%Paper: hep-th/9410139
%From: bello@cbpfsu1.cat.cbpf.br (Jose Luis Vazquez-Bello)
%Date: Wed, 19 Oct 94 12:31:26 EST
%Date (revised): Thu, 20 Oct 94 13:17:04 EST


%
%\loadamsmath
%\font\mtitle=cmmi10 at 12pt
%\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}}
%
\def\Landspec{}
%
\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
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%--------------------------------------------------------------------
%
%                            MACROS
%
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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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%
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%
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\def\fr{\formula} \def\efr{\endformula}
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\def\nfr{\nameformula}    \def\numali{\numero}
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\ifnum\draftnum=0  {\ifnum\indappcount=0%
{\ifnum\cpcount<1\xdef\spzzttrra{(\the\frmcount )}%
\else\xdef\spzzttrra{(\the\cpcount .\the\frmcount )}\fi}%
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\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}
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{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
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\def\beginpaper{\clearindappcount\clearappcount\clearcpcount
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\def\introduction{\clearindappcount\clearappcount\clearcpcount
                  \clearsubcpcount\goodbreak\null\vbox{\jump\nobreak
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\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
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:\the\ppminuti}
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\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
%
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{\global\advance\countref by 1%
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\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%
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\null
%
%--------------------------------------------------------------------
%
%                             THE    END
%
%--------------------------------------------------------------------
%



%title page
%cern and hep-th ref
\nopagenumbers{\baselineskip=12pt
\line{\hfill CBPF-NF-057194/CLAF.}
\ifdoublepage \bjump\bjump\bjump\bjump\else\vfill\fi
%title
\centerline{\capsone The BV Quantisation of Superparticles}
\sjump
\centerline{\capsone Type I and II.}
\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 discusses the BRST cohomology of superparticles type I and II.
It was used an extended super-space to construct $S0(9,1)$ superparticle
actions that lead to super-wave functions whose spinor components
satisfy $S0(9,1)$
covariant constraints. Their BRST charges were found by using BV methods,
since the models present a large number of symmetries and only close on-shell.
It is shown that the zero ghost-number cohomology class of both models
reproduce the same spectrum as that of N=1 ten dimensional super-Yang-Mills
theory. }


%\bjump
%\line{CBPF-CNPq/CLAF-94 \hfill}
%\line{October 1994 \hfill}
\sjump
\ifnum\unoduecol=2 \vfill\fi
\eject


\yespagenumbers\pageno=1
%>>>>>>>>>>>>>>>>>>
% pslash
%


\def\msh {\rlap{/}{m}}
\def\nsh {\rlap{/}{n}}
\def\psh {\rlap{/}{p}}
\def\Dslash {\rlap{/}{D}}
\def\Lamdsh {\Lambda\llap{/}}
\def\lmdash {\Lambda\llap{/}}
\def\Sigmash {\Sigma\llap{/}}
\def\Sigmsh {\Sigma\llap{/}}
\def\plislash{\Pi\llap{/}}
\def\Gammash{\Gamma\llap{/}}
%>>>>>>>>>>>>>>>>>>
\def\xmu { x^\mu}
\def\pmu { p_\mu}
\def\half {{ 1\over 2}}
\def\thetaA {\theta_A}
\def\thetaB {\theta_B}
\def\thetaC {\theta_C}
\def\phiA {\phi^A}
\def\PhiA {\Phi^A}
\def\adot {\dot a}
\def\bdot {\dot b}
\def\pplus {p^+}
\def\gammu {{\gamma^\mu}}
\def\gamnu {{\gamma^\nu}}
\def\thetaB {\theta_B}
\def\PsiA {\Psi_A}
\def\psiA {\psi^A}
\def\grad {\partial}
\def\Sigmnrs {\Sigma_{\mu\nu\rho\sigma}}
\def\Gammu {\Gamma^\mu}
\def\Gamnu {\Gamma^\nu}
\def\Lamnrs {\Lambda_{\mu\nu\rho\sigma}}
\def\starxmu {x^\star_\mu}
\def\startheta {\theta^\star}
\def\stard {d^\star}
\def\stare {e^\star}
\def\spsi {\psi^\star}
\def\sphi {\phi^\star}
\def\shatphi {{\hat\phi}^\star}
\def\svarphi {\varphi^\star}
\def\sLamdsh {{\Lamdsh}^\star}
\def\sbeta {\beta^\star}
\def\szeta {\zeta^\star}
\def\seta {\eta^\star}
\def\somega {\omega^\star}
\def\skappa {\kappa^\star}
\def\starv {v^\star}
\def\sSigmash {{\Sigmash}^\star}
\def\starc { c^\star}



Discussions of the mechanics of particles with spin shows that
these can be described by either a particle theory with local world-line
supersymmetry [\Ref{one}], or
by a local fermionic symmetry [\Ref{two}].
This was generalised to superpace, to obtain a number of
{\it spinning} superparticle theories satisfying certain constraints
whose spectrum were precisely those of the ten-dimensional supersymmetric
Yang-Mills theory.
The quantum mechanics of a free superparticle in a ten-dimensional
space-time is of interest because of its close relationship to
ten-dimensional super Yang-Mills theory, and
this corresponds to the massless sector of type I superstring theory.
The $SO(9,1)$ covariant superfield formulation of super Yang-Mills theory
which reduces to $SO(8)$ formulation can be obtained by either an $SO(9,1)$
vector or spinor superfields. The superfields are chosen to satisfy either
rotational quadratic ( \lq\lq Type I")
or linear (\lq\lq Type II") constraints that restricts their field content
to the physical propagating fields.
The constraints are imposed by an explicit projection operator, constructed
out of super-covariant derivatives, acting on unconstrained superfields.
They were explicitly given on its $SO(8)$ form in [\Ref{brink}],
and presented on its $SO(9,1)$ form in [\Ref{two}].
Although, it is well known the difficult covariant
quantisation of superparticle
models due to the large number of symmetries that
they present[\Ref{prob}],
there are
by now several formulations which can be covariantly quantized.
These superparticle theories with spectra coinciding with
that of the ten dimensional super Yang-Mills are constructed by adding
appropriate
Lagrange multiplier terms to certain superparticle actions,
some of them leading to type I or II constraints.

This letter compares the BRST spectrum of type I and II superparticle
models arising from using the BV covariant quantisation
techniques[\Ref{bv}].
The zero ghost-number BRST cohomology class of these models gives the same
physical spectrum as that of D=10, N=1 super-Yang-Mills theory.
It is also shown that both quantum actions leads to free theories.
%
%The methods of [\Ref{twox} ,\Ref{twomike}]
%are used to argue that the zero ghost-number BRST cohomology
%class in the reduced formalism is exactly the same as the zero ghost-number
%cohomology class in the full formalism with an infinite number of ghosts.
%
We begin by briefly reviewing the description of ten-dimensional
type I superparticle models [\Ref{twomike}].
A spinor wavefunction
can be obtained either from a spinning particle with local world-line
supersymmetry, or from a particle action with local fermionic symmetry.
In [\Ref{two}], it was seen that super Yang-Mills theory in ten-dimensions
is described by precisely such wavefunctions subject to certain extra
super-covariant constraints.
The quantum mechanics of the type I superparticle theory was
given in [\Ref{twomike}]. This superparticle action is formulated in an
extended ten-dimensional
superspace with coordinates $(\xmu ,\thetaA ,\phiA )$ where
$\thetaA$ and $\phiA$ are anti-commuting Majorana-Weyl spinors
%
\footnote
{$^\star$}
{A Majorana spinor $\Psi$ corresponds to a pair of Majorana-Weyl
spinors, $\PsiA$ and $\Psi^A$. The $32 \times 32$ matrices $C\gammu$
(where $C$ is the charge conjugation matrix) are block diagonal with
$16 \times 16$ blocks $\gammu^{AB}$, $\gammu_{AB}$ which are symmetric
and satisfy
$\gammu^{AB}\gamnu_{BC} + \gamnu^{AB}\gammu_{BC} =2\eta^{\mu\nu}\delta^A_C$.
In this notation the supercoordinates has components $\thetaA$,
$\theta\gammu\dot\theta = \thetaA \gammu_{AB} \dot\thetaB$,
$\psh_{AB} = p^\mu \gammu_{AB}$, etc. }.
%
The action is given by adding
$$
S_0 =\int\! d\tau \Big[ \pmu\dot\xmu +i\hat\theta\dot\theta
                         + i\hat\phi\dot\phi\Big],
\nfr{snutI}
and
$$
S_{1} =\int\! d\tau \Bigl[ -\half e p^2 + i\psi\psh d + i \Lambda\psh\hat\phi
          + \half( d \chi d - 2 \hat\phi\Gammu \chi \Gamma_\mu\psh\phi )
          +\half\hat\phi\Upsilon\hat\phi
\Bigr]\nfr{sbiprimI}
where
$\pmu$ is the momentum conjugate to the space-time coordinate $\xmu$, while
$\hat\theta =d -\psh\theta$ and $\hat\phi$ are the corresponding momentum
conjugate to the spinor coordinates $\theta$ and $\phi$, respectively.
The gauge fields $e$, $\psi$, $\Lambda$, $\chi$ and $\Upsilon$ correspond
to local symmetries, and impose the following constraints
$$
\eqalign{ p^2 =&0, \qquad \psh d =0, \qquad \psh\hat\phi =0, \cr
         \hat\phi\hat\phi &=0, \qquad
          d d + 2  \hat\phi\Gamma \Gamma \psh\phi =0.
\cr}\nfr{enconstry}
The physical
states are described by a superspace wavefunction
satisfying  $SO(9,1)$ covariant constraints[\Ref{two}]
$$\eqalign{
p^2 \Psi_A &=0, \qquad \psh_{AC}D^C \Psi_B =0, \qquad \psh^{AB}\Psi_B =0, \cr
&D^AD^B\Psi_C + 8 (\Gamma^\mu)^{E[A}(\Gamma_\mu\psh )^{B]}_{\ \ C}\Psi_E =0.
\cr}\nfr{quadra}
which leaves a superfield $\Psi_a (x^i,\theta^{\adot} )$
satisfying a quadratic projection condition
and describes precisely the $SO(8)$ constraints of ten dimensional
super Yang-Mills theory.
The covariant quantisation  of this superparticle was briefly
discussed in [\Ref{twomike}] in the gauge $e=1$ with the other gauge
fields set to zero. Covariant quantisation requires the methods of
Batalin and Vilkovisky [\Ref{bv}] since the gauge algebra only
closes on shell, and requires an infinite number of ghost fields
as the symmetries are infinitely reducible.
Following the BV formalism, it leads to a gauge-fixed quantum action
which, after field redefinitions and integrating out all non-propagating
fields, takes the form [\Ref{twomike}]
$$
S_Q =\int\! d\tau \Big[ \pmu\dot\xmu -\half p^2 + i\hat\theta\dot\theta
                       +i\hat\phi\dot\phi +\hat c\dot c +\hat\kappa\dot\kappa
                       +i\hat v\dot v + i\hat\rho\dot\rho
                       +\hat\zeta\dot\zeta \Big],
\nfr{quanty}
where
$\hat\theta = d - \psh\theta - 4\hat c\kappa$. This $S_Q$ proves to be
invariant under BRST transformations generated by the BRST charge
$$\eqalign{
Q =&\half c p^2 -Tr (\hat\rho\rho\psh\rho ) + \half i d\rho d
    + d\psh\kappa
    - 2\hat c\kappa\psh\kappa - 4 \hat c d\rho\kappa
    + 4i\hat c\hat\phi\Gammu\rho\Gamma_\mu \zeta\cr
    &+ 4i Tr (\hat v\Gammu\rho\Gamma_\mu\psh v)
    -2i \hat\phi\Gammu\rho\Gamma_\mu\psh\phi
    + \hat\phi\psh\zeta + \half i\hat\phi v\hat\phi .
\cr}
\nfr{chargeone}

The
quantum action defines a free field theory and so easy to quantize by
imposing canonical commutation relations on the operators corresponding
to the variables
$(\pmu ,\xmu ,e,\hat\theta ,\theta ,\hat\phi ,\phi ,\hat c ,c ,
\hat\kappa ,\kappa ,\hat v ,v ,\hat\rho ,\rho ,\hat\zeta ,\zeta )$.
It proves useful to choose a Fock space representation for the ghost
and define a ghost vaccum $|0>$ which is annihilated by each of the
antighosts
$(\hat\kappa |0> = 0,\hat c |0> = 0,\hat v |0> = 0,\hat\rho |0> =0,
  \hat\zeta |0> = 0 )$.
It also proves useful to define a {\it twisted} ghost vacuum $|0>_g$,
where for each ghost $g$ in the subscript, that ghost is an annihilation
operator and the corresponding anti-ghost is a creation operator.
The physical states on both twisted and untwisted Fock space
should be the same, as they are {\it dual} representations of
the same spectrum.
It is then viewed the superspace coordinates $\xmu$ ,$\hat\thetaA$ and
$\phiA$ as hermitian coordinates while $\pmu =-i\grad/\grad\xmu$,
$\hat\thetaA = \grad/\grad\thetaA$ and $\hat\phiA =\grad/\grad\phiA$.
A state of the form $\Phi (x,\theta ,\phi ) M|\Omega >$
with super-wavefunction $\Phi$, a monomial $M$ and $|\Omega >$ is considered.
The zero ghost-number cohomology class is given by $\Phi$, while $M$ is
constructed out of ghost and anti-ghost fields and $|\Omega >$
is one of the ghost ground state.
It was found then that the ghost-independent state
$\Phi (x,\theta ,\phi )|0>$ gives the physical spectrum
consisting of eight bosons and eight fermions which
form the $D=10$ super Yang-Mills multiplet together with the zero-momentum
ground state which is a supersymmetry singlet.

The general state of zero ghost-number and momentum $\pmu$ which is annhilated
by the BRST charge \chargeone\ satisfies the following conditions
$$\eqalign{
   p^2 \Phi &= 0, \qquad   \psh d\Phi = 0,
    \qquad \psh\hat\phi\Phi = 0, \cr
   \hat\phi\hat\phi \;\Phi &= 0, \qquad
   (d\; d - 8\hat\phi\Gammu\Gamma_\mu\psh\phi )\;\Phi =0. \cr}
\nfr{condiesone}
These
conditions \condiesone\
imply that the only non-vanishing parts of $\Phi (x,\theta ,\phi )$
are $\Psi_0 (x,\theta )$ and $\Psi_A (x,\theta )$. However, $\Psi_0$
is trivial unless $\pmu =0$. Thus the only non trivial part of $\Phi$
is $\Psi_A$ and satisfies precisely the covariant constraints \quadra ,
which lead to the D=10, N=1 super-Yang-Mills multiplet.
The monomial $M$ is constructed out of ghost and anti-ghost, and
to satisfy $Q^2 =0$, states of non-trivial ghost dependence should
include terms that involve $\Gammu\rho\Gamma_\mu$, which after
removing the bispinor parameters lead to the identity of $\Gamma 's$
to cancel in D=10, like in super-Yang-Mills theories appears
a term proportional to $\epsilon\psi^3$ which vanishes if supersymmetry
is to hold [\Ref{mbg}].

I shall now describe a ten-dimensional type II superparticle
model with a spinor super-wavefunction satisfying the following $S0(9,1)$
covariant linear constraints
$$\eqalign{
p^2 \Psi_A &=0, \qquad \psh_{AC}D^C \Psi_B =0, \qquad \psh^{AB}\Psi_B =0, \cr
&(\Gamma^{\mu\nu\rho\sigma})_A^{\ \ B} D^A\Psi_B =0,
\cr}\nfr{linear}
which
proves to be equivalent to constraints \quadra .
This model is also formulated in an extended ten-dimensional
superspace with coordinates
$(\xmu ,\thetaA ,\phiA )$ where $\thetaA$ and $\phiA$ are
anticommuting Majorana-Weyl spinors, and to describe super Yang-Mills
we wish to impose the extra constraints
$d^A (\Gamma^{\mu\nu\rho\sigma})^B_{\ A} =0$ and $\hat\phi\hat\phi =0$
which can be done
by adding appropriate lagrange multiplier terms ( See Ref. \Ref{two}
for further details ).

The type II superparticle action is then given by the
sum of [\Ref{two}]
$$
S_0 =\int\! d\tau \Big[ \pmu\dot\xmu +i\hat\theta\dot\theta
                         + i\hat\phi\dot\phi\Big],
\nfr{snut}
and
$$
S_{1} =\int\! d\tau \Big[ -\half e p^2 + i\psi\psh d
           + i \varphi\psh\hat\phi  + i d\Lamdsh\hat\phi
           - i\beta(\phi\hat\phi - 1) + \half\hat\phi\omega\hat\phi
           \Big] \nfr{sbiprim}
where,
as usual,
$\pmu$ is the momentum conjugate to the space-time coordinate $\xmu$,
$d^A$ is a spinor introduced so that the Grassmann coordinate $\thetaA$
has a conjugate momentum $\hat\theta^A =d^A -\psh^{AB}\thetaB$,
$\phiA$ is a new spinor coordinate and $\hat\phi_A$ is its conjugate
momentum.
The fields $e$, $\psi^A$, $\varphi_A$, $\Lambda_{\mu\nu\rho\sigma}$,
$\beta$ and $\omega^{AB}=-\omega^{BA}$ are all
Lagrange multipliers, which are
also gauge fields for corresponding local symmetries and impose the
following constraints
$$
\eqalign{ p^2 =&0, \qquad \psh d =0, \qquad \psh\hat\phi =0,\cr
         \hat\phi\hat\phi =0, \qquad
         &\phi\hat\phi - 1 =0, \qquad
         d(\Gamma^{\mu\nu\rho\sigma})\hat\phi =0.
\cr}\nfr{constry}
Physical
states are described by a superspace wavefunction
satisfying  $SO(9,1)$ covariant linear constraints \linear ,
which leaves a superfield $\Psi_a (x^i,\theta^{\adot} )$
satisfying a linear projection condition
which is precisely the $SO(8)$ constraint of the ten dimensional
super Yang-Mills theory.
%Covariant quantisation requires the techniques of
%Batalin and Vilkovisky since the gauge algebra only
%closes on shell, and it is require an infinite number of ghost fields.
%he symmetries are also infinitely reducible.
%It is found a gauge-fixed quantum action
%which, after some field redefinitions and integrating out all non-propagating
%fields, gives
The free quantum action is given by
$$
S_Q =\int\! d\tau \Big\{\pmu\dot\xmu + i\hat\theta\dot\theta
       +i\hat\phi\dot\phi +\half p^2 +\hat c\dot c
       +\hat\kappa\dot\kappa +\hat\zeta\dot\zeta
       +\hat\Sigmash\dot\Sigmash +\hat\eta\dot\eta
       +\hat v\dot v \Big\}.
\nfr{quadryx}
where
$\hat\theta = d-\psh\theta -4i\hat c\kappa_1$. This quantum action is invariant
under modified BRST transformations
which are generated by
the following conserved $(\dot Q_{BRST}=0)$ and nilpotent
$(Q^2_{BRST} =0)$ BRST charge
$$\eqalign {
 Q_{BRST} = &\half c p^2 + 2 i d\psh\kappa + 2 i \hat\phi\psh\zeta
            - 2 i d\hat\phi\Sigmash -\hat\phi v\hat\phi -\hat\phi\eta\phi
            + \hat\zeta\zeta\eta +\hat\Sigmash\Sigmash\eta \cr
          & +\hat\eta\eta\eta
            + 2 \hat v v\eta - 2 \hat c\hat\theta\kappa_2
            - 2 i \hat c\kappa\psh\kappa + i \hat\kappa\psh\kappa_2
            + 4 \hat c\hat\zeta\kappa_2\Sigmash
            + 2 i \hat\kappa\hat c\kappa_3
            + 2 i \hat v\Sigmash\psh\Sigmash .
\cr}\nfr{brstcharge}
The
quantum
action \quadryx\ is free and the superparticle type II can again be
quantized canonically by replacing each of the fields by an operator
and imposing canonical (anti-) commutation relations on the conjugate pairs
$(\pmu ,\xmu )$, $(\hat\theta ,\theta )$, $(\hat\phi ,\phi)$,
$(\hat c, c)$, $(\hat\kappa_1 ,\kappa_1)$, $(\hat\zeta_1 ,\zeta_1)$,
$(\hat\Sigmash_1 ,\Sigmash_1)$, $(\hat\eta_1 ,\eta_1)$ and
$(\hat v_1, v_1)$.
The
cohomology classes can be classified according
to their total ghost number and the physical states are taken to be
the cohomology class of some definite ghost number.
We consider
two distinct Fock space representations of the ghost system,
the {\it untwisted} one in which the ghost ground state $|0>$ is
annihilated by each of the antighosts
$(\hat\kappa_1 |0> = 0,\hat c |0> = 0,\hat v_1 |0> = 0,\hat\eta_1 |0> =0,
  \hat\Sigmash_1 |0>=0, \hat\zeta_1 |0> = 0 )$,
and the {\it twisted}
one in which antighosts are creation operators and ghosts are
annihilation operators.
It is viewed $x$, $\theta$ and $\phi$ as hermitian coordinates while
$\pmu =-i\grad/\grad\xmu$, $\hat\thetaA = \grad/\grad\thetaA$ and
$\hat\phiA =\grad/\grad\phiA$, and
consider again states of the form $\Phi (x,\theta ,\phi )\;M\;|\Omega >$
with super-wave function $\Phi$. $M$ is some monomial constructed out of
ghosts and anti-ghost and $|\Omega >$ is one of the ghost ground state.
A general state
of zero-ghost number and momentum $\pmu$ which is annhilated by the
BRST charge \brstcharge\ satisfies
$$\eqalign { &
    p^2 \Phi =0, \qquad \psh d\Phi =0, \qquad \psh\hat\phi\Phi =0,  \cr
   \hat\phi\hat\phi\Phi &=0, \qquad (\phi\hat\phi -1)\Phi=0, \qquad
   d\Gamma^{\mu\nu\rho\sigma}\hat\phi\Phi =0.
\cr}\nfr{constries}
These
implies that the only non-vanishing parts of $\Phi (x,\theta,\phi )$
are $\Psi_0 (x,\theta )$ and $\Psi_A (x,\theta )$. However,
$(\phi\hat\phi -1)\Phi = 0$ implies that $\Psi_0$ is trivial.
The only non-trivial part of the super-wave function is
$\Psi_A$ which precisely
satisfies the covariant constraints \constry\ leading
to the D=10, $N=1$ super-Yang-Mills multiplet.
The monomial
$M$ is constructed out of ghost and anti-ghosts and to satisfy
$Q^2=0$, states of non-trivial ghost dependence include terms that
involve products of $\Gamma 's$, like the term $\Sigmash\psh\Sigmash$
which after removing the multi-tensor bosonic parameters leads
also to an identity of $\Gamma 's$ in $D=10$, and needed to avoid the
propagation of these bosonic ghost fields.
Lets recall that the covariant quantisation of the $SSP2$ superparticle model
lead to the appearance of multispinor ghosts and rouse the status
of the BRST cohomology of that model [\Ref{newyork}].

\sjump

{\it acknowledgments}: I wish to thank J.A. Helayel-Neto for reading and
comments of the manuscript.

\bjump

\eject


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\endref
\ciao


