it ``Gauge'' Form of Description of Massless Fields
With Arbitrary Spin},
Sov.\,J.\,Nucl.\,Phys.\,{\bf 32} (1980) 439.
\bibitem{verlinde92a}
E.\,Verlinde,
{\it The Master Equation of String Theory},
\NPB{381}{141}{1992}.
\bibitem{vlt82a}
B.\,L.\,Voronov, P.\,M.\,Lavrov and I.\,V.\,Tyutin,
{\it Canonical Transformations and Gauge Dependence
in General Gauge Theories},
Sov.\,J.\,Nucl.\,Phys.\, {\bf 36} (1982) 292.
\bibitem{vt82a}
B.\,L.\,Voronov and I.\,V.\,Tyutin,
{\it Formulation of Gauge Theories of General Form.\,I},
\TMP{50}{218}{1982}.
\bibitem{vt82b}
B.\,L.\,Voronov and I.\,V.\,Tyutin,
{\it Formulation of Gauge Theories
of General Form.\ II.\,GaugeInvariant Renormalizability
and Renormalization Structure},
\TMP{50}{628}{1982}.
\bibitem{ward50a}
J.\,C.\,Ward,
{\it An Identity in Quantum Electrodynamics},
Phys.\,Rev.\,{\bf 78} (1950) 182.
\bibitem{weinberg67a}
S.\,Weinberg,
{\it A Model of Leptons},
\PRL{19}{1264}{1967}.
\bibitem{wb83a}
J.\,Wess and J.\,Bagger,
{\it Introduction to Supersymmetry},
(Princeton Univ.\ Press, Princeton, NJ, 1983).
\bibitem{wz71a}
J.\,Wess and B.\,Zumino,
{\it Consequences of Anomalous Ward Identities},
\PL{37B}{95}{1971}.
\bibitem{}
P.\,L.\,White,
{\it Anomaly Consistency Conditions to All Orders},
\PLB{284}{55}{1992}.
\bibitem{witten86a}
E.\,Witten,
{\it NonCommutative Geometry and String Field Theory},
\NPB{268}{253}{1986}.
\bibitem{witten86b}
E.\,Witten,
{\it Open Superstrings},
\NPB{276}{291}{1986}.
\bibitem{witten88a}
E.\,Witten,
{\it Topological Quantum Field Theory},
\CMP{117}{353}{1988}.
\bibitem{witten90a}
E.\,Witten,
{\it A Note on the Antibracket Formalism},
\MPLA{5}{487}{1990}.
\bibitem{witten92a}
E.\,Witten,
{\it On Background Independent Open String Field Theory},
\PRD{46}{5467}{1992}.
\bibitem{witten93a}
E.\,Witten,
{\it Some Computations in Background Independent
OffShell String Field Theory},
\PRD{47}{5467}{1993}.
\bibitem{ym54a}
C.\,N.\,Yang and R.\,L.\,Mills,
{\it Considerations of Isotopic Spin
and Isotopic Gauge Invariance},
\OPR{96}{191}{1954}.
\bibitem{zinnjustin75a}
J.\,ZinnJustin,
{\it Renormalization of Gauge Theories},
in Trends in Elementary Particle Theory,
edited by H.\,Rollnik and K.\,Dietz,
Lecture Notes in Physics, Vol {\bf 37},
(SpringerVerlag, Berlin, 1975).
\bibitem{zinnjustin89a}
J.\,ZinnJustin,
{\it Quantum Field Theory and Critical Phenomena},
(Oxford University Press, Oxford, England, 1989).
%(QC174.45.Z56 1989).
\bibitem{zumino83a}
B.\,Zumino,
{\it Chiral Anomalies and Differential Geometry},
in Relativity, Groups and Topology,
edited by B.\,S.\,deWitt and R.\,Stora,
(NorthHolland, Amsterdam, 1983),
and references therein.
%(QC173.6.E26 1983).
\bibitem{zwiebach90a}
B.\,Zwiebach,
{\it Quantum Closed Strings from Minimal Area},
\IJMPA{5}{275}{1990};
{\it Consistency of Closed String Polyhedra from Minimal Area},
\PLB{241}{343}{1990};
{\it How Covariant Closed String Theory Solves
a Minimal Area Problem},
\CMP{136}{83}{1991}.
\bibitem{zwiebach93a}
B.\,Zwiebach,
{\it Closed String Field Theory: An Introduction},
MIT preprint MITCTP2206,
(May, 1993),
hepth/9305026.
\bibitem{zwiebach93b}
B.\,Zwiebach,
{\it Closed String Field Theory:
Quantum Action and the BV Master Equation},
\NPB{390}{33}{1993}.
\end{thebibliography}
\vfill\eject
\end{document}