Classical and Quantum Chaos in Fundamental Field Theories
wilmington.tex
\documentclass[12pt,psfig]{dcds2}

%\input{psfig}
%----------------------------------------------
\def\fileversion{v1.20a}% was \def\fileversion{v1.20}%
\def\filedate{21.6.94}%  was \def\filedate{26.1.94}%
%%%
%%% from a suggestion by eijkhout@csrd.uiuc.edu to allow
%%% loading as a style file:
\edef\epsfigRestoreAt{\catcode`@=\number\catcode`@\relax}%
\catcode`\@=11\relax
\ifx\undefined\@makeother                % -pks-
\def\@makeother#1{\catcode`#1=12\relax}  % -pks-
\fi                                      % -pks-
\immediate\write16{Document style option `epsfig', \fileversion\space
<\filedate> (edited by SPQR + pks)}% was <\filedate> (edited by SPQR)}%
%%%--------------------------------------------------------------------
%%% psfig merged with EPSF = epsfig Release 1
%%% 1.0 November 91
%%% 1.1 Dec 91  fixed some bugs
%%% 1.2 Jan 92  fixed bug that it never stopped reading PS file until
%%%             it reached the end! merged in changes from psfig1.8
%%%             (plain TeX definition of \fbox, new code for rotation)
%%% 1.3 Jan 92  separated out \special code so that dvitops can be
%%%             supported. use \psfigdriver{dvitops}%
%%% 1.4 Jan 92  fixed dvitops. need to input psfig at different point
%%%      in dvips from dvitops. define \psfig@start and \psfig@end to cope
%%%      with this
%%% 1.5 Jan 92  force use of psfig method if using dvitops
%%% 1.6 May 92 fixed bugs
%%% 1.7 Sep 92 added support for emtex, sort of. reads a .bb file
%%%            and imports .pcx file. Boolean flag \epsfigdos used
%%%             to indicate that we cannot have more than one . in the
%%%             suffix (ie we will look at file.bb not file.eps.bb)
%%%
%%% 1.8 Sep 92 added support for OzTeX; fixed bugs in dvips support
%%% (including clipping to bounding box); removed use of dimen100.
%%%
%%% 1.9 Dec 92 cleaned up documentation again
%%% 1.10 Mar 93 cleaned up for plain TeX
%%% 1.11 Mar 93 merged in changes by Peter Galko (supplied by CERN) for
%%%             use with Textures
%%% 1.12 Apr 93 actually *tested* the Textures additions, and so rewrote
%%%            from scratch (borrowed an idea from psbox)
%%% 1.13 May 93 allow for file names with _ or $ in them in \psdraft, and
%%%             fix problem that occurred with missing BoundingBox
%%% 1.14 May 93 allow for .z suffix
%%% 1.15 July 93 allow for .gz suffix
%%%              use \leavevmode as appropriate
%%%              fix plain \TeX mode 
%%%             (fixes from Peter-Klaus Schilling <R02SCH@DSYIBM.DESY.DE>)
%%% 1.16 August 93 [peta@mcrcim.mcgill.edu (Peter Whaite)]
%%%        -- Implement Perl scripts to do compressed file inclusion and bb
%%%           generation. 
%%%        -- Add \pscompress.
%%%        -- Tidy up comments.
%%% 1.17 September 93 SPQR
%%%     better code for literal name printing
%%%     allowed for spaces around , and =
%%% 1.18 17.9.93
%%%   updates to epsbb and epsfig from Peter W
%%% 1.19 14.11.93
%%%   mistaken placement of calculation of EPS@Height resulted in errors
%%% 1.20 defaults to no compression
%--------------------------------------------------------
%%%
%%% code rearranged by Sebastian Rahtz (spqr@uk.ac.york.minster)
%%% from work by
%%% Trevor Darrell; Tom Rokicki; Daniel Smith; H. Payne, Robert Russell
%%%
%%% This is a version of Darrell's `psfig' macros which are designed
%%% to be used with dvips or dvitops, and the `epsf' macros (these are
%%% automaticaly included by the style file). I did this because the
%%% \special commands used by `epsf' appear to be more robust and
%%% portable than the raw \specials used by `psfig', but I preferred
%%% the psfig interface. This system preserves the psfig interface,
%%% but mainly uses epsf macros to carry out the job. The user
%%% interface is identical to \psfig, but you just use \epsfig
%%% instead. Rotation uses the original psfig macros: I could not work
%%% out how to do rotation with the epsf macros.
%%%
%%% The same applies to `rheight' and `rwidth'
%%%
%%% If you *really* want old psfig behaviour regardless, use \psfig. It is
%%% still there. Some of the features have gone, however. \figurepath
%%% seemed redundant, since it could be controlled by the TEXINPUTS
%%% variable anyway.
%%%
%%% As an added feature, if FILE cannot be found, a file called FILE.bb
%%% is searched for. This should simply contain a %%BoundingBox line. A perl
%%% script "epsbb" is provided to generate these automatically.  It is
%%% assumed that if this exists, then the actual postscript is in UNIX
%%% compress format in FILE.Z or in GNU gzip format in FILE.gz or FILE.z.
%%%
%%% The macros then read the BB from FILE.bb, but insert "`epsfig FILE`" in
%%% the  output for dvips to interpret (don't worry, it knows about this).
%%% "Epsfig" is another perl script which  searches the TEXINPUTS path for
%%% FILE or FILE.{gz,z,Z}.
%%%
%%% As an added bonus if you say \pscompress the -c switch will be passed to
%%% epsfig, causing automatic FILE.bb generation and compression the first
%%% time dvips is run.  Files in the system directories will NOT be
%%% compressed.
%%% If you are *not* on a Unix system and use this, deliberately or by
%%% mischance, you will have problems!
%%%
%%% This file can be included in a dumped format, or used as a LaTeX
%%% style file, or used in plain TeX. It does not load itself twice (I hope)
%%%
%--------------------------------------------------------------------
%--------------------------------------------------------------------
%%% previous notes. this copy of psfig.tex inherited from versions 1.8
%--------------------------------------------------------------------
%%% All psfig/tex software, documentation, and related files
%%% in this distribution of psfig/tex are
%%% Copyright 1987, 1988, 1991 Trevor J. Darrell
%
%%% Permission is granted for use and non-profit distribution of psfig/tex
%%% providing that this notice be clearly maintained. The right to
%%% distribute any portion of psfig/tex for profit or as part of any commercial
%%% product is specifically reserved for the author(s) of that portion.
%%%
%%% Thanks to Greg Hager (GDH) and Ned Batchelder for their contributions
%%% to this project.
%%%--------------------------------------------------------------------------
%%% GDH 7/26/87 -- changed so that it first looks in the local directory,
%%% then in a specified global directory for the ps file.
%%% SPQR 12/91 removed that last change. it follows TEXINPUTS anyway
%%% RPR 6/25/91 -- changed so that it defaults to user-supplied name if
%%% boundingbox info is specified, assuming graphic will be created by
%%% print time.
%%% SPQR 12/91 removed that last change. use LaTeX error message instead
%%% SPQR 12/91 -- `no file found' produces LaTeX error message.
%%%                If you recover from that the file name is set in draft
%%%                mode in place of the file.
%%%
%%% Modified by J. Daniel Smith on 9 October 1990 to accept the
%%% %%BoundingBox: comment with or without a space after the colon.  Stole
%%% file reading code from Tom Rokicki's EPSF.TEX file (see below).
%%% --- SPQR 12.91 removed most of this to use epsf directly
%%%
%%% More modifications by J. Daniel Smith on 29 March 1991 to allow the
%%% the included PostScript figure to be rotated.  The amount of
%%% rotation is specified by the "angle=" parameter of the \epsfig command.
%%%
%%% Modified by Robert Russell on June 25, 1991 to allow users to specify
%%% .ps filenames which don't yet exist, provided they explicitly provide
%%% boundingbox information via the \epsfig command. Note: This will only work
%%% if the "file=" parameter follows all four "bb???=" parameters in the
%%% command. This is due to the order in which psfig interprets these params.
%%% ---- SPQR 12.91 removed all this
%%%
%%% 3 Jul 1991  JDS     check if file already read in once
%%%
%-----------------------------USAGE---------------------------------------
% From: gvr@cs.brown.edu (George V. Reilly)
%
% \psdraft      draws an outline box, but doesn't include the figure
%               in the DVI file.  Useful for previewing.
%
% \psfull       includes the figure in the DVI file (default).
%
% \psscalefirst width= or height= specifies the size of the figure
%               before rotation.
% \psrotatefirst (default) width= or height= specifies the size of the
%                figure after rotation.  Asymetric figures will
%                appear to shrink.
%
% usage: \epsfig{file=, figure=, height=, width=,
%                       bbllx=, bblly=, bburx=, bbury=,
%                       rheight=, rwidth=, clip=, angle=, silent=}%
%
%       "file" is the filename.  If no path name is specified and the
%               file is not found in the current directory,
%               it will be looked for in directory \psfigurepath.
%       "figure" is a synonym for "file".
%       By default, the width and height of the figure are taken from
%               the BoundingBox of the figure.
%       If "width" is specified, the figure is scaled so that it has
%               the specified width.  Its height changes proportionately.
%       If "height" is specified, the figure is scaled so that it has
%               the specified height.  Its width changes proportionately.
%       If both "width" and "height" are specified, the figure is scaled
%               anamorphically.
%       "bbllx", "bblly", "bburx", and "bbury" control the PostScript
%               BoundingBox.
%       "rheight" and "rwidth" are the reserved height and width
%               of the figure, i.e., how big TeX actually thinks
%               the figure is.  They default to "width" and "height".
%       The "clip" option ensures that no portion of the figure will
%               appear outside its BoundingBox.  "clip=" is a switch and
%               takes no value, but the `=' must be present.
%       The "angle" option specifies the angle of rotation (degrees, ccw).
%       The "silent" option makes \psfig work silently.
%
%--------------------------------------------------------------------------
\newcount\EPS@Height \newcount\EPS@Width \newcount\EPS@xscale
\newcount\EPS@yscale
\def\psfigdriver#1{%
  \bgroup\edef\next{\def\noexpand\tempa{#1}}%
    \uppercase\expandafter{\next}%
    \def\LN{DVITOLN03}%
    \def\DVItoPS{DVITOPS}%
    \def\DVIPS{DVIPS}%
    \def\emTeX{EMTEX}%
    \def\OzTeX{OZTEX}%
    \def\Textures{TEXTURES}%
    \global\chardef\fig@driver=0
    \ifx\tempa\LN
        \global\chardef\fig@driver=0\fi
    \ifx\tempa\DVItoPS
        \global\chardef\fig@driver=1\fi
    \ifx\tempa\DVIPS
        \global\chardef\fig@driver=2\fi
    \ifx\tempa\emTeX
        \global\chardef\fig@driver=3\fi
    \ifx\tempa\OzTeX
        \global\chardef\fig@driver=4\fi
    \ifx\tempa\Textures
        \global\chardef\fig@driver=5\fi
  \egroup
\def\psfig@start{}%
\def\psfig@end{}%
\def\epsfig@gofer{}%
\ifcase\fig@driver
% case 0 ln03
\typeout{WARNING! ****
 no specials for LN03 psfig}%
%
\or % case 1: dvitops
\def\psfig@start{}%
\def\psfig@end{\special{dvitops: import \@p@sfilefinal \space
\@p@swidth sp \space \@p@sheight sp \space fill}%
\if@clip \typeout{Clipping not supported}\fi
\if@angle \typeout{Rotating not supported}\fi
}%
\let\epsfig@gofer\psfig@end
%
\or %case2 dvips
\def\psfig@start{\special{ps::[begin]  \@p@swidth \space \@p@sheight \space%
        \@p@sbbllx \space \@p@sbblly \space%
        \@p@sbburx \space \@p@sbbury \space%
        startTexFig \space }%
        \if@clip
                \if@verbose
                        \typeout{(clipped to BB) }%
                \fi
                \special{ps:: doclip \space }%
        \fi
        \if@angle              % moved after \if@clip ... \fi -pks-
                \special {ps:: \@p@sangle \space rotate \space}
        \fi
        \special{ps: plotfile \@p@sfilefinal \space }%
        \special{ps::[end] endTexFig \space }%
}%
\def\psfig@end{}%
\def\epsfig@gofer{\if@clip
                        \if@verbose
                           \typeout{(clipped to BB)}%
                        \fi
                        \epsfclipon
                  \fi
                  \epsfsetgraph{\@p@sfilefinal}%
}%
%
\or % case 3, emTeX
\typeout{WARNING. You must have a .bb info file with the Bounding Box
  of the pcx file}%
\def\psfig@start{}%
% when emtex imports a pcx file, it is *below* where we are now.
% so raise the box
\def\psfig@end{\typeout{pcx import of \@p@sfilefinal}%
\if@clip \typeout{Clipping not supported}\fi
\if@angle \typeout{Rotating not supported}\fi
\raisebox{\@p@srheight sp}{\special{em: graph \@p@sfilefinal}}}%
\def\epsfig@gofer{}%
%
\or % case 4, OzTeX
\def\psfig@start{}%
\def\psfig@end{%
\EPS@Width\@p@swidth
\EPS@Height\@p@sheight
\divide\EPS@Width by 65781  % convert sp to bp
\divide\EPS@Height by 65781
\special{epsf=\@p@sfilefinal
\space
width=\the\EPS@Width
\space
height=\the\EPS@Height
}%
\if@clip \typeout{Clipping not supported}\fi
\if@angle \typeout{Rotating not supported}\fi
}%
\let\epsfig@gofer\psfig@end
\or % case 5, Textures
\def\psfig@end{
% what is scaling of picture
         \EPS@Width=\@bbw  
         \divide\EPS@Width by 1000
         \EPS@xscale=\@p@swidth \divide \EPS@xscale by \EPS@Width
         \EPS@Height=\@bbh  
         \divide\EPS@Height by 1000
         \EPS@yscale=\@p@sheight \divide \EPS@yscale by\EPS@Height
% take largest calculated scale
  \ifnum\EPS@xscale>\EPS@yscale\EPS@xscale=\EPS@yscale\fi
\if@clip
   \if@verbose
      \typeout{(clipped to BB)}%
   \fi
   \epsfclipon
\fi
\special{illustration \@p@sfilefinal\space scaled \the\EPS@xscale}%
}%
\def\psfig@start{}%
\let\epsfig\psfig
\else
\typeout{WARNING. *** unknown  driver - no psfig}%
\fi
}%
% copies of bits of LaTeX we need....
\newdimen\ps@dimcent
%
% \fbox is defined in latex.tex; so if \fbox is undefined, assume that
% we are not in LaTeX.
% Perhaps this could be done better???
\ifx\undefined\fbox
% \fbox code from modified slightly from LaTeX
\newdimen\fboxrule
\newdimen\fboxsep
\newdimen\ps@tempdima
\newbox\ps@tempboxa
\fboxsep = 0pt
\fboxrule = .4pt
\long\def\fbox#1{\leavevmode\setbox\ps@tempboxa\hbox{#1}\ps@tempdima\fboxrule
    \advance\ps@tempdima \fboxsep \advance\ps@tempdima \dp\ps@tempboxa
   \hbox{\lower \ps@tempdima\hbox
  {\vbox{\hrule height \fboxrule
          \hbox{\vrule width \fboxrule \hskip\fboxsep
          \vbox{\vskip\fboxsep \box\ps@tempboxa\vskip\fboxsep}\hskip
                 \fboxsep\vrule width \fboxrule}%
                 \hrule height \fboxrule}}}}%
\fi
\ifx\@ifundefined\undefined
\long\def\@ifundefined#1#2#3{\expandafter\ifx\csname
  #1\endcsname\relax#2\else#3\fi}%
\fi
%
\@ifundefined{typeout}%
{\gdef\typeout#1{\immediate\write\sixt@@n{#1}}}%
{\relax}%
%
%%% check to see if macros already loaded in (maybe some other file says
%%% "\input epsfig") ...
\@ifundefined{epsfig}{}{\typeout{EPSFIG --- already loaded}\endinput}%
%
%%%
%%% we need Rokicki's EPSF macros anyway, unless they are already loaded
%
\@ifundefined{epsfbox}{\input epsf}{}%
%
%% SPQR 12.91 handling of errors using standard LaTeX error
%% mechanism. In case we are plain TeX we first define the
%% error routines...
\ifx\undefined\@latexerr
        \newlinechar`\^^J
        \def\@spaces{\space\space\space\space}%
        \def\@latexerr#1#2{%
        \edef\@tempc{#2}\expandafter\errhelp\expandafter{\@tempc}%
        \typeout{Error. \space see a manual for explanation.^^J
         \space\@spaces\@spaces\@spaces Type \space H <return> \space for
         immediate help.}\errmessage{#1}}%
\fi
%------------------------
%% a couple of LaTeX error messages
\def\@whattodo{You tried to include a PostScript figure which
cannot be found^^JIf you press return to carry on anyway,^^J
The failed name will be printed in place of the figure.^^J
or type X to quit}%
\def\@whattodobb{You tried to include a PostScript figure which
has no^^Jbounding box, and you supplied none.^^J
If you press return to carry on anyway,^^J
The failed name will be printed in place of the figure.^^J
or type X to quit}%
%------------------------
%
%------------------------------------------------------------------------
%------------------------------------------------------------------------
%%% @psdo control structure -- similar to Latex @for.
%%% I redefined these with different names so that psfig can
%%% be used with TeX as well as LaTeX, and so that it will not
%%% be vunerable to future changes in LaTeX's internal
%%% control structure,
%
\def\@nnil{\@nil}%
\def\@empty{}%
\def\@psdonoop#1\@@#2#3{}%
\def\@psdo#1:=#2\do#3{\edef\@psdotmp{#2}\ifx\@psdotmp\@empty \else
    \expandafter\@psdoloop#2,\@nil,\@nil\@@#1{#3}\fi}%
\def\@psdoloop#1,#2,#3\@@#4#5{\def#4{#1}\ifx #4\@nnil \else
       #5\def#4{#2}\ifx #4\@nnil \else#5\@ipsdoloop #3\@@#4{#5}\fi\fi}%
\def\@ipsdoloop#1,#2\@@#3#4{\def#3{#1}\ifx #3\@nnil
       \let\@nextwhile=\@psdonoop \else
      #4\relax\let\@nextwhile=\@ipsdoloop\fi\@nextwhile#2\@@#3{#4}}%
\def\@tpsdo#1:=#2\do#3{\xdef\@psdotmp{#2}\ifx\@psdotmp\@empty \else
    \@tpsdoloop#2\@nil\@nil\@@#1{#3}\fi}%
\def\@tpsdoloop#1#2\@@#3#4{\def#3{#1}\ifx #3\@nnil
       \let\@nextwhile=\@psdonoop \else
      #4\relax\let\@nextwhile=\@tpsdoloop\fi\@nextwhile#2\@@#3{#4}}%
%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% file reading stuff from epsf.tex
%%%   EPSF.TEX macro file:
%%%   Written by Tomas Rokicki of Radical Eye Software, 29 Mar 1989.
%%%   Revised by Don Knuth, 3 Jan 1990.
%%%   Revised by Tomas Rokicki to accept bounding boxes with no
%%%      space after the colon, 18 Jul 1990.
%%%   Portions modified/removed for use in PSFIG package by
%%%      J. Daniel Smith, 9 October 1990.
%%%   Just the bit which knows about (atend) as a BoundingBox
%
%%%    hacked back a bit by SPQR 12/91
%
\long\def\epsfaux#1#2:#3\\{\ifx#1\epsfpercent
   \def\testit{#2}\ifx\testit\epsfbblit
        \@atendfalse
        \epsf@atend #3 . \\%
        \if@atend
           \if@verbose
                \typeout{epsfig: found `(atend)'; continuing search}%
           \fi
        \else
                \epsfgrab #3 . . . \\%
                \epsffileokfalse\global\no@bbfalse
                \global\epsfbbfoundtrue
        \fi
   \fi\fi}%
%
%%% Determine if the stuff following the %%BoundingBox is `(atend)'
%%% J. Daniel Smith.  Copied from \epsf@grab above.
%
\def\epsf@atendlit{(atend)}
%
\def\epsf@atend #1 #2 #3\\{%
   \def\epsf@tmp{#1}\ifx\epsf@tmp\empty
      \epsf@atend #2 #3 .\\\else
   \ifx\epsf@tmp\epsf@atendlit\@atendtrue\fi\fi}%
 
 
%%% End of file reading stuff from epsf.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% trigonometry stuff from "trig.tex"
\chardef\trig@letter = 11
\chardef\other = 12
 
\newif\ifdebug %%% turn me on to see TeX hard at work ...
\newif\ifc@mpute %%% don't need to compute some values
\newif\if@atend
\c@mputetrue % but assume that we do
 
\let\then = \relax
\def\r@dian{pt }%
\let\r@dians = \r@dian
\let\dimensionless@nit = \r@dian
\let\dimensionless@nits = \dimensionless@nit
\def\internal@nit{sp }%
\let\internal@nits = \internal@nit
\newif\ifstillc@nverging
\def \Mess@ge #1{\ifdebug \then \message {#1} \fi}%
 
{ %%% Things that need abnormal catcodes %%%
        \catcode `\@ = \trig@letter
        \gdef \nodimen {\expandafter \n@dimen \the \dimen}%
        \gdef \term #1 #2 #3%
               {\edef \t@ {\the #1}%%% freeze parameter 1 (count, by value)
                \edef \t@@ {\expandafter \n@dimen \the #2\r@dian}%
                                   %%% freeze parameter 2 (dimen, by value)
                \t@rm {\t@} {\t@@} {#3}%
               }%
        \gdef \t@rm #1 #2 #3%
               {{%
                \count 0 = 0
                \dimen 0 = 1 \dimensionless@nit
                \dimen 2 = #2\relax
                \Mess@ge {Calculating term #1 of \nodimen 2}%
                \loop
                \ifnum  \count 0 < #1
                \then   \advance \count 0 by 1
                        \Mess@ge {Iteration \the \count 0 \space}%
                        \Multiply \dimen 0 by {\dimen 2}%
                        \Mess@ge {After multiplication, term = \nodimen 0}%
                        \Divide \dimen 0 by {\count 0}%
                        \Mess@ge {After division, term = \nodimen 0}%
                \repeat
                \Mess@ge {Final value for term #1 of
                                \nodimen 2 \space is \nodimen 0}%
                \xdef \Term {#3 = \nodimen 0 \r@dians}%
                \aftergroup \Term
               }}%
        \catcode `\p = \other
        \catcode `\t = \other
        \gdef \n@dimen #1pt{#1} %%% throw away the ``pt''
}%
 
\def \Divide #1by #2{\divide #1 by #2} %%% just a synonym
 
\def \Multiply #1by #2%%% allows division of a dimen by a dimen
       {{%%% should really freeze parameter 2 (dimen, passed by value)
        \count 0 = #1\relax
        \count 2 = #2\relax
        \count 4 = 65536
        \Mess@ge {Before scaling, count 0 = \the \count 0 \space and
                        count 2 = \the \count 2}%
        \ifnum  \count 0 > 32767 %%% do our best to avoid overflow
        \then   \divide \count 0 by 4
                \divide \count 4 by 4
        \else   \ifnum  \count 0 < -32767
                \then   \divide \count 0 by 4
                        \divide \count 4 by 4
                \else
                \fi
        \fi
        \ifnum  \count 2 > 32767 %%% while retaining reasonable accuracy
        \then   \divide \count 2 by 4
                \divide \count 4 by 4
        \else   \ifnum  \count 2 < -32767
                \then   \divide \count 2 by 4
                        \divide \count 4 by 4
                \else
                \fi
        \fi
        \multiply \count 0 by \count 2
        \divide \count 0 by \count 4
        \xdef \product {#1 = \the \count 0 \internal@nits}%
        \aftergroup \product
       }}%
 
\def\r@duce{\ifdim\dimen0 > 90\r@dian \then   % sin(x) = sin(180-x)
                \multiply\dimen0 by -1
                \advance\dimen0 by 180\r@dian
                \r@duce
            \else \ifdim\dimen0 < -90\r@dian \then  % sin(x) = sin(360+x)
                \advance\dimen0 by 360\r@dian
                \r@duce
                \fi
            \fi}%
 
\def\Sine#1%
       {{%
        \dimen 0 = #1 \r@dian
        \r@duce
        \ifdim\dimen0 = -90\r@dian \then
           \dimen4 = -1\r@dian
           \c@mputefalse
        \fi
        \ifdim\dimen0 = 90\r@dian \then
           \dimen4 = 1\r@dian
           \c@mputefalse
        \fi
        \ifdim\dimen0 = 0\r@dian \then
           \dimen4 = 0\r@dian
           \c@mputefalse
        \fi
%
        \ifc@mpute \then
                % convert degrees to radians
                \divide\dimen0 by 180
                \dimen0=3.141592654\dimen0
%
                \dimen 2 = 3.1415926535897963\r@dian %%% a well-known constant
                \divide\dimen 2 by 2 %%% we only deal with -pi/2 : pi/2
                \Mess@ge {Sin: calculating Sin of \nodimen 0}%
                \count 0 = 1 %%% see power-series expansion for sine
                \dimen 2 = 1 \r@dian %%% ditto
                \dimen 4 = 0 \r@dian %%% ditto
                \loop
                        \ifnum  \dimen 2 = 0 %%% then we've done
                        \then   \stillc@nvergingfalse
                        \else   \stillc@nvergingtrue
                        \fi
                        \ifstillc@nverging %%% then calculate next term
                        \then   \term {\count 0} {\dimen 0} {\dimen 2}%
                                \advance \count 0 by 2
                                \count 2 = \count 0
                                \divide \count 2 by 2
                                \ifodd  \count 2 %%% signs alternate
                                \then   \advance \dimen 4 by \dimen 2
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\textwidth=5.5 true in
 \textheight= 8.4 true in

  \topmargin -20pt



\setcounter{page}{1}
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\title[Classical and Quantum Chaos in Fundamental Field Theories]
 {Classical and Quantum Chaos in Fundamental Field Theories}

 \subjclass{Primary: 70H05, 81T25
            Secondary: }
 \keywords{Gauge theories, classical chaos, quantum chaos}

\author[Harald Markum and Rainer Pullirsch]
       {}


\def\cal#1{{\fam2#1}}

\begin{document}
\maketitle


\centerline{\scshape Harald Markum and Rainer Pullirsch}
\medskip

{\footnotesize \centerline{} \centerline{
Atominstitut der \"osterreichischen Universit\"aten}
\centerline{ Technische Universit\"at Wien}
\centerline{ Wiedner Hauptstra\ss e 8-10}
\centerline{ A-1040 Vienna, Austria}
 }


\begin{quote}{\normalfont\fontsize{8}{10}\selectfont
{\bfseries Abstract.}
An investigation of classical chaos and quantum chaos in 
gauge fields and fermion fields, respectively, is presented
for (quantum) electrodynamics.
We analyze the leading Lyapunov exponents of U(1)
gauge field configurations on a $12^3$ lattice
which are initialized by Monte Carlo simulations.
We find that configurations in the strong coupling phase are
substantially more chaotic than in the deconfinement phase.
Considering the quantum case, 
complete eigenvalue spectra of the Dirac operator in
quenched $4d$ compact QED are studied on $8^3 \times 4$ and $8^3 \times 6$
lattices. We investigate the behavior of the nearest-neighbor
spacing distribution $P(s)$ as a measure of the fluctuation
properties of the eigenvalues in the strong coupling and the
Coulomb phase. In both phases we find agreement with the Wigner
surmise of the unitary ensemble of random-matrix theory
indicating quantum chaos.
\par}
\end{quote}

\section{Lyapunov exponents in Minkowskian U(1) gauge theory}

\subsection{Classical chaotic dynamics from Monte Carlo initial states}

Cha\-otic dynamics in general is characterized by the
spectrum of Lyapunov exponents. These exponents, if they are positive,
reflect an exponential divergence of initially adjacent configurations.
In case of symmetries inherent in the Hamiltonian of the system
there are corresponding zero values of these exponents. Finally
negative exponents belong to irrelevant directions in the phase
space: perturbation components in these directions die out
exponentially. Pure gauge fields on the lattice show a characteristic
Lyapunov spectrum consisting of one third of each kind of
exponents~\cite{BOOK}.
Assuming this general structure of the Lyapunov spectrum we
investigate presently its magnitude only, namely the maximal
value of the Lyapunov exponent, $L_{{\rm max}}$.

The general definition of the Lyapunov exponent is based on a
distance measure $d(t)$ in phase space,
\begin{equation}
L := \lim_{t\rightarrow\infty} \lim_{d(0)\rightarrow 0}
\frac{1}{t} \ln \frac{d(t)}{d(0)}.
\end{equation}
In case of conservative dynamics the sum of all Lyapunov exponents
is zero according to Liouville's theorem, $\sum L_i = 0$.
We utilize the gauge invariant distance measure consisting of
the local differences of energy densities between two $3d$ field configurations
on the lattice:
\begin{equation}
d : = \frac{1}{N_P} \sum_P\nolimits \, \left| {\rm tr} U_P - {\rm tr} U'_P \right|.
\end{equation}
Here the symbol $\sum_P$ stands for the sum over all $N_P$ plaquettes,
so this distance is bound in the interval $(0,2N)$ for the group
SU(N). $U_P$ and $U'_P$ are the plaquette variables, constructed from
the basic link variables $U_{x,i}$,
\begin{equation}
U_{x,i} = \exp \left( aA_{x,i}^cT^c \right)\: ,
\end{equation}
located on lattice links pointing from the position $x=(x_1,x_2,x_3)$ to
$x+ae_i$. The generators of the group are
$T^c = -ig\tau^c/2$ with $\tau^c$ being the Pauli matrices
in case of SU(2) and $A_{x,i}^c$ is the vector potential.
The elementary plaquette variable is constructed for a plaquette with a
corner at $x$ and lying in the $ij$-plane as
$U_{x,ij} = U_{x,i} U_{x+i,j} U^{\dag}_{x+j,i} U^{\dag}_{x,j}$.
It is related to the magnetic field strength $B_{x,k}^c$:
\begin{equation}
U_{x,ij} = \exp \left( \varepsilon_{ijk} a B_{x,k}^c T^c \right).
\end{equation}
The electric field strength $E_{x,i}^c$ is related to the canonically conjugate
momentum $P_{x,i} = \dot{U}_{x,i}$ via
\begin{equation}
E^c_{x,i} = \frac{2a}{g^3} {\rm tr} \left( T^c \dot{U}_{x,i} U^{\dag}_{x,i} \right).
\end{equation}

The Hamiltonian of the lattice gauge field system can be casted into
the form
\begin{equation}
H = \sum \left[ \frac{1}{2} \langle P, P \rangle \, + \,
 1 - \frac{1}{4} \langle U, V \rangle \right].
\end{equation}
Here the scalar product stands for
$\langle A, B \rangle = \frac{1}{2} {\rm tr} (A B^{\dag} )$.
The staple variable $V$ is a sum of triple products of elementary
link variables closing a plaquette with the chosen link $U$.
This way the Hamiltonian is formally written as a sum over link
contributions and $V$ plays the role of the classical force
acting on the link variable $U$. 

Initial conditions chosen randomly with a given average magnetic energy
per plaquette have been investigated in past years~\cite{Biro94}.
We prepare the initial field configurations
from a standard four dimensional Euclidean Monte Carlo program on
a $12^3\times 4$ lattice varying the gauge coupling $g$~\cite{SU2}.
We relate such four dimensional Euclidean
lattice field configurations to Minkow\-skian momenta and fields
for the three dimensional Hamiltonian simulation
by identifying a fixed time slice of the four dimensional lattice.

\begin{figure}[t]
\centerline{{\psfig{figure=lyaptime.eps,width=6cm}}\hspace{5mm}
{\psfig{figure=lyaptimedc.eps,width=6cm}}}
\caption{
  Exponentially diverging distance in real time of initially adjacent U(1) field
  configurations on a $12^3$ lattice prepared at $\beta=0.9$ in the
  confinement (left) and at $\beta=1.1$ in the Coulomb
  phase (right).
\label{fig1}
 }
\end{figure}

\subsection{Chaos, confinement and continuum limit}

We start the presentation of our results with a characteristic example
of the time evolution of the distance between initially adjacent
configurations. An initial state prepared by a standard four dimensional
Monte Carlo simulation is evolved according to the classical Hamiltonian dynamics
in real time. Afterwards this initial state is rotated locally by
group elements which are chosen randomly near to the unity.
The time evolution of this slightly rotated configuration is then
pursued and finally the distance between these two evolutions
is calculated at the corresponding times.
A typical exponential rise of this distance followed by a saturation
can be inspected in Fig.~\ref{fig1} from an example of U(1) gauge theory
for two values of $\beta=1/g^2$ in the confinement phase and in the Coulomb phase.
While the saturation is an artifact of
the compact distance measure of the lattice, the exponential rise
(the linear rise of the logarithm)
can be used for the determination of the leading Lyapunov exponent.
The left plot exhibits that in the confinement phase the
field has larger Lyapunov exponents than in the Coulomb phase 
shown in the right plot.

The main result of the present study is the dependence of the leading
Lyapunov exponent $L_{{\rm max}}$ on the inverse coupling strength $\beta$,
displayed in Fig.~\ref{fig2}.
As expected the strong coupling phase
is more chaotic.  The transition reflects
the critical coupling to the Coulomb phase.

\begin{figure}[t]
\centerline{\psfig{figure=lyapall.eps,width=6cm}}
\caption{Lyapunov exponents of 100 U(1) field configurations as a function
         of coupling.
\label{fig2}
 }
\end{figure}

\begin{figure}[t]
\centerline{\psfig{figure=lyapenerav.eps,width=6cm}}
\caption{
  Average maximal Lyapunov exponent as a function of the
  scaled average energy per plaquette $ag^2E$. The U(1) gauge theory
  shows an approximately quadratic behavior in the weak coupling regime.
\label{fig3}
 }
\end{figure}

An interesting result concerning the continuum limit can be viewed from Fig.~\ref{fig3}
which shows the energy dependence of the Lyapunov exponents for the U(1) theory.
One observes an approximately quadratic relation in the weak coupling regime.
From scaling arguments one expects a functional relationship between
the Lyapunov exponent and the energy \cite{BOOK,SCALING}
\begin{equation}
L(a) \propto a^{k-1} E^{k}(a) ,
\label{scaling}
\end{equation}
with the exponent $k$ being crucial for the continuum limit of the
classical field theory. A value of $k < 1$ leads to a
divergent Lyapunov exponent, while $k > 1$ yields a vanishing $L$ in
the continuum. The case $k = 1$ is special allowing for a finite non-zero
Lyapunov exponent. Our analysis of the scaling relation (\ref{scaling})
gives evidence that the classical compact U(1) lattice gauge theory
has $k \approx 2$ and with $L(a) \to 0$ a regular continuum theory. 

\section{Quantum chaos in compact Euclidean QED}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Manifestation of quantum chaos} 

The fluctuation properties of the eigenvalues of Dirac operator
for quantum chromodynamics (QCD) on a lattice in Euclidean space-time
have attracted much attention in the past few years. In 
Ref.~\cite{Verb95} it was first shown for SU(2) lattice gauge theory 
that certain features of the spectrum of the Dirac operator are described
by random-matrix theory (RMT). In particular the
so-called nearest-neighbor spacing distribution $P(s)$, i.e. the distribution
of the spacings $s$ of adjacent eigenvalues on the ``unfolded'' scale,
agrees with the Wigner surmise of RMT. According to the 
Bohigas-Giannoni-Schmit conjecture \cite{Bohi84}, quantum systems whose
classical counterparts are chaotic have a $P(s)$ given by RMT whereas systems
whose classical counterparts are integrable obey a Poisson distribution
$P(s)=e^{-s}$. Therefore, the specific form of $P(s)$ is often taken as a
criterion for ``quantum chaos''. However, there is no accepted proof of the
Bohigas-Giannoni-Schmit conjecture yet. The field of quantum chaos is
still developing and there are many open conceptual problems \cite{riew}.
Applying this conjecture it was recently demonstrated that QCD is chaotic, both
in the confinement and the quark gluon plasma phase \cite{Pull98}.
 
A number of interesting results have been established for
chaotic dynamics in classical gauge theories. Lattice gauge
theories are chaotic as classical Hamiltonian dynamical systems 
\cite{Biro94}. Furthermore, it was found that the leading Lyapunov exponent
of SU(2) Yang-Mills field configurations indicates that configurations
corresponding to
the deconfinement phase are chaotic although they are less chaotic
than in the strong coupling phase at finite temperature \cite{SU2}.
The scaling of the maximal Lyapunov exponent in the classical 
continuum limit was studied in Ref.~\cite{SCALING}: It was
suggested that Abelian gauge theories behave regularly in the continuum limit
whereas non-Abelian gauge theories are chaotic in the continuum,
although the exact scaling relation is still an open problem.
Chaos to order transitions were observed in a spatially
homogeneous SU(2) Yang-Mills-Higgs system and in a spatially homogeneous
SU(2) Yang-Mills Chern-Simons Higgs system \cite{Sala97,Mukk97}.
In Ref.~\cite{Sala97} a chaos to order transition was also seen
on the quantum level, i.e. a smooth transition from a Wigner
to a Poisson distribution was found. 
A transition in $P(s)$ from Wigner to Poisson behavior was further
observed at the metal-insulator transition of the Anderson model
\cite{Alts88}.
Further, the suppression of the characteristic manifestations
of dynamical chaos by quantum fluctuations was analyzed in 
the context of spatially homogeneous scalar electrodynamics \cite{Mati97}
and for a $0+1$-dimensional space-time $N$-component $\phi^4$ theory
in the presence of an external field \cite{Case98}.
These chaos to order transitions were seen in spatially
homogeneous models and not for the full classical field theory.
The relationship to properties of the quantum field theory is an
interesting issue.

Here we focus on the Dirac operator for quenched $4d$ compact 
quantum electrodynamics (QED)
to search for the possible existence of a transition from chaotic to
regular behavior in Abelian lattice gauge
theories. In particular, we are interested in the 
nearest-neighbor spacing distribution of the eigenvalues of the Dirac 
operator across the phase transition from the strong coupling to the 
Coulomb phase. In the strong coupling region Abelian as well as non-Abelian 
lattice gauge theories are in a confined phase \cite{Wils74}. 
For compact QED this means that for couplings 
$\beta < \beta_c \approx 1.01$ the electron is confined.
However, when crossing the phase transition the conventional Coulomb phase is
observed.  It is an interesting question if the difference between
the Coulomb phase in QED and the quark-gluon plasma phase in QCD has an
influence on the level repulsion of the corresponding Dirac spectra.


\subsection{Quantum chaos of fermion fields}

We generated gauge field configurations using the standard Wilson plaquette
action for U(1) gauge theory,
\begin{equation}
S_G(U_l)=\beta\sum_P\nolimits \, (1-\cos \Theta_P) \; ,
\end{equation}
where $U_l\equiv U_{x,\mu}=\exp(i\Theta_{x,\mu})$, with $\Theta_{x,\mu}\in
[-\pi,\pi)$, are the field variables defined on the links $l\equiv(x,\mu)$.
The plaquette angles are $\Theta_P = \Theta_{x,\mu}+\Theta_{x+\hat{\mu},\nu}
-\Theta_{x+\hat{\nu},\mu}-\Theta_{x,\nu}$. We simulated $8^3 \times 4$
and $8^3 \times 6$ lattices at various values of the inverse gauge
coupling $\beta=1/g^2$
both in the strong coupling and the Coulomb phase. Typically we discarded
the first 10000 sweeps for reaching equilibrium and produced 20 
independent configurations separated by 1000 sweeps for 
each $\beta$. Because of the spectral ergodicity property of RMT one can
replace ensemble averages by spectral averages \cite{Guhr98} if one is
only interested in the bulk properties. Thus a few independent
configurations are sufficient to compute $P(s)$.

On the Euclidean lattice the Dirac operator $/\!\!\!\!D=/\!\!\!\partial+ig\: /\!\!\!\!A$ 
for staggered fermions 
\begin{equation}
M_{x,x'}=\frac{1}{2}\sum\limits_{\mu=1}^{4}\eta_{x\mu} \left( 
\delta_{x+\hat{\mu},x'} U_{x,\mu} - \delta_{x-\hat{\mu},x'} 
U_{x,\mu}^{\dagger} \right) 
\end{equation}
is anti-Hermitian so that all 
eigenvalues are imaginary. For convenience we denote them by 
$i\lambda_n$ and refer to the $\lambda_n$ as the eigenvalues in the 
following. Because of $\{/\!\!\!\!D,\gamma_5\}=0$ the $\lambda_n$ 
occur in pairs of opposite sign.  All spectra were checked against 
the analytical sum rules
\begin{equation}
\sum_{n} \lambda_n = 0 \qquad {\rm and} \qquad
\sum_{\lambda_n>0} \lambda_n^2 = V \:,
\end{equation}
where $V$ is the lattice volume.

\begin{figure}[t]
\centerline{{\psfig{figure=86b090.eps,width=6cm}}\hspace{5mm}
{\psfig{figure=86b110.eps,width=6cm}}}
  \caption{\label{fig4} Nearest-neighbor spacing distribution $P(s)$
       of the Dirac operator for compact U(1) theory in the strong
       coupling phase for $\beta=0.90$ (left) and in the Coulomb
       phase for $\beta=1.10$ (right).
       The histogram represents the lattice data on an $8^3 \times 6$
       lattice averaged over 20 independent configurations. The full
       curve is the Wigner distribution of
       Eq.~(\protect\ref{ue}) for the unitary ensemble of RMT.
       For comparison the Poisson distribution
       $P(s) = e^{-s}$ is also indicated by the dashed line.}
\end{figure}

\begin{figure}[t]
  \centerline{\hbox{
  \psfig{figure=free.eps,width=6cm}
  }}
  \caption{\label{fig5} Nearest-neighbor spacing distribution $P(s)$
       of the analytically calculated eigenvalues of
       Eq.~(\protect\ref{free}) for a free Dirac operator on a
       $53\times 47 \times 43 \times 41$ lattice (histogram)
       compared with the Poisson distribution $P(s) = e^{-s}$
       (solid line).}
\end{figure}

In RMT one has to distinguish between different universality classes
which are determined by the symmetries of the system. So far the 
classification for the QED Dirac operator has not been done. Our
calculations show that in the case of the staggered $4d$ compact QED Dirac matrix
the appropriate ensemble is the unitary ensemble. Although from a 
mathematical point of view this is the simplest one, the RMT result
for the nearest-neighbor spacing distribution is still rather complicated.
It can be expressed in terms of so-called prolate spheroidal functions,
see Ref.~\cite{Meht91} where $P(s)$ has also been tabulated. A good
approximation to $P(s)$ is provided by the Wigner surmise for the 
unitary ensemble
\begin{equation}
  \label{ue}
  P(s)=\frac{32}{\pi^2}\,s^2\,e^{-\frac{4}{\pi}s^2}\:.
\end{equation}

We have simulated $8^3 \times 4$ lattices at $\beta=0,\:0.90,\:0.95,\:1.00,\:
1.05,\:1.10,\:1.50$ and $8^3 \times 6$ lattices at $\beta=0.90,\:1.10,\:
1.50$. All results are similar to those selected for the plots.
The left plot in Fig.~\ref{fig4} shows the 
nearest-neighbor spacing distribution $P(s)$ for $\beta=0.90$ in the
confined phase averaged over 20 independent configurations on the
$8^3 \times 6$ lattice compared with the Wigner surmise for the unitary
ensemble of RMT of Eq.~(\ref{ue}). Good agreement is found. According
to the Bohigas-Giannoni-Schmit conjecture this means the system can be
regarded as chaotic in the strong coupling region. The right plot
in Fig.~\ref{fig4} shows the nearest-neighbor spacing distribution 
$P(s)$ for $\beta=1.10$ in the Coulomb phase again averaged over 
20 independent configurations and compared
with the Wigner surmise (\ref{ue}).
The agreement of the lattice data with the RMT predictions is
interpreted as a signal that quantum chaos survives the phase transition.
We find no deviation up to the maximum coupling considered, $\beta = 1.50$. 

In the strong coupling phase the result holds down to $\beta=0$.
Therefore, we tend to interpret our, as well as previous 
\cite{Pull98,Verb95}, results in the sense that the disorder of 
the gauge field configurations \cite{Biro94,SU2} is responsible
for the chaotic characteristics of the spectrum of the Dirac operator.
In contrast 
to that: The free fermion theory is non-chaotic and the
corresponding
nearest-neighbor spacing distribution obeys a Poisson distribution.
This is illustrated in Fig.~\ref{fig5} where $P(s)$ is 
obtained from the analytical eigenvalues of the free 
Dirac operator on a $53\times 47\times 43 \times 41$ lattice:
\begin{equation}
\label{free}
a^2\lambda^2 = \sum\limits_{\mu=1}^4 \sin^2 \left( \frac{2\pi n_{\mu}}{L_{\mu}}
\right)\:.
\end{equation}
Here $a$ is the lattice constant, $L_{\mu}$ is the number of lattice sites
in $\mu$-direction, and $n_{\mu}=0,...,L_{\mu}-1$. We used an asymmetric 
lattice with $L_{\mu}$ being primes and restricted the range to
$(L_{\mu}-1)/2$ instead of $L_{\mu}-1$ in each direction to avoid 
degeneracies of the free spectrum.

\section{Conclusion}

In the underlying article we performed a comparison of classical
chaos and quantum chaos in fundamental field theories of physics,
exemplified for the U(1) theory of electrodynamics. This is not a 
direct comparison, however, since it deals with the gauge field
in classical theory and with fermions in the quantum case. It
turned out that the classical U(1) field is chaotic in the confinement
phase with decreasing Lyapunov exponents towards the Coulomb phase.
A scaling analysis indicates a regular continuum theory as one expects 
from the Maxwell equations. On the other hand, our investigation of
the quantized fermion field fulfills the criterion for quantum
chaos both in the confinement and Coulomb phase. A scaling analysis
was not possible for the quantum case (due to the lack of a 
$\beta$-function) which could cover the transition to a regular
theory. Nevertheless, the free Dirac operator, in absence
of a covariant derivative and minimal gauge coupling, exhibits
regular behavior.

It would be interesting to study the direct counterpart of 
the classical gauge field after quantization.
A similarly accurate determination
of the eigenvalue spectrum of the gauge sector necessitates
to construct the corresponding Fock space and to diagonalize
high-dimensional matrices which seems to be out of reach
for $4 d$ QED/QCD. On the other hand, chaos studies of the
classical limit of the fermion field would also be of 
interest but have not yet been attempted.

{\bf Acknowledgments.  }
This work has been supported by the Austrian National Scientific Fund under
the project FWF P14435-TPH.
We thank Bernd A. Berg, Tam\'as S. Bir\'o, Natascha H\"ormann 
and Wolfgang Sakuler for previous cooperations.

\begin{thebibliography}{99}
\bibitem{BOOK}  T.S. Bir\'o, S.G. Matinyan, and B. M\"uller,
        {\it Chaos and Gauge Field Theory} (World Scientific,
        Singapore, 1995).
        %%CITATION = NONE;%%

\bibitem{Biro94} T.S. Bir\'o, C. Gong, B. M\"uller, and A. Trayanov,
  Int. J. Mod. Phys. C5 (1994) 113.
                    %%CITATION = NUCL-TH 9306002;%%

\bibitem{SU2}  T.S. Bir\'o, M. Feurstein, and H. Markum, APH Heavy Ion
        Physics 7 (1998) 235;
        %%CITATION = HEP-LAT 9711002;%%
        T.S. Bir\'o, N. H\"ormann, H. Markum, and R. Pullirsch,
        Nucl. Phys. B (Proc. Suppl.) 86 (2000) 403;
        %%CITATION = HEP-PH 9909309;%%
        H. Markum, R. Pullirsch, and W. Sakuler, hep-lat/0201001;
        %%CITATION = HEP-LAT 0201001;%%
        hep-lat/0205003;
        %%CITATION = HEP-LAT 0205003;%%
        hep-lat/0209039.
        %%CITATION = HEP-LAT 0209039;%%

\bibitem{SCALING}  L. Casetti, R. Gatto, and M. Pettini, J. Phys. A32 (1999) 3055;
                   %%CITATION = CHAO-DYN 9810033;%%
                   H.B. Nielsen, H.H. Rugh, and S.E. Rugh, ICHEP96 1603, hep-th/9611128;
                   %%CITATION = HEP-TH 9611128;%%
                   B. M\"uller, chao-dyn/9607001;
                   %%CITATION = CHAO-DYN 9607001;%%
                   H.B. Nielsen, H.H. Rugh, and S.E. Rugh, chao-dyn/9605013.
                   %%CITATION = CHAO-DYN 9605013;%%

\bibitem{Verb95} M.A. Halasz and J.J.M. Verbaarschot, Phys. Rev. 
                 Lett. 74 (1995) 3920;
                 %%CITATION = HEP-LAT 9501025;%%
                 M.A. Halasz, T. Kalkreuter, and J.J.M.
                 Verbaarschot, Nucl. Phys. B (Proc. Suppl.) 53 (1997) 266.
                 %%CITATION = HEP-LAT 9607042;%%

\bibitem{Bohi84} O. Bohigas, M.-J. Giannoni, and C. Schmit,
                 Phys. Rev. Lett. 52 (1984) 1;
                 %%CITATION = PRLTA,52,1;%%
                 O. Bohigas and M.-J. Giannoni, Springer Lect. Notes Phys. 209
                 (1984) 1.
                 %%CITATION = NONE;%%

\bibitem{riew} Reviews on the subject of quantum chaos are: 
  M.C. Gutzwiller, {\it Chaos in Classical and Quantum Mechanics}
  (Springer, New York, 1990);
  %%CITATION = NONE;%%
  Proceedings of the LII Session of the Les Houches School, {\it Chaos
  and Quantum Physics}, edited by M.-J. Giannoni and A. Voros 
  (North-Holland, Amsterdam, 1991);
  %%CITATION = NONE;%%
  {\it Quantum Chaos: Between Order and Disorder}, edited by G. Casati and
  B.V. Chirikov (Cambridge Univ. Press, Cambridge, 1995).
  %%CITATION = NONE;%%

\bibitem{Pull98} R. Pullirsch, K. Rabitsch, T. Wettig, and
  H. Markum, Phys. Lett. B427 (1998) 119;
  %%CITATION = HEP-PH 9803285;%%
  H. Markum, R. Pullirsch, and T. Wettig, Phys. Rev. Lett. 83 (1999) 484;
  %%CITATION = HEP-LAT 9906020;%%
  B.A. Berg, H. Markum, and R. Pullirsch, Phys. Rev. D59 (1999) 097504.
  %%CITATION = HEP-LAT 9812010;%%

\bibitem{Sala97} L. Salasnich, Mod. Phys. Lett. A12 (1997) 1473.
  %%CITATION = QUANT-PH 9706025;%%

\bibitem{Mukk97} C. Mukku, M.S. Sriram, J. Segar, B.A. Bambah, and 
  S. Lakshmibala, J. Phys. A30 (1997) 3003.
  %%CITATION = HEP-TH 9610071;%%

\bibitem{Alts88} B.L. Al'tshuler, I.Kh. Zharekeshev, S.A.
  Kotochigova, and B.I. Shklovski$\breve{\i}$,
  Zh. Eksp. Teor. Fiz. 94 (1988) 343 [Sov. Phys. JETP 67 (1988) 625];
  %%CITATION = NONE;%%
  B.I. Shklovski$\breve{\i}$, B. Shapiro, B.R. Sears, P. Lambrianides,
  and H.B. Shore, Phys. Rev. B47 (1993) 11487.
  %%CITATION = NONE;%%

\bibitem{Mati97} S.G. Matinyan and B. M\"uller, Phys. Rev. Lett. 78 (1997) 2515.
  %%CITATION = NONE;%%

\bibitem{Case98} L. Casetti, R. Gatto, and M. Modugno,
                 Phys. Rev. E57 (1998) 1223.
                 %%CITATION = HEP-TH 9707054;%%

\bibitem{Wils74} K.G. Wilson, Phys. Rev. D10 (1974) 2445.
  %%CITATION = NONE;%%

\bibitem{Guhr98}  T. Guhr, A. M\"uller-Groeling, and H.A. Weidenm\"uller,
  Phys. Rep. 299 (1998) 189;
  %%CITATION = COND-MAT 9707301;%%
  J.B. French, P.A. Mello, and A. Pandey, Phys. Lett. 80B (1978) 17.
  %%CITATION = NONE;%%

\bibitem{Meht91} M.L. Mehta, {\it Random Matrices}, 2nd ed. (Academic
  Press, San Diego, 1991).
  %%CITATION = NONE;%%


\end{thebibliography}
\medskip

\end{document}

nlin0303057.extracted_bib
\begin{thebibliography}{99}
\bibitem{BOOK}  T.S. Bir\'o, S.G. Matinyan, and B. M\"uller,
        {\it Chaos and Gauge Field Theory} (World Scientific,
        Singapore, 1995).
        %%CITATION = NONE;%%

\bibitem{Biro94} T.S. Bir\'o, C. Gong, B. M\"uller, and A. Trayanov,
  Int. J. Mod. Phys. C5 (1994) 113.
                    %%CITATION = NUCL-TH 9306002;%%

\bibitem{SU2}  T.S. Bir\'o, M. Feurstein, and H. Markum, APH Heavy Ion
        Physics 7 (1998) 235;
        %%CITATION = HEP-LAT 9711002;%%
        T.S. Bir\'o, N. H\"ormann, H. Markum, and R. Pullirsch,
        Nucl. Phys. B (Proc. Suppl.) 86 (2000) 403;
        %%CITATION = HEP-PH 9909309;%%
        H. Markum, R. Pullirsch, and W. Sakuler, hep-lat/0201001;
        %%CITATION = HEP-LAT 0201001;%%
        hep-lat/0205003;
        %%CITATION = HEP-LAT 0205003;%%
        hep-lat/0209039.
        %%CITATION = HEP-LAT 0209039;%%

\bibitem{SCALING}  L. Casetti, R. Gatto, and M. Pettini, J. Phys. A32 (1999) 3055;
                   %%CITATION = CHAO-DYN 9810033;%%
                   H.B. Nielsen, H.H. Rugh, and S.E. Rugh, ICHEP96 1603, hep-th/9611128;
                   %%CITATION = HEP-TH 9611128;%%
                   B. M\"uller, chao-dyn/9607001;
                   %%CITATION = CHAO-DYN 9607001;%%
                   H.B. Nielsen, H.H. Rugh, and S.E. Rugh, chao-dyn/9605013.
                   %%CITATION = CHAO-DYN 9605013;%%

\bibitem{Verb95} M.A. Halasz and J.J.M. Verbaarschot, Phys. Rev. 
                 Lett. 74 (1995) 3920;
                 %%CITATION = HEP-LAT 9501025;%%
                 M.A. Halasz, T. Kalkreuter, and J.J.M.
                 Verbaarschot, Nucl. Phys. B (Proc. Suppl.) 53 (1997) 266.
                 %%CITATION = HEP-LAT 9607042;%%

\bibitem{Bohi84} O. Bohigas, M.-J. Giannoni, and C. Schmit,
                 Phys. Rev. Lett. 52 (1984) 1;
                 %%CITATION = PRLTA,52,1;%%
                 O. Bohigas and M.-J. Giannoni, Springer Lect. Notes Phys. 209
                 (1984) 1.
                 %%CITATION = NONE;%%

\bibitem{riew} Reviews on the subject of quantum chaos are: 
  M.C. Gutzwiller, {\it Chaos in Classical and Quantum Mechanics}
  (Springer, New York, 1990);
  %%CITATION = NONE;%%
  Proceedings of the LII Session of the Les Houches School, {\it Chaos
  and Quantum Physics}, edited by M.-J. Giannoni and A. Voros 
  (North-Holland, Amsterdam, 1991);
  %%CITATION = NONE;%%
  {\it Quantum Chaos: Between Order and Disorder}, edited by G. Casati and
  B.V. Chirikov (Cambridge Univ. Press, Cambridge, 1995).
  %%CITATION = NONE;%%

\bibitem{Pull98} R. Pullirsch, K. Rabitsch, T. Wettig, and
  H. Markum, Phys. Lett. B427 (1998) 119;
  %%CITATION = HEP-PH 9803285;%%
  H. Markum, R. Pullirsch, and T. Wettig, Phys. Rev. Lett. 83 (1999) 484;
  %%CITATION = HEP-LAT 9906020;%%
  B.A. Berg, H. Markum, and R. Pullirsch, Phys. Rev. D59 (1999) 097504.
  %%CITATION = HEP-LAT 9812010;%%

\bibitem{Sala97} L. Salasnich, Mod. Phys. Lett. A12 (1997) 1473.
  %%CITATION = QUANT-PH 9706025;%%

\bibitem{Mukk97} C. Mukku, M.S. Sriram, J. Segar, B.A. Bambah, and 
  S. Lakshmibala, J. Phys. A30 (1997) 3003.
  %%CITATION = HEP-TH 9610071;%%

\bibitem{Alts88} B.L. Al'tshuler, I.Kh. Zharekeshev, S.A.
  Kotochigova, and B.I. Shklovski$\breve{\i}$,
  Zh. Eksp. Teor. Fiz. 94 (1988) 343 [Sov. Phys. JETP 67 (1988) 625];
  %%CITATION = NONE;%%
  B.I. Shklovski$\breve{\i}$, B. Shapiro, B.R. Sears, P. Lambrianides,
  and H.B. Shore, Phys. Rev. B47 (1993) 11487.
  %%CITATION = NONE;%%

\bibitem{Mati97} S.G. Matinyan and B. M\"uller, Phys. Rev. Lett. 78 (1997) 2515.
  %%CITATION = NONE;%%

\bibitem{Case98} L. Casetti, R. Gatto, and M. Modugno,
                 Phys. Rev. E57 (1998) 1223.
                 %%CITATION = HEP-TH 9707054;%%

\bibitem{Wils74} K.G. Wilson, Phys. Rev. D10 (1974) 2445.
  %%CITATION = NONE;%%

\bibitem{Guhr98}  T. Guhr, A. M\"uller-Groeling, and H.A. Weidenm\"uller,
  Phys. Rep. 299 (1998) 189;
  %%CITATION = COND-MAT 9707301;%%
  J.B. French, P.A. Mello, and A. Pandey, Phys. Lett. 80B (1978) 17.
  %%CITATION = NONE;%%

\bibitem{Meht91} M.L. Mehta, {\it Random Matrices}, 2nd ed. (Academic
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  %%CITATION = NONE;%%


\end{thebibliography}