%Paper: 9202057
%From: app@cuphyf.phys.columbia.edu (Alexios Polychronakos)
%Date: Mon, 17 Feb 92 15:48:26 EST
%Date (revised): Mon, 17 Feb 92 16:20:35 EST


%%%%%%%%%%THIS IS MACROS AND JNL%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def \half {\textstyle {1 \over 2}}
\def \t {{\rm tr}}
\def \p {{\tilde \pi}}
\def \I {{\tilde I}}

\font\twelverm=cmr10  scaled 1200   \font\twelvei=cmmi10  scaled 1200
\font\twelvesy=cmsy10 scaled 1200   \font\twelveex=cmex10 scaled 1200
\font\twelvebf=cmbx10 scaled 1200   \font\twelvesl=cmsl10 scaled 1200
\font\twelvett=cmtt10 scaled 1200   \font\twelveit=cmti10 scaled 1200
\font\twelvesc=cmcsc10 scaled 1200  \font\twelvesf=cmss10 scaled 1200
\skewchar\twelvei='177   \skewchar\twelvesy='60

\font\tensc=cmcsc10

%  Define \...point macros to change fonts and spacings consistently

\def\twelvepoint{\normalbaselineskip=12.4pt plus 0.1pt minus 0.1pt
  \abovedisplayskip 12.4pt plus 3pt minus 9pt
  \belowdisplayskip 12.4pt plus 3pt minus 9pt
  \abovedisplayshortskip 0pt plus 3pt
  \belowdisplayshortskip 7.2pt plus 3pt minus 4pt
  \smallskipamount=3.6pt plus1.2pt minus1.2pt
  \medskipamount=7.2pt plus2.4pt minus2.4pt
  \bigskipamount=14.4pt plus4.8pt minus4.8pt
  \def\rm{\fam0\twelverm}          \def\it{\fam\itfam\twelveit}%
  \def\sl{\fam\slfam\twelvesl}     \def\bf{\fam\bffam\twelvebf}%
  \def\mit{\fam 1}                 \def\cal{\fam 2}%
  \def\sc{\twelvesc}               \def\tt{\twelvett}
  \def\sf{\twelvesf}
  \textfont0=\twelverm   \scriptfont0=\tenrm   \scriptscriptfont0=\sevenrm
  \textfont1=\twelvei    \scriptfont1=\teni    \scriptscriptfont1=\seveni
  \textfont2=\twelvesy   \scriptfont2=\tensy   \scriptscriptfont2=\sevensy
  \textfont3=\twelveex   \scriptfont3=\twelveex  \scriptscriptfont3=\twelveex
  \textfont\itfam=\twelveit
  \textfont\slfam=\twelvesl
  \textfont\bffam=\twelvebf \scriptfont\bffam=\tenbf
  \scriptscriptfont\bffam=\sevenbf
  \normalbaselines\rm}

%       tenpoint

\def\tenpoint{\normalbaselineskip=12pt plus 0.1pt minus 0.1pt
  \abovedisplayskip 12pt plus 3pt minus 9pt
  \belowdisplayskip 12pt plus 3pt minus 9pt
  \abovedisplayshortskip 0pt plus 3pt
  \belowdisplayshortskip 7pt plus 3pt minus 4pt
  \smallskipamount=3pt plus1pt minus1pt
  \medskipamount=6pt plus2pt minus2pt
  \bigskipamount=12pt plus4pt minus4pt
  \def\rm{\fam0\tenrm}          \def\it{\fam\itfam\tenit}%
  \def\sl{\fam\slfam\tensl}     \def\bf{\fam\bffam\tenbf}%
  \def\sc{\tensc}             \def\mit{\fam 1}%
  \def\cal{\fam 2}%
  \textfont0=\tenrm   \scriptfont0=\sevenrm   \scriptscriptfont0=\fiverm
  \textfont1=\teni    \scriptfont1=\seveni    \scriptscriptfont1=\fivei
  \textfont2=\tensy   \scriptfont2=\sevensy   \scriptscriptfont2=\fivesy
  \textfont3=\tenex   \scriptfont3=\tenex     \scriptscriptfont3=\tenex
  \textfont\itfam=\tenit
  \textfont\slfam=\tensl
  \textfont\bffam=\tenbf \scriptfont\bffam=\sevenbf
  \scriptscriptfont\bffam=\fivebf
  \normalbaselines\rm}

\let\bigtype=\twelvepoint
\let\medtype=\tenpoint

%%
%%      Various internal macros
%%

\def\beginlinemode{\endmode
  \begingroup\parskip=0pt \obeylines\def\\{\par}\def\endmode{\par\endgroup}}
\def\beginparmode{\endmode
  \begingroup \def\endmode{\par\endgroup}}
\let\endmode=\par
{\obeylines\gdef\
{}}
\def\singlespace{\baselineskip=\normalbaselineskip}
\def\oneandathirdspace{\baselineskip=\normalbaselineskip
  \multiply\baselineskip by 4 \divide\baselineskip by 3}
\def\oneandahalfspace{\baselineskip=\normalbaselineskip
  \multiply\baselineskip by 3 \divide\baselineskip by 2}
\def\doublespace{\baselineskip=\normalbaselineskip \multiply\baselineskip by 2}
\def\triplespace{\baselineskip=\normalbaselineskip \multiply\baselineskip by 3}
\newcount\firstpageno
\firstpageno=2
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\footline={\ifnum\pageno<\firstpageno{\hfil}\else{\hfil\twelverm\folio\hfil}\fi}
\def\toppageno{\global\footline={\hfil}\global\headline
  ={\ifnum\pageno<\firstpageno{\hfil}\else{\hfil\twelverm\folio\hfil}\fi}}
\let\rawfootnote=\footnote              % We must set the footnote style
\def\footnote#1#2{{\rm\singlespace\parindent=0pt\parskip=0pt
  \rawfootnote{#1}{#2\hfill\vrule height 0pt depth 6pt width 0pt}}}
\def\raggedcenter{\leftskip=4em plus 12em \rightskip=\leftskip
  \parindent=0pt \parfillskip=0pt \spaceskip=.3333em \xspaceskip=.5em
  \pretolerance=9999 \tolerance=9999
  \hyphenpenalty=9999 \exhyphenpenalty=9999 }
\def\dateline{\rightline{\ifcase\month\or
  January\or February\or March\or April\or May\or June\or
  July\or August\or September\or October\or November\or December\fi
  \space\number\year}}
\def\received{\vskip 3pt plus 0.2fill
 \centerline{\sl (Received\space\ifcase\month\or
  January\or February\or March\or April\or May\or June\or
  July\or August\or September\or October\or November\or December\fi
  \qquad, \number\year)}}

%%
%%      Page layout, margins, font and spacing (feel free to change)
%%

\hsize=6.5truein
\hoffset=0.0truein
\vsize=8.9truein
\voffset=0.0truein
\parskip=\medskipamount
\def\\{\cr}
\twelvepoint            % selects twelvepoint fonts (cf. \tenpoint)
\doublespace            % selects double spacing for main part of paper (cf.
                        %       \singlespace, \oneandahalfspace)
\overfullrule=0pt       % delete the nasty little black boxes for overfull box

%%
%%      This used to be timestamp.tex
%%

\newcount\timehour
\newcount\timeminute
\newcount\timehourminute
\def\daytime{\timehour=\time\divide\timehour by 60
  \timehourminute=\timehour\multiply\timehourminute by-60
  \timeminute=\time\advance\timeminute by \timehourminute
  \number\timehour:\ifnum\timeminute<10{0}\fi\number\timeminute}
\def\today{\number\day\space\ifcase\month\or Jan\or Feb\or Mar
  \or Apr\or May\or Jun\or Jul\or Aug\or Sep\or Oct\or
  Nov\or Dec\fi\space\number\year}
\def\timestamp{\daytime\space\today}

\def\draft{\leftline{DRAFT \timestamp}}  %  "Draft", Timestamp

%%
%%      The user definitions for major parts of a paper (feel free to change)
%%

\def\uftp#1{
 \rightline{\rm UFTP--#1}}      % Preprint number at upper right of title page

\def\cutp#1{
 \rightline{\rm CU--TP--#1}}

\def\ufift#1{
 \rightline{\rm HEP--#1}}       % Preprint number at upper right of title page

\def\title                      %  Title on title page
  {\null\vskip 3pt plus 0.2fill
   \beginlinemode \doublespace \raggedcenter \bf}

\def\author                     %  Author(s) name(s)  on title page
  {\vskip 3pt plus 0.2fill \beginlinemode
   \doublespace \raggedcenter}

\def\affil                      % Affiliations (can intermix with \author)
  {\vskip 3pt plus 0.1fill \beginlinemode
   \oneandahalfspace \raggedcenter \it}

\def\abstract                   % Begin abstract
  {\vskip 3pt plus 0.3fill \beginparmode \narrower
   \oneandahalfspace {\it  Abstract}:\  }

\def\endtopmatter               % End title page, begin body of paper
  {\endpage                     %       This subsumes \body
   \body}

\def\body                       % Begin text body;  can be used to end
  {\beginparmode}               % \title, \author, \affil, \abstract,
                                % \reference, or \figurecaption modes

\def\head#1{                    % Head;  NOTE enclose the text in {}
  \goodbreak\vskip 0.4truein    %  e.g., \head{I. Introduction}
  {\immediate\write16{#1}
   \raggedcenter {\sc #1} \par }
   \nobreak\vskip 0truein\nobreak}

\def\subhead#1{                 % Subhead;  NOTE enclose the text in {}
  \vskip 0.25truein             % e.g., \subhead{A. History of the Problem}
  {\raggedcenter {\it #1} \par}
   \nobreak\vskip 0truein\nobreak}

\def\beneathrel#1\under#2{\mathrel{\mathop{#2}\limits_{#1}}}

\def\refto#1{$^{#1}$}           % For references in text as superscript

\def\references                 % Begin references -- basic format is Phys Rev
  {\head{References}            % i.e., volume, page, year (space after
% commas).
   \beginparmode
   \frenchspacing \parindent=0pt    %\leftskip=1truecm ?
   %\parskip=8pt plus 3pt
   \parskip=0pt \everypar{\hangindent=20pt\hangafter=1}}

%%\gdef\refis#1{\indent\hbox to 0pt{\hss#1.~~}\ignorespaces}    % Ref list n.

\gdef\refis#1{\item{#1.\ }}                     % Ref list numbers.

\gdef\journal#1,#2,#3,#4.{              % Journal reference.  Comma sets
    {\sl #1~}{\bf #2}, #3 (#4).}                % off: name, vol, page, year

\def\refstylepr{
  \gdef\refto##1{${\,}^{##1}$}
  \gdef\refis##1{${\,}^{\item{##1}}$}
  \gdef\journal##1,##2,##3,##4 {
    \rm ##1 {\bf ##2}, ##3 (##4).}}
\def\refstylenp{                % Nucl Phys(or Phys Lett) ref style: V, Y, P
  \gdef\refto##1{ [##1]}                        % Reference in text []
  \gdef\refis##1{\item{##1)\ }}                 % Reference list numbers)
  \gdef\journal##1,##2,##3,##4 {                % Journal reference
     {\sl ##1~}{\bf ##2~}(##3)~##4. }}

\def\refstyleprnp{              % Input like pr, output like np!!
  \gdef\refto##1{ [##1]}                        % Reference in text []
  \gdef\refis##1{\item{##1)\ }}                 % Reference list numbers)
  \gdef\journal##1, ##2, ##3, 1##4##5##6{       % Journal reference
    {\sl ##1~}{\bf ##2~}(1##4##5##6) ##3}}

\def\pr{\journal Phys. Rev. }
\def\pra{\journal Phys. Rev. A }
\def\prb{\journal Phys. Rev. B }
\def\prc{\journal Phys. Rev. C }
\def\prd{\journal Phys. Rev. D }
\def\prl{\journal Phys. Rev. Lett. }
\def\jmp{\journal J. Math. Phys. }
\def\rmp{\journal Rev. Mod. Phys. }
\def\cmp{\journal Comm. Math. Phys. }
\def\np{\journal Nucl. Phys. }
\def\npb{\journal Nucl. Phys. B }
\def\pl{\journal Phys. Lett. }
\def\apj{\journal Astrophys. Jour. }
\def\apjl{\journal Astrophys. Jour. Lett. }
\def\annp{\journal Ann. Phys. (N.Y.) }
\def\endreferences{\body}

\def\figurecaptions             % Begin figure captions
  {\endpage
   \beginparmode
   \head{Figure Captions}
%  \parskip=24pt plus 3pt \everypar={\hangindent=4em}
}

\def\endfigurecaptions{\body}

\def\endpage                    %  Eject a page
  {\vfill\eject}

\def\endpaper                   %  Ways to say goodbye
  {\endmode\vfill\supereject}
\def\endjnl
  {\endpaper}
\def\endit
  {\endpaper\end}


%%
%%      Various little user definitions
%%

\def\ref#1{Ref.~#1}                     %       for inline references
\def\Ref#1{Ref.~#1}                     %       ditto
\def\[#1]{[\cite{#1}]}
\def\cite#1{{#1}}
\def\Equation#1{Equation~(#1)}          % For citation of equation numbers
\def\Equations#1{Equations~(#1)}        %       ditto
\def\Eq#1{Eq.~(#1)}                     %       ditto
\def\Eqs#1{Eqs.~(#1)}                   %       ditto
\def\(#1){(\call{#1})}
\def\call#1{{#1}}
\def\taghead#1{}
\def\frac#1#2{{#1 \over #2}}
\def\half{{\frac 12}}
\def\third{{\frac 13}}
\def\fourth{{\frac 14}}
\def\12{{1\over2}}
\def\eg{{\it e.g.,\ }}
\def\Eg{{\it E.g.,\ }}
\def\ie{{\it i.e.,\ }}
\def\Ie{{\it I.e.,\ }}
\def\etal{{\it et al.\ }}
\def\etc{{\it etc.\ }}
\def\via{{\it via\ }}
\def\cf{{\it cf.\ }}
\def\sla{\raise.15ex\hbox{$/$}\kern-.57em}
\def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill}
\def\twiddle{\lower.9ex\rlap{$\kern-.1em\scriptstyle\sim$}}
\def\bigtwiddle{\lower1.ex\rlap{$\sim$}}
\def\gtwid{\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}}}
\def\ltwid{\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}}
\def\square{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt
   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt}
\def\tdot#1{\mathord{\mathop{#1}\limits^{\kern2pt\ldots}}}
\def\super#1{$^{#1}$}
\def\pmb#1{\setbox0=\hbox{#1}%
  \kern-.025em\copy0\kern-\wd0
  \kern  .05em\copy0\kern-\wd0
  \kern-.025em\raise.0433em\box0 }
\def\qed{\vrule height 1.2ex width 0.5em}
\def\dalemb{{\cal t}}                           %%dalembertian, used to be \box
\def\grad{\nabla}
\def\const{{\rm const}}
\def\s{{\,\rm s}}
\def\cm{{\,\rm cm}}
\def\g{{\,\rm g}}
\def\coeff#1#2{\textstyle{#1\over #2}}
\def\partder#1#2{{{\partial #1}\over{\partial #2}}}
\def\bra#1{\left\langle #1\right|}
\def\ket#1{\left| #1\right\rangle}
\def\vev#1{\left\langle #1\right\rangle}
\def\Tr{\mathop{\rm Tr}\nolimits}
\def\Mpc{{\,h^{-1}\rm Mpc}}
\def\gev{{\,\rm GeV}}
\def\ev{{\,\rm eV}}
\def\ApJ{{\it Ap. J.}}
\def\ApJLett{{\it Ap. J. $($Letters$)$}}
\def\ApJSuppl{{\it Ap. J. Suppl.}}
\def\Nature{{\it Nature}}
\def\PRD{{\it Phys. Rev. D}}
\def\MNRAS{{\it M. N. R. A. S.}}
\def\uf{Department of Physics\\University of Florida\\Gainesville, FL 32611}
\def\cugrant{This research was supported in part by the United States
Department of Energy under contract DE-AC02-76ER02271.}
\def\crgrant{This research was supported in part by the United States
Department of Energy under contracts DE-AC02-76ER02271 and
FG05-86-ER40272.}
%%
%%              EQNORDER.TEX                    11/05/85        Doug E.
%%
%%      This macro package is intended for use with JNL.
%%      It will automatically order and sort the equations in a paper
%%      by order of appearance.  To use, say \input eqnorder
%%      after \input jnl (and after all definitions of \eqno etc.,
%%      but before any use of \eqno etc.  Use \() to cite equations
%%      in the text.  Use \eqno() or \tag to put the numbers on displayed
%%      equations; or use &() with \eqalignno{} as explained in the TeXBOOK.
%%
%%      EQNORDER depends on the macro \() to refer to equations in the
%%      text; use it as Equation \() or Eq. \() or Eqs. \(), etc.
%%      EQNORDER also contains a macro \call{} which can be used to refer
%%      equations; e.g., ``Equation \call{19} blah...'' will produce
%%      output ``Equation 19 blah''.  Multiple citations must be separated
%%      by commas.  E.g., \(24,26,27) and \call{3,7} are legal.  A sequence
%%      of equation numbers can be referred to by, e.g., \(3-7) which means
%%      the same as (3,4,5,6,7).
%%
%%      Equation ``numbers'' can actually be any alphanumeric string;
%%      e.g., equation \tag Schroedinger $$ can be referred to by
%%      \(Schroedinger).  In fact, if you expect to renumber the equations,
%%      it is actually easier and less confusing to tag them with names
%%      rather than numbers.
%%
%%      There is one big rule:  You cannot refer to an equation before you
%%      display it.  There is a limited loophole:  You can refer to the
%%      first, second or third equation number just below where you are
%%      as \(+1), \(+2), or \(+3).  In the same way, the equation number
%%      just above can be called \(0), and the three preceding numbers
%%      \(-1), \(-2), \(-3).
%%
%%      TeX keeps the equation number as a count \tagnumber.  \tagnumber
%%      is initially 0, and it is incremented by 1 just BEFORE it is used
%%      to tag a displayed equation.  You are free to reset \tagnumber,
%%      which you can do just by writing e.g. \tagnumber=23.
%%
%%      If you label a displayed equation with a null number, \tag $$
%%      or \eqno() or &(), an incremented \tagnumber will be generated for
%%      the equation, but the only way to refer to that equation is
%%      via the \(+n) or \(0) or \(-n) notation.
%%
%%      In long papers, the author often numbers the equations anew
%%      in each section in the style \tag 6.1 $$, \tag 6.2 $$, and so
%%      forth;  the equations are then referred to by \(6.1) etc.
%%      One reason for doing this is to minimize chaos when equations
%%      have to be renumbered --- but this is what EQNORDER already does!
%%      If you still want to use such a style, just declare \taghead{6.}
%%      for example at the beginning of Section 6.  The effect of \taghead
%%      is to reset \tagnumber to 0, and to save the argument of \taghead
%%      to so that it can be put in front of each equation number in the
%%      output.
%%
%%      Sometimes a sequence of displayed equations is labelled with
%%      the same number (e.g., 25) and then sublabeled a,b,c,d...  Use
%%      the form \tag 25 a$$, \tag 25 b$$,... or \eqno(25 a)$$,
%%      \eqno(25 b)$$,... to put the numbers on such a sequence;  note
%%      the space.  Such equations can be referred to in the text either
%%      as \(25) or \(25 a).  Also such constructions as \(25 a,25 b,26)
%%      are legal.  Again, note carefully the position of the space.  The
%%      effect of the space is to mark the end of the equation number that
%%      TeX keeps track of;  the following string (a or b or ...) is just
%%      put out without modification.  Thus constructions like \tag 25 ' $$
%%      and \tag 25 '' $$ are legal.
%%
%%      If you have your own pet macros to call equations such as, e.g.,
%%      \def\eqnpet#1{($#1$))}, you can bring it to the attention of
%%      REFORDER so all \eqnpet's will be properly calld simply by
%%      declaring ``\callall\eqnpet'' after \eqnpet is defined and
%%      after \input eqnorder.  This has the effect of redefining the macro
%%      as e.g., \def\eqnpet#1{($\call{#1}$)}.  (Such \callall'ed macros
%%      must have exactly one argument #1, as in \eqnpet.)

\catcode`@=11
\newcount\tagnumber\tagnumber=0

\immediate\newwrite\eqnfile
\newif\if@qnfile\@qnfilefalse
\def\write@qn#1{}
\def\writenew@qn#1{}
\def\w@rnwrite#1{\write@qn{#1}\message{#1}}
\def\@rrwrite#1{\write@qn{#1}\errmessage{#1}}

\def\taghead#1{\gdef\t@ghead{#1}\global\tagnumber=0}
\def\t@ghead{}

\expandafter\def\csname @qnnum-3\endcsname
  {{\t@ghead\advance\tagnumber by -3\relax\number\tagnumber}}
\expandafter\def\csname @qnnum-2\endcsname
  {{\t@ghead\advance\tagnumber by -2\relax\number\tagnumber}}
\expandafter\def\csname @qnnum-1\endcsname
  {{\t@ghead\advance\tagnumber by -1\relax\number\tagnumber}}
\expandafter\def\csname @qnnum0\endcsname
  {\t@ghead\number\tagnumber}
\expandafter\def\csname @qnnum+1\endcsname
  {{\t@ghead\advance\tagnumber by 1\relax\number\tagnumber}}
\expandafter\def\csname @qnnum+2\endcsname
  {{\t@ghead\advance\tagnumber by 2\relax\number\tagnumber}}
\expandafter\def\csname @qnnum+3\endcsname
  {{\t@ghead\advance\tagnumber by 3\relax\number\tagnumber}}

\def\equationfile{%
  \@qnfiletrue\immediate\openout\eqnfile=\jobname.eqn%
  \def\write@qn##1{\if@qnfile\immediate\write\eqnfile{##1}\fi}
  \def\writenew@qn##1{\if@qnfile\immediate\write\eqnfile
    {\noexpand\tag{##1} = (\t@ghead\number\tagnumber)}\fi}
}

\def\callall#1{\xdef#1##1{#1{\noexpand\call{##1}}}}
\def\call#1{\each@rg\callr@nge{#1}}

\def\each@rg#1#2{{\let\thecsname=#1\expandafter\first@rg#2,\end,}}
\def\first@rg#1,{\thecsname{#1}\apply@rg}
\def\apply@rg#1,{\ifx\end#1\let\next=\relax%
\else,\thecsname{#1}\let\next=\apply@rg\fi\next}

\def\callr@nge#1{\calldor@nge#1-\end-}
\def\callr@ngeat#1\end-{#1}
\def\calldor@nge#1-#2-{\ifx\end#2\@qneatspace#1 %
  \else\calll@@p{#1}{#2}\callr@ngeat\fi}
\def\calll@@p#1#2{\ifnum#1>#2{\@rrwrite{Equation range #1-#2\space is bad.}
\errhelp{If you call a series of equations by the notation M-N, then M and
N must be integers, and N must be greater than or equal to M.}}\else%
 {\count0=#1\count1=#2\advance\count1
by1\relax\expandafter\@qncall\the\count0,%
  \loop\advance\count0 by1\relax%
    \ifnum\count0<\count1,\expandafter\@qncall\the\count0,%
  \repeat}\fi}

\def\@qneatspace#1#2 {\@qncall#1#2,}
\def\@qncall#1,{\ifunc@lled{#1}{\def\next{#1}\ifx\next\empty\else
  \w@rnwrite{Equation number \noexpand\(>>#1<<) has not been defined yet.}
  >>#1<<\fi}\else\csname @qnnum#1\endcsname\fi}

\let\eqnono=\eqno
\def\eqno(#1){\tag#1}
\def\tag#1$${\eqnono(\displayt@g#1 )$$}

\def\aligntag#1\endaligntag
  $${\gdef\tag##1\\{&(##1 )\cr}\eqalignno{#1\\}$$
  \gdef\tag##1$${\eqnono(\displayt@g##1 )$$}}

\let\eqalignnono=\eqalignno

\def\eqalignno#1{\displ@y \tabskip\centering
  \halign to\displaywidth{\hfil$\displaystyle{##}$\tabskip\z@skip
    &$\displaystyle{{}##}$\hfil\tabskip\centering
    &\llap{$\displayt@gpar##$}\tabskip\z@skip\crcr
    #1\crcr}}

\def\displayt@gpar(#1){(\displayt@g#1 )}

\def\displayt@g#1 {\rm\ifunc@lled{#1}\global\advance\tagnumber by1
        {\def\next{#1}\ifx\next\empty\else\expandafter
        \xdef\csname @qnnum#1\endcsname{\t@ghead\number\tagnumber}\fi}%
  \writenew@qn{#1}\t@ghead\number\tagnumber\else
        {\edef\next{\t@ghead\number\tagnumber}%
        \expandafter\ifx\csname @qnnum#1\endcsname\next\else
        \w@rnwrite{Equation \noexpand\tag{#1} is a duplicate number.}\fi}%
  \csname @qnnum#1\endcsname\fi}

\def\ifunc@lled#1{\expandafter\ifx\csname @qnnum#1\endcsname\relax}

\let\@qnend=\end\gdef\end{\if@qnfile
\immediate\write16{Equation numbers written on []\jobname.EQN.}\fi\@qnend}

\catcode`@=12

%% DEBUG
%%\def\see#1 {\expandafter\show\csname#1\endcsname}
%%
%%              REFORDER.TEX                    6/7/85  Doug E.
%%
%%      This macro package is intended for use with JNL.
%%      It will automatically order and sort the references in a paper
%%      by order of first citation.(!!)  To use, say \input reforder
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%%%%%%%%%%%%%%PAPER STARTS HERE%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\cutp{551}
\title{Exchange Operator Formalism for Integrable Systems of Particles}
\author{\bf Alexios P. Polychronakos}
\affil{Pupin Physics Laboratories, Columbia University,
New York, NY 10027}
\abstract{We formulate one dimensional many body integrable systems in
terms of a new set of phase space variables involving exchange operators.
The hamiltonian in these variables assumes a decoupled form.
This greatly simplifies the derivation of the conserved charges and
the proof of their commutativity at the quantum level.}


\endtopmatter

\body
\baselineskip=20pt

In one spatial dimension a class of integrable many-body systems is
known, referred to as the Calogero-Sutherland-Moser systems$^{1-3}$. They
constitute of many identical nonrelativistic particles interacting
through two-body potentials of the inverse square type and its
generalizations, namely the inverse sine square and the Weierstrass
two-body potentials. These models are related to root systems of
$A_n$ algebras$^4$. Corresponding systems related to root systems of
other algebras exist, but their two-body potentials are not translationally
and/or permutation invariant$^5$. We will restrict ourselves to the
$A_n$ systems. For a comprehensive review of these systems see ref$.$ 5.

Many of the above systems admit a matrix formulation$^{5,6}$. Using this
formulation, a generalization of these systems was found recently
where the particles also feel external potentials of particular types$^7$.
These systems, apart from their purely mathematical interest, are
also of significant physical interest, since they are relevant to
fractional statistics and anyons$^8$, spin chain models$^9$, soliton wave
propagation$^{10}$ and, indirectly, to nonperturbative two-dimensional
quantum gravity$^{11}$.

The purpose of this paper is to present an ``exchange operator"
formalism for these systems which renders their integrable structure
explicit. Specifically, we will write generalized momentum operators
in terms of which the integrals of motion assume a ``decoupled" form.
This will allow for an easy proof of commutativity at the quantum level.

Let $\{ x_i , p_i \}$, $i=1, \dots N$ be the coordinates and momenta
of $N$ one-dimensional quantum mechanical particles, obeying canonical
commutation relations, and let $M_{ij}$ be the particle permutation
operators, obeying
$$
M_{ij} = M_{ji} = M_{ij}^\dagger \,\,,\,\,\,\, M_{ij}^2 = 1
\eqno(a1)$$
$$
M_{ij} A_j = A_i M_{ij} \,\,,\,\,\,\,
M_{ij} A_k = A_k M_{ij} \,\,,\,\,\,\, {\rm for} ~ k \neq i,j
\eqno(a)$$
where $A_i$ is any operator (including $M_{ij}$ themselves) carrying
one or more particle indices. Then define the ``coupled" momentum operators
$$
\pi_i = p_i + i \sum_{j \neq i} V_{ij} M_{ij} \,\,,\,\,\,\,
V_{ij} \equiv V( x_i - x_j )
\eqno(b)$$
with $V(x)$ an as yet undetermined function. Note that the $\pi_i$ are
``good" one-particle operators, that is they satisfy \(a), since the
remaining particle indices in \(b) appear in a permutation symmetric way.
If we impose the hermiticity condition on $\pi_i$
$$
\pi_i = \pi_i^\dagger
\eqno(c)$$
then $V(x)$ must obey
$$
V(x)^\dagger = - V(-x)
\eqno(d)$$
Consider now a hamiltonian for the system which takes a free form
in terms of $\pi_i$'s, that is,
$$
H = \half \sum_i \pi_i^2
\eqno(e)$$
In terms of the original phase space variables, $H$ takes the form
$$
H = \half \sum_i p_i^2 +
\half \sum_{i \neq j} \left [ i V_{ij} ( p_i + p_j ) M_{ij}
+ V^\prime_{ij} M_{ij} + V_{ij}^2 \right ]
- {1 \over 6} \sum_{i \neq j \neq k \neq i} V_{ijk} M_{ijk}
\eqno(f)$$
In the above, $V^\prime (x)$ is the derivative of $V(x)$ and we
defined
$$
V_{ijk} = V_{ij} V_{jk} + V_{jk} V_{ki} + V_{ki} V_{ij}
\eqno(g)$$
$M_{ijk}$ is the generator of cyclic permutations in three indices,
that is,
$$
M_{ijk} = M_{jki} = M_{kij} = M_{jik}^\dagger = M_{ij} M_{jk}
\eqno(h)$$
If we demand that the above expression for $H$ become the sum of an
ordinary kinetic term and potential terms, the terms linear in $p_i$
should drop, and this will happen if
$$
V(-x) = -V(x)
\eqno(k)$$
Finally, if we want the above hamiltonian to contain only two-body
potentials, the function $V(x)$ should satisfy
$$
V(x) V(y) + V(y) V (z) + V(z) V(x) = W(x) + W(y) + W(z)
\,\,,\,\,\,\, {\rm for} ~ x+y+z = 0
\eqno(l)$$
where $W(x)$ is a new symmetric function. $H$ takes then the form
$$
H = \half \sum_i p_i^2 +
\sum_{i < j} \left [ V_{ij}^2  + V^\prime_{ij} M_{ij}
- W_{ij} \sum_{k \neq i,j} M_{ijk} \right ]
\eqno(m)$$
and the commutator of $\pi$'s is evaluated to be
$$
[ \pi_i , \pi_j ] = \sum_{k \neq i,j} V_{ijk} [ M_{ijk} - M_{jik} ]
\eqno(n)$$

Eq$.$ \(l) is a well-known
functional equation for $V$ which also emerges as a condition for
factorizability of the ground state of many-body systems$^3$. It can be
readily solved through a small-$x$ expansion and all its solutions are
available$^{12}$. Here we consider in sequence the solutions of most interest.

Assume first that $W(x) = 0$. Then \(l) is solved by
$$
V(x) = {l \over x}
\eqno(p)$$
with $l$ a real parameter, and the hamiltonian takes the form
$$
H = \half \sum_i p_i^2 + \sum_{i > j} {l (l - M_{ij} ) \over ( x_i
- x_j )^2 }
\eqno(q)$$
Define the totally symmetric quantities
$$
I_n = \sum_i \pi_i^n
\eqno(o)$$
Since in this case $V_{ijk} = 0$ we see from \(n) that the $\pi_i$
{\it commute} and
therefore the $I_n$ also commute. Moreover, since they commute with
all $M_{ij}$, their projections in the bosonic or fermionic subspaces of
the Hilbert space also commute. In these subspaces the $M_{ij}$ simply
become $\pm 1$ and $\half I_2 = H$ becomes the hamiltonian of a set
of particles interacting through inverse square potentials of strength
$l(l \mp 1)$. Further, the higher quantities $I_n$ projected in these
subspaces become the integrals of motion of the above hamiltonian.
Therefore these integrals commute in these subspaces.

To show that these integrals commute in the full Hilbert space it
suffices to notice that they are local operators, since they involve
derivatives of at most $n$-th degree. To know their action on the
wavefunction at any point it suffices to know the wavefunction in a small
neighborhood around that point. Therefore, the fact that they commute
cannot depend on global information on the wavefunction, namely its
symmetry or antisymmetry. Thus, if they commute for bosonic or fermionic
states they must commute unconditionally.

In the case $W(x) = \,$constant, the solution for $V$ is
$$
V(x) = {l \cot ax} \,\,,\,\,\,\, {\rm or}\,\,\,\, V(x) = {l \coth ax}
\eqno(pp)$$
depending on the sign of the constant. Choosing the positive sign,
and making $a=1$ by appropriate choice of units, we have in subspaces
of definite symmetry
$$
H = \half \sum_i p_i^2 + \sum_{i<j} {l(l \mp 1) \over \sin^2 ( x_i
- x_j )} - l^2 {N(N^2 -1) \over 6}
\eqno(qq)$$
This is the Sutherland model of particles interacting through inverse
sine square potentials. In this case $V_{ijk} = l^2$ and thus the
$\pi_i$'s do not commute. To show the existence of conserved quantities,
define the new operators
$$
\p_i = \pi_i + l \sum_{j \neq i} M_{ij}
\eqno(r)$$
which are also ``good" one-particle operators and obey the
commutation relations
$$
[ \p_i , \p_j ] = 2l ( \p_i M_{ij} - M_{ij} \p_i )
\eqno(s)$$
The corresponding conserved quantities $\I_n$ constructed
from $\p_i$ can be shown to commute as follows:
$$
\eqalign{
[ \p_i^n , \p_j ] &= \sum_{\alpha = 0}^{n-1} \, \p_i^\alpha
[ \p_i , \p_j ] \, \p_i^{n-\alpha -1} \cr
&= 2l ( \p_i^n M_{ij} - M_{ij} \p_i^n ) = 2l ( M_{ij} \p_j^n -
\p_j^n M_{ij} ) \cr }
\eqno(t)$$
and thus
$$
\eqalign{
[ \I_n , \I_m ] &= \sum_{i,j} [ \p_i^n , \p_j^m ]
= \sum_{i,j} \sum_{\alpha = 0}^{m-1} \p_j^\alpha \, [ \p_i^n ,
\p_j ] \p_j^{m-\alpha -1} \cr
&= 2l \sum_{i,j} \sum_{\alpha = 0}^{m-1} \left( \p_j^\alpha M_{ij}
\p_j^{m+n-\alpha -1} - \p_j^{\alpha +n} M_{ij} \p_j^{m-\alpha- 1}
\right) \cr
&= 2l \sum_{i,j} \left( \sum_{\alpha = 0}^{m-1} - \sum_{\alpha = n}^{m+n-1}
\right) \p_j^\alpha M_{ij} \p_j^{m+n-\alpha -1} \cr }
\eqno(u)$$
Antisymmetrizing \(u) explicitly in $n$ and $m$, we get
$$
[ \I_n , \I_m ] = l \sum_{i,j} \left( \sum_{\alpha = 0}^{m-1} -
\sum_{\alpha = n}^{m+n-1} - \sum_{\alpha = 0}^{n-1} +
\sum_{\alpha = m}^{m+n-1} \right) \p_j^\alpha M_{ij}
\p_j^{m+n-\alpha -1} = 0
\eqno(v)$$
Therefore the $\I_n$ commute. In subspaces of definite symmetry,
on the other hand, they reduce to combinations of $I_n$, e.g.,
$$
\I_1 = I_1 \mp l N(N-1) \,\,,\,\,\,\,
\I_2 = I_2 \mp 2l (N-1) I_1 + l^2 N(N-1)^2 \,\,,\,\,\,\,{\rm e.t.c.}
\eqno(w)$$
Therefore the $I_n$ commute as well. By repeating the above argument,
or simply by analytic continuation, we can also deal with the $\coth ax$
solution in \(pp) which corresponds to the inverse hyperbolic sine
square potential.

A singular solution of \(l) for $W(x)$ a negative constant is
$$
V(x) = l \, {\rm sign} (x)
\eqno(ww)$$
which leads to the well-known system of particles with mutual
delta-function potentials$^{13}$. It can be treated as above, with
some extra care to possible singularities.

Finally, consider the operators
$$
h_i = ( \pi_i + i \omega x_i ) ( \pi_i - i \omega x_i )
\equiv a_i^\dagger a_i
\eqno(x)$$
for $V(x)$ as in \(p). Using the
commutativity of the $\pi_i$ in this case as well as
$$
[ x_i , \pi_j ] = i \delta_{ij} \bigl( 1 + l \sum_{k \neq i} M_{ik}
\bigr) - i (1- \delta_{ij} ) l M_{ij}
\eqno(yy)$$
we find
$$
[ a_i , a_j ] = [ a_i^\dagger , a_j^\dagger ] = 0
$$
$$
[ a_i , a_j^\dagger ] = -2l \omega M_{ij} ~~~ {\rm (for ~ i \neq j)}
\eqno(y)$$
and thus
$$
[ h_i , h_j ] = -2l \omega ( h_i M_{ij} - M_{ij} h_i )
\eqno(z)$$
We observe that the commutation relations of the $h_i$ are similar to the
ones of $\p_i$ in \(s). Therefore the quantities $I_n$ defined now
$$
I_n = \sum_i h_i^n
\eqno(aa)$$
can be shown to commute in a way similar to the one for the $\I_n$.
In particular, the hamiltonian $H = \half I_1$ in the bosonic or fermionic
subspace becomes
$$
H = \half \sum_i p_i^2 + \sum_{i > j} {l (l \mp 1 )
\over ( x_i - x_j )^2 } + \half \sum_i \omega^2 x_i^2
-N {\omega \over 2} \pm l {N(N-1) \over 2} \omega
\eqno(bb)$$
This is the Calogero model of harmonic plus inverse square potentials,
and we have derived its integrals of motion.

Notice that the constant terms appearing in \(qq) and \(bb) are the
negative of the ground state energy of the corresponding hamiltonians,
thus shifting the ground state energy to zero. In fact we can easily
find the ground state wavefunction noticing that the above $H$'s are
positive definite and thus if we can find states $\psi_{_S}$ and
$\psi_{_C}$ satisfying
$$
\pi_i \psi_{_S} = 0 \,\,,\,\,\,\, {\rm or} \,\,\,\,
a_i \psi_{_C} = 0
\eqno(cc)$$
these will be the ground state. Taking them further to be bosonic,
they thus must satisfy
$$
{\partial_i \psi_{_S} \over \psi_{_S}} = \sum_{j \neq i} l \cot ( x_i - x_j )
\eqno(dd)$$
or
$$
{\partial_i \psi_{_C} \over \psi_{_C}} = - \omega x_i + \sum_{j \neq i}
{l \over x_i - x_j }
\eqno(ee)$$
respectively. By integrating \(dd) and \(ee) we easily find the
Sutherland and Calogero ground state wavefunctions
$$
\psi_{_S} = \prod_{i<j} | \sin ( x_i - x_j ) |^l \,\,,\,\,\,\,
\psi_{_C} = \prod_{i<j} | x_i - x_j |^l \, e^{- \half \omega \sum_i x_i^2}
\eqno(ff)$$

In conclusion, we see that the above formalism identifies a better
set of phase space ``momentum" variables, which allow for an effortless
and relatively straightforward derivation of the integrability of
these systems. It is also remarkable that the above proofs work
directly in the quantum regime (the exchange operators $M_{ij}$
have no classical counterpart), thus circumventing the operator
ordering problems encountered when constructing the quantum
integrals of motion starting from the classical standpoint. It is
hoped that this formalism will provide an easy proof of the quantum
integrability of the systems recently found in ref$.$ 7, or
even that it will lead to as yet uncovered new integrable systems.
We hope to come back to these issues in a future publication.

This research was supported in part by a United States Department
of Energy grant.


\bigskip
\bigskip
\centerline {\bf REFERENCES}

\noindent
\item{$^1$}
F. Calogero, {\it J. Math. Phys.} {\bf 10}, 2191 and 2197 (1969) and
{\bf 12}, 419 (1971).

\item{$^2$}
J. Moser, {\it Adv. Math.} {\bf 16}, 1 (1975); F. Calogero, {\it Lett.
Nuovo Cim.} {\bf 13}, 411 (1975); F. Calogero and C. Marchioro, {\it Lett.
Nuovo Cim.} {\bf 13}, 383 (1975).

\item{$^3$}
B. Sutherland, {\it Phys. Rev.} {\bf A4}, 2019 (1971) and {\bf A5},
1372 (1972); {\it Phys. Rev. Lett.} {\bf 34}, 1083 (1975).

\item{$^4$}
M.A. Olshanetsky and A.M. Perelomov, {\it Invent. Math.} {\bf 37},
93 (1976).

\item{$^5$}
M.A. Olshanetsky and A.M. Perelomov, {\it Phys. Rep.} {\bf 71}, 314 (1981)
and {\bf 94}, 6 (1983).

\item{$^6$}
D. Kazdan, B. Kostant and S. Sternberg, {\it Comm. Pure Appl.
Math.} {\bf 31}, 481 (1978).

\item{$^7$}
A.P. Polychronakos, {\it Phys. Lett.} {\bf B266}, 29 (1991); Columbia
preprints CU-TP-527, July 1991, to appear in Phys. Lett. B and
CU-TP-537, October 1991, to appear in Phys. Lett. B.

\item{$^8$}
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