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\begin{center}
{\Large  \bf Quantum Effects of an Extra Compact
 Dimension on the Wave Function of the Universe}

\vspace{5mm}

\end{center}

\vspace{5 mm}

\begin{center}
{\bf Guowen Peng\footnote[1]{Department of Physics, 
Dalian University of Technology,
Dalian, 116024, People's  Republic of China.},
Hongya Liu\footnotemark[1]$^{,}$\footnote[2]
{\noindent Corresponding author, E-mail: hyliu@dlut.edu.cn.
}, and Qiuyang Zhang\footnotemark[1]}

\vspace{3mm}

\end{center}

\vspace{1cm}

\begin{center}
{\bf Abstract}
\end{center}
\baselineskip 23pt
We extend the direct quantum approach of the standard FRW cosmology 
from $4D$ to $5D$ and obtain a Hamiltonian formulation for a wave-like
 $5D$ FRW cosmology. Using a late-time approximation we isolate out a 
$y$-part from the full wave function of the $5D$ Universe. Then we find that 
the compactness of the fifth dimension $y$ yields a quantized spectrum for 
the momentum $P_{5}$ along the fifth dimension, and we show that the 
whole space-part of the wave function of the $5D$ Universe satisfies a 
two-dimensional Schr\"{o}dinger equation.

\vspace{1cm}
{\noindent \bf Key words:} Quantum cosmology; 
 Hamiltonian formulaiton.
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\section{Introduction}

Quantum cosmology has received extensive studies (Wheeler et al. 1964;
Novello et al. 1995; Anchordoqui et al. 2000). Recently, a direct quantum approach was used to derive
Hamiltonian formulation of the standard Friedmann-Robertson-Walker (FRW)
cosmological models (Elbaz et al. 1997; Novello et al. 1996). The method is
based on assumptions of validity of the classical FRW equations and a
transformation from a pair of physical variables $(\rho ,\theta )$ (the
density of matter and the inverse of the Hubble radius), which describe the
dynamics of a spatially homogeneous and isotropic perfect fluid, to another
pair of canonical variables $(q,p)$ (roughly the radius of the Universe and
its rate of change). In this way, a canonical description of quantum FRW
cosmology was obtained. In this paper, we wish to extend this procedure to a 
$5D$ FRW cosmology and discuss quantum effects of the fifth compact
dimension on the wave function of the Universe.

The arrangement of this paper is as follows. In section 2, we study a class
of $5D$ wave-like cosmological solutions and derive Hamiltonian formulation
of the $5D$ FRW\ cosmology. In section 3, we use a late-time approximation
to separate the $5D$ wave function of the Universe and isolate out the part
of the fifth dimension. In section 4, we study quantization of the wave
function due to the compactness of the fifth dimension. Section 5 is a
conclusion.

\section{Hamiltonian Formulation of 5D Cosmology}

We consider a FRW type $5D$ metric

\begin{equation}
dS^{2}=-B^{2}(t,y)dt^{2}+A^{2}(t,y)(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega
^{2})+C^{2}(t,y)dy^{2}\;,  \tag{2.1}
\end{equation}
where $d\Omega ^{2}\equiv d\theta ^{2}+\sin ^{2}\theta d\varphi ^{2}$ and
the coordinates $x^{A}=(t,r\theta \varphi ,y)$ (Here and throughout this
paper, lowercase Greek letters run $0,123$ and uppercase Latin letters run $%
0,123,5$). Using the $4D$ part of the $5D$ metric (2.1) we can calculate all
the non-vanishing components of the $4D$ Einstein tensor $^{(4)}G_{\alpha
\beta },$ which are as follows: 
\begin{equation}
^{(4)}G_{0}^{0}=3\frac{\dot{A}^{2}}{A^{2}B^{2}}+\frac{3k}{A^{2}}\;, 
\tag{2.2}
\end{equation}
\begin{equation}
^{(4)}G_{1}^{1}=\;^{(4)}G_{2}^{2}=\;^{(4)}G_{3}^{3}=-2\frac{\ddot{A}}{AB^{2}}%
-\frac{\dot{A}^{2}}{A^{2}B^{2}}-\frac{k}{A^{2}}+2\frac{\dot{A}\dot{B}}{AB^{3}%
}\;.  \tag{2.3}
\end{equation}
where a dot denotes partial derivative with respect to time $t$.

It is known that solutions which are empty in $5D$ may have matter in $4D$
(Wesson 1999; Overduin \& Wesson 1997). Therefore, one can define an induced 
$4D$ energy-momentum tensor $T_{\alpha \beta \text{ }}$as 
\begin{equation}
T_{\alpha \beta }\equiv \;^{(4)}G_{\alpha \beta }\;.  \tag{2.4}
\end{equation}
It is found that this $T_{\alpha \beta }$ can take the form of a perfect
fluid, 
\begin{equation}
T_{\alpha \beta }=(\rho +P)u_{\alpha }u_{\beta }+Pg_{\alpha \beta }\;, 
\tag{2.5}
\end{equation}
where $\rho $ and $P$ are the energy density and pressure of the induced
matter, respectively.

Now we let the equation of state being 
\begin{equation}
P=\gamma \rho \;,  \tag{2.6}
\end{equation}
where $\gamma =0$ and $\gamma =\frac{1}{3}$ represent matter-dominated and
radiation-dominated eras, respectively. Then substituting (2.2), (2.3),
(2.5) and (2.6) in (2.4), we obtain the well-known Friedmann equation 
\begin{equation}
3\frac{\dot{A}^{2}}{A^{2}}=B^{2}\rho -\frac{3kB^{2}}{A^{2}}\;,  \tag{2.7}
\end{equation}
and the Raychaudhuri one 
\begin{equation}
3\frac{\ddot{A}}{A}-3\frac{\dot{A}\dot{B}}{AB}+\frac{B^{2}}{2}(1+3\gamma
)\rho =0\;.  \tag{2.8}
\end{equation}
The conservation law for the energy-momentum $T_{\alpha \beta }$ gives us
another equation 
\begin{equation}
\dot{\rho}+3(1+\gamma )\frac{\dot{A}}{A}\rho =0\;.  \tag{2.9}
\end{equation}

An exact $5D$ cosmological solution satisfying the empty $5D$ equations $%
R_{AB}=0$ is (Liu \& Wesson 1994) 
\begin{equation}
dS^2=-B^2(u)dt^2+A^2(u)(dr^2+r^2d\Omega ^2)+B^2(u)dy^2\;,  \tag{2.10}
\end{equation}
where $k$ is chosen to be zero,

\begin{equation}
\left. 
\begin{array}{c}
A(u)=(hu)^{\frac 1{2+3\gamma }}\;, \\ 
\;\;\;\;B(u)=(hu)^{-\frac{1+3\gamma }{2(2+3\gamma )}}\;,
\end{array}
\right.   \tag{2.11}
\end{equation}
and $u\equiv t-y.$ The energy density $\rho $ and pressure $P$ are 
\begin{equation}
\rho =\frac{3h^2}{(2+3\gamma )^2}A^{-3(1+\gamma )},\;\;P=\gamma \rho \;. 
\tag{2.12}
\end{equation}
This is a wave-like cosmological solution which can be interpreted as a
shock wave propagating along the fifth dimension (Wesson et al. 2000).

Now let us return to equations (2.7)-(2.9). There are three basic equations
from which we want to derive the Hamiltonian formulation. Note that since $%
A=A(u)$ and $u=t-y,$ we have $\dot{A}=\partial A/\partial t=dA/du.$
Therefore, by using a transformation 
\begin{equation}
\tilde{u}=\int B(u)du\;,  \tag{2.13}
\end{equation}
we can reduce equations (2.7)-(2.9) to 
\begin{equation}
\frac{3\stackrel{\ast }{A}^{2}}{A^{2}}=\rho \;,  \tag{2.14}
\end{equation}
\begin{equation}
\frac{3\stackrel{\ast \ast }{A}}{A}+\frac{1+3\gamma }{2}\rho =0\;, 
\tag{2.15}
\end{equation}
\begin{equation}
\stackrel{\ast }{\rho }+3(1+\gamma )\frac{\stackrel{\ast }{A}}{A}\rho =0\;, 
\tag{2.16}
\end{equation}
where an asterisk denotes derivative with respect to $\tilde{u}.$ These
three equations (2.14)-(2.16) are of the same forms as in the standard $4D$
FRW solutions algebraically. Therefore, we can follow the procedure given by
Elbaz et al. (Elbaz et al. 1997; see also Novello et al. 1996) to derive the
canonical formulation of the cosmology.

Firstly, we introduce an expansion parameter $\Theta ,\Theta $ $\equiv 3%
\frac{\stackrel{*}{A}}A$, which brings (2.14)-(2.16) to 
\begin{equation}
\frac 13\Theta ^2=\rho \;,  \tag{2.17}
\end{equation}
\begin{equation}
\stackrel{\ast }{\Theta }+\frac 13\Theta ^2+\frac 12(1+3\gamma )\rho =0\;, 
\tag{2.18}
\end{equation}
\begin{equation}
\stackrel{\ast }{\rho }+(1+\gamma )\rho \Theta =0\;.  \tag{2.19}
\end{equation}
Then we choose a new set of canonical variables $(q,p)$ with

\begin{equation}
\left. 
\begin{array}{c}
q=b\rho ^{-1/(3+3\gamma )}\;, \\ 
\;\;\;\;p=\frac 13b\Theta \rho ^{-1/(3+3\gamma )}\;,
\end{array}
\right.   \tag{2.20}
\end{equation}
where $b$ is an arbitrary constant. In this way, we arrive at the following
Hamiltonian 
\begin{equation}
\mathcal{H}(q,p)=\frac 12p^2-\frac 16b^{3(1+\gamma )}q^{-(1+3\gamma )}\;. 
\tag{2.21}
\end{equation}
This Hamiltonian describes a particle with momentum $p$ in a potential $%
V(q)=-\frac 16b^{3(1+\gamma )}q^{-(1+3\gamma )}$. Using relations
(2.17)-(2.21) we can verify that the two Hamilton equations $\stackrel{*}{q}%
=\partial \mathcal{H}/\partial p$ and $\stackrel{*}{p}=-\partial \mathcal{H}%
/\partial q$ hold. Be aware that here the time coordinate is not $t$ but $%
\tilde{u}$, that is, $\stackrel{*}{q}\equiv dq/d\tilde{u}$ and $\stackrel{*}{%
p}\equiv dp/d\tilde{u}$. This will make a difference between the two
canonical formulations of the $4D$ and $5D$ cosmologies as we will see in
next section.

\section{Wave Function and Late-Time Approximation}

From the Hamiltonian (2.21) we can employ the standard quantization
procedure to write the corresponding Schr\"{o}dinger equation. Because the
time coordinate in the Hamiltonian formulation is $\tilde{u}$ , so the
time-dependent wave function is of the form 
\begin{equation}
\Psi (q,\tilde{u})=e^{-i\tilde{E}\tilde{u}}\varphi (q)  \tag{3.1}
\end{equation}
The correspondence principle, $p\rightarrow -i\frac{\partial }{\partial q}$,
gives a stationary Schr\"{o}dinger equation for the stationary wave function 
$\varphi (q)$ as

\begin{equation}
\frac{1}{2}\frac{\partial ^{2}}{\partial q^{2}}\varphi (q)+[\tilde{E}+\frac{1%
}{6}b^{3(1+\gamma )}q^{-(1+3\gamma )}]\varphi (q)=0\;.  \tag{3.2}
\end{equation}
In (3.1) and (3.2) $\tilde{E}$ is the associated eigenvalue of the operator $%
\mathcal{\hat{H}}$, $\tilde{E}=<$ $\Psi (q,\tilde{u}$ $)|\mathcal{\hat{H}}%
|\Psi (q,\tilde{u}$ $)>$ . Quantized bound states with negative $\tilde{E}$
of the Schr\"{o}dinger equation (3.2) were studied (Novello et al. 1996;
Mongan 1999, 2000, 2001). Their results can also be used here.

Since $\tilde{u}$ is not the proper time of the cosmic fluid, we are not
sure if we can interpret $\tilde{E}$ as the energy. Therefore, we wish to
look for the relation between $\tilde{u}$ and the proper time. Now let us
consider the $5D$ metric (2.10), from which we see that the $2D$
line-element in the $t$ - $y$ plane can be written as 
\begin{equation}
ds^{2}=-B^{2}(u)dud(t+y)\;.  \tag{3.3}
\end{equation}
Thus by defining

\begin{equation}
U\equiv \int B^{2}(u)du\;,V\equiv t+y\;.  \tag{3.4}
\end{equation}
we get $ds^{2}=-dUdV.$ Then let 
\begin{equation}
U\equiv T-\lambda Y\;,V\equiv T+\lambda Y\;,  \tag{3.5}
\end{equation}
where $\lambda $ is a constant to be determined later, we get $%
ds^{2}=-dT^{2}+\lambda ^{2}dY^{2}.$ Therefore we find that $T$ is the proper
time. Substituting (2.11) in (2.13) and (3.4), we find 
\begin{equation}
h\tilde{u}=\frac{2(2+3\gamma )}{3(1+\gamma )}(hu)^{\frac{3(1+\gamma )}{%
2(2+3\gamma )}}\;,  \tag{3.6}
\end{equation}
\begin{equation}
hU=(2+3\gamma )(hu)^{\frac{1}{2+3\gamma }}\;.  \tag{3.7}
\end{equation}
From these two equations we obtain

\begin{equation}
\tilde{u}=W_{\gamma }U^{\frac{3(1+\gamma )}{2}}=W_{\gamma }(T-\lambda Y)^{%
\frac{3(1+\gamma )}{2}}\;,  \tag{3.8}
\end{equation}
where 
\begin{equation}
W_{\gamma }\equiv \frac{2}{3(1+\gamma )}(\frac{h}{2+3\lambda })^{\frac{%
1+3\gamma }{2}}  \tag{3.9}
\end{equation}

Now let us consider a short period in later times of the Universe. This
means that we choose $T=0$ as the beginning of the Universe (on the $Y=0$
hypersurface) and write $T=T_0+\tau $ with $|\tau |\ll T_0$. Then (3.8) gives
\begin{equation}
\left. 
\begin{array}{c}
\tilde{u}=W_\gamma T_0^{\frac{3(1+\gamma )}2}(1+\frac{\tau -\lambda Y}{T_0}%
)^{\frac{3(1+\gamma )}2} \\ 
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\approx W_\gamma T_0^{\frac{3(1+\gamma )}2}+%
\frac{3(1+\gamma )}2T_0^{\frac{1+3\gamma }2}(\tau -\lambda Y)\;.
\end{array}
\right.   \tag{3.10}
\end{equation}

Using this relation we find that, up to a constant factor, the wave function 
$\Psi (q,\tilde{u})$ in (3.1) becomes 
\begin{equation}
\Psi (q,Y,\tau )\approx e^{-iE\tau }e^{iP_5Y}\varphi (q)\;.  \tag{3.11}
\end{equation}
where
\begin{equation}
\left. 
\begin{array}{c}
E\equiv \chi \tilde{E}\;, \\ 
\;P_5\equiv \chi \tilde{E}\lambda \;,
\end{array}
\right.   \tag{3.12}
\end{equation}

and 
\begin{equation}
\chi =\frac{3(1+\gamma )W_\gamma }2T_0^{(1+3\gamma )/2}=(\frac{hT_0}{%
2+3\gamma })^{\frac{1+3\gamma }2}\;.  \tag{3.13}
\end{equation}
Thus we successfully have obtained an approximate wave function $\Psi
(q,Y,\tau )$ as shown in (3.11), in which the three variables $\tau $, $q$
and $Y$ were separated.

\section{Quantization With a Compact Fifth Dimension}

The wave function $\Psi (q,Y,\tau )$ in (3.11) is in a form with three
separated variables $\tau ,q$ and $Y.$ Denote the $Y$-part of $\Psi $ as $%
\Phi (Y),$ $\Phi (Y)=e^{iP_{5}Y}$, then the momentum operator $\hat{P}_{5}=-i%
\frac{\partial }{\partial Y}$ gives 
\begin{equation}
\hat{P}_{5}\Phi =P_{5}\Phi \;.  \tag{4.1}
\end{equation}
So $P_{5}$ is the eigenvalue of the momentum along the fifth dimension.

Now we suppose the fifth dimension to be a circle with a radius $R$, i.e., $%
Y=R\phi $. So we have $\Phi =\Phi (\phi )=e^{iP_{5}R\phi }.$ The boundary
condition requires that $\Phi (\phi )$ must be periodic with period $2\pi $.
It follows that we must have

\begin{equation}
P_{5}=\frac{n}{R},\;\;n=0,\pm 1,\pm 2,...\;\text{.}  \tag{4.2}
\end{equation}
In this way, the fifth momentum $P_{5\text{ }}$and the corresponding wave
function $\Phi (\phi )$ are quantized.

The momentum operator $\hat{P}_{5\text{ }}=-i\frac \partial {\partial Y}$,
acting on Eq.(4.1) from the left hand side, gives 
\begin{equation}
\frac 12\frac{\partial ^2}{\partial Y^2}\Phi (Y)+\frac{n^2}{2R^2}\Phi
(Y)=0\;.  \tag{4.3}
\end{equation}
Let $\psi (q,Y)\equiv \varphi (q)\Phi (Y)$, then equations (3.2) and (4.3)
give 
\begin{equation}
\frac 12(\frac{\partial ^2}{\partial q^2}+\frac{\partial ^2}{\partial Y^2}%
)\psi (q,Y)+(\tilde{E}+\frac{n^2}{2R^2}+\frac 16b^{3(1+\gamma
)}q^{-(1+3\gamma )})\psi (q,Y)=0\;.  \tag{4.4}
\end{equation}
So we obtain a stationary two-dimensional Schr\"{o}dinger equation.

\section{Conclusion}

In this paper we have obtained a Hamiltonian formulation for a $5D$
cosmology. This Hamiltonian leads immediately to a stationary
one-dimensional Schr\"{o}dinger equation (3.2). Thus the corresponding full
wave function can describe quantized states of a $5D$ Universe. By using a
late-time approximation we have successfully separated the wave function
into three parts,$\Psi (q,Y,\tau )\approx e^{-iE\tau }e^{iP_{5}Y}\varphi (q)$
, corresponding to the proper time $\tau $, the variable $q$ (which
describes the radius of the Universe) and the fifth coordinate $Y$,
respectively. Notice that this wave function just valid in later times of the
Universe with $|\tau |\ll T_{0}$ , where $T_{0}$ is the age of the Universe
at that time. Then by assuming the compact fifth dimension to be a circle,
we obtain a quantized spectrum for the momentum $P_{5}$ in the fifth
direction. Be aware that the $q$-part of the wave function, $\varphi (q)$,
satisfies the one-dimensional Schr\"{o}dinger equation (3.2), for which
bound states were given (Novello et al. 1996). Using their results, as well
as the spectrum for $P_{5}$, we can obtain bound states for the whole
space-part of the wave function, $\psi (q,Y)=e^{iP_{5}Y}\varphi (q)$, which
satisfies the two-dimensional Schr\"{o}dinger equation (4.4). Further
studies are needed.

\noindent \textbf{Acknowledgments}

This work was supported by NSF of P. R. China under grant 19975007.

\bigskip

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\bibitem{Wessonbook}  Wesson, P.S., 1999, \textit{Space, Time, Matte}r
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\end{thebibliography}

\end{document}

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  \else\graffile{#1}{#3}{#4}{#5}%
  \fi
 }%
%
\def\addtoLaTeXparams#1{%
    \edef\LaTeXparams{\LaTeXparams #1}}%
%
% JCS -  added a switch BoxFrame that can 
% be set by including X in the frame params.
% If set a box is drawn around the frame.

\newif\ifBoxFrame \BoxFramefalse
\newif\ifOverFrame \OverFramefalse
\newif\ifUnderFrame \UnderFramefalse

\def\BOXTHEFRAME#1{%
   \hbox{%
      \ifBoxFrame
         \frame{#1}%
      \else
         {#1}%
      \fi
   }%
}


\def\doFRAMEparams#1{\BoxFramefalse\OverFramefalse\UnderFramefalse\readFRAMEparams#1\end}%
\def\readFRAMEparams#1{%
 \ifx#1\end%
  \let\next=\relax
  \else
  \ifx#1i\dispkind=\z@\fi
  \ifx#1d\dispkind=\@ne\fi
  \ifx#1f\dispkind=\tw@\fi
  \ifx#1t\addtoLaTeXparams{t}\fi
  \ifx#1b\addtoLaTeXparams{b}\fi
  \ifx#1p\addtoLaTeXparams{p}\fi
  \ifx#1h\addtoLaTeXparams{h}\fi
  \ifx#1X\BoxFrametrue\fi
  \ifx#1O\OverFrametrue\fi
  \ifx#1U\UnderFrametrue\fi
  \ifx#1w
    \ifnum\draft=1\wasdrafttrue\else\wasdraftfalse\fi
    \draft=\@ne
  \fi
  \let\next=\readFRAMEparams
  \fi
 \next
 }%
%
%Macro for In-line graphics object
%   \IFRAME{ contentswidth (scalar)  }               %#1
%          { contentsheight (scalar) }               %#2
%          { vertical shift when in-line (scalar) }  %#3
%          { draft name }                            %#4
%          { body }                                  %#5
%          { caption}                                %#6


\def\IFRAME#1#2#3#4#5#6{%
      \bgroup
      \let\QCTOptA\empty
      \let\QCTOptB\empty
      \let\QCBOptA\empty
      \let\QCBOptB\empty
      #6%
      \parindent=0pt%
      \leftskip=0pt
      \rightskip=0pt
      \setbox0 = \hbox{\QCBOptA}%
      \@tempdima = #1\relax
      \ifOverFrame
          % Do this later
          \typeout{This is not implemented yet}%
          \show\HELP
      \else
         \ifdim\wd0>\@tempdima
            \advance\@tempdima by \@tempdima
            \ifdim\wd0 >\@tempdima
               \textwidth=\@tempdima
               \setbox1 =\vbox{%
                  \noindent\hbox to \@tempdima{\hfill\GRAPHIC{#5}{#4}{#1}{#2}{#3}\hfill}\\%
                  \noindent\hbox to \@tempdima{\parbox[b]{\@tempdima}{\QCBOptA}}%
               }%
               \wd1=\@tempdima
            \else
               \textwidth=\wd0
               \setbox1 =\vbox{%
                 \noindent\hbox to \wd0{\hfill\GRAPHIC{#5}{#4}{#1}{#2}{#3}\hfill}\\%
                 \noindent\hbox{\QCBOptA}%
               }%
               \wd1=\wd0
            \fi
         \else
            %\show\BBB
            \ifdim\wd0>0pt
              \hsize=\@tempdima
              \setbox1 =\vbox{%
                \unskip\GRAPHIC{#5}{#4}{#1}{#2}{0pt}%
                \break
                \unskip\hbox to \@tempdima{\hfill \QCBOptA\hfill}%
              }%
              \wd1=\@tempdima
           \else
              \hsize=\@tempdima
              \setbox1 =\vbox{%
                \unskip\GRAPHIC{#5}{#4}{#1}{#2}{0pt}%
              }%
              \wd1=\@tempdima
           \fi
         \fi
         \@tempdimb=\ht1
         \advance\@tempdimb by \dp1
         \advance\@tempdimb by -#2%
         \advance\@tempdimb by #3%
         \leavevmode
         \raise -\@tempdimb \hbox{\box1}%
      \fi
      \egroup%
}%
%
%Macro for Display graphics object
%   \DFRAME{ contentswidth (scalar)  }               %#1
%          { contentsheight (scalar) }               %#2
%          { draft label }                           %#3
%          { name }                                  %#4
%          { caption}                                %#5
\def\DFRAME#1#2#3#4#5{%
 \begin{center}
     \let\QCTOptA\empty
     \let\QCTOptB\empty
     \let\QCBOptA\empty
     \let\QCBOptB\empty
     \ifOverFrame 
        #5\QCTOptA\par
     \fi
     \GRAPHIC{#4}{#3}{#1}{#2}{\z@}
     \ifUnderFrame 
        \nobreak\par #5\QCBOptA
     \fi
 \end{center}%
 }%
%
%Macro for Floating graphic object
%   \FFRAME{ framedata f|i tbph x F|T }              %#1
%          { contentswidth (scalar)  }               %#2
%          { contentsheight (scalar) }               %#3
%          { caption }                               %#4
%          { label }                                 %#5
%          { draft name }                            %#6
%          { body }                                  %#7
\def\FFRAME#1#2#3#4#5#6#7{%
 \begin{figure}[#1]%
  \let\QCTOptA\empty
  \let\QCTOptB\empty
  \let\QCBOptA\empty
  \let\QCBOptB\empty
  \ifOverFrame
    #4
    \ifx\QCTOptA\empty
    \else
      \ifx\QCTOptB\empty
        \caption{\QCTOptA}%
      \else
        \caption[\QCTOptB]{\QCTOptA}%
      \fi
    \fi
    \ifUnderFrame\else
      \label{#5}%
    \fi
  \else
    \UnderFrametrue%
  \fi
  \begin{center}\GRAPHIC{#7}{#6}{#2}{#3}{\z@}\end{center}%
  \ifUnderFrame
    #4
    \ifx\QCBOptA\empty
      \caption{}%
    \else
      \ifx\QCBOptB\empty
        \caption{\QCBOptA}%
      \else
        \caption[\QCBOptB]{\QCBOptA}%
      \fi
    \fi
    \label{#5}%
  \fi
  \end{figure}%
 }%
%
%
%    \FRAME{ framedata f|i tbph x F|T }              %#1
%          { contentswidth (scalar)  }               %#2
%          { contentsheight (scalar) }               %#3
%          { vertical shift when in-line (scalar) }  %#4
%          { caption }                               %#5
%          { label }                                 %#6
%          { name }                                  %#7
%          { body }                                  %#8
%
%    framedata is a string which can contain the following
%    characters: idftbphxFT
%    Their meaning is as follows:
%             i, d or f : in-line, display, or floating
%             t,b,p,h   : LaTeX floating placement options
%             x         : fit contents box to contents
%             F or T    : Figure or Table. 
%                         Later this can expand
%                         to a more general float class.
%
%
\newcount\dispkind%

\def\makeactives{
  \catcode`\"=\active
  \catcode`\;=\active
  \catcode`\:=\active
  \catcode`\'=\active
  \catcode`\~=\active
}
\bgroup
   \makeactives
   \gdef\activesoff{%
      \def"{\string"}
      \def;{\string;}
      \def:{\string:}
      \def'{\string'}
      \def~{\string~}
      %\bbl@deactivate{"}%
      %\bbl@deactivate{;}%
      %\bbl@deactivate{:}%
      %\bbl@deactivate{'}%
    }
\egroup

\def\FRAME#1#2#3#4#5#6#7#8{%
 \bgroup
 \@ifundefined{bbl@deactivate}{}{\activesoff}
 \ifnum\draft=\@ne
   \wasdrafttrue
 \else
   \wasdraftfalse%
 \fi
 \def\LaTeXparams{}%
 \dispkind=\z@
 \def\LaTeXparams{}%
 \doFRAMEparams{#1}%
 \ifnum\dispkind=\z@\IFRAME{#2}{#3}{#4}{#7}{#8}{#5}\else
  \ifnum\dispkind=\@ne\DFRAME{#2}{#3}{#7}{#8}{#5}\else
   \ifnum\dispkind=\tw@
    \edef\@tempa{\noexpand\FFRAME{\LaTeXparams}}%
    \@tempa{#2}{#3}{#5}{#6}{#7}{#8}%
    \fi
   \fi
  \fi
  \ifwasdraft\draft=1\else\draft=0\fi{}%
  \egroup
 }%
%
% This macro added to let SW gobble a parameter that
% should not be passed on and expanded. 

\def\TEXUX#1{"texux"}

%
% Macros for text attributes:
%
\def\BF#1{{\bf {#1}}}%
\def\NEG#1{\leavevmode\hbox{\rlap{\thinspace/}{$#1$}}}%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% macros for user - defined functions
\def\func#1{\mathop{\rm #1}}%
\def\limfunc#1{\mathop{\rm #1}}%

%
% miscellaneous 
%\long\def\QQQ#1#2{}%
\long\def\QQQ#1#2{%
     \long\expandafter\def\csname#1\endcsname{#2}}%
%\def\QTP#1{}% JCS - this was changed becuase style editor will define QTP
\@ifundefined{QTP}{\def\QTP#1{}}{}
\@ifundefined{QEXCLUDE}{\def\QEXCLUDE#1{}}{}
%\@ifundefined{Qcb}{\def\Qcb#1{#1}}{}
%\@ifundefined{Qct}{\def\Qct#1{#1}}{}
\@ifundefined{Qlb}{\def\Qlb#1{#1}}{}
\@ifundefined{Qlt}{\def\Qlt#1{#1}}{}
\def\QWE{}%
\long\def\QQA#1#2{}%
%\def\QTR#1#2{{\em #2}}% Always \em!!!
%\def\QTR#1#2{\mbox{\begin{#1}#2\end{#1}}}%cb%%%
\def\QTR#1#2{{\csname#1\endcsname #2}}%(gp) Is this the best?
\long\def\TeXButton#1#2{#2}%
\long\def\QSubDoc#1#2{#2}%
\def\EXPAND#1[#2]#3{}%
\def\NOEXPAND#1[#2]#3{}%
\def\PROTECTED{}%
\def\LaTeXparent#1{}%
\def\ChildStyles#1{}%
\def\ChildDefaults#1{}%
\def\QTagDef#1#2#3{}%
%
% Macros for style editor docs
\@ifundefined{StyleEditBeginDoc}{\def\StyleEditBeginDoc{\relax}}{}
%
% Macros for footnotes
\def\QQfnmark#1{\footnotemark}
\def\QQfntext#1#2{\addtocounter{footnote}{#1}\footnotetext{#2}}
%
% Macros for indexing.
\def\MAKEINDEX{\makeatletter\input gnuindex.sty\makeatother\makeindex}%	
\@ifundefined{INDEX}{\def\INDEX#1#2{}{}}{}%
\@ifundefined{SUBINDEX}{\def\SUBINDEX#1#2#3{}{}{}}{}%
\@ifundefined{initial}%  
   {\def\initial#1{\bigbreak{\raggedright\large\bf #1}\kern 2\p@\penalty3000}}%
   {}%
\@ifundefined{entry}{\def\entry#1#2{\item {#1}, #2}}{}%
\@ifundefined{primary}{\def\primary#1{\item {#1}}}{}%
\@ifundefined{secondary}{\def\secondary#1#2{\subitem {#1}, #2}}{}%
%
%
\@ifundefined{ZZZ}{}{\MAKEINDEX\makeatletter}%
%
% Attempts to avoid problems with other styles
\@ifundefined{abstract}{%
 \def\abstract{%
  \if@twocolumn
   \section*{Abstract (Not appropriate in this style!)}%
   \else \small 
   \begin{center}{\bf Abstract\vspace{-.5em}\vspace{\z@}}\end{center}%
   \quotation 
   \fi
  }%
 }{%
 }%
\@ifundefined{endabstract}{\def\endabstract
  {\if@twocolumn\else\endquotation\fi}}{}%
\@ifundefined{maketitle}{\def\maketitle#1{}}{}%
\@ifundefined{affiliation}{\def\affiliation#1{}}{}%
\@ifundefined{proof}{\def\proof{\noindent{\bfseries Proof. }}}{}%
\@ifundefined{endproof}{\def\endproof{\mbox{\ \rule{.1in}{.1in}}}}{}%
\@ifundefined{newfield}{\def\newfield#1#2{}}{}%
\@ifundefined{chapter}{\def\chapter#1{\par(Chapter head:)#1\par }%
 \newcount\c@chapter}{}%
\@ifundefined{part}{\def\part#1{\par(Part head:)#1\par }}{}%
\@ifundefined{section}{\def\section#1{\par(Section head:)#1\par }}{}%
\@ifundefined{subsection}{\def\subsection#1%
 {\par(Subsection head:)#1\par }}{}%
\@ifundefined{subsubsection}{\def\subsubsection#1%
 {\par(Subsubsection head:)#1\par }}{}%
\@ifundefined{paragraph}{\def\paragraph#1%
 {\par(Subsubsubsection head:)#1\par }}{}%
\@ifundefined{subparagraph}{\def\subparagraph#1%
 {\par(Subsubsubsubsection head:)#1\par }}{}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% These symbols are not recognized by LaTeX
\@ifundefined{therefore}{\def\therefore{}}{}%
\@ifundefined{backepsilon}{\def\backepsilon{}}{}%
\@ifundefined{yen}{\def\yen{\hbox{\rm\rlap=Y}}}{}%
\@ifundefined{registered}{%
   \def\registered{\relax\ifmmode{}\r@gistered
                    \else$\m@th\r@gistered$\fi}%
 \def\r@gistered{^{\ooalign
  {\hfil\raise.07ex\hbox{$\scriptstyle\rm\text{R}$}\hfil\crcr
  \mathhexbox20D}}}}{}%
\@ifundefined{Eth}{\def\Eth{}}{}%
\@ifundefined{eth}{\def\eth{}}{}%
\@ifundefined{Thorn}{\def\Thorn{}}{}%
\@ifundefined{thorn}{\def\thorn{}}{}%
% A macro to allow any symbol that requires math to appear in text
\def\TEXTsymbol#1{\mbox{$#1$}}%
\@ifundefined{degree}{\def\degree{{}^{\circ}}}{}%
%
% macros for T3TeX files
\newdimen\theight
\def\Column{%
 \vadjust{\setbox\z@=\hbox{\scriptsize\quad\quad tcol}%
  \theight=\ht\z@\advance\theight by \dp\z@\advance\theight by \lineskip
  \kern -\theight \vbox to \theight{%
   \rightline{\rlap{\box\z@}}%
   \vss
   }%
  }%
 }%
%
\def\qed{%
 \ifhmode\unskip\nobreak\fi\ifmmode\ifinner\else\hskip5\p@\fi\fi
 \hbox{\hskip5\p@\vrule width4\p@ height6\p@ depth1.5\p@\hskip\p@}%
 }%
%
\def\cents{\hbox{\rm\rlap/c}}%
\def\miss{\hbox{\vrule height2\p@ width 2\p@ depth\z@}}%
%\def\miss{\hbox{.}}%        %another possibility 
%
\def\vvert{\Vert}%           %always translated to \left| or \right|
%
\def\tcol#1{{\baselineskip=6\p@ \vcenter{#1}} \Column}  %
%
\def\dB{\hbox{{}}}%                 %dummy entry in column 
\def\mB#1{\hbox{$#1$}}%             %column entry
\def\nB#1{\hbox{#1}}%               %column entry (not math)
%
%\newcount\notenumber
%\def\clearnotenumber{\notenumber=0}
%\def\note{\global\advance\notenumber by 1
% \footnote{$^{\the\notenumber}$}}
%\def\note{\global\advance\notenumber by 1
\def\note{$^{\dag}}%
%
%

\def\newfmtname{LaTeX2e}
\def\chkcompat{%
   \if@compatibility
   \else
     \usepackage{latexsym}
   \fi
}

\ifx\fmtname\newfmtname
  \DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
  \DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
  \DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
  \DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
  \DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
  \DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl}
  \DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc}
  \chkcompat
\fi

%
% Greek bold macros
% Redefine all of the math symbols 
% which might be bolded	 - there are 
% probably others to add to this list

\def\alpha{\Greekmath 010B }%
\def\beta{\Greekmath 010C }%
\def\gamma{\Greekmath 010D }%
\def\delta{\Greekmath 010E }%
\def\epsilon{\Greekmath 010F }%
\def\zeta{\Greekmath 0110 }%
\def\eta{\Greekmath 0111 }%
\def\theta{\Greekmath 0112 }%
\def\iota{\Greekmath 0113 }%
\def\kappa{\Greekmath 0114 }%
\def\lambda{\Greekmath 0115 }%
\def\mu{\Greekmath 0116 }%
\def\nu{\Greekmath 0117 }%
\def\xi{\Greekmath 0118 }%
\def\pi{\Greekmath 0119 }%
\def\rho{\Greekmath 011A }%
\def\sigma{\Greekmath 011B }%
\def\tau{\Greekmath 011C }%
\def\upsilon{\Greekmath 011D }%
\def\phi{\Greekmath 011E }%
\def\chi{\Greekmath 011F }%
\def\psi{\Greekmath 0120 }%
\def\omega{\Greekmath 0121 }%
\def\varepsilon{\Greekmath 0122 }%
\def\vartheta{\Greekmath 0123 }%
\def\varpi{\Greekmath 0124 }%
\def\varrho{\Greekmath 0125 }%
\def\varsigma{\Greekmath 0126 }%
\def\varphi{\Greekmath 0127 }%

\def\nabla{\Greekmath 0272 }
\def\FindBoldGroup{%
   {\setbox0=\hbox{$\mathbf{x\global\edef\theboldgroup{\the\mathgroup}}$}}%
}

\def\Greekmath#1#2#3#4{%
    \if@compatibility
        \ifnum\mathgroup=\symbold
           \mathchoice{\mbox{\boldmath$\displaystyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\textstyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\scriptstyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\scriptscriptstyle\mathchar"#1#2#3#4$}}%
        \else
           \mathchar"#1#2#3#4% 
        \fi 
    \else 
        \FindBoldGroup
        \ifnum\mathgroup=\theboldgroup % For 2e
           \mathchoice{\mbox{\boldmath$\displaystyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\textstyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\scriptstyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\scriptscriptstyle\mathchar"#1#2#3#4$}}%
        \else
           \mathchar"#1#2#3#4% 
        \fi     	    
	  \fi}

\newif\ifGreekBold  \GreekBoldfalse
\let\SAVEPBF=\pbf
\def\pbf{\GreekBoldtrue\SAVEPBF}%
%

\@ifundefined{theorem}{\newtheorem{theorem}{Theorem}}{}
\@ifundefined{lemma}{\newtheorem{lemma}[theorem]{Lemma}}{}
\@ifundefined{corollary}{\newtheorem{corollary}[theorem]{Corollary}}{}
\@ifundefined{conjecture}{\newtheorem{conjecture}[theorem]{Conjecture}}{}
\@ifundefined{proposition}{\newtheorem{proposition}[theorem]{Proposition}}{}
\@ifundefined{axiom}{\newtheorem{axiom}{Axiom}}{}
\@ifundefined{remark}{\newtheorem{remark}{Remark}}{}
\@ifundefined{example}{\newtheorem{example}{Example}}{}
\@ifundefined{exercise}{\newtheorem{exercise}{Exercise}}{}
\@ifundefined{definition}{\newtheorem{definition}{Definition}}{}


\@ifundefined{mathletters}{%
  %\def\theequation{\arabic{equation}}
  \newcounter{equationnumber}  
  \def\mathletters{%
     \addtocounter{equation}{1}
     \edef\@currentlabel{\theequation}%
     \setcounter{equationnumber}{\c@equation}
     \setcounter{equation}{0}%
     \edef\theequation{\@currentlabel\noexpand\alph{equation}}%
  }
  \def\endmathletters{%
     \setcounter{equation}{\value{equationnumber}}%
  }
}{}

%Logos
\@ifundefined{BibTeX}{%
    \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
                 T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}}{}%
\@ifundefined{AmS}%
    {\def\AmS{{\protect\usefont{OMS}{cmsy}{m}{n}%
                A\kern-.1667em\lower.5ex\hbox{M}\kern-.125emS}}}{}%
\@ifundefined{AmSTeX}{\def\AmSTeX{\protect\AmS-\protect\TeX\@}}{}%
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% NOTE: The rest of this file is read only if amstex has not been
% loaded.  This section is used to define amstex constructs in the
% event they have not been defined.
%
%
\ifx\ds@amstex\relax
   \message{amstex already loaded}\makeatother\endinput% 2.09 compatability
\else
   \@ifpackageloaded{amstex}%
      {\message{amstex already loaded}\makeatother\endinput}
      {}
   \@ifpackageloaded{amsgen}%
      {\message{amsgen already loaded}\makeatother\endinput}
      {}
\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%
%
%  Macros to define some AMS LaTeX constructs when 
%  AMS LaTeX has not been loaded
% 
% These macros are copied from the AMS-TeX package for doing
% multiple integrals.
%
\let\DOTSI\relax
\def\RIfM@{\relax\ifmmode}%
\def\FN@{\futurelet\next}%
\newcount\intno@
\def\iint{\DOTSI\intno@\tw@\FN@\ints@}%
\def\iiint{\DOTSI\intno@\thr@@\FN@\ints@}%
\def\iiiint{\DOTSI\intno@4 \FN@\ints@}%
\def\idotsint{\DOTSI\intno@\z@\FN@\ints@}%
\def\ints@{\findlimits@\ints@@}%
\newif\iflimtoken@
\newif\iflimits@
\def\findlimits@{\limtoken@true\ifx\next\limits\limits@true
 \else\ifx\next\nolimits\limits@false\else
 \limtoken@false\ifx\ilimits@\nolimits\limits@false\else
 \ifinner\limits@false\else\limits@true\fi\fi\fi\fi}%
\def\multint@{\int\ifnum\intno@=\z@\intdots@                          %1
 \else\intkern@\fi                                                    %2
 \ifnum\intno@>\tw@\int\intkern@\fi                                   %3
 \ifnum\intno@>\thr@@\int\intkern@\fi                                 %4
 \int}%                                                               %5
\def\multintlimits@{\intop\ifnum\intno@=\z@\intdots@\else\intkern@\fi
 \ifnum\intno@>\tw@\intop\intkern@\fi
 \ifnum\intno@>\thr@@\intop\intkern@\fi\intop}%
\def\intic@{%
    \mathchoice{\hskip.5em}{\hskip.4em}{\hskip.4em}{\hskip.4em}}%
\def\negintic@{\mathchoice
 {\hskip-.5em}{\hskip-.4em}{\hskip-.4em}{\hskip-.4em}}%
\def\ints@@{\iflimtoken@                                              %1
 \def\ints@@@{\iflimits@\negintic@
   \mathop{\intic@\multintlimits@}\limits                             %2
  \else\multint@\nolimits\fi                                          %3
  \eat@}%                                                             %4
 \else                                                                %5
 \def\ints@@@{\iflimits@\negintic@
  \mathop{\intic@\multintlimits@}\limits\else
  \multint@\nolimits\fi}\fi\ints@@@}%
\def\intkern@{\mathchoice{\!\!\!}{\!\!}{\!\!}{\!\!}}%
\def\plaincdots@{\mathinner{\cdotp\cdotp\cdotp}}%
\def\intdots@{\mathchoice{\plaincdots@}%
 {{\cdotp}\mkern1.5mu{\cdotp}\mkern1.5mu{\cdotp}}%
 {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}%
 {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}}%
%
%
%  These macros are for doing the AMS \text{} construct
%
\def\RIfM@{\relax\protect\ifmmode}
\def\text{\RIfM@\expandafter\text@\else\expandafter\mbox\fi}
\let\nfss@text\text
\def\text@#1{\mathchoice
   {\textdef@\displaystyle\f@size{#1}}%
   {\textdef@\textstyle\tf@size{\firstchoice@false #1}}%
   {\textdef@\textstyle\sf@size{\firstchoice@false #1}}%
   {\textdef@\textstyle \ssf@size{\firstchoice@false #1}}%
   \glb@settings}

\def\textdef@#1#2#3{\hbox{{%
                    \everymath{#1}%
                    \let\f@size#2\selectfont
                    #3}}}
\newif\iffirstchoice@
\firstchoice@true
%
%    Old Scheme for \text
%
%\def\rmfam{\z@}%
%\newif\iffirstchoice@
%\firstchoice@true
%\def\textfonti{\the\textfont\@ne}%
%\def\textfontii{\the\textfont\tw@}%
%\def\text{\RIfM@\expandafter\text@\else\expandafter\text@@\fi}%
%\def\text@@#1{\leavevmode\hbox{#1}}%
%\def\text@#1{\mathchoice
% {\hbox{\everymath{\displaystyle}\def\textfonti{\the\textfont\@ne}%
%  \def\textfontii{\the\textfont\tw@}\textdef@@ T#1}}%
% {\hbox{\firstchoice@false
%  \everymath{\textstyle}\def\textfonti{\the\textfont\@ne}%
%  \def\textfontii{\the\textfont\tw@}\textdef@@ T#1}}%
% {\hbox{\firstchoice@false
%  \everymath{\scriptstyle}\def\textfonti{\the\scriptfont\@ne}%
%  \def\textfontii{\the\scriptfont\tw@}\textdef@@ S\rm#1}}%
% {\hbox{\firstchoice@false
%  \everymath{\scriptscriptstyle}\def\textfonti
%  {\the\scriptscriptfont\@ne}%
%  \def\textfontii{\the\scriptscriptfont\tw@}\textdef@@ s\rm#1}}}%
%\def\textdef@@#1{\textdef@#1\rm\textdef@#1\bf\textdef@#1\sl
%    \textdef@#1\it}%
%\def\DN@{\def\next@}%
%\def\eat@#1{}%
%\def\textdef@#1#2{%
% \DN@{\csname\expandafter\eat@\string#2fam\endcsname}%
% \if S#1\edef#2{\the\scriptfont\next@\relax}%
% \else\if s#1\edef#2{\the\scriptscriptfont\next@\relax}%
% \else\edef#2{\the\textfont\next@\relax}\fi\fi}%
%
%
%These are the AMS constructs for multiline limits.
%
\def\Let@{\relax\iffalse{\fi\let\\=\cr\iffalse}\fi}%
\def\vspace@{\def\vspace##1{\crcr\noalign{\vskip##1\relax}}}%
\def\multilimits@{\bgroup\vspace@\Let@
 \baselineskip\fontdimen10 \scriptfont\tw@
 \advance\baselineskip\fontdimen12 \scriptfont\tw@
 \lineskip\thr@@\fontdimen8 \scriptfont\thr@@
 \lineskiplimit\lineskip
 \vbox\bgroup\ialign\bgroup\hfil$\m@th\scriptstyle{##}$\hfil\crcr}%
\def\Sb{_\multilimits@}%
\def\endSb{\crcr\egroup\egroup\egroup}%
\def\Sp{^\multilimits@}%
\let\endSp\endSb
%
%
%These are AMS constructs for horizontal arrows
%
\newdimen\ex@
\ex@.2326ex
\def\rightarrowfill@#1{$#1\m@th\mathord-\mkern-6mu\cleaders
 \hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill
 \mkern-6mu\mathord\rightarrow$}%
\def\leftarrowfill@#1{$#1\m@th\mathord\leftarrow\mkern-6mu\cleaders
 \hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill\mkern-6mu\mathord-$}%
\def\leftrightarrowfill@#1{$#1\m@th\mathord\leftarrow
\mkern-6mu\cleaders
 \hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill
 \mkern-6mu\mathord\rightarrow$}%
\def\overrightarrow{\mathpalette\overrightarrow@}%
\def\overrightarrow@#1#2{\vbox{\ialign{##\crcr\rightarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}%
\let\overarrow\overrightarrow
\def\overleftarrow{\mathpalette\overleftarrow@}%
\def\overleftarrow@#1#2{\vbox{\ialign{##\crcr\leftarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}%
\def\overleftrightarrow{\mathpalette\overleftrightarrow@}%
\def\overleftrightarrow@#1#2{\vbox{\ialign{##\crcr
   \leftrightarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}%
\def\underrightarrow{\mathpalette\underrightarrow@}%
\def\underrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil
  $\crcr\noalign{\nointerlineskip}\rightarrowfill@#1\crcr}}}%
\let\underarrow\underrightarrow
\def\underleftarrow{\mathpalette\underleftarrow@}%
\def\underleftarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil
  $\crcr\noalign{\nointerlineskip}\leftarrowfill@#1\crcr}}}%
\def\underleftrightarrow{\mathpalette\underleftrightarrow@}%
\def\underleftrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th
  \hfil#1#2\hfil$\crcr
 \noalign{\nointerlineskip}\leftrightarrowfill@#1\crcr}}}%
%%%%%%%%%%%%%%%%%%%%%

% 94.0815 by Jon:

\def\qopnamewl@#1{\mathop{\operator@font#1}\nlimits@}
\let\nlimits@\displaylimits
\def\setboxz@h{\setbox\z@\hbox}


\def\varlim@#1#2{\mathop{\vtop{\ialign{##\crcr
 \hfil$#1\m@th\operator@font lim$\hfil\crcr
 \noalign{\nointerlineskip}#2#1\crcr
 \noalign{\nointerlineskip\kern-\ex@}\crcr}}}}

 \def\rightarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
  $#1\copy\z@\mkern-6mu\cleaders
  \hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
  \mkern-6mu\mathord\rightarrow$}
\def\leftarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
  $#1\mathord\leftarrow\mkern-6mu\cleaders
  \hbox{$#1\mkern-2mu\copy\z@\mkern-2mu$}\hfill
  \mkern-6mu\box\z@$}


\def\projlim{\qopnamewl@{proj\,lim}}
\def\injlim{\qopnamewl@{inj\,lim}}
\def\varinjlim{\mathpalette\varlim@\rightarrowfill@}
\def\varprojlim{\mathpalette\varlim@\leftarrowfill@}
\def\varliminf{\mathpalette\varliminf@{}}
\def\varliminf@#1{\mathop{\underline{\vrule\@depth.2\ex@\@width\z@
   \hbox{$#1\m@th\operator@font lim$}}}}
\def\varlimsup{\mathpalette\varlimsup@{}}
\def\varlimsup@#1{\mathop{\overline
  {\hbox{$#1\m@th\operator@font lim$}}}}

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\def\tfrac#1#2{{\textstyle {#1 \over #2}}}%
\def\dfrac#1#2{{\displaystyle {#1 \over #2}}}%
\def\binom#1#2{{#1 \choose #2}}%
\def\tbinom#1#2{{\textstyle {#1 \choose #2}}}%
\def\dbinom#1#2{{\displaystyle {#1 \choose #2}}}%
\def\QATOP#1#2{{#1 \atop #2}}%
\def\QTATOP#1#2{{\textstyle {#1 \atop #2}}}%
\def\QDATOP#1#2{{\displaystyle {#1 \atop #2}}}%
\def\QABOVE#1#2#3{{#2 \above#1 #3}}%
\def\QTABOVE#1#2#3{{\textstyle {#2 \above#1 #3}}}%
\def\QDABOVE#1#2#3{{\displaystyle {#2 \above#1 #3}}}%
\def\QOVERD#1#2#3#4{{#3 \overwithdelims#1#2 #4}}%
\def\QTOVERD#1#2#3#4{{\textstyle {#3 \overwithdelims#1#2 #4}}}%
\def\QDOVERD#1#2#3#4{{\displaystyle {#3 \overwithdelims#1#2 #4}}}%
\def\QATOPD#1#2#3#4{{#3 \atopwithdelims#1#2 #4}}%
\def\QTATOPD#1#2#3#4{{\textstyle {#3 \atopwithdelims#1#2 #4}}}%
\def\QDATOPD#1#2#3#4{{\displaystyle {#3 \atopwithdelims#1#2 #4}}}%
\def\QABOVED#1#2#3#4#5{{#4 \abovewithdelims#1#2#3 #5}}%
\def\QTABOVED#1#2#3#4#5{{\textstyle 
   {#4 \abovewithdelims#1#2#3 #5}}}%
\def\QDABOVED#1#2#3#4#5{{\displaystyle 
   {#4 \abovewithdelims#1#2#3 #5}}}%
%
% Macros for text size operators:

%JCS - added braces and \mathop around \displaystyle\int, etc.
%
\def\tint{\mathop{\textstyle \int}}%
\def\tiint{\mathop{\textstyle \iint }}%
\def\tiiint{\mathop{\textstyle \iiint }}%
\def\tiiiint{\mathop{\textstyle \iiiint }}%
\def\tidotsint{\mathop{\textstyle \idotsint }}%
\def\toint{\mathop{\textstyle \oint}}%
\def\tsum{\mathop{\textstyle \sum }}%
\def\tprod{\mathop{\textstyle \prod }}%
\def\tbigcap{\mathop{\textstyle \bigcap }}%
\def\tbigwedge{\mathop{\textstyle \bigwedge }}%
\def\tbigoplus{\mathop{\textstyle \bigoplus }}%
\def\tbigodot{\mathop{\textstyle \bigodot }}%
\def\tbigsqcup{\mathop{\textstyle \bigsqcup }}%
\def\tcoprod{\mathop{\textstyle \coprod }}%
\def\tbigcup{\mathop{\textstyle \bigcup }}%
\def\tbigvee{\mathop{\textstyle \bigvee }}%
\def\tbigotimes{\mathop{\textstyle \bigotimes }}%
\def\tbiguplus{\mathop{\textstyle \biguplus }}%
%
%
%Macros for display size operators:
%

\def\dint{\mathop{\displaystyle \int}}%
\def\diint{\mathop{\displaystyle \iint }}%
\def\diiint{\mathop{\displaystyle \iiint }}%
\def\diiiint{\mathop{\displaystyle \iiiint }}%
\def\didotsint{\mathop{\displaystyle \idotsint }}%
\def\doint{\mathop{\displaystyle \oint}}%
\def\dsum{\mathop{\displaystyle \sum }}%
\def\dprod{\mathop{\displaystyle \prod }}%
\def\dbigcap{\mathop{\displaystyle \bigcap }}%
\def\dbigwedge{\mathop{\displaystyle \bigwedge }}%
\def\dbigoplus{\mathop{\displaystyle \bigoplus }}%
\def\dbigodot{\mathop{\displaystyle \bigodot }}%
\def\dbigsqcup{\mathop{\displaystyle \bigsqcup }}%
\def\dcoprod{\mathop{\displaystyle \coprod }}%
\def\dbigcup{\mathop{\displaystyle \bigcup }}%
\def\dbigvee{\mathop{\displaystyle \bigvee }}%
\def\dbigotimes{\mathop{\displaystyle \bigotimes }}%
\def\dbiguplus{\mathop{\displaystyle \biguplus }}%
%
%Companion to stackrel
\def\stackunder#1#2{\mathrel{\mathop{#2}\limits_{#1}}}%
%
%
% These are AMS environments that will be defined to
% be verbatims if amstex has not actually been 
% loaded
%
%
\begingroup \catcode `|=0 \catcode `[= 1
\catcode`]=2 \catcode `\{=12 \catcode `\}=12
\catcode`\\=12 
|gdef|@alignverbatim#1\end{align}[#1|end[align]]
|gdef|@salignverbatim#1\end{align*}[#1|end[align*]]

|gdef|@alignatverbatim#1\end{alignat}[#1|end[alignat]]
|gdef|@salignatverbatim#1\end{alignat*}[#1|end[alignat*]]

|gdef|@xalignatverbatim#1\end{xalignat}[#1|end[xalignat]]
|gdef|@sxalignatverbatim#1\end{xalignat*}[#1|end[xalignat*]]

|gdef|@gatherverbatim#1\end{gather}[#1|end[gather]]
|gdef|@sgatherverbatim#1\end{gather*}[#1|end[gather*]]

|gdef|@gatherverbatim#1\end{gather}[#1|end[gather]]
|gdef|@sgatherverbatim#1\end{gather*}[#1|end[gather*]]


|gdef|@multilineverbatim#1\end{multiline}[#1|end[multiline]]
|gdef|@smultilineverbatim#1\end{multiline*}[#1|end[multiline*]]

|gdef|@arraxverbatim#1\end{arrax}[#1|end[arrax]]
|gdef|@sarraxverbatim#1\end{arrax*}[#1|end[arrax*]]

|gdef|@tabulaxverbatim#1\end{tabulax}[#1|end[tabulax]]
|gdef|@stabulaxverbatim#1\end{tabulax*}[#1|end[tabulax*]]


|endgroup
  

  
\def\align{\@verbatim \frenchspacing\@vobeyspaces \@alignverbatim
You are using the "align" environment in a style in which it is not defined.}
\let\endalign=\endtrivlist
 
\@namedef{align*}{\@verbatim\@salignverbatim
You are using the "align*" environment in a style in which it is not defined.}
\expandafter\let\csname endalign*\endcsname =\endtrivlist




\def\alignat{\@verbatim \frenchspacing\@vobeyspaces \@alignatverbatim
You are using the "alignat" environment in a style in which it is not defined.}
\let\endalignat=\endtrivlist
 
\@namedef{alignat*}{\@verbatim\@salignatverbatim
You are using the "alignat*" environment in a style in which it is not defined.}
\expandafter\let\csname endalignat*\endcsname =\endtrivlist




\def\xalignat{\@verbatim \frenchspacing\@vobeyspaces \@xalignatverbatim
You are using the "xalignat" environment in a style in which it is not defined.}
\let\endxalignat=\endtrivlist
 
\@namedef{xalignat*}{\@verbatim\@sxalignatverbatim
You are using the "xalignat*" environment in a style in which it is not defined.}
\expandafter\let\csname endxalignat*\endcsname =\endtrivlist




\def\gather{\@verbatim \frenchspacing\@vobeyspaces \@gatherverbatim
You are using the "gather" environment in a style in which it is not defined.}
\let\endgather=\endtrivlist
 
\@namedef{gather*}{\@verbatim\@sgatherverbatim
You are using the "gather*" environment in a style in which it is not defined.}
\expandafter\let\csname endgather*\endcsname =\endtrivlist


\def\multiline{\@verbatim \frenchspacing\@vobeyspaces \@multilineverbatim
You are using the "multiline" environment in a style in which it is not defined.}
\let\endmultiline=\endtrivlist
 
\@namedef{multiline*}{\@verbatim\@smultilineverbatim
You are using the "multiline*" environment in a style in which it is not defined.}
\expandafter\let\csname endmultiline*\endcsname =\endtrivlist


\def\arrax{\@verbatim \frenchspacing\@vobeyspaces \@arraxverbatim
You are using a type of "array" construct that is only allowed in AmS-LaTeX.}
\let\endarrax=\endtrivlist

\def\tabulax{\@verbatim \frenchspacing\@vobeyspaces \@tabulaxverbatim
You are using a type of "tabular" construct that is only allowed in AmS-LaTeX.}
\let\endtabulax=\endtrivlist

 
\@namedef{arrax*}{\@verbatim\@sarraxverbatim
You are using a type of "array*" construct that is only allowed in AmS-LaTeX.}
\expandafter\let\csname endarrax*\endcsname =\endtrivlist

\@namedef{tabulax*}{\@verbatim\@stabulaxverbatim
You are using a type of "tabular*" construct that is only allowed in AmS-LaTeX.}
\expandafter\let\csname endtabulax*\endcsname =\endtrivlist

% macro to simulate ams tag construct


% This macro is a fix to eqnarray
\def\@@eqncr{\let\@tempa\relax
    \ifcase\@eqcnt \def\@tempa{& & &}\or \def\@tempa{& &}%
      \else \def\@tempa{&}\fi
     \@tempa
     \if@eqnsw
        \iftag@
           \@taggnum
        \else
           \@eqnnum\stepcounter{equation}%
        \fi
     \fi
     \global\tag@false
     \global\@eqnswtrue
     \global\@eqcnt\z@\cr}


% This macro is a fix to the equation environment
 \def\endequation{%
     \ifmmode\ifinner % FLEQN hack
      \iftag@
        \addtocounter{equation}{-1} % undo the increment made in the begin part
        $\hfil
           \displaywidth\linewidth\@taggnum\egroup \endtrivlist
        \global\tag@false
        \global\@ignoretrue   
      \else
        $\hfil
           \displaywidth\linewidth\@eqnnum\egroup \endtrivlist
        \global\tag@false
        \global\@ignoretrue 
      \fi
     \else   
      \iftag@
        \addtocounter{equation}{-1} % undo the increment made in the begin part
        \eqno \hbox{\@taggnum}
        \global\tag@false%
        $$\global\@ignoretrue
      \else
        \eqno \hbox{\@eqnnum}% $$ BRACE MATCHING HACK
        $$\global\@ignoretrue
      \fi
     \fi\fi
 } 

 \newif\iftag@ \tag@false
 
 \def\tag{\@ifnextchar*{\@tagstar}{\@tag}}
 \def\@tag#1{%
     \global\tag@true
     \global\def\@taggnum{(#1)}}
 \def\@tagstar*#1{%
     \global\tag@true
     \global\def\@taggnum{#1}%  
}

% Do not add anything to the end of this file.  
% The last section of the file is loaded only if 
% amstex has not been.



\makeatother
\endinput

thmsupp.tex
%theorem support
%Copyright (C) 1994-95 TCI Software Research
\typeout{TCI Theorem Support for Scientific Word <1 July 1994>.}
\typeout{NOTICE:  This macro file is NOT proprietary and may be 
freely copied and distributed.}
%

\def\newfmtname{LaTeX2e}

\def\chkcompat{%
  \if@compatibility
  \else
    \DeclareOption{thmsa}{\input thmsa.sty}
    \DeclareOption{thmsb}{\input thmsb.sty}
    \DeclareOption{thmsc}{\input thmsc.sty}
    \DeclareOption{thmsd}{\input thmsd.sty}
    \ProcessOptions
  \fi
}

\ifx\fmtname\newfmtname  
  \chkcompat
\fi
\endinput

gr-qc0111062.extracted_bib
\begin{thebibliography}{99}
\bibitem{Elbaz}  Elbaz, E., Novello, M., Salim, J.M., Motta da Silva, M.C.
and Klippert, R. (1997). \textit{Hamiltonian Formulation of FRW Equations of
Cosmology,} General Relativity and Gravitation, \textbf{29, }481-487.

\bibitem{Liu}  Liu, Hongya and Wesson, Paul S. (1994). \textit{Cosmological
Solution and their Effective Properties of Matter in Kaluza-Klein Theory, }%
International Journal of Modern Physics D\textbf{3} 627-637.

\bibitem{Mongan1}  Mongan, T.R.(1999), \textit{N + 1 dimensional quantum
mechanical model for a closed universe,} International Journal of
Theoretical Physics \textbf{38} ,1521-1529.

\bibitem{Mongan2}  Mongan, T.R.(2000). \textit{Inflation, extra dimensions
and Omega sub Lambda approximately =1, }International Journal of Theoretical
Physics \textbf{39,} 99-104.

\bibitem{Mongan3}  Mongan, T.R.(2001). \textit{A simple Quantum Cosmology}. 
\textit{gr-qc/0103021}.

\bibitem{Novello1}  Novello, M., Salim, J.M., Motta da Silva, M.C., Joras,
S.E. and Klippert, R.(1995). \textit{Minimal closed set of observables in
the theory of cosmological perturbations. }Phys. Rev. D \textbf{51}, 450; 
\textit{Minimal closed set of observables in the theory of cosmological
perturbations. II. Vorticity and gravitational waves. }Phys. Rev. D \textbf{52}%
, 730.

\bibitem{Novello2}  Novello, M., Salim, J.M., Motta da Silva, M.C. and
Klippert, R. (1996). \textit{Canonical formulation of standard cosmology:
Direct quantum approach, } Physical Review D \textbf{54}, 6202-6205.

\bibitem{Overdiun}  Overduin, J.M., and Wesson, P.S.(1997). \textit{%
Kaluza-Klein gravity,} Physics Reports,\textit{\ }\textbf{283}, 303-380.

\bibitem{Wesson}  Wesson, P.S., Liu, H. and Seahra, S.S.(2000). \textit{The
big bang as a higher-dimensional shock wave,} Astronomy and Astrophysics 
\textbf{358}, 425-427.

\bibitem{Wessonbook}  Wesson, P.S., 1999, \textit{Space, Time, Matte}r
(Singapore: World Scientific).

\bibitem{Wheeler}  Wheeler, J.A., DeWitt, M.C., and DeWitt, B.S. (1964). 
\textit{In Relativity, Groups and Topology, }DeWitt, B., DeWitt, C., eds.
(Gordon Breach, New York).
\bibitem{Anchordoqui}  Anchordoqui, L., Nunez, C., and Olsen, K.(2000).\textit{
Quantum Cosmology and AdS/CFT, } JHEP \textbf{0100}, 050.
\end{thebibliography}