%Paper: hep-th/9411107
%From: bello@cbpfsu1.cat.cbpf.br (Jose Luis Vazquez-Bello)
%Date: Tue, 15 Nov 94 15:19:30 EDT


%%%%%%%%%%This requires the PANDA.TEX macropackage
%
%\input panda.tex
%-------------------------------------------------------------------
%
%                                      Andrea PASQUINUCCI, 1988
%              PANDA.TEX               S.I.S.S.A., Trieste, Italy
%                                      (Revised 1991, Princeton, USA)
%
%--------------------------------------------------------------------
%
%    These are TEX macros. They work with PLAIN TEX (the basis
%    version of TEX). The only problem can be with the double-page
%    format since it depends on the type of postscript and postscript
%    laserwriters you use, so I cannot guarantee that the double-page
%    format will work properly. Double-page MUST be printed in
%    LANDSCAPE orientation. (You shouldn't have troubles with fonts;
%    if you do, please let me know.)
%
%--------------------------------------------------------------------
%
%                     INTERACTIVE SECTION
%
%--------------------------------------------------------------------
%
\def\standardrisposta{s }\def\reducedrisposta{r }
\def\mplarisposta{mpla }
\def\doublerisposta{d }\def\cartarisposta{e }\def\amsrisposta{y }
\newcount\ingrandimento \newcount\sinnota \newcount\dimnota
\newcount\unoduecol \newdimen\collhsize \newdimen\tothsize
\newdimen\fullhsize \newcount\controllorisposta \sinnota=1
\newskip\infralinea  \global\controllorisposta=0
%
\message{ ********    Welcome to PANDA macros (Plain TeX, AP, 1991)}
\message{ ******** }
\message{       You'll have to answer a few questions in lowercase.}
\message{>  Do you want it in double-page (d), reduced (r)}
\message{or standard format (s) ? }\read-1 to\risposta
\message{>  Do you want it in USA A4 (u) or EUROPEAN A4 (e)}
\message{paper size ? }\read-1 to\srisposta
\message{>  Do you have AMSFonts 2.0 (math) fonts (y/n) ? }
\read-1 to\arisposta
%%--------------------------------------------------------------------
%
%             END INTERACTIVE SECTION - PAGE FORMATTING
%
%--------------------------------------------------------------------
%
%\def\risposta{d } \def\srisposta{u } \def\arisposta{n }
%
\ifx\risposta\standardrisposta \ingrandimento=1200
\message{>> This will come out UNREDUCED << }
\dimnota=2 \unoduecol=1 \global\controllorisposta=1 \fi
\ifx\risposta\reducedrisposta \ingrandimento=1095 \dimnota=1
\unoduecol=1  \global\controllorisposta=1
\message{>> This will come out REDUCED << } \fi
\ifx\risposta\doublerisposta \ingrandimento=1000 \dimnota=2
\unoduecol=2  \global\controllorisposta=1 \special{landscape}
\message{>> You must print this in LANDSCAPE orientation << } \fi
\ifx\risposta\mplarisposta \ingrandimento=1000 \dimnota=1
\message{>> Mod. Phys. Lett. A format << }
\unoduecol=1 \global\controllorisposta=1 \fi
\ifnum\controllorisposta=0  \ingrandimento=1200
\message{>>> ERROR IN INPUT, I ASSUME STANDARD UNREDUCED FORMAT <<< }
\dimnota=2 \unoduecol=1 \fi
%
\magnification=\ingrandimento
%
%--------------------------------------------------------------------
%
%                        PARAMETERS SETTING
%
%  You can modify these parameters at your will (and resposability)
%--------------------------------------------------------------------
%
\newdimen\eucolumnsize \newdimen\eudoublehsize \newdimen\eudoublevsize
\newdimen\uscolumnsize \newdimen\usdoublehsize \newdimen\usdoublevsize
\newdimen\eusinglehsize \newdimen\eusinglevsize \newdimen\ussinglehsize
\newskip\standardbaselineskip \newdimen\ussinglevsize
\newskip\reducedbaselineskip \newskip\doublebaselineskip
%
\eucolumnsize=12.0truecm    % column h-size for european doublepage
                            % (12.0treucm default)
\eudoublehsize=25.5truecm   % sheet h-size for european duoblepage
                            % (25.5treucm default)
\eudoublevsize=6.5truein    % sheet v-size for european doublepage
                            % (6.5treuin default  or 17truecm?)
\uscolumnsize=4.4truein     % column h-size for american doublepage
                            % (4.4treuin default)
\usdoublehsize=9.4truein    % sheet h-size for american duoblepage
                            % (9.4treuin default)
\usdoublevsize=6.8truein    % sheet v-size for american doublepage
                            % (6.8treuin default)
\eusinglehsize=6.5truein    % sheet h-size for european singlepage
                            % (6.5truein default)
\eusinglevsize=24truecm     % sheet v-size for european singlepage
                            % (24truecm default)
\ussinglehsize=6.5truein    % sheet h-size for american singlepage
                            % (6.5truein default)
\ussinglevsize=8.9truein    % sheet v-size for american singlepage
                            % (8.9truein default)
\standardbaselineskip=16pt  % baselineskip for standard format
                            % (16pt default)
\reducedbaselineskip=14pt   % baselineskip for reduced format
                            % (14pt default)
\doublebaselineskip=12pt    % baselineskip for doublepage format
                            % (12pt default)
%
%  \Portoffset and \Landoffset define the horizontal and vertical
%  offsets respectively for portrait and landscape modes. Example:
%  \def\Portoffset{\voffset=.4truein\hoffset=.125truein}
%
\def\Portoffset{}
\def\Landoffset{}
%
\ifx\risposta\mplarisposta \def\Portoffset{\hoffset=1.8truecm} \fi
%
%  \Landspec defines the \special command that sets the printer
%  to landscape mode without need to specify it directly in the
%  TeX to postscript translator (the command is site dependent).
%  Example: \def\Landspec{\special{ps: landscape}}
%
%
\tolerance=10000
\parskip 0pt plus 2pt  \leftskip=0pt \rightskip=0pt
%
%   Do not modify anything of what follows
%                       (unless you know what you are doing!)
%----------------------------------------------------------------------
%
\ifx\risposta\standardrisposta \infralinea=\standardbaselineskip \fi
\ifx\risposta\reducedrisposta  \infralinea=\reducedbaselineskip \fi
\ifx\risposta\doublerisposta   \infralinea=\doublebaselineskip \fi
\ifx\risposta\mplarisposta     \infralinea=13pt \fi
\ifnum\controllorisposta=0    \infralinea=\standardbaselineskip \fi
%
\ifx\risposta\doublerisposta   \Landoffset \else \Portoffset \fi
%
\ifx\risposta\doublerisposta \ifx\srisposta\cartarisposta
\tothsize=\eudoublehsize \collhsize=\eucolumnsize
\vsize=\eudoublevsize  \else  \tothsize=\usdoublehsize
\collhsize=\uscolumnsize \vsize=\usdoublevsize \fi \else
\ifx\srisposta\cartarisposta \tothsize=\eusinglehsize
\vsize=\eusinglevsize \else  \tothsize=\ussinglehsize
\vsize=\ussinglevsize \fi \collhsize=4.4truein \fi
\ifx\risposta\mplarisposta \tothsize=5.0truein
\vsize=7.8truein \collhsize=4.4truein \fi
%
%--------------------------------------------------------------------
%
%                            FONTS
%
%--------------------------------------------------------------------
%
\newcount\contaeuler \newcount\contacyrill \newcount\contaams
\font\ninerm=cmr9  \font\eightrm=cmr8  \font\sixrm=cmr6
\font\ninei=cmmi9  \font\eighti=cmmi8  \font\sixi=cmmi6
\font\ninesy=cmsy9  \font\eightsy=cmsy8  \font\sixsy=cmsy6
\font\ninebf=cmbx9  \font\eightbf=cmbx8  \font\sixbf=cmbx6
\font\ninett=cmtt9  \font\eighttt=cmtt8  \font\nineit=cmti9
\font\eightit=cmti8 \font\ninesl=cmsl9  \font\eightsl=cmsl8
\skewchar\ninei='177 \skewchar\eighti='177 \skewchar\sixi='177
\skewchar\ninesy='60 \skewchar\eightsy='60 \skewchar\sixsy='60
\hyphenchar\ninett=-1 \hyphenchar\eighttt=-1 \hyphenchar\tentt=-1
%
\def\bfmath{\cmmib}                 % math italic bold \bfmath
\font\tencmmib=cmmib10  \newfam\cmmibfam  \skewchar\tencmmib='177
\def\bfcal{\cmbsy}                  % math bold (cal) symbols
\font\tencmbsy=cmbsy10  \newfam\cmbsyfam  \skewchar\tencmbsy='60
\def\scaps{\cmcsc}                 % small caps (uppercase)
\font\tencmcsc=cmcsc10  \newfam\cmcscfam
%
\ifnum\ingrandimento=1095 \font\rmone=cmr10 at 10.95pt
\font\rmtwo=cmr10 at 13.145pt \font\rmthree=cmr10 at 15.774pt
\font\bfone=cmbx10 at 10.95pt \font\bftwo=cmbx10 at 13.145pt
\font\bfthree=cmbx10 at 15.77pt
\font\slone=cmsl10 at 10.95pt \font\sltwo=cmsl10 at 13.145pt
\font\slthree=cmsl10 at 15.77pt
\font\capsone=cmcsc10 at 10.95pt \font\capstwo=cmcsc10 at 13.145pt
\font\capsthree=cmcsc10 at 15.77pt
\else  \font\rmone=cmr10 at 12pt
\font\rmtwo=cmr10 at 14.4pt \font\rmthree=cmr10 at 17.28pt
\font\bfone=cmbx10 at 12pt \font\bftwo=cmbx10 at 14.4pt
\font\bfthree=cmbx10 at 17.28pt
\font\slone=cmsl10 at 12pt \font\sltwo=cmsl10 at 14.4pt
\font\slthree=cmsl10 at 17.28pt
\font\capsone=cmcsc10 at 12pt \font\capstwo=cmcsc10 at 14.4pt
\font\capsthree=cmcsc10 at 17.28pt\fi
%
\def\chapterfont#1{\xdef\ttaarr{#1}}
\def\sectionfont#1{\xdef\ppaarr{#1}}
\def\ttaarr{\bf}		% chapter titles' font
\def\ppaarr{\sl}		% section titles' font
\def\drftfont{\rm}
%%
\def\loadextratext{
\font\bfsl=cmbxsl10                % boldface slanted
\font\tenu=cmu10                   % unslanted text italic
\font\bfit=cmbxti10                % boldface italic
\font\ttsmcap=cmtcsc10             % typewriter small caps
\font\ttslan=cmsltt10              % slanted typewriter
\font\ttital=cmitt10               % italic typewriter
\font\tttex=cmtex10                % TeX typewriter
\font\ttvariable=cmvtt10           % var-width typewriter
\font\sserif=cmss10                % sans serif
\font\sserit=cmssi10               % sans serif italic
\font\eightss=cmssq8               % sans serif quotation (almost sl)
\font\eightssi=cmssqi8             % sans serif quot italic(ditto sl)
\font\sserbf=cmssbx10              % sans serif bold extended
\font\sserdcond=cmssdc10           % sans serif demi cond
\font\fibbo=cmfi10                 % Fibonacci
\font\fibbobf=cmfib8               % bold Fibonacci
\font\dunhill=cmdunh10             % Dunhill style
\font\ffuy=cmff10                  % funny font
\font\inchhigh=cminch        }     % inch-high caps (enormous)
%
%   AMS fonts (this works only if you have at least the 2.0
%              version of AMSFonts, otherwise say no)
%
\newfam\eufmfam \newfam\msamfam \newfam\msbmfam \newfam\eufbfam
%%
\def\Loadeulerfonts{\global\contaeuler=1 \ifx\arisposta\amsrisposta
\font\teneufm=eufm10              %  \eufm   Gothic (or Euler)
\font\eighteufm=eufm8 \font\nineeufm=eufm9 \font\sixeufm=eufm6
\font\seveneufm=eufm7  \font\fiveeufm=eufm5
\font\teneufb=eufb10              %  \eufb   Bold Gothic (or Euler)
\font\eighteufb=eufb8 \font\nineeufb=eufb9 \font\sixeufb=eufb6
\font\seveneufb=eufb7  \font\fiveeufb=eufb5
\font\teneurm=eurm10              %  \eurm   Roman Gothic (or Euler)
\font\eighteurm=eurm8 \font\nineeurm=eurm9
\font\teneurb=eurb10              %  \eurb   Roman Bold Gothic
\font\eighteurb=eurb8 \font\nineeurb=eurb9
\font\teneusm=eusm10              %  \eusm   Slanted Capital Gothic
\font\eighteusm=eusm8 \font\nineeusm=eusm9
\font\teneusb=eusb10              %\eusb Slanted Capital Bold Gothic
\font\eighteusb=eusb8 \font\nineeusb=eusb9
\else \def\eufm{\tt} \def\eufb{\tt} \def\eurm{\tt} \def\eurb{\tt}
\def\eusm{\tt} \def\eusb{\tt}    \fi}
\def\loadeuler{\Loadeulerfonts\tenpoint}
\def\loadamsmath{\global\contaams=1 \ifx\arisposta\amsrisposta
\font\tenmsam=msam10 \font\ninemsam=msam9 \font\eightmsam=msam8
\font\sevenmsam=msam7 \font\sixmsam=msam6 \font\fivemsam=msam5
\font\tenmsbm=msbm10 \font\ninemsbm=msbm9 \font\eightmsbm=msbm8
\font\sevenmsbm=msbm7 \font\sixmsbm=msbm6 \font\fivemsbm=msbm5
\else \def\msbm{\bf} \fi \def\Bbb{\msbm} \def\symbl{\msam} \tenpoint}
\def\loadcyrill{\global\contacyrill=1 \ifx\arisposta\amsrisposta
\font\tenwncyr=wncyr10 \font\ninewncyr=wncyr9 \font\eightwncyr=wncyr8
\font\tenwncyb=wncyr10 \font\ninewncyb=wncyr9 \font\eightwncyb=wncyr8
\font\tenwncyi=wncyr10 \font\ninewncyi=wncyr9 \font\eightwncyi=wncyr8
\else \def\cyrill{\sl} \def\cyrilb{\sl} \def\cyrili{\sl} \fi\tenpoint}
%%
\ifx\arisposta\amsrisposta
\font\sevenex=cmex7               %  reduced math symbols
\font\eightex=cmex8  \font\nineex=cmex9
\font\ninecmmib=cmmib9   \font\eightcmmib=cmmib8
\font\sevencmmib=cmmib7 \font\sixcmmib=cmmib6
\font\fivecmmib=cmmib5   \skewchar\ninecmmib='177
\skewchar\eightcmmib='177  \skewchar\sevencmmib='177
\skewchar\sixcmmib='177   \skewchar\fivecmmib='177
\font\ninecmbsy=cmbsy9    \font\eightcmbsy=cmbsy8
\font\sevencmbsy=cmbsy7  \font\sixcmbsy=cmbsy6
\font\fivecmbsy=cmbsy5   \skewchar\ninecmbsy='60
\skewchar\eightcmbsy='60  \skewchar\sevencmbsy='60
\skewchar\sixcmbsy='60    \skewchar\fivecmbsy='60
\font\ninecmcsc=cmcsc9    \font\eightcmcsc=cmcsc8     \else
\def\cmmib{\fam\cmmibfam\tencmmib}\textfont\cmmibfam=\tencmmib
\scriptfont\cmmibfam=\tencmmib \scriptscriptfont\cmmibfam=\tencmmib
\def\cmbsy{\fam\cmbsyfam\tencmbsy} \textfont\cmbsyfam=\tencmbsy
\scriptfont\cmbsyfam=\tencmbsy \scriptscriptfont\cmbsyfam=\tencmbsy
\scriptfont\cmcscfam=\tencmcsc \scriptscriptfont\cmcscfam=\tencmcsc
\def\cmcsc{\fam\cmcscfam\tencmcsc} \textfont\cmcscfam=\tencmcsc \fi
%
%   attenzione: strano fenomeno \**point sembra fare un \hskip 48pt ?
\catcode`@=11
\newskip\ttglue
\gdef\tenpoint{\def\rm{\fam0\tenrm}
  \textfont0=\tenrm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm
  \textfont1=\teni \scriptfont1=\seveni \scriptscriptfont1=\fivei
  \textfont2=\tensy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy
  \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
  \def\mcal{\fam2 \tensy}  \def\mmit{\fam1 \teni}
  \textfont\itfam=\tenit \def\it{\fam\itfam\tenit}
  \textfont\slfam=\tensl \def\sl{\fam\slfam\tensl}
  \textfont\ttfam=\tentt \scriptfont\ttfam=\eighttt
  \scriptscriptfont\ttfam=\eighttt  \def\tt{\fam\ttfam\tentt}
  \textfont\bffam=\tenbf \scriptfont\bffam=\sevenbf
  \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\tenbf}
     \ifx\arisposta\amsrisposta    \ifnum\contaeuler=1
  \textfont\eufmfam=\teneufm \scriptfont\eufmfam=\seveneufm
  \scriptscriptfont\eufmfam=\fiveeufm \def\eufm{\fam\eufmfam\teneufm}
  \textfont\eufbfam=\teneufb \scriptfont\eufbfam=\seveneufb
  \scriptscriptfont\eufbfam=\fiveeufb \def\eufb{\fam\eufbfam\teneufb}
  \def\eurm{\teneurm} \def\eurb{\teneurb} \def\eusm{\teneusm}
  \def\eusb{\teneusb}    \fi    \ifnum\contaams=1
  \textfont\msamfam=\tenmsam \scriptfont\msamfam=\sevenmsam
  \scriptscriptfont\msamfam=\fivemsam \def\msam{\fam\msamfam\tenmsam}
  \textfont\msbmfam=\tenmsbm \scriptfont\msbmfam=\sevenmsbm
  \scriptscriptfont\msbmfam=\fivemsbm \def\msbm{\fam\msbmfam\tenmsbm}
     \fi      \ifnum\contacyrill=1     \def\cyrill{\tenwncyr}
  \def\cyrilb{\tenwncyb}  \def\cyrili{\tenwncyi}         \fi
  \textfont3=\tenex \scriptfont3=\sevenex \scriptscriptfont3=\sevenex
  \def\cmmib{\fam\cmmibfam\tencmmib} \scriptfont\cmmibfam=\sevencmmib
  \textfont\cmmibfam=\tencmmib  \scriptscriptfont\cmmibfam=\fivecmmib
  \def\cmbsy{\fam\cmbsyfam\tencmbsy} \scriptfont\cmbsyfam=\sevencmbsy
  \textfont\cmbsyfam=\tencmbsy  \scriptscriptfont\cmbsyfam=\fivecmbsy
  \def\cmcsc{\fam\cmcscfam\tencmcsc} \scriptfont\cmcscfam=\eightcmcsc
  \textfont\cmcscfam=\tencmcsc \scriptscriptfont\cmcscfam=\eightcmcsc
     \fi            \tt \ttglue=.5em plus.25em minus.15em
  \normalbaselineskip=12pt
  \setbox\strutbox=\hbox{\vrule height8.5pt depth3.5pt width0pt}
  \let\sc=\eightrm \let\big=\tenbig   \normalbaselines
  \baselineskip=\infralinea  \rm}
\gdef\ninepoint{\def\rm{\fam0\ninerm}
  \textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
  \textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei
  \textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
  \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
  \def\mcal{\fam2 \ninesy}  \def\mmit{\fam1 \ninei}
  \textfont\itfam=\nineit \def\it{\fam\itfam\nineit}
  \textfont\slfam=\ninesl \def\sl{\fam\slfam\ninesl}
  \textfont\ttfam=\ninett \scriptfont\ttfam=\eighttt
  \scriptscriptfont\ttfam=\eighttt \def\tt{\fam\ttfam\ninett}
  \textfont\bffam=\ninebf \scriptfont\bffam=\sixbf
  \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\ninebf}
     \ifx\arisposta\amsrisposta  \ifnum\contaeuler=1
  \textfont\eufmfam=\nineeufm \scriptfont\eufmfam=\sixeufm
  \scriptscriptfont\eufmfam=\fiveeufm \def\eufm{\fam\eufmfam\nineeufm}
  \textfont\eufbfam=\nineeufb \scriptfont\eufbfam=\sixeufb
  \scriptscriptfont\eufbfam=\fiveeufb \def\eufb{\fam\eufbfam\nineeufb}
  \def\eurm{\nineeurm} \def\eurb{\nineeurb} \def\eusm{\nineeusm}
  \def\eusb{\nineeusb}     \fi   \ifnum\contaams=1
  \textfont\msamfam=\ninemsam \scriptfont\msamfam=\sixmsam
  \scriptscriptfont\msamfam=\fivemsam \def\msam{\fam\msamfam\ninemsam}
  \textfont\msbmfam=\ninemsbm \scriptfont\msbmfam=\sixmsbm
  \scriptscriptfont\msbmfam=\fivemsbm \def\msbm{\fam\msbmfam\ninemsbm}
     \fi       \ifnum\contacyrill=1     \def\cyrill{\ninewncyr}
  \def\cyrilb{\ninewncyb}  \def\cyrili{\ninewncyi}         \fi
  \textfont3=\nineex \scriptfont3=\sevenex \scriptscriptfont3=\sevenex
  \def\cmmib{\fam\cmmibfam\ninecmmib}  \textfont\cmmibfam=\ninecmmib
  \scriptfont\cmmibfam=\sixcmmib \scriptscriptfont\cmmibfam=\fivecmmib
  \def\cmbsy{\fam\cmbsyfam\ninecmbsy}  \textfont\cmbsyfam=\ninecmbsy
  \scriptfont\cmbsyfam=\sixcmbsy \scriptscriptfont\cmbsyfam=\fivecmbsy
  \def\cmcsc{\fam\cmcscfam\ninecmcsc} \scriptfont\cmcscfam=\eightcmcsc
  \textfont\cmcscfam=\ninecmcsc \scriptscriptfont\cmcscfam=\eightcmcsc
     \fi            \tt \ttglue=.5em plus.25em minus.15em
  \normalbaselineskip=11pt
  \setbox\strutbox=\hbox{\vrule height8pt depth3pt width0pt}
  \let\sc=\sevenrm \let\big=\ninebig \normalbaselines\rm}
\gdef\eightpoint{\def\rm{\fam0\eightrm}
  \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
  \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei
  \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
  \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
  \def\mcal{\fam2 \eightsy}  \def\mmit{\fam1 \eighti}
  \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}
  \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}
  \textfont\ttfam=\eighttt \scriptfont\ttfam=\eighttt
  \scriptscriptfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}
  \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf
  \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}
     \ifx\arisposta\amsrisposta   \ifnum\contaeuler=1
  \textfont\eufmfam=\eighteufm \scriptfont\eufmfam=\sixeufm
  \scriptscriptfont\eufmfam=\fiveeufm \def\eufm{\fam\eufmfam\eighteufm}
  \textfont\eufbfam=\eighteufb \scriptfont\eufbfam=\sixeufb
  \scriptscriptfont\eufbfam=\fiveeufb \def\eufb{\fam\eufbfam\eighteufb}
  \def\eurm{\eighteurm} \def\eurb{\eighteurb} \def\eusm{\eighteusm}
  \def\eusb{\eighteusb}       \fi    \ifnum\contaams=1
  \textfont\msamfam=\eightmsam \scriptfont\msamfam=\sixmsam
  \scriptscriptfont\msamfam=\fivemsam \def\msam{\fam\msamfam\eightmsam}
  \textfont\msbmfam=\eightmsbm \scriptfont\msbmfam=\sixmsbm
  \scriptscriptfont\msbmfam=\fivemsbm \def\msbm{\fam\msbmfam\eightmsbm}
     \fi       \ifnum\contacyrill=1     \def\cyrill{\eightwncyr}
  \def\cyrilb{\eightwncyb}  \def\cyrili{\eightwncyi}         \fi
  \textfont3=\eightex \scriptfont3=\sevenex \scriptscriptfont3=\sevenex
  \def\cmmib{\fam\cmmibfam\eightcmmib}  \textfont\cmmibfam=\eightcmmib
  \scriptfont\cmmibfam=\sixcmmib \scriptscriptfont\cmmibfam=\fivecmmib
  \def\cmbsy{\fam\cmbsyfam\eightcmbsy}  \textfont\cmbsyfam=\eightcmbsy
  \scriptfont\cmbsyfam=\sixcmbsy \scriptscriptfont\cmbsyfam=\fivecmbsy
  \def\cmcsc{\fam\cmcscfam\eightcmcsc} \scriptfont\cmcscfam=\eightcmcsc
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\global\contaeuler=0 \global\contacyrill=0 \global\contaams=0
%
%--------------------------------------------------------------------
%%                            MACROS
%
%--------------------------------------------------------------------
%
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\gdef\fullline#1{\hbox to\fullhsize{\hskip\leftskip{#1}%
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%
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\gdef\vfootnote#1{\insert\footins\bgroup
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\count\footins=1000  \dimen\footins=8in
\catcode`@=12
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\tenpoint
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%%
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\def\nttsymb{\ifcase\nmbnota \or\dag\or\sharp\or\ddag\or\star\or
\natural\or\flat\or\clubsuit\or\diamondsuit\or\heartsuit
\or\spadesuit\fi}   \clearnmbnota
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\ifnum\tipbnota=1 {\rm\nttlett} \else {\ifnum\tipbnota=2
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%
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\gdef\figure{\global\advance\nbmfig by 1
      {\rm fig. \the\nbmfig}}   \clearnbmfig
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%
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\def\fr{\formula} \def\efr{\endformula}
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\def\nameformula#1{\global\advance\frmcount by 1%
\ifnum\draftnum=0  {\ifnum\indappcount=0%
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\else\xdef\spzzttrra{(\applett .\the\frmcount )}\fi}%
\else\xdef\spzzttrra{(#1)}\fi%
\expandafter\xdef\csname#1\endcsname{\spzzttrra}
\eqno \ifnum\draftnum=0 {\ifnum\indappcount=0
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\def\nfr{\nameformula}    \def\numali{\numero}
\def\nameali#1{\global\advance\frmcount by 1%
\ifnum\draftnum=0  {\ifnum\indappcount=0%
{\ifnum\cpcount<1\xdef\spzzttrra{(\the\frmcount )}%
\else\xdef\spzzttrra{(\the\cpcount .\the\frmcount )}\fi}%
\else\xdef\spzzttrra{(\applett .\the\frmcount )}\fi}%
\else\xdef\spzzttrra{(#1)}\fi%
\expandafter\xdef\csname#1\endcsname{\spzzttrra}
  \ifnum\draftnum=0  {\ifnum\indappcount=0
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  {\hbox{\rm (\applett .\the\frmcount )}} \fi} \else (#1) \fi}
\clearfrmcount
%%
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\newcount\subcpcount \def\clearsubcpcount{\global\subcpcount=0}
\newcount\appcount \def\clearappcount{\global\appcount=0}
\newcount\indappcount \def\clearindappcount{\indappcount=0}
\newcount\sottoparcount \def\clearsottoparcount{\sottoparcount=0}
\def\numsubcp{\subcpcount}
\def\applett{\ifcase\appcount  \or {A}\or {B}\or {C}\or
{D}\or {E}\or {F}\or {G}\or {H}\or {I}\or {J}\or {K}\or {L}\or
{M}\or {N}\or {O}\or {P}\or {Q}\or {R}\or {S}\or {T}\or {U}\or
{V}\or {W}\or {X}\or {Y}\or {Z}\fi
             \ifnum\appcount<0
    \message{>>  ERROR: counter \appcount out of range <<}\fi
             \ifnum\appcount>26
   \message{>>  ERROR: counter \appcount out of range <<}\fi}
\clearappcount  \clearindappcount
\newcount\connttrre  \def\clearconnttrre{\global\connttrre=0}
\newcount\countref  \def\clearcountref{\global\countref=0}
\clearcountref
%
\def\chapter#1{\global\advance\cpcount by 1 \clearfrmcount
                 \goodbreak\null\vbox{\jump\nobreak
                 \clearsubcpcount\clearindappcount
                 \itemitem{\ttaarr\the\cpcount .\qquad}{\ttaarr #1}
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\def\section#1{\global\advance\subcpcount by 1 \goodbreak\null
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                \else{\itemitem{\ppaarr\applett .\the\subcpcount\quad
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\clearsubcpcount
\def\appendix#1{\global\advance\appcount by 1 \clearfrmcount
                  \goodbreak\null\vbox{\jump\nobreak
                  \global\advance\indappcount by 1 \clearsubcpcount
                  \itemitem{\ttaarr App.\applett\ }{\ttaarr #1}
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\clearappcount \clearindappcount
\def\references{\goodbreak\null\vbox{\jump\nobreak
   \itemitem{}{\ttaarr References} \nobreak\jump\sjump}\nobreak}
\def\beginpaper{\clearindappcount\clearappcount\clearcpcount
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\def\introduction{\clearindappcount\clearappcount\clearcpcount
                  \clearsubcpcount\goodbreak\null\vbox{\jump\nobreak
  \itemitem{}{\ttaarr Introduction} \nobreak\jump\sjump}\nobreak}
\clearcpcount\clearcountref
\def\acknowledgements{\goodbreak\null\vbox{\jump\nobreak
\itemitem{ }{\ttaarr Acknowledgements} \nobreak\jump\sjump}\nobreak}
\def\setchap#1{\ifnum\indappcount=0{\ifnum\subcpcount=0%
\xdef\spzzttrra{\the\cpcount}%
\else\xdef\spzzttrra{\the\cpcount .\the\subcpcount}\fi}
\else{\ifnum\subcpcount=0 \xdef\spzzttrra{\applett}%
\else\xdef\spzzttrra{\applett .\the\subcpcount}\fi}\fi
\expandafter\xdef\csname#1\endcsname{\spzzttrra}}
%%
\newcount\draftnum \newcount\ppora   \newcount\ppminuti
\global\ppora=\time   \global\ppminuti=\time
\global\divide\ppora by 60  \draftnum=\ppora
\multiply\draftnum by 60    \global\advance\ppminuti by -\draftnum
\global\draftnum=0
\def\droggi{\number\day /\number\month /\number\year\ \the\ppora
:\the\ppminuti}
\def\draftmode#1{\headline={\hskip-48pt \eightpoint
\drftfont{\bf DRAFT}\hfill
{}~~~ --- ~~~ \hfill #1 \hfill ~~~ --- ~~~  \hfill\droggi}
\global\draftnum=1} \global\draftnum=0
\def\draftcomment#1{\ifnum\draftnum=0 \relax \else
{\ {\bf ***}\ #1\ {\bf ***}\ }\fi} \def\drafthead{\draftnum=0}
%
%     Maximum number of references = 200
%     boxes 50 -> 250 reserved for references
%
\catcode`@=11
\gdef\Ref#1{\expandafter\ifx\csname @rrxx@#1\endcsname\relax%
{\global\advance\countref by 1%
\ifnum\countref>200%
\message{>>> ERROR: maximum number of references exceeded <<<}%
\expandafter\xdef\csname @rrxx@#1\endcsname{0}\else%
\expandafter\xdef\csname @rrxx@#1\endcsname{\the\countref}\fi}\fi%
\ifnum\draftnum=0 \csname @rrxx@#1\endcsname \else#1\fi}
\gdef\beginref{\ifnum\draftnum=0  \gdef\Rref{\fairef}
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\ifx\risposta\mplarisposta \ninepoint \fi
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{\global\conttemp=0
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\clearcountref \clearconnttrre
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%
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%%
\def\ghost#1{\smash{\phantom{\smash{#1}}} }
\def\slashchar#1{\setbox0=\hbox{$#1$} \dimen0=\wd0
     \setbox1=\hbox{/} \dimen1=\wd1 \ifdim\dimen0>\dimen1
      \rlap{\hbox to \dimen0{\hfil/\hfil}} #1 \else
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\def\square{\hbox{{$\sqcup$}\llap{$\sqcap$}}}  \def\grad{\nabla}
\def\del{\partial}  \def\alfa{\alpha} \def\ro{\rho}
\def\bull{\vrule height .9ex width .8ex depth -.1ex }
\def\frac#1#2{{\textstyle{#1 \over #2}}}
\def\smallfrac#1#2{{\scriptstyle {#1 \over #2}}}
\def\half{\ifinner {\scriptstyle {1 \over 2}}\else {1 \over 2} \fi}
\def\bra#1{\langle#1\vert}  \def\ket#1{\vert#1\rangle}
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\def\simle{\rlap{\raise 2pt \hbox{$<$}}{\lower 2pt \hbox{$\sim$}}}
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\limits^{\hbox{\fiverm (--)}}\!\null}}
\def\nunubar{\parenbar{\nu}} \def\ppbar{\parenbar{p}}
\def\buildchar#1#2#3{{\null\!\mathop{#1}\limits^{#2}_{#3}\!\null}}
\def\overcirc#1{\buildchar{#1}{\circ}{}}
\def\vbigl{\mathopen\vbig}\def\vbigm{\mathrel\vbig}
\def\vbigr{\mathclose\vbig}
\def\vbig#1#2{{\vbigd@men=#2\divide\vbigd@men by 2%
\hbox{$\left#1\vbox to \vbigd@men{}\right.\n@space$}}}
\def\Leftcases#1{\smash{\vbigl\{{#1}}}
\def\Rightcases#1{\smash{\vbigr\}{#1}}}
\def\noblackbox{\overfullrule=0pt}
\def\yesblackbox{\overfullrule=5pt}
\null
%
%--------------------------------------------------------------------
%
%                             THE    END
%
%--------------------------------------------------------------------
%
%title page
%cern and hep-th ref
\nopagenumbers{\baselineskip=12pt
\line{\hfill CBPF-NF-064/94}
%\line{\hfill hep-th/9410xxx}
\ifdoublepage \bjump\bjump\bjump\bjump\else\vfill\fi
%title
\centerline{\capsone Sp(2) Covariant Quantisation of General}
\sjump
\centerline{\capsone Gauge Theories.}
\bjump
%author
\centerline{\scaps {Jos\'e-Luis V\'azquez-Bello}}
\sjump
%address
\centerline{\sl CBPF-CNPq/CLAF Centro Brasileiro de Pesquisas Fisicas}
\centerline{\sl Rua Dr. Xavier Sigaud 150, CEP. 22290}
\centerline{\sl  Rio de  Janeiro - RJ, BRASIL.}
\sjump
%e-mail
\centerline{ {\sl e-mail:} bello@cbpfsu7.cat.cbpf.br}
\vfill
\ifnum\unoduecol=2 \eject\null\vfill\fi
%abstract
\centerline{\capsone abstract }
\sjump
\noindent
{This letter studies the Sp(2) covariant quantisation of gauge theories.
The geometrical interpretation of gauge theories in terms of quasi principal
fibre bundles $Q(M_S ,G_S)$ is reviewed. It is then described the Sp(2)
algebra of ordinary Yang-Mills theory. A consistent formulation of covariant
lagrangian quantisation for general gauge theories based on Sp(2) BRST
symmetry is established. The original $N=1$, ten dimensional superparticle
is considered as an example of infinitely reducible gauge algebras,
and given explicitly its Sp(2) BRST invariant action. }

\sjump
\ifnum\unoduecol=2 \vfill\fi
\eject
\yespagenumbers\pageno=1


\def\scm{M}
\def\scn{N}
\def\scmn{MN}
\def\dth{\partial_\theta}
\def\bdth{\partial_{\bar\theta}}
\def\gmi{g^{-1}}
\def\cala{\cal A}
\def\half{{1\over2}}
\def\twelfe{{1\over {12}}}
\def\del{\partial}
\def\delmu{\partial_\mu}
%----------------------------------
\def\xmu {x^\mu}
\def\pmu {p_\mu}
\def\Amu {A_\mu}
\def\gammu {\gamma^\mu}
\def\psh {\rlap{/}{p}}
\def\thetaa {{\theta^a}}
\def\thetab {{\theta^b}}
\def\thetac {{\theta^c}}
\def\thetai {{\theta^i}}
\def\ginv {g^{-1}}
\def\varri { {\delta_r}\over {\delta\Phi^A} }
\def\varle { {\delta_l}\over {\delta\Phi^\star_A} }
\def\varrid {{ {\delta  }\over {\delta\Phi^A} }}
\def\varlea {{ {\delta_l}\over {\delta\Phi^\star_{Aa}} }}
\def\varrist  {{ {\delta_r}\over {\delta\Phi^{\star\star}_A} }}
\def\notequiv{\rlap{/}{=}}



\chapter {Gauge theories in terms of quasi-principal fibre bundles.}

Gauge theories have a nice geometrical interpretation in terms of connections
on a principal fibre bundle (pfb) $P(M,G)$, where $M$ is the base space-time
manifold and G is the gauge group [\Ref{bon1},\Ref{bon2}, \Ref{hoyos},
\Ref{rogers}].
However, quantisation of gauge theories requires the introduction
of fields $(c^n_m ,\pi^n_m)$. It would be then desirable to have
a formalism where those extra fields fit into some representation of a
larger group and all the fields are components of a superfield.
This is a step in the direction of recovering a geometrical interpretation
of quantum gauge theories. The main ingredients in the construction of
geometrical quasi-principal fibre bundles
(qpfb) are a space-time base manifold $M$, a gauge group $G$,
an extended superspace manifold $M_S$ which is obtained by adding
two extra Grassmann variables $\theta^a$ $(a=1,2)$ to $M$, in the case of
$Sp(2)$ symmetry, and a supergroup $G_S$.
The construction is performed basically in three steps
[\Ref{bon1},\Ref{bon2}, \Ref{hoyos}].
It starts with a pfb $P(M,G)$ and extend the gauge group $G$
to a supergroup $G_S$. The composition of $G$ with a Grassmann algebra $B$
prolongs $P(M,G)$ to a pfb $P'(M,G_S )$.
The most general supergroup $G_S$ can be represented in matrix form. In
particular, $OSp(N/M)$ groups are represented by block matrices of the form
$$
\pmatrix { A & E \cr
           C & D \cr}
\nfr{matrix}
where
$A$, $D$ are $(N\times N)$ and $(M\times M)$ matrices whose elements
are taken from the even part of the Grassmann algebra $B$ constructed
over a complex vector space $W$, whilst $E$, $C$ are $(N\times M)$ and
$(N\times M)$ rectangular matrices whose elements belong to the
odd part of $B$.
Next, it is enlarged the base space manifold $M$ to a superspace $M_S$ in
$P'(M,G_S )$ by adding Grassmann variables. At this stage, a pfb
$P''(M_S ,G_S )$ is obtained. Finally, the pfb $P'' (M_S G_S )$ is transform
into a quasi-principal fibre bundle $Q(M_S ,G_S )$.
For instance,
given a one-form valued function $\alpha (x) = \Amu d\xmu$ on $M$ this
induces a connection $\omega$ on the pfb $P(M,G)$. Then a one-form valued
function $\alpha '$ on $M_S$ is found by
$$
\alpha ' (x,\thetaa )= \ginv\Amu d\xmu g + \ginv dg
\nfr{qalpha}
where
$g = g(\xmu ,\thetaa )$, $(a=1,2,\dots )$ which
induces a connection $\omega '$ on the qpfb $Q(M_S,G_S)$
[\Ref{bon1},\Ref{bon2},\Ref{hoyos},\Ref{rogers}].



\chapter{The $Sp(2)$ BRST Algebra of Yang-Mills Theory.}


It has
been realized for some time [\Ref{baul},\Ref{our},\Ref{tieg},\Ref{mieg}] that
a geometrical construction can be useful for the discussion of BRST and
anti-BRST symmetry.
The idea is to use a superspace with coordinates
$Z^M = (x^\mu ,\theta^a)$, where $(a=1,2)$ and $\theta^a$ is an
anti-commuting scalar coordinate
and the BRST generators $s^a$ are realized as differential
operators on superspace,
$s^a={\partial\over{\partial\thetaa}}$, so that $s^a s^b + s^b s^a =0$
holds automatically
\footnote{$^1$}
{It
is usually defined a bosonic operator
$\sigma = \half\epsilon_{ab} s^a s^b$ where $\epsilon_{ab}$ is
the symplectic invariant form of $Sp(2)$, so that
$\epsilon_{ab} = -\epsilon_{ba}$,  $\epsilon^{ab}\epsilon_{bc}=\delta_{ac}$
and $\epsilon_{12}=1$. The generator $\sigma$ is invariant under  $Sp(2)$
and satisfies $s^a \sigma = 0 $. The Sp(2) generators
$\sigma^i$ $(i=\pm ,0)$ and the fermionic charges
$s^a$ together form an algebra which is a contraction of
$OSp(1, 1/2)$ and denoted as $ISp(2)$
[\Ref{espies}].}. For example, in
Yang-Mills theory the gauge potential $A^i_\mu$ and the Faddeev-Popov
ghost $(c^a)^i$ (where $i$ is an adjoint group index) can be
combined into a super-gauge field ${\cal A}_M^i (Z)$ whose lowest
order components are
${\cal A}_{\scm}^i (Z)\big\vert_{\thetaa =0}
=({\cal A}_\mu^i ,{\cal A}^i_\thetaa )\big\vert_{\thetaa =0}
=(A_\mu^i ,{c^a}^i )$.
Then the standard Yang-Mills BRST transformations arise
from imposing the constraints
${\cal F}_{\mu\thetaa}^i=0$, ${\cal F}^i_{\thetaa\thetab}=0$
on the superfield strength ${\cal F}^i_{\scmn}$ [\Ref{baul},\Ref{our}].
This gives an elegant geometrical description of BRST and anti-BRST symmetry.

Let us review the construction
of gauge theories in the superspace with
coordinates $Z^{\scm}=(x^\mu ,\thetaa )$, which gives a geometric
formulation of $Sp(2)$ BRST symmetry.
We consider matter fields $\Phi^i (x,\thetaa )$ and
a gauge potential
${\cal A}^i_{\scm}(x,\thetaa )
=\bigl ({\cal A}^i_\mu (x,\thetaa ),{\cal A}^i_\thetaa (x,\thetaa )\bigl )$.
These can be used to  define a covariant derivative
$$
{\cal D}_{\scm}\Phi^i
=\partial_{\scm}\Phi^i -{(T^k)^i}_j{\cal A}^k_{\scm}\Phi^j
\nfr{covariant}
and the field strength
$${\cal F}^i_{\scmn}
=\partial_{\scm}{\cal A}^i_{\scn}
-(-1)^{\scmn}\partial_{\scn}{\cal A}^i_{\scm}
+{f^i}_{jk}{\cal A}^j_{\scm}{\cal A}^k_{\scn}
\nfr{strength}
where
$(-1)^{\scmn}$ is 1 unless both $M$ and $N$ are indices referring
to anti-commuting coordinates, in which case it is $-1$.
The gauge potential ${\cal A}_{\scm}$ contains more
component fields than the physical gauge and ghost fields
and so, as in supersymmetric theories, constraints should be imposed
on the field strength ${\cal F}$. Appropriate constraints are
[\Ref{baul},\Ref{our}]
$${\cal F}_{\thetaa\thetab} =0,\qquad {\cal F}_{\mu\thetaa} =0.
\nfr{constraints}
These
can be written
more explicitly
as
$$\delmu{\cala}_\thetaa -\partial_\thetaa {\cala}_\mu
                     +[{\cala}_\mu ,{\cala}_\thetaa ]=0\nfr{acala}
$$\partial_\thetaa{\cala}_\thetaa
           +{1\over 2}[{\cala}_\thetaa ,{\cala}_\thetaa ]=0\nfr{bcala}
$$\epsilon^{ab}\Big(\partial_\thetaa {\cala}_\thetab
      + [{\cala}_\thetaa,{\cala}_\thetab ]\Big) =0.
\nfr{ecala}
Defining
the component expansions
$${\cala}_\mu (x,\thetaa )
             =A_\mu (x)+\thetaa \Lambda_{a\mu} (x)
                           +\thetaa\thetab \Omega_{ab\mu} (x)\nfr{aexp}
$${\cala}^b_\thetaa (x,\thetaa )
         =c^b (x) + \thetaa \Upsilon^b_a (x) + \thetaa\thetac \omega^b_{ac}
(x),\nfr{thexp}
however, it was found that
if $A_\mu$, $c^a$, $\pi$ are identified with the gauge, ghost
(anti-ghost) and auxiliary fields respectively then the supergauge fields
have the expansions
\footnote{$^\star$}
{In Ref [\Ref{our}], it was obtained explicitly a geometrical formulation
of BRST and anti-BRST symmetries and given the field content of
$\Lambda$, $\Upsilon$, $\Omega$ and $\omega$. The components in the
expansion can also be read as conditions on the mapping of the coordinates
$\phi^i$ of the fibres over $\{ {\cal U}_i \}$ (covering set of $M_G$) and
expressed as cocycle conditions.}
$$
{\cala}_\mu = A_\mu + \thetaa (s^a A_\mu ) + \thetaa\thetab (s^a s^b A_\mu )
\nfr{aexpan}
$${\cala}^b_\thetaa = c^b + \thetaa (s^a c^b ) + \thetaa\thetac (s^a s^c c^b )
.\nfr{thexpan}
The
BRST and anti-BRST generators $s^a$ are
then identified with the superspace differential operators $\partial_\thetaa$
[\Ref{baul},\Ref{our}], and the complete set of BRST and anti-BRST
transformations are given by [\Ref{ore} ,\Ref{gaume} ,\Ref{spirid} ]
$$
\eqalign{
s^a \phi^i &= R^i_\alpha c^{\alpha a}, \qquad
s^a c^{\alpha b} =\epsilon^{ab} \pi^\alpha - \half f^\alpha_{\beta\gamma}
c^{\beta a} c^{\gamma b}, \cr
s^a \pi^\alpha &= \half f^\alpha_{\beta\gamma}\pi^\beta c^{\gamma a}
-\twelfe (f^\alpha_{\beta\gamma} f^\beta_{\delta\tau} +
f^\alpha_{\delta\tau ,i} R^i_\gamma )c^{\delta a}c^{\tau e}
\epsilon_{eb}c^{\gamma b},\cr}
\nfr{setbrst}
where
the generators $R^i_\alpha$ for the gauge field $A^i_\mu$
read off from $s^a A^i_\mu = (D_\mu c^a )^i$, and $\pi^\alpha$
is an auxiliary field which connects ghost and antighost sectors.
%
%The BRST generators $s^a$ satisfy $s^a s^b + s^b s^a = 0 $.
%A representation of
%this algebra on the fields $A_\mu$, $c^a$, $\pi^a_n$ is given as usual by
%[\Ref{one} ,\Ref{eleven} ,\Ref{twelve} ]
%$$s^a A_\mu =\delmu c^a + [A_\mu,  c^a ], \qquad
% s^a c^b = -\half f^b_{pq} c^p c^q \nfr{onebrst}
%$$\epsilon_{ab} (s^a c^b + s^b c^a)= -\epsilon_{ab} c^a c^b ,
%\qquad \epsilon_{12}s^1\pi^a_n =s\pi^a_n = 0.\nfr{twobrst}
%

For
the matter fields $\Phi^i (x,\thetaa )$, we impose the constraint
$${\cal D}_\thetaa\Phi^i
     =\partial_\thetaa \Phi^i - {(T^k )^i}_j{\cal A}^k_\thetaa\Phi^j = 0
\nfr{constrmatter}
which
implies
$$\Phi^i = \psi^i + \thetaa (s^a \psi^i ),\nfr{matterx}
and the BRST and anti-BRST transformations again corresponds to
translations in the $\thetaa$ direction with
$\partial\over{\partial\thetaa}$ realized as differential operators
on the extended space manifold $M_S$.


\chapter{Sp(2) formalism for General gauge theories.}


Consider a
general gauge theory with classical fields $A^i (\xmu)$
$(i=1,2,\dots ,n)$ and classical action $S_0 (A^i )$. The action is
invariant under gauge transformations
$$
\delta A^i = R^i_\alpha \xi^\alpha,\nfr{gaugeinv}
where
$\xi^\alpha$ is the local gauge parameter.
The Noether equations are given by
$$
S_0,i \;\; R^i_\alpha = 0 ,\qquad \alpha =1,2,\dots ,m\qquad (0\leq m\leq n),
\nfr{noether}
and
the generators of the gauge transformations satisfy
$$
R^i_{\alpha ,j} R^j_\beta - (-)^{\alpha\beta} R^i_{\beta ,j}R^j_{\alpha}
   =-R^i_{\gamma} {f^\gamma}_{\alpha\beta} - S_{0,j} M^{ij}_{\alpha\beta}.
\nfr{generat}

To construct a covariant lagrangian formalism for general gauge theories
either with open ($M^{ij}_{\alpha\beta} \notequiv 0$) or closed algebras, and
based on $Sp(2)$ BRST symmetry, it is needed to enlarge the base manifold
$M$ to $M_S$. It is then defined a superspace $M_S$ to include classical
fields
$\Phi^A$ and $Sp(2)$ doublets of anti-fields $\Phi^\star_A$,
$\Phi^{\star\star}_A$
[\Ref{lavrovx},\Ref{lavrov},\Ref{hull}].
The properties of these fields and anti-fields are
$\epsilon (\Phi^A ) =\epsilon_A$, $\epsilon (\Phi^\star_A ) =\epsilon_A + 1$,
$\epsilon (\Phi^{\star\star}_A ) =\epsilon_A$,
$gh (\Phi^\star_A ) = (-)^a - gh (\Phi^A )$ and
$gh (\Phi^{\star\star}_A ) = - gh (\Phi^A )$. An extended Poisson
superbracket is defined by
$$
(F,G) = {\varri}F{\varle}G -(-)^{\epsilon_F\epsilon_G} {\varri}G{\varle}F,
\nfr{poisson}
where
${\epsilon_F}$, ${\epsilon_G}$ denotes the Grassmann parity of the $F$, $G$
functions on $M_S$, and left (right) derivatives are understood
with respect to anti-fields (fields) unless otherwise stated.
The extended anti-bracket
\poisson\ satisfies
$$
\epsilon\Big( (F,G)\Big) = \epsilon (F) + \epsilon (G) + 1
\nfr{propone}
$$gh \Big( (F,G)^a\Big) = - (-)^a + gh (F) + gh (G), \qquad a=1,2
\nfr{proptwo}
$$ (F,G) = - (-)^{\epsilon_F\epsilon_G} (G,F) \nfr{propthree}
and
$$(-)^{\epsilon_F}{\epsilon_G} \big( (F,G), H\big) + [cycl.\;perm\;\;
(F,G,H)] = 0. \nfr{propfour}

A bosonic
action functional $S = S(\Phi^A ,\Phi^\star_A ,\Phi^{\star\star}_A )$
is constructed on $M_S$. This action satisfy the following {\it generating
equation}
$$
\bar\Delta^a\exp^{{i\over\hbar}
S(\Phi^A ,\Phi^\star_A ,\Phi^{\star\star}_A )} = 0, \qquad  (a=1,2),
\nfr{generating}
together
with the boundary condition
$$
S(\Phi^A ,\Phi^\star_A ,\Phi^{\star\star}_A )
\Big\vert_{\Phi^\star_A =\Phi^{\star\star}_A =0} = S (\Phi_A ).
\nfr{boundary}
The
operator $\bar\Delta^a$ is defined by
$$
\bar\Delta^a = \Delta^a  + (i/\hbar ) V^a, \qquad
\Delta^a = (-)^{\epsilon_A} \varrid \varlea
\qquad
V^a = \epsilon^{ab}\Phi^\star_{Ab} \varrist .
\nfr{operators}
The
algebra of operators \operators\ satisfy the important property
\footnote{$^2$}
{A supercommutative, associative algebra $\cal A$ equipped with an extended
Poisson anti-bracket structure plus a nilpotent property it is known
as a BV-algebra, or coboundary Gersterhaber algebra (CGA) [\Ref{galgebra}].}
$$
\bar\Delta^{\{a} \bar\Delta^{b\}} = 0.
\nfr{nilpotency}
The solution
to the generating equation \generating\ is given as a power series
of the Planck constant
$$
S(\Phi^A ,\Phi^\star_A ,\Phi^{\star\star}_A ) = \sum^{+\infty}_{n=0}
{\hbar}^n S_{(n)},\nfr{power}
where
the classical approximation $S_{(0)}$ satisfies
$$
\half ( S_{(0)},S_{(0)} )^a + V^a S_{(0)} = 0.
\nfr{classy}
For a theory
in which the $Sp(2)$ algebra closes off-shell
the classical solution $S_{(0)}$ takes the form
$$
S_{(0)} = S_0 + \Phi^\star_{Aa} s^a\Phi^A + \half
\Phi^{\star\star}_A \epsilon_{ab} s^as^b \Phi^A
          + F_{AB} \epsilon_{ab} s^a \Phi^A s^b\Phi^B .
\nfr{solzero}
For more complicated
theories like superparticles or superstrings,
$S_{(0)}$ has terms of higher order in the fields
$\Phi^\star_A$, $\Phi^{\star\star}_A$ to compensate those terms
which makes the $Sp(2)$ algebra to close on-shell.
The classical solution to the generating equation
is $Sp(2)$ BRST invariant under modified BRST generators $\tilde s^a$
which satisfy $\tilde s^a \tilde s^b + \tilde s^b \tilde s^a = 0$.


\chapter{Orthosymplectic Structure of the Original BSC Superparticle.}


The original BSC superparticle $S_{BSC}$ and further models are
known to yield the same spectrum as that of $D=10$, $N=1$ super-Yang-Mills
theory [\Ref{elself}].
It is used here as an illustrative example to construct its $Sp(2)$
covariant lagrangian, since the model has an infinitely reducible
algebra. The BSC superparticle action is given by [\Ref{bscone}]
$$
S_0 = \int d\tau [ \pmu\dot x^\mu - i \theta{\psh}\dot\theta
                   - \half e p^2 ].
\nfr{bscparticle}
This
action describes a particle with world-line parametrized by $\tau$
moving through a ten-dimensional $N=1$ superspace with coordinates
$(\xmu ,\theta_A )$. The superparticle action $S_{BSC}$ is invariant
under a $10$ dimensional super-Poincar\'e symmetry
$$
\delta\theta =\epsilon ,\qquad \delta\xmu = i\epsilon\Gamma^\mu\theta ,
\nfr{poincare}
together
with world-line reparametrisations and a local fermionic symmetry
$$\eqalign{\delta\theta =& \psh\kappa ,\qquad
\delta e = 4i\kappa\dot\theta +\dot\xi ,\cr
&\delta\xmu = i\theta\Gamma^\mu\psh\kappa + \xi\pmu .
\cr}\nfr{fermionic}
The Grassmann spinor $\kappa_A$ parametrizes
the local symmetry while $\xi$ parametrizes a linear combination of
world-line
diffeomorphisms and a local {\it trivial} local symmetry.
To construct a covariant $Sp(2)$ orthosymplectic structure for this model,
it is required the formalism of the previuos section since the
classical infinitely reducible gauge algebra $\cal A$
closes on-shell.
It is then defined a superspace $M_S$ to include the classical fields
$\Phi^A = (\xmu ,\pmu ,e,\theta_A )$ and $Sp(2)$ doublets of
anti-fields $\Phi^\star_A$, $\Phi^{\star\star}_A$.
The classical approximation $S_{(0)}$ which satisfies \classy\ is
given by
$$
S_{(0)} = S_{BSC} + S_1 + S_2 + S_3 ,\nfr{orthobsc}
where
$S_{BSC}$ is the classical action of the original superparticle and
$S_1$, $S_2$, and $S_3$ are
$$
\eqalign{
S_1 =\int d\tau [&\theta^\star_a\psh\kappa^a_1 + e^\star_a
(4i\kappa^a_1\dot\theta +\dot c^a )
+ \kappa^\star_{nab} ( (-)^n\psh )(f^{ab}_{\ c}\kappa^c_{n+1}
+\epsilon^{ab}\pi_n ) \cr
&+ x^\star_{\mu a} (i\theta\gamma^\mu\psh\kappa^a_1 + \pmu c)
+ c^\star_{da} (-2i f^{ad}_{rs} \kappa^r_1\psh\kappa^s_1
+\epsilon^{ad}\pi )], \cr}
\nfr{sone}
$$
\eqalign{
S_2 =\int d\tau [&\theta^{\star\star}(-p^2\epsilon_{ab}f^{ab}_{\ c}\kappa^c_2 )
+ e^{\star\star} (-4i\epsilon_{ab}f^{ab}_{\ c}\kappa^c_2\psh\dot\theta
+ 2i\epsilon_{ab}\kappa^a_1\dot{\psh}\kappa^b_1 ) \cr
&+ \kappa^{\star\star}_{n r} ( -p^2 )(\epsilon_{ab}f^{br}_{\ c}
f^{ac}_{\ s}\kappa^s_{n+2} + f^{br}_{\ b}\pi_{n+1}  )  \cr
&+ x^{\star\star}_{\mu} (ip^2\epsilon_{ab} )
( f^{ab}_{\ c}\theta\gammu\kappa^c_2 - \kappa^a_1\gammu\kappa^b_1 )  \cr
&+ c^{\star\star}_e (-4ip^2 )(\epsilon_{ab} f^{be}_{rs}f^{as}_{\ c}
\kappa^r_1\kappa^c_2 + f^{be}_{rb} \kappa^r_1\pi_1 )],
\cr}\nfr{stwo}
and
$$
\eqalign{
S_3 =\int d\tau &\half e^\star_a
[\theta^{\star}_b( -\kappa^c_2 )( 2f^{ab}_{\ c} + \epsilon^{ab}
\epsilon_{rs} f^{rs}_{\ c} ) \cr
&+ x^\star_{\mu b} ( 2i\kappa^a_1\gammu\kappa^b_1
+ i\epsilon^{ab}\epsilon_{rs}\kappa^r_1\gammu\kappa^s_1
- i\theta\gammu\kappa^c_2 ( 2 f^{ab}_{\ c} + \epsilon^{ab}\epsilon_{rs}
f^{rs}_{\ c} ) ) \cr
&+ \kappa^{\star}_{nAb} ( -\kappa^c_{n+2} (\epsilon^{ab}\epsilon_{ps}
f^{sA}_{\ q}f^{pq}_{\ c} + 2 f^{Ab}_{\ s}f^{as}_{\ c} )
-\pi_{n+1} ( 2\epsilon^{as} f^{Ab}_{\ s} + \epsilon^{ab} f^{pA}_{\ p} ) )\cr
&- 4ic^{\star}_{Ab} ( \kappa^r_1\kappa^c_2
(\epsilon^{ab}\epsilon_{ps} f^{sA}_{rq}f^{pq}_{\ c}
+ 2f^{bA}_{rs} f^{as}_{\ c} ) + \kappa^r_1 \pi_1
(\epsilon^{ab} f^{bA}_{rs} + \epsilon^{ab} f^{pA}_{qp} ) )].
\cr}\nfr{sthree}


\sjump

{\it acknowledgments}: I would like to thank S.P. Sorella and C.
Arag{\~ a}o de Carvalho for useful discussions and comments.
I am also grateful to Prof. J.J. Giambiagi for estimulating conversations.}

\bjump\vfill
\eject

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\sjump
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\endref
\ciao



