%Paper: hep-th/9307051
%From: PARANJ@LPS.UMONTREAL.CA
%Date: Wed, 7 Jul 1993 15:50:31 -0400 (EDT)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Just strip off the mail header and tex the remaining,
%this will create one dvi file, the file "deli.tex" and the file
%"delipics.ps".  Send the dvi file to your printer with appropriate
%conversion to postscript. The other files created are unecessary.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This collection of files was produced with CERN psbox package
% To decompose and tex it:
%-save this with a filename CONTAINING ONLY LETTERS and a .TEX
% extension (say, JOINTFIL.TEX), in some uncrowded directory;
%-make sure you can \input psbox.tex (version>=1.3);
% (else ftp cs.nyu.edu(=128.122.140.24):pub/TeX/psbox/, then get
% and tex the file psboxall.tex; more info in psbREAD.ME)
%-tex JOINTFIL.TEX using Plain, or LaTeX, or whatever is needed by
% the first file in the joining (after splitting JOINTFIL.TEX into
% it's constituents, TeX will try to process it as it stands).
%Beginning-Of-File-Named:deli
%
%       %%%%%%%    %%%%%        %%%%%%    %%%%%   %     %
%       %      %  %             %     %  %     %   %   %
%       %      %  %             %     %  %     %    % %
%       %%%%%%%    %%%%%        %%%%%%   %     %     %
%       %               %       %     %  %     %    % %
%       %               %       %     %  %     %   %   %
%       %         %%%%%%        %%%%%%    %%%%%   %     %
%
%       By Jean Orloff
%       Comments & suggestions by e-mail: ORLOFF@surya11.cern.ch
%       No modification of this file allowed if not e-sent to me.
%
% WHAT IS IT:
% psbox is a set of machine-independent TeX macros to
% 1) allow (Encapsulated) PostScript figure inclusion in all versions
%    of TeX (Plain, LaTeX) on all machines using a PostScript printer
% 2) facilitate the communication (e-mail, ftp, ...) of all the files
%    (text, macros, figs) needed to reproduce a TeX document by grouping
%    them together into a single, TeXable file.
% For more info, get the file pub/TeX/psbox/PSBOXALL.TEX by anonymous
%     ftp from cs.nyu.edu(=128.122.140.24)
%
% History:
%  1.34  \readfilename=final fix for all filename scans; try \psforptips
%  1.33: corrects \psnewinput for LaTeX (still fails if fname=a{b}c)
%  1.32: corrects \psfordvialw and adds .TEX to PSBOXALL(!)
%  1.31: adds \psfordvialw(?)
%  1.30: adds \splitfile & \joinfiles for multi-file management
%  1.24: fix error handling & add \psonlyboxes
%  1.22: makes \drawingBox \global for use in Phyzzx
%  1.21: accepts %%BoundingBox: (atend)
%  1.20: tries to add \psfordvitps for the TeXPS package.
%  1.10: adds \psforoztex, error handling...
%2345678 1 2345678 2 2345678 3 2345678 4 2345678 5 2345678 6 2345678 7 23456789
%
% Checking version no to avoid multiple loadings
\def\temp{1.34}%
\let\tempp=\relax
\expandafter\ifx\csname psboxversion\endcsname\relax
  \message{PSBOX(\temp) loading}%
\else
    \ifdim\temp cm>\psboxversion cm
      \message{PSBOX(\temp) loading}%
    \else
      \message{PSBOX(\psboxversion) is already loaded: I won't load
        PSBOX(\temp)!}%
      \let\temp=\psboxversion
      \let\tempp=\endinput
    \fi
\fi
\tempp
\let\psboxversion=\temp
\catcode`\@=11
% Every macro likes a little privacy...
%
%Trying to tame the variety of \special commands for Postscript: the
%  universal internal command \PSspeci@l##1##2 takes ##1 to be the
%  filename and ##2 to be the integer scale factor*1000 (as for usual
%   TeX \scale commands)
%
\def\psfortextures{%     For TeXtures on the Macintosh
%-----------------
\def\PSspeci@l##1##2{%
\special{illustration ##1\space scaled ##2}%
}}%
\def\psfordvitops{%      For the DVItoPS converter on IBM mainframes
%----------------
\def\PSspeci@l##1##2{%
\special{dvitops: import ##1\space \the\drawingwd \the\drawinght}%
}}%
\def\psfordvips{%      For DVIPS converter on VAX, UNIX and PC's
%--------------
\def\PSspeci@l##1##2{%
%    \special{/@scaleunit 1000 def}% never read dox without trying!
\d@my=0.1bp \d@mx=\drawingwd \divide\d@mx by\d@my% BUG! for large \drawingwd
\special{PSfile=##1\space llx=\psllx\space lly=\pslly\space%
urx=\psurx\space ury=\psury\space rwi=\number\d@mx
}}}%
\def\psforoztex{%        For the OzTeX shareware on the Macintosh
%--------------
\def\PSspeci@l##1##2{%
\special{##1 \space
      ##2 1000 div dup scale
      \number-\psllx\space \number-\pslly\space translate
}}}%
\def\psfordvitps{%       From the UNIX TeXPS package, vers.>3.12
%---------------
% Convert a dimension into the number \psn@sp (in scaled points)
\def\psdimt@n@sp##1{\d@mx=##1\relax\edef\psn@sp{\number\d@mx}}
\def\PSspeci@l##1##2{%
% psfig.psr contains the def of "startTexFig": if you can locate it
% and include the correct pathname, it should work
\special{dvitps: Include0 "psfig.psr"}% contains def of "startTexFig"
\psdimt@n@sp{\drawingwd}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\drawinght}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psllx bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\pslly bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psurx bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psury bp}
\special{dvitps: Literal "\psn@sp\space startTexFig\space"}
\special{dvitps: Include1 "##1"}
\special{dvitps: Literal "endTexFig\space"}
}}%
\def\psfordvialw{%   Try for dvialw, a UNIX public domain
%---------------
\def\PSspeci@l##1##2{
\special{language "PostScript",
position = "bottom left",
literal "  \psllx\space \pslly\space translate
  ##2 1000 div dup scale
  -\psllx\space -\pslly\space translate",
include "##1"}
}}%
\def\psforptips{%   For MS-DOS; LUOMA@brandeis.bitnet
%---------------
\def\PSspeci@l##1##2{{
\d@mx=\psurx bp
\advance \d@mx by -\psllx bp
\divide \d@mx by 1000\multiply\d@mx by \xscale
\incm{\d@mx}
\let\tmpx\dimincm
\d@my=\psury bp
\advance \d@my by -\pslly bp
\divide \d@my by 1000\multiply\d@my by \xscale
\incm{\d@my}
\let\tmpy\dimincm
\d@mx=-\psllx bp
\divide \d@mx by 1000\multiply\d@mx by \xscale
\d@my=-\pslly bp
\divide \d@my by 1000\multiply\d@my by \xscale
\at(\d@mx;\d@my){\special{ps:##1 x=\tmpx, y=\tmpy}}
}}}%
\def\psonlyboxes{%     Draft-like behaviour if none of the others works
%---------------
\def\PSspeci@l##1##2{%
\at(0cm;0cm){\boxit{\vbox to\drawinght
  {\vss\hbox to\drawingwd{\at(0cm;0cm){\hbox{({\tt##1})}}\hss}}}}
}}%
\def\psloc@lerr#1{%
\let\savedPSspeci@l=\PSspeci@l%
\def\PSspeci@l##1##2{%
\at(0cm;0cm){\boxit{\vbox to\drawinght
  {\vss\hbox to\drawingwd{\at(0cm;0cm){\hbox{({\tt##1}) #1}}\hss}}}}
\let\PSspeci@l=\savedPSspeci@l% restore normal output for other figs!
}}%
%\def\psfor...  add your own!
%
% Some common defs
%
\newread\pst@mpin
\newdimen\drawinght\newdimen\drawingwd
\newdimen\psxoffset\newdimen\psyoffset
\newbox\drawingBox
\newcount\xscale \newcount\yscale \newdimen\pscm\pscm=1cm
\newdimen\d@mx \newdimen\d@my
\newdimen\pswdincr \newdimen\pshtincr
\let\ps@nnotation=\relax
{\catcode`\|=0 |catcode`|\=12 |catcode`|%=12 |catcode`~=12
|catcode`#=12 |catcode`*=14
|xdef|backslashother{\}*
|xdef|percentother{%}*
|xdef|tildeother{~}*
|xdef|sharpother{#}*
}%
% useful to display special chars in \tt; fails for \,#,%
\def\R@moveMeaningHeader#1:->{}%
\def\uncatcode#1{%
\edef#1{\expandafter\R@moveMeaningHeader\meaning#1}}%
%
\def\execute#1{#1}% NOT stupid: cs in #1 are then identified BEFORE execution
\def\psm@keother#1{\catcode`#112\relax}% borrowed from latex
\def\executeinspecs#1{%
\execute{\begingroup\let\do\psm@keother\dospecials\catcode`\^^M=9#1\endgroup}}%
\def\@mpty{}%
% \if\matchin#1#2<=> \iftrue if #1 contains #2, <=>\iffalse otherwise:
% \if\matchexpin: idem, but #1 & #2 are first fully expanded (no \if
% inside!)
% \tmpa & \tmpb contain what's before and after the occurence of #2
\def\matchexpin#1#2{
  \fi%
%\message{(#1>#2)}
  \edef\tmpb{{#2}}%
  \expandafter\makem@tchtmp\tmpb%
  \edef\tmpa{#1}\edef\tmpb{#2}%
  \expandafter\expandafter\expandafter\m@tchtmp\expandafter\tmpa\tmpb\endm@tch%
  \if\match%
}%
\def\matchin#1#2{%
  \fi%
  \makem@tchtmp{#2}%
  \m@tchtmp#1#2\endm@tch%
  \if\match%
}%
\def\makem@tchtmp#1{\def\m@tchtmp##1#1##2\endm@tch{%
  \def\tmpa{##1}\def\tmpb{##2}\let\m@tchtmp=\relax%
  \ifx\tmpb\@mpty\def\match{YN}%
  \else\def\match{YY}\fi%
}}%
% converts any dimen in cm, with 1E-4 cm precision
\def\incm#1{{\psxoffset=1cm\d@my=#1
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \xdef\dimincm{\number\d@mx.}
  \advance\d@my by -\number\d@mx cm
  \multiply\d@my by 100
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \edef\dimincm{\dimincm\number\d@mx}
  \advance\d@my by -\number\d@mx cm
  \multiply\d@my by 100
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \xdef\dimincm{\dimincm\number\d@mx}
}}%
%
%  \ReadPSize{PSfilename} reads the dimensions of a PostScript drawing
%      and stores it in \drawinght(wd)
\newif\ifNotB@undingBox
\newhelp\PShelp{Proceed: you'll have a 5cm square blank box instead of
your graphics (Jean Orloff).}%
\def\s@tsize#1 #2 #3 #4\@ndsize{
  \def\psllx{#1}\def\pslly{#2}%
  \def\psurx{#3}\def\psury{#4}%  needed by a crazyness of dvips!
  \ifx\psurx\@mpty\NotB@undingBoxtrue% this is not a valid one!
  \else
    \drawinght=#4bp\advance\drawinght by-#2bp
    \drawingwd=#3bp\advance\drawingwd by-#1bp
%  !Units related by crazy factors as bp/pt=72.27/72 should be BANNED!
  \fi
  }%
\def\sc@nBBline#1:#2\@ndBBline{\edef\p@rameter{#1}\edef\v@lue{#2}}%
\def\g@bblefirstblank#1#2:{\ifx#1 \else#1\fi#2}%
{\catcode`\%=12
\xdef\B@undingBox{%%BoundingBox}}%
%% is not a true comment in PostScript, even if % is!
\def\ReadPSize#1{
 \readfilename#1\relax
 \let\PSfilename=\lastreadfilename
 \openin\pst@mpin=#1\relax
 \ifeof\pst@mpin \errhelp=\PShelp
   \errmessage{I haven't found your postscript file (\PSfilename)}%
   \psloc@lerr{was not found}%
   \s@tsize 0 0 142 142\@ndsize
   \closein\pst@mpin
 \else
% each entry in \GlobalInputList should be unique
   \if\matchexpin{\GlobalInputList}{, \lastreadfilename}%
   \else\xdef\GlobalInputList{\GlobalInputList, \lastreadfilename}%
     \immediate\write\psbj@inaux{\lastreadfilename,}%
   \fi%
   \loop
     \executeinspecs{\catcode`\ =10\global\read\pst@mpin to\n@xtline}%
     \ifeof\pst@mpin
       \errhelp=\PShelp
       \errmessage{(\PSfilename) is not an Encapsulated PostScript File:
           I could not find any \B@undingBox: line.}%
       \edef\v@lue{0 0 142 142:}%
       \psloc@lerr{is not an EPSFile}%
       \NotB@undingBoxfalse
     \else
       \expandafter\sc@nBBline\n@xtline:\@ndBBline
       \ifx\p@rameter\B@undingBox\NotB@undingBoxfalse
         \edef\t@mp{%
           \expandafter\g@bblefirstblank\v@lue\space\space\space}%
         \expandafter\s@tsize\t@mp\@ndsize
       \else\NotB@undingBoxtrue
       \fi
     \fi
   \ifNotB@undingBox\repeat
   \closein\pst@mpin
 \fi
\message{#1}%
}%
%
% \psboxto(xdim;ydim){psfilename}: you specify the dimensions and
%    TeX uniformly scales to fit the largest one. If xdim=0pt, the
%    scale is fully determined by ydim and vice versa.
%    Notice: psboxes are a real vboxes; couldn't take hbox otherwise all
%    indentation and all cr's would be interpreted as spaces (hugh!).
%
\def\psboxto(#1;#2)#3{\vbox{%
   \ReadPSize{#3}%
   \advance\pswdincr by \drawingwd
   \advance\pshtincr by \drawinght
   \divide\pswdincr by 1000
   \divide\pshtincr by 1000
   \d@mx=#1
   \ifdim\d@mx=0pt\xscale=1000
         \else \xscale=\d@mx \divide \xscale by \pswdincr\fi
   \d@my=#2
   \ifdim\d@my=0pt\yscale=1000
         \else \yscale=\d@my \divide \yscale by \pshtincr\fi
   \ifnum\yscale=1000
         \else\ifnum\xscale=1000\xscale=\yscale
                    \else\ifnum\yscale<\xscale\xscale=\yscale\fi
              \fi
   \fi
   \divide\drawingwd by1000 \multiply\drawingwd by\xscale
   \divide\drawinght by1000 \multiply\drawinght by\xscale
   \divide\psxoffset by1000 \multiply\psxoffset by\xscale
   \divide\psyoffset by1000 \multiply\psyoffset by\xscale
   \global\divide\pscm by 1000
   \global\multiply\pscm by\xscale
   \multiply\pswdincr by\xscale \multiply\pshtincr by\xscale
   \ifdim\d@mx=0pt\d@mx=\pswdincr\fi
   \ifdim\d@my=0pt\d@my=\pshtincr\fi
   \message{scaled \the\xscale}%
 \hbox to\d@mx{\hss\vbox to\d@my{\vss
   \global\setbox\drawingBox=\hbox to 0pt{\kern\psxoffset\vbox to 0pt{%
      \kern-\psyoffset
      \PSspeci@l{\PSfilename}{\the\xscale}%
      \vss}\hss\ps@nnotation}%
   \global\wd\drawingBox=\the\pswdincr
   \global\ht\drawingBox=\the\pshtincr
   \global\drawingwd=\pswdincr
   \global\drawinght=\pshtincr
   \baselineskip=0pt
   \copy\drawingBox
 \vss}\hss}%
  \global\psxoffset=0pt
  \global\psyoffset=0pt
  \global\pswdincr=0pt
  \global\pshtincr=0pt % These are local to one figure
  \global\pscm=1cm %should not be necessary
}}%
%
% \psboxscaled{scalefactor*1000}{PSfilename} allows to bypass the
%   rounding errors of TeX integer divisions for situations where the
%   TeX box should fit the original BoundingBox with a precision
%   better
%   than 1/1000.
%
\def\psboxscaled#1#2{\vbox{%
  \ReadPSize{#2}%
  \xscale=#1
  \message{scaled \the\xscale}%
  \divide\pswdincr by 1000 \multiply\pswdincr by \xscale
  \divide\pshtincr by 1000 \multiply\pshtincr by \xscale
  \divide\psxoffset by1000 \multiply\psxoffset by\xscale
  \divide\psyoffset by1000 \multiply\psyoffset by\xscale
  \divide\drawingwd by1000 \multiply\drawingwd by\xscale
  \divide\drawinght by1000 \multiply\drawinght by\xscale
  \global\divide\pscm by 1000
  \global\multiply\pscm by\xscale
  \global\setbox\drawingBox=\hbox to 0pt{\kern\psxoffset\vbox to 0pt{%
     \kern-\psyoffset
     \PSspeci@l{\PSfilename}{\the\xscale}%
     \vss}\hss\ps@nnotation}%
  \advance\pswdincr by \drawingwd
  \advance\pshtincr by \drawinght
  \global\wd\drawingBox=\the\pswdincr
  \global\ht\drawingBox=\the\pshtincr
  \global\drawingwd=\pswdincr
  \global\drawinght=\pshtincr
  \baselineskip=0pt
  \copy\drawingBox
  \global\psxoffset=0pt
  \global\psyoffset=0pt
  \global\pswdincr=0pt
  \global\pshtincr=0pt % These are local to one figure
  \global\pscm=1cm
}}%
%
%  \psbox{PSfilename} makes a TeX box having the minimal size to
%      enclose the picture
\def\psbox#1{\psboxscaled{1000}{#1}}%
%------------------------------------------------------
%  \joinfiles file1, file2, ...n \into joinedfilename .
%     makes one file out of many
%  \splitfile joinedfilename
%     the opposite
\newif\ifn@teof\n@teoftrue
\newif\ifc@ntrolline
\newif\ifmatch
\newread\j@insplitin
\newwrite\j@insplitout
\newwrite\psbj@inaux
\immediate\openout\psbj@inaux=psbjoin.aux
\immediate\write\psbj@inaux{\string\joinfiles}%
\immediate\write\psbj@inaux{\jobname,}%
%
% INPUT REDEFINITION
%
% works if #1 is a single character
\def\toother#1{\ifcat\relax#1\else\expandafter%
  \toother@ux\meaning#1\endtoother@ux\fi}%
\def\toother@ux#1 #2#3\endtoother@ux{\def\tmp{#3}%
  \ifx\tmp\@mpty\def\tmp{#2}\let\next=\relax%
  \else\def\next{\toother@ux#2#3\endtoother@ux}\fi%
\next}%
%
% \readfilename defs:
%
\let\readfilenamehook=\relax
\def\re@d{\expandafter\re@daux}% spares typing 10 \expandafter's...
\def\re@daux{\futurelet\nextchar\stopre@dtest}%
\def\re@dnext{\xdef\lastreadfilename{\lastreadfilename\nextchar}%
  \afterassignment\re@d\let\nextchar}%
\def\stopre@d{\egroup\readfilenamehook}%
\def\stopre@dtest{%
  \ifcat\nextchar\relax\let\nextread\stopre@d
  \else
    \ifcat\nextchar\space\def\nextread{%
      \afterassignment\stopre@d\chardef\nextchar=`}%
    \else\let\nextread=\re@dnext
      \toother\nextchar
      \edef\nextchar{\tmp}%
    \fi
  \fi\nextread}%
\def\readfilename{\bgroup%
  \let\\=\backslashother \let\%=\percentother \let\~=\tildeother
  \let\#=\sharpother \xdef\lastreadfilename{}%
  \re@d}%
%
% redefines \input using \readfilename
%
\xdef\GlobalInputList{\jobname}%
\def\psnewinput{%
  \def\readfilenamehook{% each entry in \GlobalInputList should be unique
    \if\matchexpin{\GlobalInputList}{, \lastreadfilename}%
    \else\xdef\GlobalInputList{\GlobalInputList, \lastreadfilename}%
      \immediate\write\psbj@inaux{\lastreadfilename,}%
    \fi%
    \ps@ldinput\lastreadfilename\relax%
    \let\readfilenamehook=\relax%
  }\readfilename%
}%
\expandafter\ifx\csname @@input\endcsname\relax    % then Plain
  \immediate\let\ps@ldinput=\input\def\input{\psnewinput}%
\else
  \immediate\let\ps@ldinput=\@@input
  \def\@@input{\psnewinput}%
\fi%
%
\def\nowarnopenout{%
 \def\warnopenout##1##2{%
   \readfilename##2\relax
   \message{\lastreadfilename}%
   \immediate\openout##1=\lastreadfilename\relax}}%
\def\warnopenout#1#2{%
 \readfilename#2\relax
 \def\t@mp{TrashMe,psbjoin.aux,psbjoint.tex,}\uncatcode\t@mp
 \if\matchexpin{\t@mp}{\lastreadfilename,}%
 \else
   \immediate\openin\pst@mpin=\lastreadfilename\relax
   \ifeof\pst@mpin
     \else
     \errhelp{If the content of this file is so precious to you, abort (ie
press x or e) and rename it before retrying.}%
     \errmessage{I'm just about to replace your file named \lastreadfilename}%
   \fi
   \immediate\closein\pst@mpin
 \fi
 \message{\lastreadfilename}%
 \immediate\openout#1=\lastreadfilename\relax}%
% % will have an unusual catcode below; use * instead
%\vbox
{\catcode`\%=12\catcode`\*=14
\gdef\splitfile#1{*
 \readfilename#1\relax
 \immediate\openin\j@insplitin=\lastreadfilename\relax
 \ifeof\j@insplitin
   \message{! I couldn't find and split \lastreadfilename!}*
 \else
   \immediate\openout\j@insplitout=TrashMe
   \message{< Splitting \lastreadfilename\space into}*
   \loop
     \ifeof\j@insplitin
       \immediate\closein\j@insplitin\n@teoffalse
     \else
       \n@teoftrue
       \executeinspecs{\global\read\j@insplitin to\spl@tinline\expandafter
         \ch@ckbeginnewfile\spl@tinline%Beginning-Of-File-Named:%\endcheck}*
       \ifc@ntrolline
       \else
         \toks0=\expandafter{\spl@tinline}*
         \immediate\write\j@insplitout{\the\toks0}*
       \fi
     \fi
   \ifn@teof\repeat
   \immediate\closeout\j@insplitout
 \fi\message{>}*
}*
\gdef\ch@ckbeginnewfile#1%Beginning-Of-File-Named:#2%#3\endcheck{*
 \def\t@mp{#1}*
 \ifx\@mpty\t@mp
   \def\t@mp{#3}*
   \ifx\@mpty\t@mp
     \global\c@ntrollinefalse
   \else
     \immediate\closeout\j@insplitout
     \warnopenout\j@insplitout{#2}*
     \global\c@ntrollinetrue
   \fi
 \else
   \global\c@ntrollinefalse
 \fi}*
\gdef\joinfiles#1\into#2{*
 \message{< Joining following files into}*
 \warnopenout\j@insplitout{#2}*
 \message{:}*
 {*
 \edef\w@##1{\immediate\write\j@insplitout{##1}}*
\w@{% This collection of files was produced with CERN psbox package}*
\w@{% To decompose and tex it:}*
\w@{%-save this with a filename CONTAINING ONLY LETTERS and a .TEX}*
\w@{% extension (say, JOINTFIL.TEX), in some uncrowded directory;}*
\w@{%-make sure you can \string\input\space psbox.tex (version>=1.3);}*
\w@{%  (else ftp cs.nyu.edu(=128.122.140.24):pub/TeX/psbox/, then get}*
\w@{%  and tex the file psboxall.tex; more info in psbREAD.ME)}*
\w@{%-tex JOINTFIL.TEX using Plain, or LaTeX, or whatever is needed by}*
\w@{%  the first file in the joining (after splitting JOINTFIL.TEX into}*
\w@{%  it's constituents, TeX will try to process it as it stands).}*
\w@{\string\input\space psbox.tex}*
\w@{\string\splitfile{\string\jobname}}*
\w@{\string\let\string\autojoin=\string\relax}*
}*
 \expandafter\tre@tfilelist#1, \endtre@t
 \immediate\closeout\j@insplitout
 \message{>}*
}*
\gdef\tre@tfilelist#1, #2\endtre@t{*
 \readfilename#1\relax
 \ifx\@mpty\lastreadfilename
 \else
   \immediate\openin\j@insplitin=\lastreadfilename\relax
   \ifeof\j@insplitin
     \errmessage{I couldn't find file \lastreadfilename}*
   \else
     \message{\lastreadfilename}*
     \immediate\write\j@insplitout{%Beginning-Of-File-Named:\lastreadfilename}*
     \executeinspecs{\global\read\j@insplitin to\oldj@ininline}*
     \loop
       \ifeof\j@insplitin\immediate\closein\j@insplitin\n@teoffalse
       \else\n@teoftrue
         \executeinspecs{\global\read\j@insplitin to\j@ininline}*
         \toks0=\expandafter{\oldj@ininline}*
         \let\oldj@ininline=\j@ininline
         \immediate\write\j@insplitout{\the\toks0}*
       \fi
     \ifn@teof
     \repeat
   \immediate\closein\j@insplitin
   \fi
   \tre@tfilelist#2, \endtre@t
 \fi}*
}%
% To be put at the end of a file, for making a tar-like file containing
%   everything it used.
\def\autojoin{%
 \immediate\write\psbj@inaux{\string\into{psbjoint.tex}}%
 \immediate\closeout\psbj@inaux
 \expandafter\joinfiles\GlobalInputList\into{psbjoint.tex}%
}%
%----------------------------------------------------------------
%  Annotations & Captions etc...
%
%
% \centinsert{anybox} is just a centered \midinsert, but is included as
%    people barely use the original inserts from TeX.
%
\def\centinsert#1{\midinsert\line{\hss#1\hss}\endinsert}%
\def\psannotate#1#2{\vbox{%
  \def\ps@nnotation{#2\global\let\ps@nnotation=\relax}#1}}%
\def\pscaption#1#2{\vbox{%
   \setbox\drawingBox=#1
   \copy\drawingBox
   \vskip\baselineskip
   \vbox{\hsize=\wd\drawingBox\setbox0=\hbox{#2}%
     \ifdim\wd0>\hsize
       \noindent\unhbox0\tolerance=5000
    \else\centerline{\box0}%
    \fi
}}}%
% for compatibility with older versions, but \psfig is a bad name!
%\def\psfig#1#2#3{\pscaption{\psannotate{#1}{#2}}{#3}}
%\def\psfigurebox#1#2#3{\pscaption{\psannotate{\psbox{#1}}{#2}}{#3}}
%
% \at(#1;#2)#3 puts #3 at #1-higher and #2-right of the current
%    position without moving it (to be used in annotations).
\def\at(#1;#2)#3{\setbox0=\hbox{#3}\ht0=0pt\dp0=0pt
  \rlap{\kern#1\vbox to0pt{\kern-#2\box0\vss}}}%
%
% \gridfill(ht;wd) makes a 1cm*1cm grid of ht by wd whose lower-left
%   corner is the current point
\newdimen\gridht \newdimen\gridwd
\def\gridfill(#1;#2){%
  \setbox0=\hbox to 1\pscm
  {\vrule height1\pscm width.4pt\leaders\hrule\hfill}%
  \gridht=#1
  \divide\gridht by \ht0
  \multiply\gridht by \ht0
  \gridwd=#2
  \divide\gridwd by \wd0
  \multiply\gridwd by \wd0
  \advance \gridwd by \wd0
  \vbox to \gridht{\leaders\hbox to\gridwd{\leaders\box0\hfill}\vfill}}%
%
% Useful to measure where to put annotations
\def\fillinggrid{\at(0cm;0cm){\vbox{%
  \gridfill(\drawinght;\drawingwd)}}}%
%
% \textleftof\anybox: Sample text\endtext
%   inserts "Sample text" on the left of \anybox ie \vbox, \psbox.
%   \textrightof is the symmetric (not documented, too uggly)
% Welcome any suggestion about clean wraparound macros from
%   TeXhackers reading this
%
\def\textleftof#1:{%
  \setbox1=#1
  \setbox0=\vbox\bgroup
    \advance\hsize by -\wd1 \advance\hsize by -2em}%
\def\textrightof#1:{%
  \setbox0=#1
  \setbox1=\vbox\bgroup
    \advance\hsize by -\wd0 \advance\hsize by -2em}%
\def\endtext{%
  \egroup
  \hbox to \hsize{\valign{\vfil##\vfil\cr%
\box0\cr%
\noalign{\hss}\box1\cr}}}%
%
% \frameit{\thick}{\skip}{\anybox}
%    draws with thickness \thick a box around \anybox, leaving \skip of
%    blank around it. eg \frameit{0.5pt}{1pt}{\hbox{hello}}
% \boxit{\anybox} is a shortcut.
\def\frameit#1#2#3{\hbox{\vrule width#1\vbox{%
  \hrule height#1\vskip#2\hbox{\hskip#2\vbox{#3}\hskip#2}%
        \vskip#2\hrule height#1}\vrule width#1}}%
\def\boxit#1{\frameit{0.4pt}{0pt}{#1}}%
%
%
\catcode`\@=12 % cs containing @ are unreachable
%
% CUSTOMIZE YOUR DEFAULT DRIVER:
%    Uncomment the line corresponding to your TeX system:
%\psfortextures%     For TeXtures on the Macintosh
%\psforoztex   %     For OzTeX shareware on the Macintosh
%\psfordvitops %     For the DVItoPS converter for TeX on IBM mainframes
 \psfordvips   %     For DVIPS converter on VAX and UNIX
%\psfordvitps  %     For dvitps from TeXPS package under UNIX
%\psfordvialw  %     For dvialw, UNIX public domain
%\psonlyboxes  %     Blank Boxes (when all else fails).
\splitfile{\jobname}
\let\autojoin=\relax
%%%%%%%%%%%%%%%%%%%%%%%
%%article starts here%%
%%%%%%%%%%%%%%%%%%%%%%%
\magnification=1200
%\baselineskip=24pt
\hsize=16.5truecm
\vsize=24truecm
\def\undertext#1{$\underline{\smash{\hbox{#1}}}$}
\hyphenation{Di-rac}
\hyphenation{fer-mi-ons}
\centerline{\bf Delicacies of the Mass Perurbation}
\centerline{\bf in the Schwinger Model on a Circle}
\vskip.5truecm
\centerline{M. B. Paranjape}
\centerline{ Laboratoire de physique nucl\'eaire, D\'epartement de physique,
Universit\'e de Montr\'eal,}
\centerline{C. P. 6128 succ. ``A", Montr\'eal,
Qu\'ebec, Canada, H3C 3J7}
\vskip.5truecm
\centerline{\bf Abstract}
\vskip.5truecm
\noindent
The Hilbert bundle for the massless fermions of the Schwinger model on a
circle,
over the space of gauge field configurations, is topologically non-trivial
(twisted).  The corresponding bundle for massive fermions is topologically
trivial (periodic).  Since the structure of the
fermionic Hilbert bundle changes discontinuously the possibility of perturbing
in the mass is thrown into doubt.  In this article, we show that a direct
application of the anti-adiabatic theorem of Low, allows the structure of the
massless theory to be dynamically preserved in the strong coupling limit,
${e\over m}>>1$.
This justifies the use of perturbation theory in the bosonized version of the
model, in this limit.
\vskip.5truecm
\centerline{\bf Introduction}
\vskip.5truecm\par\noindent
The massless Schwinger model on a circle was studied by Manton$^1$ and by
Hosotani and Hetrick$^2$.  It was pointed out that proper treatment of the
Wilson loop variable, the only dynamical degree of freedom in the gauge field,
was necessary to obtain the correct spectrum, that of a free massive scalar
theory.  The initial investigation of the massive Schwinger model$^3$, albeit
on
the line, often used perturbation in the mass.  In Ref. 4, the effects of
adding
a fermionic mass terms were investigated, primarily in the limit of large
fermion mass, or equivalently, weak coupling, ${e\over m}\rightarrow 0$.  We
direct the reader to Ref. 4 for a review and for more detailed references.  It
was found that the structure of the fermionic Hilbert bundle over the space of
gauge field configurations is drastically changed with the addition of a
fermionic mass.  This then indicates that the analyticity of the $m\rightarrow
0$ limit should be questioned.  In this article we examine the delicate nature
of the massless limit in more detail.  We show that it is indeed correct to
perturb in the mass about the twisted Hilbert bundle of the massless case.  The
dynamics induce transitions that cause the Hilbert bundle of the massive theory
to become twisted.  Basically, the fermionic energy levels that pass through an
exactly degenerate point in the massless case, pass through a quasi-degenerate
point in the massive case.  If the driving frequency is much larger than the
gap, the system can behave anti-adiabatically.  Of course if the frequency is
much smaller than the gap, it is the adiabatic theorem which governs the
dynamics, and the fermionic Hilbert bundle remains trivial.  Isler, Schmid and
Truegenburger$^5$ studied in detail the effects of mass on the Klein effect,
i.e., anomalous chiral charge production, due to {\it local} electric fields,
on
the line.  We are essentially considering anomalous chiral charge production
due
to {\it spatially constant} electric fields on a circle, which are not
permitted
on the line. \vskip.5truecm\par\noindent  The plan of this paper is as follows.
First we elaborate the structure of the Hilbert bundle for the massless and the
massive case.  Then we solve the relevant, external field Dirac equation to
demonstrate adiabatic behaviour of the weak coupling limit (large mass), and
the
anti-adiabatic behaviour of the strong coupling limit (small mass).  Finally we
show that the relevant driving frequencies indeed satisfy the appropriate
conditions in the two limits to permit perturbation about the corresponding
Hilbert bundles.   \vskip.5truecm
\centerline{\bf Hilbert Bundles}
\vskip.5truecm\par\noindent
We record here some results from Ref. 4, that we will find necessary.  The
classical Hamiltonian for the massive Schwinger model on a circle, of length
$2\pi L$, in the temporal gauge $A_0=0$, is given by
$$
{\cal H}=\int dx{1\over 2}(\dot v(x,t))^2+\Psi^\dagger
(x,t)\left(-i\gamma_5(\partial_x+iev(x,t))+ m\gamma^0\right)\Psi (x,t).\eqno(1)
$$
The only dynamical degree of freedom is the Wilson loop variable
$$
v={1\over 2\pi L}\int_0^{2\pi L}dxv(x).\eqno(2)
$$
Quantizing in the Schr\"odinger picture and eliminating the non-dynamical gauge
modes
with the Gauss law, yields the quantum Hamiltonian
$$
{\hat H}=-{1\over 4\pi L}{d^2\over dv^2}+{\hat
H}_F^0+evQ_5+Le^2v^2+{-1\over 2\pi L} \sum_{p\in Z\atop p\ne 0}{1\over
2}e^2L^2{j^0(p)j^0(-p)\over p^2}\eqno(3)
$$
where$^{4,6}$ ${\hat H}_F^0$ is the free Hamiltonian
$$
{\hat H}_F^0={1\over L}\sum_{p\in Z}\sqrt{p^2+(mL)^2}(a_p^\dagger
a_p+b_p^\dagger
b_p),\eqno(4)
$$
$Q_5$ is the axial charge and $j^0(p)$ is the charge density, see Ref. 4 or 6
for details.
We can diagonalize the fermionic Hamiltonian for the zero momentum sector
excluding the Coulomb energy term, exactly, which provides the vacuum energy.
Now comes an important divergence between the massive and the massless cases.
The massive case yields $$
{\hat H}_F={\hat H}_F^0+evQ_5+Le^2v^2=\sum_{p\in Z}\sqrt{({p\over
L}+ev)^2+m^2}\left(\bar a_p^\dagger  (v)\bar a_p(v)+\bar b_p^\dagger   (v)\bar
b_p(v)\right)-g(v) \eqno(5)
$$
with
$$
g(v)={-1\over  L}{2mL\over \pi}\sum_{n=1}^\infty{K_1(\pi nmL)\over n}(\cos
(2\pi
neLv)-1)\eqno(6)
$$
which is a smooth, periodic function of $v$, with period ${1\over eL}$.  The
limit
$m\rightarrow 0$ is not smooth.  We get
$$
{{\rm lim}\atop m\rightarrow 0}-g(v)={(eL)^2\over L}\left( v-{1\over
2eL}\right)^2-{1\over 4L} \eqno(7)
$$
on the domain $v\in [0,{1\over eL})$.  We could consider $-g(v)$ as periodic
with
discontinuous derivative at $0$, this is not, however, what we expect.  There
is no
reason to believe that the model is not smooth in $v$.  We must look in more
detail at
the massless case.  The limit $m\rightarrow 0$ of the Hamiltonian (5) is well
defined for each $v$ in the open set $(0,{1\over eL})$.  For $v=0$ or $v=
{1\over eL}$, however, the massless limit is not continuous.
\vskip.5truecm\par\noindent
The Dirac equation in the presence of a constant external gauge field is
trivially
solvable. For
$$
i\partial_t \Psi (x,t)=h(x,t)\Psi (x,t)=\left(-i\gamma_5(\partial_x
+iev)+m\gamma^0\right) \Psi (x,t)\eqno(8)
$$
the solutions are given by$^6$
$$
\eqalign{
\psi_\pm(x,p,v,m,L)&={e^{i{px\over L}}\over \sqrt{2\pi L}}\psi_\pm (p,v,m,L)\cr
&={e^{i{px\over L}}\over\sqrt{\left( 2(\sqrt{({p\over
L}-ev)^2+m^2})(\sqrt{({p\over L}-ev)^2+m^2}\mp ({p\over
L}-ev))\right)}}\times\cr
&\times\pmatrix{m\cr\cr\pm\sqrt {({p\over L}-ev)^2+m^2}-({p\over L}-ev)}}
\eqno(9)
$$
with corresponding energies $E_\pm (p,v,m,L)$,
$$
E_\pm (p,v,m,L)=\pm\sqrt {({p\over L}-ev)^2+m^2}.\eqno(10)
$$
\vskip.5truecm\par\noindent
The massless limit of the first quantized eigenfunctions for all $v\in
[0,{1\over eL})$
yields the correct, chiral eigenstates,
$$
{{\rm lim}\atop m\rightarrow
0}\psi^{\pm}(p,m,v,L)=\pmatrix{\theta(\pm(p+eLv))\cr
\theta(\mp(p+eLv))}=\cases {\pmatrix{1\cr 0}&chirality $+$\cr\pmatrix{0\cr
1}&chirality $-$},\eqno(11)
$$
while for $p+eLv=0$, for example for $v=0$, $p=0$, we have
$$
{{\rm lim}\atop m\rightarrow 0}\psi^{\pm}(p=0,m,v=0,L)={1\over\sqrt 2}
\pmatrix{1\cr\pm 1} \eqno(12)
$$
or for $v={1\over eL}$, $p=-1$, we have
$$
{{\rm lim}\atop m\rightarrow 0}\psi^{\pm}(p=-1,m,v={1\over eL},L)={1\over\sqrt
2}\pmatrix
{1\cr\pm 1}. \eqno(13)
$$
Due to the discontinuous behaviour in $v$, we do not expect the corresponding
annihilation and creation operators of the second quantized theory to be smooth
periodic
functions of $v$.  Indeed, as is well know by now, the massless Schwinger model
exhibits
the phenomena of spectral flow.  As $v$ increases from $0$ to ${1\over eL}$,
one positive
chirality state crosses zero energy from negative to positive, while one
negative
chirality state crosses zero energy from positive to negative.  The resulting
state
contains a chiral fermion--anti-fermion pair.  The limit from the massive
theory
automatically yields the state with this chiral pair removed, since the massive
theory exhibits no spectral flow.  Thus it is not surprising that periodicity
of
the massive theory is lost for the massless case.
\vskip.5truecm\par\noindent
In the massive case we have the unitary transformation
$$
\eqalign{
{\bar a}_p(v)&=V{\bar a}_{p+1}(v-{1\over eL})\cr
{\bar b}_p^\dagger (v)&=V^\dagger{\bar b}_{p+1}^\dagger (v-{1\over
eL})}\eqno(14)
$$
which relates annihilation and creation operators around the circle in $v$.
$V$
is actually independent of $v$ and relates the free operators in the following
way
$$
\eqalign{
a_p&=V^\dagger
\langle\psi^0_+(p)|\left(|\psi^0_+(p+1)\rangle a_{p+1}+|\psi^0_-(p+1)\rangle
b^\dagger_{p+1}\right)V\cr b^\dagger_p&=V^\dagger
\langle\psi^0_-(p)|\left(|\psi^0_+(p+1)\rangle a_{p+1}+|\psi^0_-(p+1)\rangle
b^\dagger_{p+1}\right)V,}\eqno(15)
$$
where $\psi^0_\pm (p)$ are the eigenfunctions at $v=0$.
In the massless limit this yields
$$
\eqalign{
a_p&=V^\dagger a_{p+1}\quad p\ne 0\cr
b^\dagger_p&=V^\dagger b^\dagger_{p+1}\quad p\ne 0}\eqno(16)
$$
but
$$
\eqalign{
a_0&=V^\dagger b^\dagger_1\cr
b^\dagger_0&=V^\dagger a_1.}\eqno(17)
$$
Thus for the massive case, $V$ does not mix annihilation operators with
creation
operators and vice versa, correspondingly, the vacuum is annihilated by the
same
annihilation operators at $v=0$ and at $v={1\over eL}$.  For the massless
case, however, it does mix annihilation and creation operators, exactly for
those
states which cross through zero, therefore the vacuum is not invariant.  Thus
in
the massive case, the vacuum is periodic, up to a phase, while in the massless
case it suffers a non-trivial unitary transformation which has the effect of
removing a chiral pair.  In either case the unitary transformation on the
fermionic Hilbert space is a representation of the topologically non-trivial
gauge transformation
$$
\psi(x,t)\rightarrow e^{i{x\over L}}\psi(x,t)\eqno(18)
$$
for the first quantized levels.  For the massless case we obtain a genuinely
twisted fermionic Hilbert bundle, while in the massive case the bundle is
topologically trivial, there are no topologically non-trivial complex line
bundles over the circle.
\vskip.5truecm\par\noindent
The inclusion of the Coulomb term is unimportant to the Hilbert bundle
structure
it does not involve any gauge field dependence.  The Coulomb term creates
fermion--anti-fermion pairs at zero total momentum which is, of course, of
great
importance for the full solution of the massless theory.
\vskip.5truecm\par\noindent
The discontinuity of the Hilbert bundles throws into
doubt the continuity of the massless limit of the massive Schwinger model, and
hence, perturbation theory in the mass.  Our aim here is to demonstrate that
mass perturbation is in fact without problem.  The kinematics of the massless
theory are reproduced by the dynamics of the massive theory, allowing for
smooth
behaviour as the mass tends to zero.  We will examine this in complete detail
in
the next section at the first quantized level.
\vskip.5truecm
\centerline{\bf First Quantized Theory}
\vskip.5truecm\par\noindent
The solutions (9) of the Dirac equation (8) have energies distributed along
the positive and negative branches of a hyperbola, at the integers, $p$,
shifted
by $eLv\in (0,1)$, see Figure 1.  As $v$ varies from $0$ to ${1\over eL}$,
the energies simply
permute by shifting along the hyperbola, the negative and positive branches
maintaining their identity.   In the massless limit, the hyperbola assumes its
asymptotes, which cross at $p=0$.  Chirality, the eigenvalue of $\gamma_5$, is
a
conserved quantum number in this case and the states now maintain their
identities along the lines $E={p\over L}$ for positive chirality and along
$E=-{p\over L}$ for  negative chirality.  Evidently negative energy
states and positive energy states do not maintain their own identities as $v$
increases from $0$ to ${1\over eL}$.
\centinsert{\pscaption{
\boxit{\psboxscaled{800}{delipics.ps}}}
{{\bf Figure 1}: Spectrum for the massive and massless cases, with $eLv=.2$,
$m=2.5$.}}
\vskip.5truecm\par\noindent
The topologically non-trivial gauge transformation (18) relates the wave
function at $v=0$ and $v={1\over eL}$.  This gauge transformation exactly
permutes the wave functions, for a given integer $p$ with that defined for the
integer $p+1$, along the branches of the hyperbola in the massive case,
while along the asymptotes in the massless case.  The corresponding unitary
transformation at the second quantized level (15) mixes creation and
annihilation
operators in the massless theory, but does not in the massive theory.  Hence
the
massive vacuum is invariant, while the massless vacuum is not.
\vskip.5truecm\par\noindent
Consider now the Dirac equation for a time
dependent, spatially constant external field $v(t)$.  We have
$$ i\partial_t
\Psi (x,t)=\left(-i\gamma_5(\partial_x +iev(t))+m\gamma^0\right) \Psi
(x,t)\eqno(19)
$$
Taking the spatial Fourier transform we see that the system separates into an
infinite set of independent two level systems indexed by the spatial momentum
$p$,
$$
i\partial_t \Psi (p,t)=\left(\gamma_5({p\over L}+ev(t) )+m\gamma^0\right) \Psi
(p,t)
\eqno(20)
$$
The time dependent instantaneous hamiltonian can be diagolnalized as follows
$$
\left(\gamma_5({p\over L}+ev(t) )+m\gamma^0\right)\psi_\pm
(p,v(t),m,L)=\pm\sqrt{({p\over L}+ev(t) )^2+m^2}
\psi_\pm (p,v(t),m,L)\eqno(21)
$$
with $\psi_\pm (p,v(t),m,L)$ given exactly as in (9).  Then expanding $\Psi
(p,t)$ in terms of the instantaneous eigenstates  $\psi_\pm (p,v(t),m,L)$
$$
\Psi (p,t)=a_+\psi_+ (p,v(t),m,L)+a_-\psi_- (p,v(t),m,L)\eqno(22)
$$
gives the two level system
$$
\left(i\partial_t-\left(\sqrt{({p\over L}+ev(t)
)^2+m^2}\sigma^3+{me\over2}{{\dot v(t)}\over ({p\over
L}-ev(t))^2+m^2}\sigma^2\right)\right)\pmatrix {a_+\cr a_-}=0,\eqno(23)
$$
where the $\sigma^i$ are the Pauli matrices.
Remarkably this system of equations can be exactly solved for certain time
dependence
$v(t)$, which we will use below.  The first term, proportional to $\sigma^3$,
is just the
instantaneous energy, while the second term proportional to $\sigma^2$ can
effect
transitions between instantaneous positive and negative energy states.
\vskip.5truecm\par\noindent
We can without loss of generality we consider the case $p=0$ as each two level
system behaves in the same way.  Now we imagine $v$ starting at large negative
values and increasing at a typical rate to large positive values.  Then for
$|ev|>>m$ we have approximately
$$
i\partial_t-{1\over L}\left( |evL|\sigma^3+\left({m\over ev}\right){L{\dot
v}(t)\over 2v}\sigma^2\right)\pmatrix{a_+\cr a_-}\approx  0.\eqno(24)
$$
Thus for ${L{\dot v}(t)\over v}\le o(1)$ the first term dominates and
transitions are negligible, we are in the adiabatic regime for either strong or
weak
coupling.  For $|ev|<<m$, however, we have approximately
$$
i\partial_t-{1\over L}\left( mL\sigma^3+{e\over m}{L{\dot v}(t)\over
2}\sigma^2\right)\pmatrix{a_+\cr a_-}\approx  0.\eqno(25)
$$
Now we obtain the adiabatic limit for ${L{\dot v}(t)\over v}<<{m\over ev}$.
Then in the strong coupling limit, ${m\over e}\rightarrow 0$, this inequality
can
be violated.  Once we obtain the situation ${m\over ev}<<{L{\dot v}(t)\over v}$
as ${m\over e}\rightarrow 0$, the adiabatic theorem does not apply.
Indeed it is the anti-adiabatic theorem of Low$^7$ which applies here, the
system behaves anti-adiabatically, the probability of transition instead of
being
negligible, becomes arbitrarily close to one.
\vskip.5truecm\par\noindent
We can elaborate the situation by taking the following time dependence for
$v(t)$,
$$
v(t)=\cases {-{m\over
e}{\alpha\over\sqrt{1-\alpha^2}}&$t<-{\alpha\over\omega}$\cr
             {m\over e}{\omega t\over\sqrt{1-(\omega
t)^2}}&$-{\alpha\over\omega}\le
              t\le{\alpha\over\omega}$\cr
              {m\over e}{\alpha\over\sqrt{1-\alpha^2}}&${\alpha\over\omega}<t$}
\eqno(26)
$$
with $\alpha$ some fixed dimensionless parameter between $0$ and $1$.  We get
$$
\eqalign{
&i\partial_t\pmatrix{a_+\cr a_-}=\cr
&-\left( {m\theta
(-t-{\alpha\over\omega})\over\sqrt{1-\alpha^2}}\sigma^3+{m\left(\theta
(t+{\alpha\over\omega})-\theta (t-{\alpha\over\omega})\right)\over
\sqrt{1-(\omega
t)^2}}\left(\sigma^3+{\omega\over 2m}\sigma^2\right)+{m\theta
(t-{\alpha\over\omega})\over\sqrt{1-\alpha^2}}\sigma^3\right)}\pmatrix{a_+\cr
a_-}   \eqno(27)
$$
which has the solution
$$
\pmatrix{a_+(t)\cr
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
a_-(t)}=\left(\cases{e^{-i(t+{\alpha\over\omega}){m\over\sqrt{1-\alpha^2}}\sigma^3}
&$t<-{\alpha\over\omega}$\cr
e^{-i{m\over\omega}\left(\sin^{-1}(\omega
t)+\sin^{-1}(\alpha)\right)\left(\sigma^3+{\omega\over 2m}\sigma^2\right)}&
$-{\alpha\over\omega}\le t\le{\alpha\over\omega}$\cr
e^{-i(t+{\alpha\over\omega}){m\over\sqrt{1-\alpha^2}}\sigma^3}
e^{-i{m\over\omega}\left(\sin^{-1}(\alpha)\right)\left(\sigma^3+{\omega\over
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
2m}\sigma^2\right)}&${\alpha\over\omega}<t$}\right)\pmatrix{a_+(-{\alpha\over\omega})\cr
a_-(-{\alpha\over\omega})},\eqno(28)
$$
with initial condition specified at $t=-{\alpha\over\omega}$.  For
$|t|>{\alpha\over\omega}$ there are no transitions, $v(t)$ is constant.  All
transitions occur between $-{\alpha\over\omega}$ and ${\alpha\over\omega}$,
thus
the transiton probability is determined by
$$
\pmatrix{a_+({\alpha\over\omega})\cr
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
a_-({\alpha\over\omega})}=e^{-i{2m\over\omega}\sin^{-1}(\alpha)\left(\sigma^3+{\omega\over
2m}\sigma^2\right)}\pmatrix{a_+(-{\alpha\over\omega})\cr
a_-(-{\alpha\over\omega})}.\eqno(29)
$$
As ${\omega\over m}\rightarrow 0$, we obtain the adiabatic limit,
$$
\pmatrix{a_+({\alpha\over\omega})\cr
a_-({\alpha\over\omega})}\rightarrow
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
e^{-i{2m\over\omega}\sin^{-1}(\alpha)\sigma^3}\pmatrix{a_+(-{\alpha\over\omega})\cr
a_-(-{\alpha\over\omega})}, \eqno(30)
$$
and there are negligible transitions.  As ${\omega\over m}\rightarrow\infty$,
however, we
get
$$
\pmatrix{a_+({\alpha\over\omega})\cr
a_-({\alpha\over\omega})}\rightarrow
e^{-i\sin^{-1}(\alpha)\sigma^2}\pmatrix{a_+(-{\alpha\over\omega})\cr
a_-(-{\alpha\over\omega})}. \eqno(31)
$$
Now if we further consider the massless limit (strong coupling), i.e., ${m\over
e}\rightarrow 0$, we must adjust $\alpha$ so that the change in
$ev$ is much larger than $m$, $e\Delta
v=e\left(v\left({\alpha\over\omega}\right)-
v\left(-{\alpha\over\omega}\right)\right)>>m$.  But
$$
e\Delta v=2{m\over
e}{\alpha\over\sqrt{1-\alpha^2}}>>m\eqno(32)
$$
implies we must take $\alpha\rightarrow 1$ as ${m\over e}\rightarrow 0$.
Hence  $\sin^{-1}(\alpha)\rightarrow{\pi\over 2}$ and
$$
{{\rm lim}\atop{{\omega\over m}\rightarrow\infty\atop{e\over
m}\rightarrow\infty}}
\pmatrix{a_+({\alpha\over\omega})\cr
a_-({\alpha\over\omega})}=e^{-i\sigma^2}\pmatrix{a_+(-{\alpha\over\omega})\cr
a_-(-{\alpha\over\omega})}=\pmatrix{-a_-(-{\alpha\over\omega})\cr
a_+(-{\alpha\over\omega})},\eqno(33)
$$
i.e., the transition probability is one.  The interpretation of this result is
quite straightforward.  The instantaneous fermionic eigenstates are essentially
chiral eigenstates at $t=-{\alpha\over\omega}$.  Then if $t$ increases to
${\alpha\over\omega}$, with $\omega\rightarrow\infty$, the sudden approximation
becomes relevant.  The eigenstates are essentially frozen as they were at
$t=-{\alpha\over\omega}$, but now continue to evolve according to the dynamics
of the Hamiltonian at $t={\alpha\over\omega}$.  Hence chirality is preserved.
A
previously negative energy, chirality plus wavefunction now evolves as a
positive energy, chirality plus wave function while a previously positive
energy, chirality minus wavefunction now evolves as a negative energy,
chirality
minus  wave function.  This is exactly as what occurs for the massless theory.
Therefore even though the Hilbert bundle of the massive fermions over the space
of gauge fields is  topologically trivial, it is appropriate to perturb about
the topologically twisted Hilbert bundle of the massless theory.
\vskip.5truecm
\centerline{\bf Driving Frequencies}
\vskip.5truecm\par\noindent
It remains to verify that the dynamics of the gauge fields do indeed satisfy
the appropriate condition on ${\omega\over m}$ in the strong and weak coupling
limit.  As elaborated in Ref. 4, the problem for the  single, dynamical gauge
degree of freedom in the weak coupling limit, ${e\over m}\rightarrow 0$,
corresponds to the one dimensional quantum mechanics system on a circle of
circumference $1\over eL$,
$$
({-1\over  4\pi L}{d^2\over  dv^2}-g(v))\psi (v)={\cal E}^0\psi (v),\eqno(34)
$$
with $g(v)$ given in (6).  Even this problem is not exactly solvable, however,
we
can extract enough relevant information in approximation.  Truncating the
series
at the first term, we obtain $$
-g(v)\approx -{4mL\over L\pi}K_1(2\pi mL)sin^2(\pi eLv)\eqno(35)
$$
which has is minimum at $v={1\over 2eL}$.  The harmonic approximation here
gives
$$
-g(v)\approx -{4mL\over L\pi}K_1(2\pi mL)\left(1-(\pi eL)^2(v-{1\over
2eL})^2+\cdots\right).\eqno(36)
$$
This harmonic approximation has for its ground state wave function
$$
\psi_0(v)\sim{\cal N}e^{-{2\pi L2e\sqrt{mlK_1(2\pi mL)}\over 2} (v-{1\over
2eL})^2} \eqno(37)
$$
which has a spread $\sigma$ approximately given by
$$
\eqalign{
\sigma &\sim{1\over (2\pi eL\sqrt{mlK_1(2\pi mL)})^{1\over 2}}\cr
&={2\over \pi e LS_0^{1\over 2}}}.\eqno(38)
$$
$S_0$ is the Euclidean action of the classical instanton which mediates
tunneling
around the circle, see Ref. 4 for details.  In the weak coupling limit,
$S_0\rightarrow\infty$.  Thus the ratio of $\sigma$ to the size of the circle,
$$
{\sigma\over\left({1\over eL}\right)}={2\over \pi S_0^{1\over 2}}\rightarrow 0.
\eqno(39)
$$
Therefore the spread of the wave function is negligible compared to the size of
the circle and the harmonic approximation is reasonable.  The relevant
frequency
then is just
$$
\omega = 2e\sqrt{mlK_1(2\pi mL)},\eqno(40)
$$
which gives a reasonable measure of $\dot v$.  Hence
$$
{{\dot v}\over m}\sim 2{e\over m}\sqrt{mlK_1(2\pi mL)}<<1\eqno(41)
$$
for weak coupling and the adiabatic treatment is valid.  The relevant Hilbert
bundle structure is the periodic, topologically trivial structure associated to
the weak coupling limit.
\vskip.5truecm\par\noindent
In the strong coupling limit, i.e., near the massless theory, the harmonic
approximation about the minimum of the potential fails to be reasonable.  The
spread of the ground state wave function surpasses the size of the circle
$$
{\sigma\over\left({1\over eL}\right)}\rightarrow\infty.\eqno(42)
$$
The size of the circle is now the determining constraint to find the typical
value of  $\dot v$.  A free particle on a circle of circumference ${1\over
eL}$, with kinetic term the same as that in (34),  has energy levels
$$
{\cal
E}_n={(2\pi eL)^2\over 4\pi L}n^2.\eqno(43)
$$
The corresponding frequency is
$$
\omega^\prime = {(2\pi eL)^2\over 4\pi L}.\eqno(44)
$$
Thus
$$L{\dot v}={(2\pi eL)^2\over 4\pi}>>{m\over e},\eqno(45)
$$
since  ${m\over e}\rightarrow 0$ in the strong coupling limit.  Then it is
clear
that the anti-adiabatic treatment is relevant and it is incorrect to neglect
the
dynamical twisting of the fermionic Hilbert bundle.  Since the transition
probability is one in the anti-adiabatic limit, we recover the Hilbert bundle
structure of the massless theory.  As shown in Ref. 1, the bosonization process
inherently takes into account the twisted nature of the fermionic Hilbert
bundle.  Hence analyses which involve perturbation theory in the mass are
properly done in the bosonized version, such as in Ref. 3.
\vskip.5truecm\par\noindent
In conclusion, we have shown that it is correct to perturb in the mass, about
the
topologically non-trivial (twisted) fermionic Hilbert bundle structure of the
massless theory, even though the Hilbert bundle structure of the massive
Schwinger model on a circle is topologically trivial (periodic).  The twisted
structure is dynamically maintained for small mass (strong coupling) following
the anti-adiabatic theorem of Low$^7$.
\vskip.5truecm
\centerline{\bf Acknowlegements}
\vskip.5truecm\par\noindent
We thank R. B. MacKenzie for
useful discussions and the McGill High Energy Physics Theory Group, the
preparation of a seminar for which, illicited the preceding enquiry.  This work
supported in part by NSERC of Canada and FCAR of Qu\'ebec.
\vskip.5truecm
\centerline{\bf References}
\vskip.5truecm
\par\noindent
1.  N. S. Manton, Annals of Physics,{\bf 159}, 220, (1985).
\par\noindent
2.  Y. Hosotani and J. Hetrick, Phys. Lett. {\bf B230}, 88, (1989); Phys. Rev.
{\bf D38}, 2621, (1988).
\par\noindent
3.  S. Coleman, R. Jackiw, L. Susskind, Annal of Physics, {\bf 93}, 267,
(1975);S. Coleman, ibid, {\bf 101}, 239, (1976).
4.  M. B. Paranjape and R. Ross, Phys. Rev. {\bf D}, in press.
\par\noindent
\par\noindent
5.  K. Isler, C. A. Truegenberger and C. Schmid, Nucl. Phys.,{\bf B314}, 269,
(1989).
\par\noindent
6.  M. B. Paranjape, Phys. Rev. {\bf D40}, 540, (1989).
\par\noindent
7.  F. Low, Phys. Rev. Lett. {\bf 63}, 2322, (1989).
\autojoin
\end









%Beginning-Of-File-Named:delipics.ps
%!PS-Adobe-2.0 EPSF-2.0
%%BoundingBox: 0 0 504 504
%%Creator: Mathematica
%%EndComments
/Mfontsize 10 def
/Mlmarg .5 72 mul def
/Mrmarg .5 72 mul def
/Mbmarg .5 72 mul def
/Mtmarg .5 72 mul def
/Mwidth 7 72 mul def
/Mheight 7 72 mul def
/Mtransform {  } bind def
/Mnodistort true def
/Mpstart {
MathPictureStart
} bind def
/Mpend {
MathPictureEnd
} bind def
/Mscale {
0 1 0 1
5 -1 roll
MathScale
} bind def
/Plain
/Courier findfont
def
/Bold
/Courier-Bold findfont
def
/Italic
/Courier-Oblique findfont
def
/MathPictureStart {
gsave
Mtransform
Mlmarg
Mbmarg
translate
/Mtmatrix
matrix currentmatrix
def
Plain
Mfontsize scalefont
setfont
} bind def
/MathPictureEnd {
grestore
showpage
} bind def
/Mdot {
moveto
0 0 rlineto
stroke
} bind def
/Mtetra {
moveto
lineto
lineto
lineto
fill
} bind def
/Metetra {
moveto
lineto
lineto
lineto
closepath
gsave
fill
grestore
0 setgray
stroke
} bind def
/Mistroke {
flattenpath
0 0 0
{
4 2 roll
pop pop
}
{
4 -1 roll
2 index
sub dup mul
4 -1 roll
2 index
sub dup mul
add sqrt
4 -1 roll
add
3 1 roll
}
{
stop
}
{
stop
}
pathforall
pop pop
currentpoint
stroke
moveto
currentdash
3 -1 roll
add
setdash
} bind def
/Mfstroke {
stroke
currentdash
pop 0
setdash
} bind def
/Msboxa {
newpath
5 -1 roll
Mvboxa
pop
6 -1 roll
5 -1 roll
4 -1 roll
Msboxa1
5 -3 roll
Msboxa1
[
7 -2 roll
2 copy
[
3 1 roll
10 -1 roll
9 -1 roll
]
6 1 roll
5 -2 roll
]
} bind def
/Msboxa1 {
sub
2 div
dup
2 index
1 add
mul
3 -1 roll
-1 add
3 -1 roll
mul
} bind def
/Mvboxa {
gsave
newpath
[ true
3 -1 roll
{
Mbbox
5 -1 roll
{
0
5 1 roll
}
{
7 -1 roll
exch sub
(m) stringwidth pop
.3 mul
sub
7 1 roll
6 -1 roll
4 -1 roll
Mmin
3 -1 roll
5 index
add
5 -1 roll
4 -1 roll
Mmax
4 -1 roll
}
ifelse
false
}
forall
{ stop } if
counttomark
1 add
4 roll
]
grestore
} bind def
/Mbbox {
0 0 moveto
false charpath
flattenpath
pathbbox
newpath
} bind def
/Mmin {
2 copy
gt
{ exch } if
pop
} bind def
/Mmax {
2 copy
lt
{ exch } if
pop
} bind def
/Mshowa {
4 -2 roll
moveto
2 index
Mtmatrix setmatrix
Mvboxa
7 1 roll
6 -1 roll
5 -1 roll
4 -1 roll
Mshowa1
4 1 roll
Mshowa1
rmoveto
currentpoint
0 1
4 index length
-1 add
{
2 index
4 index
2 index
get
3 index
add
moveto
4 index
exch get
show
} for
pop pop pop pop
Mgmatrix setmatrix
} bind def
/Mshowa1 {
2 copy
add
4 1 roll
sub
mul
sub
-2 div
} bind def
/MathScale {
Mwidth
Mlmarg Mrmarg add
sub
Mheight
Mbmarg Mtmarg add
sub
0 0 moveto
1 index 0
lineto
2 copy
lineto
0 1 index
lineto
clip newpath
Mlp
translate
scale
pop pop pop pop
/Mgmatrix
matrix currentmatrix
def
} bind def
/Mlp {
3 copy
Mlpfirst
{
Mnodistort
{
Mmin
dup
} if
4 index
2 index
2 index
Mlprun
11 index
11 -1 roll
10 -4 roll
Mlp1
8 index
9 -5 roll
Mlp1
4 -1 roll
and
{ exit } if
3 -1 roll
pop pop
} loop
exch
3 1 roll
7 -3 roll
pop pop pop
} bind def
/Mlpfirst {
3 -1 roll
dup length
2 copy
-2 add
get
aload
pop pop pop
4 -2 roll
-1 add
get
aload
pop pop pop
6 -1 roll
3 -1 roll
5 -1 roll
sub
div
4 1 roll
exch sub
div
} bind def
/Mlprun {
2 copy
4 index
0 get
dup
4 1 roll
Mlprun1
3 copy
8 -2 roll
9 -1 roll
{
3 copy
Mlprun1
3 copy
11 -3 roll
/gt Mlpminmax
8 3 roll
11 -3 roll
/lt Mlpminmax
8 3 roll
} forall
pop pop pop pop
3 1 roll
pop pop
aload pop
5 -1 roll
aload pop
exch
6 -1 roll
Mlprun2
8 2 roll
4 -1 roll
Mlprun2
6 2 roll
3 -1 roll
Mlprun2
4 2 roll
exch
Mlprun2
6 2 roll
} bind def
/Mlprun1 {
aload pop
exch
6 -1 roll
5 -1 roll
mul add
4 -2 roll
mul
3 -1 roll
add
} bind def
/Mlprun2 {
2 copy
add 2 div
3 1 roll
exch sub
} bind def
/Mlpminmax {
cvx
2 index
6 index
2 index
exec
{
7 -3 roll
4 -1 roll
} if
1 index
5 index
3 -1 roll
exec
{
4 1 roll
pop
5 -1 roll
aload
pop pop
4 -1 roll
aload pop
[
8 -2 roll
pop
5 -2 roll
pop
6 -2 roll
pop
5 -1 roll
]
4 1 roll
pop
}
{
pop pop pop
} ifelse
} bind def
/Mlp1 {
5 index
3 index sub
5 index
2 index mul
1 index
le
1 index
0 le
or
dup
not
{
1 index
3 index div
.99999 mul
8 -1 roll
pop
7 1 roll
}
if
8 -1 roll
2 div
7 -2 roll
pop sub
5 index
6 -3 roll
pop pop
mul sub
exch
} bind def
%%AspectRatio: 1
MathPictureStart
% Scaling calculations
0.49066 0.09337 0.5 0.08253 [
[(-5)] 0.02381 0.4875 0 1 Msboxa
[(-4)] 0.11718 0.4875 0 1 Msboxa
[(-3)] 0.21055 0.4875 0 1 Msboxa
[(-2)] 0.30392 0.4875 0 1 Msboxa
[(-1)] 0.39729 0.4875 0 1 Msboxa
[(1)] 0.58403 0.4875 0 1 Msboxa
[(2)] 0.6774 0.4875 0 1 Msboxa
[(3)] 0.77077 0.4875 0 1 Msboxa
[(4)] 0.86415 0.4875 0 1 Msboxa
[(5)] 0.95752 0.4875 0 1 Msboxa
[(-6)] 0.47816 0.00481 1 0 Msboxa
[(-4)] 0.47816 0.16987 1 0 Msboxa
[(-2)] 0.47816 0.33494 1 0 Msboxa
[(2)] 0.47816 0.66506 1 0 Msboxa
[(4)] 0.47816 0.83013 1 0 Msboxa
[(6)] 0.47816 0.99519 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 1.001 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
%%Object: Graphics
[ ] 0 setdash
0 setgray
gsave
gsave
0.002 setlinewidth
0 0.5 moveto
1 0.5 lineto
stroke
0.02381 0.49375 moveto
0.02381 0.50625 lineto
stroke
0 setgray
[(-5)] 0.02381 0.4875 0 1 Mshowa
0.11718 0.49375 moveto
0.11718 0.50625 lineto
stroke
0 setgray
[(-4)] 0.11718 0.4875 0 1 Mshowa
0.21055 0.49375 moveto
0.21055 0.50625 lineto
stroke
0 setgray
[(-3)] 0.21055 0.4875 0 1 Mshowa
0.30392 0.49375 moveto
0.30392 0.50625 lineto
stroke
0 setgray
[(-2)] 0.30392 0.4875 0 1 Mshowa
0.39729 0.49375 moveto
0.39729 0.50625 lineto
stroke
0 setgray
[(-1)] 0.39729 0.4875 0 1 Mshowa
0.58403 0.49375 moveto
0.58403 0.50625 lineto
stroke
0 setgray
[(1)] 0.58403 0.4875 0 1 Mshowa
0.6774 0.49375 moveto
0.6774 0.50625 lineto
stroke
0 setgray
[(2)] 0.6774 0.4875 0 1 Mshowa
0.77077 0.49375 moveto
0.77077 0.50625 lineto
stroke
0 setgray
[(3)] 0.77077 0.4875 0 1 Mshowa
0.86415 0.49375 moveto
0.86415 0.50625 lineto
stroke
0 setgray
[(4)] 0.86415 0.4875 0 1 Mshowa
0.95752 0.49375 moveto
0.95752 0.50625 lineto
stroke
0 setgray
[(5)] 0.95752 0.4875 0 1 Mshowa
0.49066 0 moveto
0.49066 1 lineto
stroke
0.48441 0.00481 moveto
0.49691 0.00481 lineto
stroke
0 setgray
[(-6)] 0.47816 0.00481 1 0 Mshowa
0.48441 0.16987 moveto
0.49691 0.16987 lineto
stroke
0 setgray
[(-4)] 0.47816 0.16987 1 0 Mshowa
0.48441 0.33494 moveto
0.49691 0.33494 lineto
stroke
0 setgray
[(-2)] 0.47816 0.33494 1 0 Mshowa
0.48441 0.66506 moveto
0.49691 0.66506 lineto
stroke
0 setgray
[(2)] 0.47816 0.66506 1 0 Mshowa
0.48441 0.83013 moveto
0.49691 0.83013 lineto
stroke
0 setgray
[(4)] 0.47816 0.83013 1 0 Mshowa
0.48441 0.99519 moveto
0.49691 0.99519 lineto
stroke
0 setgray
[(6)] 0.47816 0.99519 1 0 Mshowa
grestore
grestore
0 0 moveto
1 0 lineto
1 1 lineto
0 1 lineto
closepath
clip
newpath
0 setgray
gsave
gsave
gsave
gsave
0.004 setlinewidth
0.02381 0.96137 moveto
0.06349 0.93028 lineto
0.10317 0.89986 lineto
0.14286 0.87025 lineto
0.18254 0.84169 lineto
0.22222 0.81444 lineto
0.2619 0.78889 lineto
0.30159 0.76553 lineto
0.34127 0.74497 lineto
0.38095 0.72798 lineto
0.40079 0.72109 lineto
0.42063 0.71542 lineto
0.44048 0.71105 lineto
0.4504 0.70938 lineto
0.46032 0.70807 lineto
0.47024 0.70712 lineto
0.4752 0.70678 lineto
0.47768 0.70665 lineto
0.48016 0.70654 lineto
0.48264 0.70645 lineto
0.48388 0.70642 lineto
0.48512 0.70639 lineto
0.48636 0.70637 lineto
0.4876 0.70635 lineto
0.48884 0.70634 lineto
0.49008 0.70633 lineto
0.49132 0.70633 lineto
0.49256 0.70634 lineto
0.4938 0.70635 lineto
0.49504 0.70637 lineto
0.49628 0.70639 lineto
0.49752 0.70642 lineto
0.5 0.7065 lineto
0.50248 0.70659 lineto
0.50496 0.70672 lineto
0.50992 0.70703 lineto
0.51488 0.70744 lineto
0.51984 0.70794 lineto
0.52976 0.70921 lineto
0.53968 0.71083 lineto
0.55952 0.71512 lineto
0.57937 0.72073 lineto
0.61905 0.73548 lineto
0.65873 0.75425 lineto
0.69841 0.77621 lineto
0.7381 0.80068 lineto
0.77778 0.82708 lineto
0.81746 0.85498 lineto
0.85714 0.88407 lineto
0.89683 0.91408 lineto
Mistroke
0.93651 0.94484 lineto
0.97619 0.97619 lineto
Mfstroke
grestore
grestore
grestore
gsave
gsave
gsave
0.004 setlinewidth
0.02381 0.03863 moveto
0.06349 0.06972 lineto
0.10317 0.10014 lineto
0.14286 0.12975 lineto
0.18254 0.15831 lineto
0.22222 0.18556 lineto
0.2619 0.21111 lineto
0.30159 0.23447 lineto
0.34127 0.25503 lineto
0.38095 0.27202 lineto
0.40079 0.27891 lineto
0.42063 0.28458 lineto
0.44048 0.28895 lineto
0.4504 0.29062 lineto
0.46032 0.29193 lineto
0.47024 0.29288 lineto
0.4752 0.29322 lineto
0.47768 0.29335 lineto
0.48016 0.29346 lineto
0.48264 0.29355 lineto
0.48388 0.29358 lineto
0.48512 0.29361 lineto
0.48636 0.29363 lineto
0.4876 0.29365 lineto
0.48884 0.29366 lineto
0.49008 0.29367 lineto
0.49132 0.29367 lineto
0.49256 0.29366 lineto
0.4938 0.29365 lineto
0.49504 0.29363 lineto
0.49628 0.29361 lineto
0.49752 0.29358 lineto
0.5 0.2935 lineto
0.50248 0.29341 lineto
0.50496 0.29328 lineto
0.50992 0.29297 lineto
0.51488 0.29256 lineto
0.51984 0.29206 lineto
0.52976 0.29079 lineto
0.53968 0.28917 lineto
0.55952 0.28488 lineto
0.57937 0.27927 lineto
0.61905 0.26452 lineto
0.65873 0.24575 lineto
0.69841 0.22379 lineto
0.7381 0.19932 lineto
0.77778 0.17292 lineto
0.81746 0.14502 lineto
0.85714 0.11593 lineto
0.89683 0.08592 lineto
Mistroke
0.93651 0.05516 lineto
0.97619 0.02381 lineto
Mfstroke
grestore
grestore
grestore
gsave
gsave
gsave
0.004 setlinewidth
0.02381 0.08734 moveto
0.06349 0.12241 lineto
0.10317 0.15749 lineto
0.14286 0.19257 lineto
0.18254 0.22764 lineto
0.22222 0.26272 lineto
0.2619 0.2978 lineto
0.30159 0.33287 lineto
0.34127 0.36795 lineto
0.38095 0.40302 lineto
0.42063 0.4381 lineto
0.46032 0.47318 lineto
0.5 0.50825 lineto
0.53968 0.54333 lineto
0.57937 0.57841 lineto
0.61905 0.61348 lineto
0.65873 0.64856 lineto
0.69841 0.68363 lineto
0.7381 0.71871 lineto
0.77778 0.75379 lineto
0.81746 0.78886 lineto
0.85714 0.82394 lineto
0.89683 0.85902 lineto
0.93651 0.89409 lineto
0.97619 0.92917 lineto
stroke
grestore
grestore
grestore
gsave
gsave
gsave
0.004 setlinewidth
0.02381 0.91266 moveto
0.06349 0.87759 lineto
0.10317 0.84251 lineto
0.14286 0.80743 lineto
0.18254 0.77236 lineto
0.22222 0.73728 lineto
0.2619 0.7022 lineto
0.30159 0.66713 lineto
0.34127 0.63205 lineto
0.38095 0.59698 lineto
0.42063 0.5619 lineto
0.46032 0.52682 lineto
0.5 0.49175 lineto
0.53968 0.45667 lineto
0.57937 0.42159 lineto
0.61905 0.38652 lineto
0.65873 0.35144 lineto
0.69841 0.31637 lineto
0.7381 0.28129 lineto
0.77778 0.24621 lineto
0.81746 0.21114 lineto
0.85714 0.17606 lineto
0.89683 0.14098 lineto
0.93651 0.10591 lineto
0.97619 0.07083 lineto
stroke
grestore
grestore
grestore
gsave
gsave
gsave
0.02 setlinewidth
0.04248 0.94667 Mdot
0.13585 0.87541 Mdot
0.22923 0.8098 Mdot
0.3226 0.75425 Mdot
0.41597 0.71664 Mdot
0.50934 0.70699 Mdot
0.60271 0.72887 Mdot
0.69608 0.77485 Mdot
0.78945 0.83515 Mdot
0.88282 0.9034 Mdot
0.97619 0.97619 Mdot
grestore
grestore
grestore
gsave
gsave
gsave
0.02 setlinewidth
0.04248 0.05333 Mdot
0.13585 0.12459 Mdot
0.22923 0.1902 Mdot
0.3226 0.24575 Mdot
0.41597 0.28336 Mdot
0.50934 0.29301 Mdot
0.60271 0.27113 Mdot
0.69608 0.22515 Mdot
0.78945 0.16485 Mdot
0.88282 0.0966 Mdot
0.97619 0.02381 Mdot
grestore
grestore
grestore
gsave
gsave
gsave
0.02 setlinewidth
0.04248 0.10385 Mdot
0.13585 0.18638 Mdot
0.22923 0.26891 Mdot
0.3226 0.35144 Mdot
0.41597 0.43397 Mdot
0.50934 0.51651 Mdot
0.60271 0.59904 Mdot
0.69608 0.68157 Mdot
0.78945 0.7641 Mdot
0.88282 0.84664 Mdot
0.97619 0.92917 Mdot
grestore
grestore
grestore
gsave
gsave
gsave
0.02 setlinewidth
0.04248 0.89615 Mdot
0.13585 0.81362 Mdot
0.22923 0.73109 Mdot
0.3226 0.64856 Mdot
0.41597 0.56603 Mdot
0.50934 0.48349 Mdot
0.60271 0.40096 Mdot
0.69608 0.31843 Mdot
0.78945 0.2359 Mdot
0.88282 0.15336 Mdot
0.97619 0.07083 Mdot
grestore
grestore
grestore
gsave
gsave
gsave
0.004 setlinewidth
0.30392 0.79725 moveto
0.31948 0.78888 lineto
0.33505 0.78099 lineto
0.35061 0.77363 lineto
0.36617 0.76685 lineto
0.38173 0.7607 lineto
0.39729 0.75524 lineto
0.41285 0.7505 lineto
0.42842 0.74655 lineto
0.44398 0.74343 lineto
0.45176 0.74219 lineto
0.45954 0.74117 lineto
0.46732 0.74037 lineto
0.47121 0.74006 lineto
0.4751 0.7398 lineto
0.47899 0.7396 lineto
0.48094 0.73952 lineto
0.48288 0.73946 lineto
0.48483 0.73941 lineto
0.4858 0.73939 lineto
0.48677 0.73937 lineto
0.48775 0.73936 lineto
0.48823 0.73935 lineto
0.48872 0.73935 lineto
0.4892 0.73935 lineto
0.48969 0.73935 lineto
0.49018 0.73934 lineto
0.49066 0.73934 lineto
0.49115 0.73934 lineto
0.49164 0.73935 lineto
0.49212 0.73935 lineto
0.49261 0.73935 lineto
0.49358 0.73936 lineto
0.49455 0.73937 lineto
0.49553 0.73939 lineto
0.4965 0.73941 lineto
0.49844 0.73946 lineto
0.50039 0.73952 lineto
0.50233 0.7396 lineto
0.50622 0.7398 lineto
0.51012 0.74006 lineto
0.51401 0.74037 lineto
0.52179 0.74117 lineto
0.53735 0.74343 lineto
0.55291 0.74655 lineto
0.56847 0.7505 lineto
0.58403 0.75524 lineto
0.5996 0.7607 lineto
0.61516 0.76685 lineto
0.63072 0.77363 lineto
Mistroke
0.64628 0.78099 lineto
0.66184 0.78888 lineto
0.6774 0.79725 lineto
Mfstroke
grestore
grestore
grestore
gsave
0.004 setlinewidth
0.66807 0.80042 moveto
0.6774 0.79725 lineto
0.66807 0.78392 lineto
stroke
grestore
gsave
gsave
gsave
0.004 setlinewidth
0.30392 0.20275 moveto
0.31948 0.21112 lineto
0.33505 0.21901 lineto
0.35061 0.22637 lineto
0.36617 0.23315 lineto
0.38173 0.2393 lineto
0.39729 0.24476 lineto
0.41285 0.2495 lineto
0.42842 0.25345 lineto
0.44398 0.25657 lineto
0.45176 0.25781 lineto
0.45954 0.25883 lineto
0.46732 0.25963 lineto
0.47121 0.25994 lineto
0.4751 0.2602 lineto
0.47899 0.2604 lineto
0.48094 0.26048 lineto
0.48288 0.26054 lineto
0.48483 0.26059 lineto
0.4858 0.26061 lineto
0.48677 0.26063 lineto
0.48775 0.26064 lineto
0.48823 0.26065 lineto
0.48872 0.26065 lineto
0.4892 0.26065 lineto
0.48969 0.26065 lineto
0.49018 0.26066 lineto
0.49066 0.26066 lineto
0.49115 0.26066 lineto
0.49164 0.26065 lineto
0.49212 0.26065 lineto
0.49261 0.26065 lineto
0.49358 0.26064 lineto
0.49455 0.26063 lineto
0.49553 0.26061 lineto
0.4965 0.26059 lineto
0.49844 0.26054 lineto
0.50039 0.26048 lineto
0.50233 0.2604 lineto
0.50622 0.2602 lineto
0.51012 0.25994 lineto
0.51401 0.25963 lineto
0.52179 0.25883 lineto
0.53735 0.25657 lineto
0.55291 0.25345 lineto
0.56847 0.2495 lineto
0.58403 0.24476 lineto
0.5996 0.2393 lineto
0.61516 0.23315 lineto
0.63072 0.22637 lineto
Mistroke
0.64628 0.21901 lineto
0.66184 0.21112 lineto
0.6774 0.20275 lineto
Mfstroke
grestore
grestore
grestore
gsave
0.004 setlinewidth
0.66807 0.19958 moveto
0.6774 0.20275 lineto
0.66807 0.21608 lineto
stroke
grestore
gsave
gsave
gsave
0.004 setlinewidth
0.30392 0.36795 moveto
0.31948 0.3817 lineto
0.33505 0.39546 lineto
0.35061 0.40921 lineto
0.36617 0.42297 lineto
0.38173 0.43673 lineto
0.39729 0.45048 lineto
0.41285 0.46424 lineto
0.42842 0.47799 lineto
0.44398 0.49175 lineto
0.45954 0.5055 lineto
0.4751 0.51926 lineto
0.49066 0.53301 lineto
0.50622 0.54677 lineto
0.52179 0.56052 lineto
0.53735 0.57428 lineto
0.55291 0.58803 lineto
0.56847 0.60179 lineto
0.58403 0.61555 lineto
0.5996 0.6293 lineto
0.61516 0.64306 lineto
0.63072 0.65681 lineto
0.64628 0.67057 lineto
0.66184 0.68432 lineto
0.6774 0.69808 lineto
stroke
grestore
grestore
grestore
gsave
0.004 setlinewidth
0.66807 0.69808 moveto
0.6774 0.69808 lineto
0.66807 0.68157 lineto
stroke
grestore
gsave
gsave
gsave
0.004 setlinewidth
0.30392 0.63205 moveto
0.31948 0.6183 lineto
0.33505 0.60454 lineto
0.35061 0.59079 lineto
0.36617 0.57703 lineto
0.38173 0.56327 lineto
0.39729 0.54952 lineto
0.41285 0.53576 lineto
0.42842 0.52201 lineto
0.44398 0.50825 lineto
0.45954 0.4945 lineto
0.4751 0.48074 lineto
0.49066 0.46699 lineto
0.50622 0.45323 lineto
0.52179 0.43948 lineto
0.53735 0.42572 lineto
0.55291 0.41197 lineto
0.56847 0.39821 lineto
0.58403 0.38445 lineto
0.5996 0.3707 lineto
0.61516 0.35694 lineto
0.63072 0.34319 lineto
0.64628 0.32943 lineto
0.66184 0.31568 lineto
0.6774 0.30192 lineto
stroke
grestore
grestore
grestore
gsave
0.004 setlinewidth
0.66807 0.30192 moveto
0.6774 0.30192 lineto
0.66807 0.31843 lineto
stroke
grestore
gsave
0 setgray
[(positive,massive)] 0.72409 0.8879 0 0 Mshowa
grestore
gsave
0 setgray
[(negative,massive)] 0.72409 0.1121 0 0 Mshowa
grestore
gsave
0 setgray
[(positive,massless)] 0.72409 0.58253 0 0 Mshowa
grestore
gsave
0 setgray
[(negative,massless)] 0.72409 0.41747 0 0 Mshowa
grestore
grestore
% End of Graphics
MathPictureEnd

