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%\ICRRnumber{UT-Komaba/03-8  }{April 2003}{hep-th/0304237 }
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          \par\noindent  {April  2003}\par\noindent {hep-th/0304237}  
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\Title{A Quantum Analysis on \\
Recombination Process
and Dynamics of\\ 
D-p-branes at one angle}


\Authors{{   Takeshi Sato
\footnote{tsato@hep1.c.u-tokyo.ac.jp}} \\
 \vskip 3ex
{\itshape  Institute of Physics, University of Tokyo, \\
3-8-1 Komeba, Meguro-ku, Tokyo 153-8902 Japan} \\
}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Abstract
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A quantum-mechanical technique is used within the framework of U(2) 
super-Yang-Mills theory to describe recombination
processes of two D-p-branes at one angle; 
how tachyons condense starting from certain initial conditions, 
and how (curved) shapes of 
the recombined branes develop.  
Two types of initial conditions are considered: 
branes at one angle made put from parallel ones at the initial moment, 
and the ones approaching each other slowly, 
which is important as a candidate of inflation mechanism.
An interesting behavior of the branes' shapes is shown to appear,
each of which comes to have multiple (three) extremes 
due to localization of tachyon condensation
but not the effect of compact spaces. 
Blowing-up behavior of pair-creations of stings connecting 
the recombined branes are also observed.
The above two behaviors means that the branes in the process 
have a relatively stronger tendency than naively expected,
to dissipate their energy into radiations of
gravitational and (RR and NSNS) gauge fields.
This implies that the tachyonic preheating era might be
rather short if this set-up is applied to the braneworld scenarios. 


\newpage
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\section{Introduction}
\setcounter{equation}{0}

Recombination of D-branes at angles are processes 
increasing its value recently 
from both phenomenological 
and theoretical point of view, related to the presence of
tachyonic modes which appear between the two D-branes\cite{angle}.  
In the brane inflation scenarios,
these are some of the promising candidates 
to explain the mechanism of inflation\cite{halyo}\cite{hirano1}%
\cite{ky}\cite{oneangle}\cite{int2}\cite{hirano2}\cite{tye1}%
\cite{2angles}\cite{bro}\cite{br1}.
%and recombination of two D-p-branes is one of them
%\cite{oneangle}\cite{int2}\cite{tye1}\cite{2angles}.
In addition, among various string constructions of Standard Model,
intersecting brane models are one class of hopeful
candidates\cite{bl1}\cite{chiral}\cite{bl2}\cite{getjust}\cite{higgs}%
\cite{koko1}\footnote{
For a recent review and other references, see e.g.
ref.\cite{ur1}.},
in some of which it has been proposed that
the recombination process occurs as Higgs mechanism% 
\cite{chiral}\cite{higgs}\cite{koko1} (see also \cite{bachas}\cite{koko2}). 
%although for this case  
%multiple angle cases are rather favored. 
On the other hand, this system 
and process can be regarded as a generalized 
setting of $D\bar{D}$ system and its annihilation process, 
which has been studied thoroughly\cite{sen1}\cite{sen2}\cite{tye2}. 
In this way, this is one of the most important phenomena 
to explore at present in string theory.

The recombination process of D-p-branes, however, 
especially its mechanism
how the tachyon condensation causes recombination,
have not been completely understand.
Thus, shapes of the D-p-branes after the recombination and 
and those time-evolution behavior have not also been understood 
(though schematic pictures of them have been drown in
many references).
Very recently, in ref.\cite{hn}, the authors
found  that a T-dual of Yang-Mills theory is the framework
which allows a direct connection between condensing tachyon fields and 
a relative motion of the recombined branes in the case of 
two D-strings at one angle,
and tried to presume how the shapes of the branes develop 
using a classical analysis.\footnote{Very recently, the process is also 
analyzed via tachyon effective field theory in ref.\cite{wh}.}

In order to understand precise behavior of the D-branes 
in the process, however, 
a quantum analysis is indispensable;
D-brane is fundamentally a quantum-mechanical object.   
Furthermore, the system, its tachyon sector in particular, 
is essentially a collection of inverse harmonic oscillators with  
(generically time-dependent) negative frequency-squareds,
and its time-evolution behavior depends crucially on
the initial condition for fluctuations of the system.
Thus, if one consider a system at zero temperature 
(as we will do later),
quantum fluctuations are relevant, and 
an appropriate analysis of them is necessary, which has not been 
done at least in this setting.\footnote{
In the case of inflation and other systems, see 
ref.\cite{guth}\cite{boya1}, and for a non-perturbative analysis,
see ref.\cite{linde1}.}

The purpose of this paper is to understand precise 
time-evolution behavior of D-branes
just on and after the recombination
via a quantum-mechanical analysis,
in certain cases of two D-p-branes at one angle for each $p$ ($\ge 1$)
(i.e. the simplest case among those of D-branes at angles),
and to explore implications of the results
for phenomenological scenarios where these settings are used.
To be concrete,
we consider two types of definite initial conditions for  
branes: the case (I): two D-p-branes intersecting at one angle $\theta$ 
made put from parallel ones at the initial instant, and  
the case (II): two D-p-branes at one angle $\theta$ approaching each other 
with a small relative velocity $v$.
The case (I) is one of the simplest conditions.
We first investigate the case to understand recombination 
process itself, and 
establishing the method and 
learning some knowledge of the recombination process,
we next discuss the case (II), a more practical one;
the case (II) with $p=4$ is one of the hopeful setting for 
the brane inflation scenarios%
\cite{halyo}\cite{oneangle}\cite{tye1}, so, 
we can get some qualitative feature of the inflation model 
from the analysis.
In addition, the case may be regarded as a (simpler) test case of 
the Higgs mechanism in the D-brane model intersecting
at multiple angles \cite{chiral}\cite{higgs}, where
D-branes' geometries are responsible for the physical 
parameters in the models.
So, investigating it might also lead to the 
understanding of the Higgs mechanism in the intersecting brane model. 
(Actually, there is a subtle point in case (II)
to refer to the brane's shape, which will be discussed 
in the final section. )
Practically speaking, the case (I) may seem less realistic,
but it may be regarded as a some local approximation 
to more complicated setting of curved branes, or 
a rough approximation to the case (II) with more rapid 
(but non-relativistic) motion and with a relative large angle. 
We believe that the work in this paper is also of value in that 
we describes time-dependent, i.e. 
dynamical behavior of curved branes, though for 
rather short time (e.g. less than a few hundred times $l_{s}$). 


The outline of our analysis 
is summarized in the following six paragraphs:

The framework we choose is the T-dualized version of U(2) 
super-Yang-Mills theory, as done in ref.\cite{hn};
A short distance analysis is adequate for our purpose
because we need to handle the behavior of the D-branes 
{\it just on and after} the recombination.
\footnote{
I thank K. Hashimoto for having told me 
on a part of their work ref.\cite{hn}, in progress at that time,
(before I started this work)
that the mass spectrum of fluctuations on D-branes at angles,
including tachyonic ones, 
can be deduced within the framework of Yang-Mills theory,
which I had been seeking. See ref.\cite{baal}\cite{hw}\cite{hn}.}

The fundamental set-up is;
we consider compact world-space dimensions of D-branes 
wrapping on a p-torus and keep the radii large but finite 
to avoid the branes to have an infinite mass.
We assume that $\theta$ and the string coupling $g_{s}$ 
are so small that the D-branes are heavy and initially almost rigid, 
and that the velocity $v$ can be regarded as a constant
%with as that of a constant velocity
for the case (II) (see subsection 3.1 more detail).
%\footnote{
%The force between the two branes is proportional to a positive power
%of $\theta$ and approaches to zero as $\theta \to 0$ 
%(refs.\cite{jabbari}\cite{pol} and for short distance, 
%see ref.\cite{oneangle}).}.

Under the framework and the set-up, 
we derive time-evolution behavior of the tachyonic
fluctuation for each of the two cases; 
we set an ansatz for U(2) ``background" fields 
corresponding to the initial D-p-branes, and 
consider 
fluctuations around it, including a 
complex (potentially) tachyonic one, 
which appears in off-diagonal elements of
a transverse scalar field and 
the gauge field, as discussed in ref.\cite{baal}\cite{hw}.
Using mode-expansions, and
taking terms second order in the modes
(i.e. WKB approximation), 
we get an action of a collection of harmonic oscillators,
some of which have frequency-squareds
of initially positive but later getting negative value;
constantly negative ones except at the initial moment
for tachyonic modes of the case (I), and initially positive 
but decreasing and finally getting negative ones for tachyonic 
modes of the case (II).
Thus, applying quantum mechanics,
we evaluate a typical absolute value of the tachyon field 
by using a sum of contributions of tachyonic modes.
We note that we neglect the effect of massless and massive modes
because their contribution is suppressed by 
a factor $1/\sqrt{T_{p}}=g_{s}^{1/2}$.
(To argue validity of the WKB approximation later,  
we further make a detailed study of a typical value of the tachyon
in terms of momenta which tachyon modes possess 
along world-space dimensions.)

Next, we try to extract geometric information of the branes
from the typical tachyonic fluctuation.
In this paper we 
identify the typical absolute value of the tachyon field
as its VEV.
Then, for case (I), there is 
only one scalar field which has nontrivial VEV's 
(or classical values) in its elements. 
Thus, it is a logical step 
to choose the gauge 
in which  the VEV's of the scalar field are diagonalized and 
regard the diagonal elements as the positions of 
the branes, 
as done in the conventional cases
where all of VEV's of transverse scalar fields commute.
Then, one can connect the blowing-up tachyon modes
directly to a certain relative motion between the recombined branes
as in ref.\cite{hn} (see the second footnote in subsection 2.3).
For the case (II), however, there are two scalar fields which have
nontrivial VEV's and does not commute with each other, 
so, the case is not so simple.
In section 3
we will assume that the diagonalized elements of one scalar field 
represent the positions of the branes along the direction and 
proceed the investigation, postponing the discussion of
this point in the final section. Then,
%Finally, following the above assumption, 
we can describe time-evolution behavior of 
the shapes of the recombined branes.

Our analysis shows that an interesting behavior appears;
each of their shapes will develop into the one with 
multiple (three) extremes due to localization of tachyon condensation
but not the effect of compact world-spaces.
In addition, blowing-up behavior of pair-creations of strings 
connecting the branes after recombination is also observed 
as that of the electric flux.

Finally, we discuss physical consequences of the above two interesting 
behaviors, and their implication for the brane-world phenomenology.

The organization of this paper is as follows: 
In section 2  
we discuss the case (I):  
D-p-branes at one angle initially put parallel, for each $p$.
(We deal with the case of $p\ge 2$ and that of $p=1$, 
i.e. D-strings separately for a physical reason.) 
In subsection 2.1 we present its detailed set-up and preliminaries 
for a quantum analysis.
In subsection 2.2 we evaluate 
time-evolution behavior of the typical amplitude of the 
tachyonic fluctuation using quantum mechanics, 
and discuss its properties in detail in terms of the momentum
which tachyons possess. 
Then, in subsection 2.3, 
connecting the VEV's with relative motions of recombined branes, 
we discuss time-evolution behavior of the shapes of the branes,
and describe how the electric flux as strings blows up.
In section 3, we discuss the case (II):
D-p-branes at one angle approaching slowly.
In subsection 3.1 we give its set-up and preliminaries.
In subsection 3.2 we evaluate 
time-evolution behavior of the typical amplitude of tachyons.
In subsection 3.3 we first deal with the shapes of the branes, 
and the blow-up behavior of pair-creations of strings. 
Then, we discuss their physical consequences related to the dissipation,
and finally speculate its implication for the brane-world phenomenology.
In section 4 we discuss diagonalization procedure
in the case (II), related to the non-commutativity of scalar fields  
and geometrical information of the branes, the former of which
appears in this case. 
In the appendix we give a short review of deriving 
a width of a Gaussian wave function of each mode at arbitrary time,
in terms of basis functions of its equation of motion.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Case (I); initially parallel two D-p-branes}
\setcounter{equation}{0}
\setcounter{footnote}{0}

In this section we discuss the case (I);
two D-p-branes intersecting at one angle $\theta$ 
made put from parallel ones at the initial instant.

\subsection{Preliminaries}

In this we present the set-up and preliminaries for a
quantum analysis.

We set the string coupling $g_{s}$ very small 
and consider D-p-branes in a 10-dimensional
flat target space with a metric $g_{\mu\nu}={\rm diag}(-1,1,..,1)$
and coordinates $x^{a}$ ($a=0,..,9$). We compactify all 
the space dimensions on a 9-torus with large periods $L_{a}$ for 
$a=1,\cdots,9$. 

The initial condition is that for the period $t<0$,
the two D-p-branes are parallel, and at $t=0$ they are
put into the configuration of intersecting at one angle.
The simplest one is represented in Fig.1.
Although there are multiple intersecting points in 
cases of compact dimensions, we concentrate on the behavior 
around $x=0$ in principle in this paper.
\begin{figure}[h] 
\setlength{\unitlength}{1mm}
\begin{picture}(160,40)
\put(35,17.5){\vector(1,0){75}}
\put(72.5,5){\vector(0,1){25}}
\put(35,5){\line(1,0){75}}
\put(35,30){\line(1,0){75}}
\put(35,5){\line(0,1){25}}
\put(110,5){\line(0,1){25}}
\put(35,5){\thicklines\vector(3,1){75}}
\put(35,30){\thicklines\vector(3,-1){75}}
\put(90,18){\makebox(5,5){$\theta /2$}}
\put(90,12){\makebox(5,5){$\theta /2$}}
\put(75,23){\makebox(5,5){$x_{p+1}$}}
\put(105,17){\makebox(5,5){$x_{p}$}}
\put(85,25){\makebox(5,5){$Dp$}}
\put(85,5){\makebox(5,5){$Dp$}}
\put(72.5,33){\makebox(5,5){$L_{p}$}}
\put(115,15){\makebox(5,5){$L_{p+1}$}}
\end{picture}
\caption{The D-p-branes at $t=0$ for the case (I)}
\end{figure}
The configuration cannot be distinguished from 
that of two branes each bent and touched at $x_{p+1}=0$;
the energy density and flux are the same (if the configuration 
would be possible). 
The latter configuration is unstable because 
contraction of the length or the area of the branes possible in this
case save the energy of the branes, so
the branes begins to decay into two shortened separate branes.
Their shapes are expected to be like a hyperbola.
This instability leads to the existence of a complex tachyon field 
with a mass-squared 
\beqa
m^{2}=-\theta,\label{mass1}
\eeqa 
appearing as modes of a string connecting the two D-branes
as shown in ref.\cite{angle}, and 
the condensation of the tachyons is expected to
cause the recombination process.   
We describe behavior of the two D-p-branes using 
T-dualized versions of U(2) super-Yang-Mills theory
as in ref.\cite{hw}\cite{hn}. 

For each $p$, the action for two D-p-branes derived from the 
Born-Infeld action is
%\footnote{Here we keep the term $1$  
%representing the energy density of the branes.} 
\beqa
S_{2Dp}=-T_{p} l_{s}^4 \int d^{p+1} \xi \ 
{\rm Tr} (l_{s}^{-4}+\frac{1}{4}F_{\mu\nu}F^{\mu\nu} +\frac{1}{2 
l_{s}^4}(D_{\mu}X_{i})^2
-\frac{1}{l_{s}^{8}} [X_{i},X_{j}]^{2})
\eeqa
where $T_{p}$ is the tension of a D-p-brane. 
$F^{\mu\nu}$ is the field
strength of the world-volume U(2) gauge field $A_{\mu}$ for
$\mu=0,1,..,p$. $X_{i}$ for $i=p+1,..,9$ are 
U(2) adjoint scalar fields corresponding to 
coordinates $x^{i}$
transverse to the branes. $\xi^{\mu}$ is a world-volume 
coordinate, which we embed as $x^{\mu}=\xi^{\mu} $ 
for $\mu=0,..,p$. 
We denote $x^{0}$ as $t$ and $x_{p}$ as $x$ below,
and set $l_{s}= 1 $ for convenience (and revive it when needed).  
We note that $x$ is 
parallel to neither of the branes, 
but can parametrize a world-space 
of either branes, though the energy density 
is $\sqrt{1+\beta^2}$ times larger then the parallel embedding.
To avoid the branes to have infinite masses,
we consider compact world-space dimensions 
wrapping on a p-torus of a large volume.
We also note that the period $L_{p}$ is limited as
\beqa
L_{p}<\frac{1}{\theta}\label{lcond1}
\eeqa
because the super-Yang-Mills theory is the field theory
of string theory effective for distance shorter than $l_{s}$,
which gives the constraint $L_{p}\theta < l_{s}(=1)$.

The ``background" D-p-branes for the case (I) are 
represented by the configuration:
\beqa
\begin{array}{cc}
X_{p+1}^{(0)}= \left(
\begin{array}{cc}
\beta x & 0 \\
0 & -\beta x
\end{array}
\right),
& X_{p+2}^{(0)}=\cdots =X_{9}^{(0)}=0,\ \ A_{\mu}^{(0)}=0
\end{array}\label{back1} 
\eeqa
where $\beta\equiv \tan(\theta/2)$. 
This is T-dual 
to the configuration of
two D-(p+1)-brane with a constant field strength
$F_{p,p+1} = \beta$, as discussed in ref.\cite{hw}. 
Since tachyonic fluctuations are modes of a string
connecting the two branes, 
they are expected (and will be shown) to arise in off-diagonal 
elements of $A_{\mu}$ and $X_{p+1}$.
Actually, in ref.\cite{baal}, 
tachyonic modes around the T-dualized ($F_{p,p+1} = \beta$)
background were shown to appear in off-diagonal ones of
$A_{p}$ and $A_{p+1}$ in a certain gauge,
T-dual to $X_{p+1}$ in the present case.
However, the connection via T-duality does not of course means
that the two systems are physically equivalent, and
actually, as we will show later,  
the T-duality 
has a non-trivial effect (to change
of the width of a wave function by a  $\sqrt{2}$).
Thus, we examine the spectrum from the beginning.

Denoting the fluctuations as
\begin{equation}
\begin{array}{ccc}
A_{\bar{\mu}}= \left(
\begin{array}{cc}
0 & c_{\bar{\mu}}^{*} \\
c_{\bar{\mu}} & 0
\end{array}
\right)
, &
A_{p}= \left(
\begin{array}{cc}
0 & c^{*} \\
c & 0
\end{array}
\right)
, &
X_{p+1}= \left(
\begin{array}{cc}
0 & d^{*} \\
d & 0 
\end{array}
\right) 
\end{array}\label{ansatz}
\end{equation}
(where $\bar{\mu}=0,..,p-1$),
the action of $c_{\mu}$ and $d$ 
%related to the tachyonic modes 
takes the form
\beqa
S_{2Dp}
&=&T_{p} \int d^{p+1}x [ 
-\partial_{\mu}c^{*}_{\nu}\partial^{\mu}c^{\nu} 
+|\partial_{\mu}c^{\mu}|^2 
-\partial_{\mu}d^{*}\partial^{\mu}d 
+2 i \beta x(c^{*}_{\mu}\partial^{\mu}d 
-c_{\mu} \partial^{\mu}d^{*}) \nonumber\\ 
& &-2 i \beta (c^{*}d - c d^{*})
-4(\beta x)^{2} |c_{\mu}|^{2}\nonumber\\
& &-\frac{1}{2}|c_{\mu}^{*}c_{\nu}-c_{\nu}^{*}c_{\mu}|^{2}
-|c^{*} d-c d^{*}|^{2}].
\label{fullaction1}
\eeqa
Following ref.\cite{baal}\cite{hw} where the gauge fixing condition
is $\partial_{\mu} A^{\mu}+i[A_{\mu}^{(0)},A^{\mu}]=0$,
we set the gauge $\partial_{\mu} A^{\mu}+i[X_{p+1}^{(0)},X^{p+1}]=0$,
which gives
\beqa
\partial_{\mu}c^{\mu} -2 i\beta x d=0.\label{gf1}
\eeqa
Integrated by part
and with (\ref{gf1}), the second  and fourth terms of 
(\ref{fullaction1}) are written as 
$-2 i \beta (c^{*}d - c d^{*})-4(\beta x)^{2} |d|^{2}$. 
Redefining the fields as
\beqa
\tilde{c} \equiv \frac{c-id}{\sqrt{2}} ,\ 
\tilde{d} \equiv \frac{d-ic}{\sqrt{2}}, 
\label{defctilder} 
\eeqa
the action second order in the fluctuations is written as
\beqa
S_{2Dp}|_{{\rm 2nd}}&=&  T_{p} \int d^{p+1}x [
-\tilde{c}^{*} \{ -(\partial_{\mu})^{2}
+4(\beta x)^{2} -4\beta\} \tilde{c}\nonumber\\
&-&\tilde{d}^{*}  \{ -(\partial_{\mu})^{2}
+4(\beta x)^{2} +4\beta\}\tilde{d} 
+c^{*}_{\bar{\nu}} \{ -(\partial_{\mu})^{2}
+4(\beta x)^{2} \} c^{\bar{\nu}}
].\label{secondaction1}
\eeqa
We can make a mode expansion of the fields $\tilde{c}$, $\tilde{d}$ and 
$c_{\bar{\nu}}$ with respect to $x$ 
if we find the full eigenvalues $\tilde{m}^{2}$ and 
eigenfunctions $f$ of the operator 
\beqa
[-(\partial_{x})^{2}+4(\beta x)^{2} -4 a \beta]\label{ho}
\eeqa 
where $a=1,-1,0$ for $\tilde{c}$, $\tilde{d}$ and 
$c_{\bar{\nu}}$, respectively.
Since $x$ direction is compactified,   
the eigenfunctions are expected to be mathematically complicated, 
e.g. related to theta functions
as in the T-dualized case in ref.\cite{baal}\cite{hw}.
However, if one consider a certain length of the period $L_{p}$,
one can find a set of approximate solutions easily,
as we will discuss below.

If we assume that $L_{p}$ is so large that the periodicity 
can be ignored,
$\tilde{m}^{2}$ and $f$ satisfying the equation
$[-(\partial_{x})^{2}+4(\beta x)^{2} -4 a \beta] f = \tilde{m}^{2} f$
are identified with those of a one-dimensional harmonic oscillator.
They are obtained as
\beqa 
\tilde{m}^{2}_{n}&=&2\beta(2 n -a)\label{massform}\\
f_{n}&=&\frac{1}{\sqrt{2^{n}n!}}(\frac{2\beta}{\pi})^{1/4}
e^{-\beta x^{2}}H_{n}(\sqrt{2\beta}x)
\eeqa
for each non-negative integer $n$ where $H_{n}$ are Hermite polynomials.
We can see that
the lowest mode of $\tilde{c}$ (we denote as $\tilde{c}_{0}$)
is a complex tachyon field with a mass-squared
\beqa
m^{2}=-2 \beta,
\eeqa 
which agrees with (\ref{mass1}) obtained in ref.\cite{angle} 
for a small $\theta$, while
the lowest modes of $c_{\bar{\nu}}$ are massless and the others are
massive. 
Since the wave function of 
$\tilde{c}_{0}$ is Gaussian whose square has 
a relatively broad width $\delta\equiv 1/\sqrt{2\beta}$ as 
\beqa
f_{0}(x)=(\frac{2\beta}{\pi})^{1/4} e^{-\beta x^{2}}, 
\label{gauss}
\eeqa
it is localized around $x=0$ with the width, and
so do those functions of the other modes, at least 
for not extremely higher n.
Thus, we can regard $f_{n}$ as a set of approximate solutions
of this case (with the compactified dimension $x$) if 
$L_{p}$ satisfies
\beqa
\delta \equiv \frac{1}{\sqrt{2\beta}} \ll L_{p}.\label{lcond2}
\eeqa 
(The discrepancy of $f_{n}$ with the real solutions appears, 
for example, as discontinuity of differential coefficients
of $f_{n}$ at $x=\pm L_{p}/2$, which is irrelevant if 
(\ref{lcond2}) is satisfied.)
We note that the condition (\ref{lcond2}) is compatible with
the one (\ref{lcond1}) if $\theta \ll 1$. Throughout this paper,
we consider such a range of  $L_{p}$ and use $f_{n}$ (as 
approximate eigenfunctions of (\ref{ho})) to expand the fields as
\beqa
\tilde{c}(x_{\mu})&=&\sum_{n=0}^{\infty}\tilde{c}_{n}(x_{\bar{\mu}})f_{n}(x)
\nonumber\\
\tilde{d}(x_{\mu})&=&\sum_{n=0}^{\infty}\tilde{c}_{n}(x_{\bar{\mu}})f_{n}(x)
\nonumber\\
c_{\bar{\nu}}(x_{\mu})&=&\sum_{n=0}^{\infty} 
c_{n, \bar{\nu} }(x_{\bar{\mu}})f_{n}(x).
\label{modeexp1}
\eeqa
We note that the (approximate) width of the function $f_{n}$
in this case is larger than that in the T-dualized theory
in ref.\cite{hw}\cite{baal} by  a factor $\sqrt{2}$, 
which is one of non-trivial effects of the T-duality. 
 
Substituting (\ref{modeexp1}) for the action (\ref{secondaction1}),
we get
\beqa
S_{2Dp}|_{{\rm 2nd}}&=&T_{p}\int d^{p}x
\sum_{n=0}[|\partial_{\bar{\mu}}\tilde{c}_{n}|^2 
-\tilde{m}^{2}_{n}|\tilde{c}_{n}|^{2}
+(\tilde{c}_{n}\to \tilde{d}_{n}, \ c_{\bar{\nu},n})
]\label{secondaction2}
\eeqa
due to (approximate) orthogonality of $f_{n}$.
We consider mode expansions further with respect to the world-space 
coordinates $x_{\hat{\mu}}$ ($\hat{\mu}=1,\cdots,p-1$) 
other than $x$ for $p \ge 2$. 
(In the case of D-strings we set $k^{2}=0$ below.)
Since these dimensions are compact,
the wave vectors (momenta) are 
$ k_{\hat{\mu}}=2 \pi  n_{\hat{\mu}}/L_{\hat{\mu}}$ for integers 
$n_{\hat{\mu}}$, and basis functions are given by 
\beqa
u_{k}\equiv\frac{
e^{i k_{\hat{\mu}} {x}_{\hat{\mu}}}}{\sqrt{V_{p-1}}}
\label{ufunc}
\eeqa
where 
%$ L_{\hat{\mu}}$ is a length of the $x^{\hat{\mu}}$ 
%direction of the (p-1)-torus and 
$V_{p-1}$ is the volume of (p-1)-torus.
Expanding $\tilde{c}_{n}$ and the other fields such as 
\beqa
\tilde{c}_{n}(x_{\bar{\mu}}) =\sum_{k_{\hat{\mu}}} 
\tilde{c}_{n,k_{\hat{\mu}}} (t) 
u_{k}(x_{\hat{\mu}}),
%\tilde{d} =\sum_{k_{\hat{\mu}}} \tilde{d}_{k_{\hat{\mu}}} (t) 
%u_{k}(x^{\hat{\mu}}), 
\label{expansion2}
\eeqa
we obtain the action 
\beqa
S_{2Dp}|_{{\rm 2nd}}=T_{p}\int d t
\sum_{n,k_{\hat{\mu}}} [ |\frac{d\tilde{c}_{n,k}}{dt}|^{2}
-\omega_{n,k}^{2}|c_{n,k}|^{2}
+(\tilde{c}_{n,k}\to \tilde{d}_{n, k}, \ c_{\bar{\nu},n,k})]
\label{secondaction3}.
\eeqa
where 
\beqa
\omega_{n,k}^{2}= k_{\tilde{\mu}}^2+2 \beta (2n-a).\label{omega1}
\eeqa
(Note that $a=1,-1,0$ for $\tilde{c}_{n,k} \tilde{d}_{n,k},  
c_{\bar{\nu},n,k}$, 
respectively.)
This looks much like the action of a collection of harmonic 
oscillators
with a mass $T_{p}$, although it includes modes with negative 
frequency-squareds;
in this case only the modes $\tilde{c}_{0,k}$ with 
$0 \le k^{2} \le 2\beta$
have negative $\omega^{2}$'s, while
all the other modes %of $\tilde{d}_{n\ k}$, $\tilde{d}_{n\ k}$ 
have non-negative $\omega^{2}$.
We use the action (\ref{secondaction3}) to make a quantum analysis.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{A quantum analysis on tachyon condensation in case (I)}

In this subsection we analyze in detail how the tachyon fields condense,
applying quantum mechanics to the action (\ref{secondaction3}). 
Throughout this paper we consider the system initially 
at zero temperature, 
i.e. at the 
ground state.

One of the two basic ingredients (we will use here) is
that in a harmonic oscillator with a positive constant $\omega^{2}$, 
the ground state wave function is Gaussian, and 
its variable $q$ has a typical amplitude of fluctuation
$q_{{\rm typical}}\equiv \sqrt{<q^{2}>}=1/\sqrt{2m \omega}$ where $m$ is
its mass. ($<A>$ denotes a VEV of an operator $A$.)
Another basic one is that even if $\omega^{2}$ is time-dependent and 
becomes negative afterward, time-evolution behavior of the wave function 
and hence that of $q_{{\rm typical}}$ can be deduced within WKB 
approximation,
by using its propagator or kernel, as in e.g. ref.\cite{fey}.
Taking these two into account, time-evolution behavior of the 
typical amplitude 
of each mode can be evaluated using quantum mechanics,
if we consider such a definite initial condition that 
$\omega^{2}$ of each mode is positive at the initial moment 
but will change into a negative one later.

In this case since
$\tilde{c}, \tilde{d}, c_{\bar{\nu}}$ are fields,
we have to sum (or integrate as an approximation of large periods 
$L_{\bar{\mu}}$) 
$|\tilde{c}_{n, k}|_{{\rm typical}}^{2} (f_{n})^{2}|u_{n,k}|^{2}$ 
with respect to $n$ and
the momenta $k_{\bar{\mu}}$ to get 
the typical absolute value of the tachyonic field
$|\tilde{c}(x_{\mu})|_{{\rm typical}}$
(except for the case of D-strings),
but then, the integral is divergent.
However, this is a kind of divergence that appears 
in a usual quantum field theory
and has nothing to do with the tachyon blow-up which we want to know.
So, we ignore the contribution of all the non-tachyonic modes 
and define the typical value of the amplitude of the tachyonic
fluctuation as
\beqa
|\tilde{c}(x_{\mu})|_{{\rm typical}} \equiv \{ \int_{k^{2}\le 2\beta} 
\frac{d^{p-1}{\bf k}}{(2\pi)^{p-1}}<|\tilde{c}_{0, k}(t)|^{2}> 
(f_{0})^{2}\}^{1/2}.\label{typfldef}
\eeqa
(We have replaced $\Sigma_{k}/V_{p-1}$ for 
$\int d^{p-1}{\bf k}/(2\pi)^{p-1}$. )
In this paper we regard this quantity as a VEV of $|\tilde{c}(x)|$ due 
to tachyon condensation.
We note that the typical amplitudes of the modes with non-negative 
$\omega^{2}$'s are highly suppressed by $T_{p}$ in the denominator as, 
\beqa
|\tilde{c}_{n, k}(t)|_{{\rm typical}}\sim \frac{1}{\sqrt{2T_{p}\omega}}=\sqrt{\frac{g_{s}}{2 
\omega}} \ \ (n\ge 1)
\eeqa
so the WKB approximation is quite good 
until the amplitude of the tachyonic fluctuation becomes large.

Now, let us evaluate the time-evolution behavior of $|\tilde{c}_{0, k}|$:
The initial typical amplitude of each fluctuation is:
Since it holds $\beta=0$ until $t=0$ and
the system is at the zero temperature,
the mode for each $k$ stays in the lowest energy state with 
the Gaussian wave function $\Psi(t=0,c_{0, k})$ 
whose absolute square has the width 
$\Delta_{k}(t=0)=1/\sqrt{T_{p} |k|}$ for $t\le 0$.
The typical amplitude of $|\tilde{c}_{0, k}|$
at $t=0$ is $\Delta_{k}(0)/\sqrt{2}$.
It is shown that 
the initially Gaussian wave function develops so that 
its absolute value keeps to be Gaussian
within WKB approximation.  
In addition, the time-dependent width describing the system
can be written in terms of two independent solutions 
to the equations of motion 
\beqa
\frac{d^{2} \tilde{c}_{0,k}}{dt^{2}}= 
-\omega_{0,k}^{2} \tilde{c}_{0,k}=(2\beta-k^{2})\tilde{c}_{0,k}.
\label{eom1}
\eeqa
(These are reviewed in the appendix. See ref.\cite{fey} for more detail.)
We denote the solutions as ``basis functions''.
Defining $K\equiv \sqrt{-\omega^{2}_{k}}$ ($\ge 0$ for tachyonic modes),
the basis functions in this case are
\beqa
N_{1}\cosh (Kt), \ \ N_{2} \sinh (Kt) 
\eeqa
where $N_{1}$ and $N_{2}$ are normalization constants.
Substituting them and the initial width $\Delta_{k}(0)$ 
for (\ref{formaldelta}),
the time-evolution behavior of $|\tilde{c}_{0,k}|_{{\rm typical}}$
is represented by the width of 
the absolute square value of the wave function as
\beqa
|\tilde{c}_{0,k}|_{{\rm typical}}^{2}\equiv 
\frac{(\Delta_{k}(t))^{2}}{2}=
\frac{(\Delta_{k}^{0})^{2}}{2}\cosh^{2}(Kt)
+\frac{1}{2(\Delta_{k}^{0})^{2} T_{p}^{2} K^{2}}\sinh^{2}(Kt)\nonumber\\
=\frac{1}{2 T_{p}}\{\frac{1}{|k|}
+\frac{2\beta}{2\beta-k^{2}}\sinh^{2}(\sqrt{2\beta-k^{2}}t) \}.
\label{staticsemi}
\eeqa


Thus, the typical value of $|\tilde{c}(x_{\mu})|$ defined in
(\ref{typfldef}) is obtained as 
\beqa
|\tilde{c}(x^{\mu})|_{{\rm typical}}^{2} 
\equiv  A(t)^{2}f_{0}(x)^{2}
&=&\frac{\Omega_{p-2} }{2 T_{p} (2\pi)^{p-1}}
\int_{0}^{\sqrt{2\beta}} k^{p-2}dk\nonumber\\
& &
[\frac{1}{k}+\frac{2\beta}{2\beta-k^{2}}
\sinh^{2}(\sqrt{2\beta-k^{2}}t)]
\sqrt{\frac{2\beta}{\pi}}e^{-2\beta x^{2}}\label{staticcfinal} %\\
\eeqa
for D-p-brane for $p\ge 2$,
where $k\equiv \sqrt{(k^{\hat{\mu}})^{2}}$,
$\Omega_{p-2}$ is the volume of (p-2)-sphere, and $A(t)>0$.    
This is the formula for the time-evolution of
the tachyonic fluctuation for the case (I) with $p\ge 2$, 
evaluated properly using quantum analysis.
We note that for $p=2$ the integral
of the first term need a regularization, but the term
does not have time-dependence, so 
the essential part is not affected by that.
We also note that 
$|\tilde{c}(x^{\mu})|_{{\rm typical}}$
has x-dependence through the Gaussian function $f_{0}$.
This means that the tachyon is localized around $x=0$,
though the width is relatively broad such as $\delta=1/\sqrt{\beta}$.
(This fact gives rise to an interesting phenomenon,
which we will discuss later.)
The time evolution behaviors of $A(t)\sqrt{T_{p}}$
(the time-dependent part of $\tilde{c}\sqrt{T_{p}}$) 
\footnote{
We do not fix  $T_{p}=1/g_{s}$ at this point yet, so 
cancel the dependence of $\tilde{c}$ on $T_{p}$ by multiplying it.}
for the D-2-brane and 
D-4-brane case are plotted in Fig.2 for a specific value 
of the angle $2 \beta =0.1$. 
($T^{(I)}\equiv 1/\sqrt{2\beta }(\simeq 3.162 l_{s})$
denotes a typical time scale of this decay which will be discussed soon.)
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{f2}
\caption{$\log_{10}(A/\sqrt{g_{s}})$ vs. $t/T^{(I)}$.The lower graph is 
for D-4-branes while the upper one is for D-2-branes, with $2\beta(\simeq \theta)=0.1$.}
\end{center}
\end{figure}

We can compare the result with the
behavior $|\tilde{c}_{0}|\sim e^{\sqrt{2\beta}t}$
expected naively from the classical field equation
$(\partial_{t})^{2}\tilde{c}_{0}=2\beta \tilde{c}_{0}$.
The asymptotic behavior of $|\tilde{c}(x^{\mu})|_{{\rm typical}} $
for relatively large $t$ on the basis of (\ref{staticcfinal}) is 
roughly given by
\beqa
|\tilde{c}(x^{\mu})|_{{\rm typical}} 
&\sim &
\sqrt{\frac{\Omega_{p-2}}{4T_{p}(2\pi)^{p-1}}}
(0.2\beta)^{\frac{p-2}{4}}f_{0}(x)
\frac{e^{ \sqrt{2\beta} t}}{\sqrt{t}}
%(\frac{2\beta}{\pi})^{1/4}
%e^{-\beta x^{2}}
,\label{asymp1}
\eeqa
whose exponential behavior for $t$ is consistent with the classical one.
(So, we define the coefficient $\sqrt{2\beta}\equiv \gamma_{0}$
and the typical time scale $T^{(I)}$.) 
% \equiv 1/\sqrt{2\beta}$.)
Furthermore, for a later use,
we explore momentum correction to the coefficient $\gamma_{0}$ 
because 
$\gamma_{0}^{2}\equiv 2\beta$ appear in the time-dependent part of 
$|\tilde{c}|_{{\rm typical}}$ through the form $2\beta-k^{2}$.
Let us define  an effective coefficient $\gamma_{{\rm eff}}^{(I)}
\equiv
\partial_{t}\ln|\tilde{c}_{0}(t)|_{{\rm typical}}$ (i.e. 
$|\tilde{c}|\sim e^{\gamma_{{\rm eff}}^{(I)} t})$.
%which receive a correction by contribution of 
%the momentum tachyons possess. 
Its behavior for $t$ %of $\gamma_{{\rm eff}}$ 
is given in Fig.3, by which 
we can see that the effective momentum correction to $\gamma_{0}$ is
not so large, 
about the order of 20 $\sim$ 5 percent and monotonically increasing.
(It is reasonable since the more $t$ passes, the more the tachyons with 
less momentum blow up and contribute to 
$|\tilde{c}(x^{\mu})|_{{\rm typical}} $.)
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{f3}
\caption{$\gamma_{{\rm eff}}/\gamma_{0}$ vs. $t/T^{(I)}$.}
\end{center}
\end{figure}
Actually, 
we can also estimate %using (\ref{staticcfinal})
a typical momentum of the tachyon: 
which momentum-possessing tachyon modes contribute the most to 
the blow-up. 
%and confirm the consistency of the behavior of
%$\gamma_{{\rm eff}}^{(I)} $. 
The average of their square momenta can be defined as
\beqa
<k^{2}>\equiv \int \frac{d^{p-1}{\bf k}}{(2\pi)^{p-1}} 
\Delta_{k}(t)^{2} k^{2}
/\int \frac{d^{p-1}{\bf k}}{(2\pi)^{p-1}} \Delta_{k}(t)^{2},
\label{momav}
\eeqa
since the weight for a tachyon mode with a momentum ${\bf k}$ 
in the fluctuation $|\tilde{c}(x^{\mu})|^{2}$ 
is $(\Delta_{k})^{2}/2$ in (\ref{staticsemi}).
Then, for D-4-branes with $2\beta =0.1$,
$<k^{2}>/k^{2}_{{\rm max}}$ is drown in Fig.4. 
The graph is consistent with 
the behavior of $\gamma_{{\rm eff}}$ in Fig.3 
because for relatively large $t$,
tachyons with small $k$ contribute and then, it approximately
holds that $\sqrt{2\beta-k^{2}}\approx 
\sqrt{2\beta}(1-k^{2}/2 k_{max}^{2})$.  
The typical wave length $\bar{\lambda}$ of the tachyon modes
can be estimated;
e.g.  when $k^{2}/2 k_{max}^{2}\sim 0.1$, $\bar{\lambda}\sim 60 l_{s}$.
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{f4}
\caption{$<k^2>/k^{2}_{{\rm max}}$ vs. $t/T^{(I)}$ for D-4-branes}
\end{center}
\end{figure}
Taking into account all of the results stated in this paragraph,
we define the quality $|\tilde{c}_{0,k}|_{{\rm average}}\equiv 
|\tilde{c}_{0,k_{{\rm av}}}|_{{\rm typical}} $ 
as an approximate function of $|\tilde{c}_{0,k}|_{{\rm typical}}$
where we take $k^{2}_{av}\sim 0.1 \times 2\beta$ or some other value.
If we replace the integrand (\ref{staticsemi}) of 
$|\tilde{c}(x_{\mu})|_{{\rm typical}}$ (\ref{staticcfinal})
by  
$|\tilde{c}_{0,k}|_{{\rm average}}$
to define $A_{{\rm av}}(t)$,
$A_{{\rm av}}(t)$ and $A(t)$ are plotted as Fig.5,
which shows that the approximation is not so bad.
We use this approximate function to discuss validity of 
WKB approximation later.
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{f5}
\caption{$A_{{\rm av}}(t)$ and $A(t)$  vs. $t/T^{(I)}$ for D-4-branes.
The upper is $A_{{\rm av}}(t)$.}
\end{center}
\end{figure}

In addition, we can read a tendency from (\ref{staticcfinal}) or Fig.2
that the blow-up is more
delayed for the case with a larger value of $p$.
%The physical reason is as follows:
%The frequency of tachyon modes is given as $\omega_{k}=k^{2}-2\beta$,
This fact can be explained as follows: 
the tachyonic modes with less momentum tend to blow up more rapidly,
but there are larger number of tachyonic modes 
for larger momentum.
That is, except for the case of D-strings,
for larger value of $p$,
the most ``active'' tachyonic mode which has $k=0$ is 
more smeared in some region of $k$  
and the start of the blow up becomes slower.

Finally,  %discussing $p\ge 2$ cases,  
we mention the case of $p=1$, i.e. D-strings.
We cannot apply (\ref{staticcfinal}) to the case of D-strings
because in this case 
there is no room for the tacyonic fluctuation to have a momentum $k$,
so it diverges.
However, as discussed in the next section, if 
we displace one of the D-string from the intersection point 
by a small distance $z_{0}$ ($z_{0} < \sqrt{2\beta}$),
$z_{0}$ contributes to the action as a mass parameter for $c$ and $d$,
and we have
\beqa
|\tilde{c}_{{\rm typical}}(t)|^{2}=\frac{
1}{2 T_{D1}}\{\frac{1}{|z_{0}|}
+\frac{2\beta}{2\beta-(z_{0})^{2}}
\sinh^{2}(\sqrt{2\beta-(z_{0})^{2}}t) \}f_{0}(x)^{2}.
\eeqa
For this setting of D-string case,
we can make the same discussion as that 
for $p\ge 2$ in the following. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Time-evolution of the D-p-branes' shapes and 
vast number of pair-creations of strings}
\setcounter{footnote}{0}

In this subsection we discuss time-evolution behavior 
of the D-p-branes' shapes. 
We also discuss the blowing-up behavior of pair-creations of 
stings connecting 
the recombined branes.

Using (\ref{defctilder}), we rewrite the result in terms of 
$|d|$ as\footnote{As 
to the phase of $d$, if there is an experiment
by which one could observe the phase, and when it is observed, 
it jumps into one of the allowed values. However, it does not 
affect the following discussion.} 
\beqa
|d|_{{\rm typical}} = 
\frac{|\tilde{c}|_{{\rm typical}} }{\sqrt{2}}=\frac{A(t)f_{0}(x)}{\sqrt{2}},
\label{dbehavior}
\eeqa
since $|\tilde{d}|_{{\rm typical}}$ is negligible.
We regard this quality as the absolute value of VEV of the field $d$,
which leads to the fact  
that the VEV's
of the transverse scalar U(2) field $X_{p+1}$
has the form: 
\beqa
X_{p+1}^{(0)}= \left(
\begin{array}{cc}
\beta x & d^{*} \\
d & -\beta x
\end{array}
\right).\label{xp1}
\eeqa
%where $|d|$ is given by (\ref{dbehavior}).

Let us get geometrical information on the branes 
from (\ref{xp1}). 
In this case, there is only one transverse scaler field 
which has non-trivial VEV elements, so, all the scalar fields are 
commutative with each other. 
Thus, as done in such conventional cases,
it is a logical step or a conventional procedure
to choose the gauge to diagonalize the VEV's of 
the scalar field and interpret the diagonal parts as
the positions of the two branes.\footnote{
Though the procedure of diagonalization is already 
discussed in ref.\cite{hn}, strictly speaking, 
the authors seems to use it in a way different from ours.
Our position is that
the diagonalization in this case is definitely a logical step, 
and the shape of the branes is the thing to be derived. }
$X_{p+1}^{(0)}$ is diagonalized as
\begin{equation}
\begin{array}{cc}
X^{(0){\rm diag}}_{p+1}=U^{-1} \left(
\begin{array}{cc}
\beta x & d^{*} \\
d & -\beta x
\end{array}
\right)
U &=
\left(
\begin{array}{cc}
\sqrt{(\beta x)^{2}+|d|^{2}}& 0 \\
0 & - \sqrt{(\beta x)^{2}+|d|^{2}}
\end{array}
\right) .
\end{array}\label{diagonal1}
\end{equation}
Therefore,
the formula for the shape of one of the recombined brane is
\beqa
y\equiv\sqrt{(\beta x)^{2} + |d|^{2}}=\sqrt{(\beta x)^{2} 
+ \frac{A(t)^{2}}{2}\sqrt{\frac{2\beta}{\pi}}e^{-2\beta x^{2}} }.
\label{braneshape1}
\eeqa
By (\ref{braneshape1}) 
we can make a precise analysis on the shape of the branes at arbitrary $t$ 
as long as the WKB approximation is valid,
since we know $|d|$ or $A(t)$ 
(given in (\ref{staticcfinal}) and (\ref{dbehavior})) including 
the overall factor and fine coefficients determined 
by the initial condition.

At the initial stage, the shape of each brane is approximately 
one part of hyperbola as in Fig. 6, 
which is consistent with our intuition; 
though $|d|$ is dependent on $x$ through the Gaussian
$f_{0}(x)$,
its width $1/\sqrt{\beta}$ is broad and the change of the shape
at the initial stage is limited to a region smaller than 
the width.
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{f6}
\caption{The shape of D-4-branes;
$2\beta=0.1$ and $A(t)=0.1$}
\end{center}
\end{figure}
The larger $|d|$ becomes, 
the further the positions of the two D-brane get.
That is, the VEV of (the absolute value of) 
the blowing-up tachyon field $d$ 
corresponds directly to such a  
relative motion between the recombined branes as discussed 
in ref.\cite{hn}. 
We note that if one consider the limit $|d|\to 0$,
the VEV of the scalar field $X^{(0){\rm diag}}_{p+1}$ leads to
${\rm diag}(\beta |x|,-\beta |x|) $ which is different from 
the initial background (\ref{back1}). 
This causes no problem, however; Using the gauge fixing condition 
$\partial_{\mu}c^{\mu}-2i\beta|x|d$ 
instead of (\ref{gf1}),
the second order action (\ref{secondaction1}) 
and the physical spectrum 
in $X^{(0)}_{p+1}={\rm diag}(\beta |x|,-\beta |x|) $ can 
be shown to be essentially the same as that in (\ref{massform}) 
(up to the phase of $d$). 
This fact corresponds to the fact that 
the two branes intersecting at an angle cannot 
be distinguished from  that each bent and  toughed at $x=0$.
Thus, there appears no discontinuity and the recombination 
proceeds smoothly. 

The typical time scale of the recombination 
is substantially that of tachyon condensation:
the distance between the two branes at $x=0$
exhibit an exponential blow-up, roughly written
as $|d|\sim e^{\gamma_{{\rm eff}}^{(I)} t}/\sqrt{T_{p}}$
where $\gamma_{{\rm eff}}^{(I)}$ is given in Fig.3. 
For a more detailed analysis on the time-evolution behavior,
see the previous subsection.

So far, the shape of the p-branes is within the reach of our
intuition. 
After more time passes, however, an interesting phenomenon
occurs; the shape of each brane deviates from the approximate
hyperbola and comes to have multiple (three) extremes 
due to localization of tachyon condensation, but not 
the effect of compact spaces. 
This happens for the cases of D-p-brane for $p\ge 2$,
and is also expected to happen for the case of D-strings. 
The values of $x$ giving multiple extremes of (\ref{braneshape1}) 
are formally given by
\beqa
x=0,\pm\sqrt{\frac{1}{4\beta}\ln \frac{2 A^4}{\pi\beta}}.
\eeqa 
which means that each brane's shape has multiple extremes
if it holds
\beqa
A>(\pi\beta/2)^{1/4}\equiv A_{{\rm critical}}.\label{wkbcond1}
\eeqa
The formula (\ref{braneshape1}) is deduced from the action
second order in the fluctuations,
so, all we have to do to prove that this really happens is 
to show that
higher order terms in the fluctuations in 
(\ref{fullaction1}) do not disturb this behavior.
Since only $c$ and $d$ include the tachyonic field,
we define
\beqa
S_{2}&\equiv& -T_{p}\int d^{p+1}x 2 i \beta (c^{*}d - c d^{*})\nonumber\\
S_{4}&\equiv& -T_{p}\int d^{p+1}x |c^{*}d-cd^{*}|^{2}\label{ss4}
\eeqa
as a second order term and the largest forth order term respectively,
and show $|S_{4}|\ll |S_{2}|$ below.
First, we prove the above for the cases of D-p-branes with 
$p\ge 2$, and then, discuss the effect of nonlinear terms
in the case of D-strings.

As for the case of p-branes,
if we include the non-linear terms in the equation of motion, 
the basis functions of the mode expansion
\beqa
\tilde{c}=\sum 
\tilde{c}_{n,k}(t) f_{n}(x) u_{k}(x_{\hat{\mu}})\label{modeexp3}
\eeqa
are not the solutions to the equation any more. However, 
since each of Hermite polynomials and $u_{k}$ given in (\ref{ufunc})
forms a complete set,
the expansion is still available
as a type of decomposition of 
degrees of freedom (which make clear the tachyonic 
degrees of freedom
when the fluctuations are small).
So, we substitute (\ref{modeexp3}) for $S_{2}$ and $S_{4}$
and discuss their amplitudes. 
Neglecting the contribution of non-tachyonic 
(i.e. non-blowing-up) modes,
they are written as
\beqa
S_{2}&=&-T_{p}\int dt (-2\beta)A(t)^{2}
\nonumber\\
S_{4}&=&-T_{p}\int dt \sqrt{\frac{\beta}{\pi}} 
\frac{1}{V_{p-1}}\sum_{k_{1}}\sum_{k_{2}}\sum_{k_{3}}
\tilde{c}_{0,k_{1}}^{*}\tilde{c}_{0,k_{2}}\tilde{c}_{0,k_{3}}^{*}
\tilde{c}_{0,-k_{1}-k_{2}-k_{3}}\label{s4}
\eeqa
where we have used $\sum_{k;k^{2}\le 2\beta}
<\tilde{c}_{0,k}^{*}\tilde{c}_{0,k}>=A(t)^{2}$.
We note that the factor $\sqrt{\beta/\pi}$ comes from extra normalization
constants of $f_{0}$'s which remains after x-integration, 
while $1/V_{p-1}$ arises as 
those of $u_{k}$'s.
To carry out the integrations with respect to momenta in (\ref{s4}) 
is difficult, though its numerical calculation is possible.
However, only its order estimation suffices here, which we will do
by replacing  $\tilde{c}_{0,k_{i}}$ for some $k_{i}$ by
$|\tilde{c}_{0,k}|_{{\rm average}}$ defined in the 
previous subsection.
Then, $S_{4}$ is estimated as
\beqa
S_{4}\sim-T_{p}\int dt  
\frac{\Omega_{p-2}(2\beta)^{p/2}}{(2\pi)^{p-1/2}V_{p-1}}(A_{{\rm av}}(t))^{4}
\eeqa 
where $A_{{\rm av}}(t)^{2}\equiv \sum_{k} 
|\tilde{c}_{0,k}|^{2}_{{\rm average}}$ as discussed in the 
previous subsection.
So, the condition for the validity of WKB approximation 
($|S_{4}|\ll |S_{2}|$) is
\beqa
A(t)\ll \sqrt{\frac{(2\pi)^{p-1/2} V_{p-1}}{\Omega_{p-2}(2\beta)^{(p-2)/2} }}.
\label{wkbdp}
\eeqa
Especially, in the case of D-4-brane,
\beqa
A(t)\ll \sqrt{\frac{3(2\pi)^{5/2} V_{3}}{2\cdot 2\beta }}\label{wkbd4}.
\eeqa
Here, $V_{p-1}$ is the volume of the brane except for the
direction of $x$. In particular, 
if we apply this set-up or that of the case (II)
to braneworld scenarios,
$V_{3}$ corresponds to  the volume of our world.
Since it is before the inflation at this time 
as an inflation model, it is not so trivial that
$V_{3}$ is very large, but in this case the unit length is 
string scale $l_{s}$, so it is very probable to assume
that $V_{3}$ is larger than several hundreds times $l_{s}$.
Then, (or if not so, for some small angle $\theta$)  
the inequality (\ref{wkbd4}) 
is sufficiently
compatible with the condition of WKB approximation (\ref{wkbcond1}).
For the other values of $p$,
(\ref{wkbdp}) is also compatible with (\ref{wkbcond1}) for 
a relatively large world-volume. 
Therefore, the shape of the recombined D-p-brane for $p\ge 2$ surely has multiple extremes without the effect of compact space,
an example of which is drown in Fig.7.
This happens essentially due to the localization of tachyons around $x$,
since the factor $e^{-\beta x^{2}}$ in (\ref{braneshape1}) is directly
responsible for the multiple extremes.
\begin{figure} 
\begin{center}
\includegraphics[width=8cm]{f7}
\caption{The shape of a D-brane with multiple extremes;
$2\beta=0.1$ and $A(t)=0.8$}
\end{center}
\end{figure}

For the case of D-strings,
the case is a bit more subtle, but we argue that 
the same phenomenon will happen. 
In this case 
the second and forth order actions  are written as
\beqa
S_{2}&=&-T_{p}\int dt (-2\beta)|\tilde{c}_{0}(t)|^{2}
\nonumber\\
S_{4}&=&-T_{p}\int dt \sqrt{\frac{\beta}{\pi}}|\tilde{c}_{0}(t)|^{4} 
\label{s41}
\eeqa
instead of (\ref{s4}), where we will substitute $|\tilde{c}_{0}(t)|$ for
$A(t)$. Then, the condition $|S_{4}|< |S_{2}|$ is
\beqa
A(t) < (4\pi \beta)^{1/4}.\label{wkbcond2}
\eeqa
If one consider the equation of motion for $\tilde{c}_{0}(t)$,
\beqa
\frac{d^{2} \tilde{c}_{0}(t)}{dt^{2}}= 2\beta \tilde{c}_{0}(t) -2
|\tilde{c}_{0}(t)|^{2}\tilde{c}_{0}(t),\label{eomhi}
\eeqa
(\ref{wkbcond2}) is replaced by $A(t) < (\pi \beta)^{1/4}\equiv A_{{\rm critical}}$.
Anyway, there is, a little though , a range 
for $A(t)$ to be allowed to have multiple extremes
within the WKB approximation: 
$(\pi \beta/2)^{1/4}< A(t) < (\pi \beta)^{1/4}$.
This may seem subtle, but if we solve numerically the 
equation of motion for $\tilde{c}_{0}(t)$
we can see that $|\tilde{c}_{0}(t)|$ really develop to exceed 
$A_{{\rm critical}}$.
We note that the last term in (\ref{eomhi}) appears to
disturb condensation of tachyons, but,
it continues to develop due to 
higher order corrections coming from Born-Infeld action such as
\beqa
S_{6}=T_{p}\int d^{2}x \frac{1}{4}\{|\partial_{x}d|^{2}
-(c^{*}d -cd^{*})^{2} \}^{2}.
\eeqa 
In addition,  suppose that the mode expansion (\ref{modeexp3}) decomposes 
degrees of freedom properly. 
Then, even if non-linear terms have the effect to mix 
the degrees of freedom,  
the blowing-up mode is also expected to be localized 
since all the modes including higher ones (with $ n\ge 1 $ and all $k$) are 
also localized because of the property of $f_{n}$.
Taking into account the two facts that $ \tilde{c}_{0}(t)$ continues to develop,
and that its blow-up is localized around $x=0$, it is very probable that
the shape of the D-string also comes to have multiple 
extremes because localization of 
tachyon condensation is essential to this phenomenon. 
Thus, we argue that it happens.

The physical interpretation of the interesting property is 
as follows;
the energy released via tachyon condensation pushes the recombined branes
away from each other, but
it is given to the local part of the branes around $x=0$, so, only 
the part is much accelerated.
Though the D-branes have a large tension, 
they also have a large inertia, and 
when the given energy of the local part is large,
the branes extend, surpassing the tension, to form three extremes.
That is, {\it localization of tachyon condensation
and the (large) inertia causes the shapes of branes 
with multiple extremes.}
 
The physical consequence of the property is
that in this setting, the branes have a stronger tendency than we had 
expected to dissipate its energy into radiations, 
but we will discuss this point 
in subsection 3.3.

Finally, we discuss another interesting phenomenon;
vast number of pair-creations of fundamental strings connecting 
between the two D-p-branes {\it after} recombination.
It happens around $x=0$  with the typical width $1/\beta$. 
One can find its evidence by examining the behavior of 
the electric flux on the world-volumes since electric charges on
D-branes correspond to fundamental strings\cite{cm}.
An important thing is that even after the gauge choice which
diagonalizes $X_{p+1}$ (which we denote here as ``diagonal gauge''), 
only the off-diagonal elements of the typical field strength
$F_{0p}=\partial_{t} A_{p}$ blow up, since  it holds
\beqa
\begin{array}{ccc}
F_{0p}^{(0)\ {\rm diag }}=U^{-1}\frac{1}{\sqrt{2}}
\left(
\begin{array}{cc}
0 & \partial_{t} \tilde{c}^{*} \\
\partial_{t}\tilde{c} & 0
\end{array}
\right) U
 &
=\frac{1}{\sqrt{2}}\left(
\begin{array}{cc}
0 & \partial_{t} \tilde{c}^{*} \\
\partial_{t}\tilde{c} & 0
\end{array}
\right).
\end{array}
\eeqa
This is because the gauge transformation diagonalizing $X_{p+1}$
corresponds to a ``rotation  around the axis along the vector 
${\bf \tilde{c}}$'' in the SU(2) internal vector space. 
(The vector  ${\bf \tilde{c}}$ is defined as 
$A_{p}={\bf \sigma \cdot \tilde{c}}$ where 
${\bf \sigma}$ is Pauli matrix.) 
The blow-up of the off-diagonal elements
means that fundamental strings connecting the two 
D-branes are vastly created after the recombination.
Since the true VEV of (linear) $F_{0p}$ vanishes, 
so does the total electric charge, and the
strings are considered to be pair-created.
The value of the flux is evaluated as
\beqa
|F_{0p}^{(0)}|_{{\rm typical}}\simeq \sqrt{\beta}A(t)f_{0}.
\eeqa

It seems of worth to estimate the typical number of the created strings. 
It might be possible in the case of  D-strings:
As in ref.\cite{sj}, fundamental strings attached on a D-string's 
world-volume perpendicularly, are represented by a sudden appearance of 
a constant electric flux (proportional to their number) 
on the D-string's world-line.
Thus, the derivative of the flux $F_{0x}$ 
with respect to the world-line's coordinate $x$ should correspond to
the number density of fundamental strings.
Based on this analogy, we give an estimation of the typical number of the 
fundamental strings on the recombined D-strings as
\beqa
N_{{\rm D1}} \sim \int dx 
|\partial_{x} F_{0x}|\sim F_{0x}(x=0)=\sqrt{\frac{2}{\pi}}\beta
A(t). 
\eeqa  
In the case of  D-p-branes for $p\ge 2$, taking into account of the extra 
world-volume dimensions and $u_{0,k}$, we might be able to estimate
the typical number on the world-volume
as $N_{{\rm D1}}\sqrt{V_{p-1}}$, though we do not argue that
this is true.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Case (II); two D-p-branes at one angle approaching slowly}
\setcounter{equation}{0}
\setcounter{footnote}{0}

In this section we deal with the case (II):
two almost parallel D-p-branes approaching each other 
with a small relative velocity $v$ and at a small angle $\theta$.

\subsection{Preliminaries}

In this subsection we give preliminaries for a quantum analysis 
for the case (II).

The simplest case of the initial condition is described in Fig.8;
we consider the system in which 
the center of mass is at rest, 
denoting by $z(t)$ the relative distance along $x_{9}$
before recombination.
The other basic setting is the same as those in the case (I).
We discuss recombination process of this system
via Yang-Mills theory with adjoint Higgs fields,  
concentrating on the behavior of branes 
around $x=0$ again in principle.
\begin{figure}[h] 
\setlength{\unitlength}{1mm}
\begin{picture}(160,40)
\put(4,17.5){\vector(1,0){56}}
\put(32,8){\vector(0,1){19}}
\put(4,8){\thicklines\vector(3,1){56}}
\put(4,27){\thicklines\vector(3,-1){56}}
\put(4,8){\line(1,0){56}}
\put(4,27){\line(1,0){56}}
\put(4,8){\line(0,1){19}}
\put(60,8){\line(0,1){19}}
\put(49,18){\makebox(5,5){$\theta /2$}}
\put(49,12){\makebox(5,5){$\theta /2$}}
\put(35,20){\makebox(5,5){$x_{p+1}$}}
\put(55,17){\makebox(5,5){$x$}}
\put(50,28){\makebox(5,5){$Dp$}}
\put(50,2){\makebox(5,5){$Dp$}}
\put(31,28){\makebox(5,5){$L_{p}$}}
\put(63,15){\makebox(5,5){$L_{p+1}$}}
%%%%%%%%%%%%%%%%%%%
\put(80,17.5){\vector(1,0){58}}
\put(106.5,7){\vector(0,1){23}}
\put(90,17.5){\vector(0,1){7.5}}
\put(90,17.5){\vector(0,-1){7.5}}
\put(115,24){\vector(0,-1){5}}
\put(115,11){\vector(0,1){5}}
\put(80,10){\thicklines\vector(1,0){56}}
\put(80,25){\thicklines\vector(1,0){56}}
\put(120,19){\makebox(5,5){$v/2$}}
\put(120,11){\makebox(5,5){$v/2$}}
\put(105,30){\makebox(5,5){$x_{9}$}}
\put(135,18){\makebox(5,5){$x$}}
\put(120,28){\makebox(5,5){$Dp$}}
\put(120,2){\makebox(5,5){$Dp$}}
\put(92,19){\makebox(5,5){$z(t)$}}
\end{picture}
\caption{The initial configuration of D-p-branes for case (II)}
\end{figure}

The background D-p-branes for the case (II) are 
represented by the configuration:
\beqa
\begin{array}{ccc}
X_{p+1}= \left(
\begin{array}{cc}
\beta x & 0 \\
0 & -\beta x
\end{array}
\right),
 &
X_{9}= \left(
\begin{array}{cc}
z/2 & 0 \\
0 & -z/2 
\end{array}
\right) ,
& X_{p+2}=\cdots =X_{8}=A_{\mu}=0.
\end{array}
\eeqa
Since we set $g_{s}$ and $\beta\equiv \tan(\theta/2)$ 
very small, the force between the two branes is so 
weak\cite{jabbari}\cite{pol}(and for short distance, see \cite{oneangle}) 
that the velocity is regarded as a constant, giving
\beqa
z=z_{0} -v t.
\eeqa
As in the case (I),
$\beta$  essentially corresponds to
a constant background (magnetic) field strength 
in a T-dual picture,
%since this system is a generalized version of the case (I).
while $z$ is a VEV of an adjoint Higgs field
which decreases slowly. 
U(2) gauge symmetry is spontaneously broken to
U(1) $\times$ U(1) by the constant field strength and the 
non-trivial VEV. 
We will deal with such a Yang-Mills-Higgs system below.

Since the system approaches the set-up of the case (I)
as $z$ decreases, 
potentially tachyonic modes should appear 
in the non-diagonal elements of $A_{p}$ and $X_{p+1}$ again.
In addition, however, there is a possibility
that tachyonic modes might appear in the non-diagonal elements of 
$A_{0}$ and $X_{9}$ because the two D-p-branes
approaching with a small velocity is 
T-dual to a two D-(p+1)-brane system
with a constant field strength (though it is electric),
and is also 
considered to be a Wick-rotated version of 
branes at one angle $v$\cite{pol}.
So, we denote the fluctuations, 
including those of $A_{0}$ and $X_{9}$, as
\begin{equation}
\begin{array}{ccc}
A_{\bar{\mu}}= \left(
\begin{array}{cc}
0 & c_{\mu}^{*} \\
c_{\mu} & 0
\end{array}
\right), &
X_{p+1}= \left(
\begin{array}{cc}
0 & d^{*} \\
d & 0 
\end{array}
\right) 
, &
X_{9}= \left(
\begin{array}{cc}
0 & w^{*} \\
w & 0 
\end{array}
\right) 
\end{array}\label{ansatz2}
\end{equation}
where we denote $c_{p}$ as $c$, and discuss their spectrum below.

The part of the action
second order in the fluctuations 
is
\beqa
S_{2Dp}|_{{\rm 2nd}}
&=&T_{p} \int d^{p+1}x [ 
-\partial_{\mu}c^{*}_{\nu}\partial^{\mu}c^{\nu} 
+|\partial_{\mu}c^{\mu}|^2 
-\partial_{\mu}d^{*}\partial^{\mu}d 
-\partial_{\mu}w^{*}\partial^{\mu}w\nonumber\\ 
& &-2 i \beta (c^{*}d -c d^{*})
+2 i \beta x(c^{*}_{\mu}\partial^{\mu}d 
-c_{\mu} \partial^{\mu}d^{*}) 
-iv(c_{0}^{*}w-c_{0}w^{*})\nonumber\\
& &+iz(c^{*}_{\mu}\partial^{\mu}w 
-c_{\mu} \partial^{\mu}w^{*})
-(4(\beta x)^{2}+z^{2})|c_{\mu}|^{2}-|2\beta x w-zd|^{2} 
].\label{secondaction22}
\eeqa 
We consider such a gauge fixing condition 
that it is written as $D_{\mu}A^{\mu}\equiv\partial_{\mu}
A^{\mu}+i[A_{\mu}^{(0)},A^{\mu}]=0$ in a T-dualized pure 
Yang-Mills system, which gives
\beqa
\partial_{\mu}c^{\mu}=i(2\beta x d+zw).\label{gf3}
\eeqa
Integrating by part some terms and  using (\ref{gf3}),
the action is written as
\beqa
S_{2Dp}|_{{\rm 2nd}}&=&  T_{p} \int d^{p+1}x [
-c^{*}_{\nu} \{ -(\partial_{\mu})^{2}
+4(\beta x)^{2} +z^{2}\} c_{\nu} 
- d^{*}  \{ -(\partial_{\mu})^{2}
+4(\beta x)^{2} +z^{2} \}d \nonumber\\
& &-w^{*}\{ -(\partial_{\mu})^{2}
+4(\beta x)^{2}+z^{2} \} w
-4 i \beta (c^{*}d -c d^{*})-2iv(c_{0}^{*}w-c_{0}w^{*})
].\label{secondaction23}
\eeqa
We can see that $z^{2}$ merely plays a role of a mass-squared  
for each field. In order to make a quantum analysis,
we need the fact that, when making a mode expansion,
all the modes have positive frequency-squareds at $t=0$.
Thus we set $z_{0}=0.5 l_{s}$, relatively a large value
but not to exceed the applicable range of Yang-Mills theory.
As for the last term but one in (\ref{secondaction23})
including $(c^{*}d -c d^{*})$, 
it is diagonalized by introducing $\tilde{c}$ and $\tilde{d}$ 
in (\ref{defctilder}) to produce the 
terms $ 4\beta (|\tilde{c}|^{2} -|\tilde{d}|^{2})$,
resulting in the tachyonic mass term of $\tilde{c}$ as 
(\ref{ho}) and (\ref{massform}).
When one consider a mode expansion using $f_{n}$ and $u_{k}$,
each of the modes $\tilde{c}_{0,k}$ with $k^{2}\le 2\beta$
becomes tachyonic 
when the distance $z(t)$ decreases to 
a critical one $z_{*}$
(which is determined
by the value of $k$) and blows up to cause the recombination.

In the same way, the last term in (\ref{secondaction23})
might also seem to produce such terms with $2\beta$, $c$,$d$ 
replaced by $v$,$c_{0}$, $w$, respectively, resulting in
the tachyonic modes.
However, the sign of the kinetic term of $c_{0}$
is different from that of $w$, and, if one carry out 
the diagonalization, it results in a non-local action including 
square roots of differential operators;
it is unclear whether $c_{0}$ and $w$ include tachyonic modes.
However,
suppose we consider the case with $2\beta \gg v$.
Then, even if there are tachyon modes composed of
$c_{0}$ and $w$, 
the critical distances $z'_{*}$'s of them are much shorter than
those of $\tilde{c}$, and
it is much before the postulated tachyons of 
$c_{0}$ and $w$ get ``awake"
that the tachyon modes of $\tilde{c}$ condensate and
the recombination proceeds.
Thus, we consider the parameter region $2\beta \gg v$ so that 
we can neglect the above problem, 
dealing with $c_{0}$ and $w$ as massive fields.  

Now, we derive the action like a collection of  
harmonic oscillators
by making the same mode expansion as (\ref{modeexp3}),
and substituting it for (\ref{secondaction23}):
it is obtained  as
\beqa
S_{2Dp}|_{{\rm 2nd}}=T_{p}\int d t
\sum_{n,k_{\hat{\mu}}} [ |\partial_{t} \tilde{c}_{n,k}|^{2}
-\omega_{n,k}^{2}|c_{n,k}|^{2}
+(\tilde{c}_{n,k}\to \tilde{d}_{n, k}, \ c_{\bar{\nu},n,k})]
\label{secondaction24}.
\eeqa 
where $\omega_{n,k}^{2}$ in this case is
\beqa
\omega_{n,k}^{2}= k_{\tilde{\mu}}^2+2 \beta (2n-a)+z(t)^{2}.
\label{omega2}
\eeqa
where $a=1,0,-1$ for $\tilde{c}$, $\tilde{d}$, 
$c_{\bar{\mu}}$ again. 
Among all of the modes, 
potentially tachyonic modes are 
$\tilde{c}_{0,k}$ with momentum $|k|\le \sqrt{2\beta}$,
since $z(t)$ can become zero in principle
(though the modes which are non-tachyonic at the moment 
are not effective to the blow-up).
As in the case (I), we will focus on 
the potentially tachyonic modes $\tilde{c}_{0,k}$, and 
make a quantum analysis.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{A quantum analysis on tachyon condensation in 
case (II)}

In this subsection we analyze how tachyon condenses 
in the case (II) using the the same quantum 
approach as done for the case (I).

The equations of motion for the tachyonic modes
$\tilde{c}_{0,k}$ with $|k|\le \sqrt{2\beta}$ are
\beqa
\frac{d^{2} \tilde{c}_{0,k}}{dt^{2}}= 
-\omega_{k}(t)^{2} \tilde{c}_{0,k},
\label{eom2}
\eeqa
with  frequency-squareds  
\beqa
\omega_{0,k}(t)^{2}= (z_{0}-v t)^{2} +k_{\tilde{\mu}}^2-2 \beta.
\label{omega3}
\eeqa
Since we set $z_{0} = 0.5l_{s}$ and of course
$2\beta< 0.5$,
each $\omega_{0,k}^{2}$ is positive
at the initial instant $t=0$.
As $z(t)$ decreases gradually,
so does $\omega_{0,k}^{2}$ and
the wave function $\Psi(t,\tilde{c}_{0,k})$ of 
$\tilde{c}_{0,k}$ spreads little by little.
At $t= t_{0}\equiv (z_{0}-\sqrt{2\beta})/v$, 
$\omega^{2}$ 
with the lowest momentum $k=0$ becomes zero and 
then negative, i.e.  tachyonic, and the corresponding
wave function begins to spread radically, resulting in
the blow-up of the mode.
After that, the modes with lower $k$ are also getting tachyonic  
and start to blow up one after another, leading to the 
tachyon condensation.  
We define
$t=t_{*}(k)\equiv (z_{0}-\sqrt{2\beta-k^{2} })/v$
which gives
$\omega_{0,k}(t_{*})=0$
and rewrite the $\omega_{k}^{2}$ using $t_{*}$,
for a later use, as
\beqa
\omega_{k}^{2}(t)= \alpha_{k} (t-t_{*})-v^{2}(t-t_{*})^{2}
\label{omega22}
\eeqa
where we denote the coefficient $\alpha_{k} \equiv \sqrt{8 v^{2} \beta'}$
and $\beta'\equiv \beta-k^{2}/2$.

Let us confirm the outline of the evaluation of 
$|\tilde{c}_{0,k}|_{{\rm typical}}$: 
Assuming the zero temperature,
the initial wave function $\Psi(t=0,\tilde{c}_{0,k})$ of 
$\tilde{c}_{0,k}$ 
is Gaussian, resulting in  
$|\tilde{c}_{0,k}(0)|_{{\rm typical}}=\Delta_{k}(0)/\sqrt{2}$.
In this case   
$|\Psi(t,\tilde{c}_{0,k})|$
is shown to keep to be Gaussian within the WKB approximation,
described by the width $\Delta_{k}(t)$
\cite{fey} (and review in the appendix).
Thus, all we have to do now is to find two solutions of (\ref{eom2}) and
substitute it for the formula (\ref{formaldelta}).

Let us discuss solutions to (\ref{eom2}). 
Since formal exact solutions  to it
are a bit difficult to deal with 
for our purpose, 
we consider approximate solutions in the following way:
if $t$ approaches $t_{*}$, the second term quadratic in $t-t_{*}$ of
(\ref{omega22}) becomes so small as to be neglected compared to
the first term linear in $t-t_{*}$. (We denote this time $t=t_{1}$.) 
Thus, for $t>t_{1}$, two Airy functions can be used as basis functions. 
On the other hand, taking a closer look will let one notice that 
(\ref{eom2}) has the same form  as the Schrodinger equation 
in one dimension with relatively slowly-changing potential
(if we replace $t$ by $x$). 
So, when $\omega^{2}_{k}(t)$ is relatively large, 
we can apply ``WKB approximation''
to get two approximate solutions of (\ref{eom2}).
Thus, if the approximation can be applied until the time
$t=t_{1}$, we can follow the time-evolution of 
$\Psi(t,\tilde{c}_{0,k})$ and  $\Delta_{k}(t)$
by using the WKB-approximated solutions until $t=t_{1}$, and then
letting the Airy functions take over the role of basis functions. 
We take this prescription to evaluate their 
time-evolution.\footnote{
As more time passes, the second term in (\ref{omega22})
cannot be neglected again.  In our setting, however,
recombination proceeds  much before that, so, 
we do not consider the region.}
The appropriate parameter region that 
the WKB approximation is applicable until $t_{1}$ will be deduced below:

Suppose $\omega^{2}(t)$ is relatively large.
two WKB-approximated solutions of (\ref{eom2}) are obtained as
\beqa
\phi_{1}(t)=\frac{N_{1}'}{\omega^{1/2}}\cos(\int^{t}_{t_{1}}dt'
\omega(t)),\nonumber\\
\phi_{2}(t)=\frac{N_{2}'}{\omega^{1/2}}\sin(\int^{t}_{t_{1}}dt'
\omega(t))\label{basis1}
\eeqa
where $N_{i}'$ are normalization constants.
The WKB approximation is valid  as far as
\beqa
v\ll \omega_{0,k}^{2}{\rm \ \ and \ \ } 
|\frac{\partial\omega_{0,k}}{\partial t}|\ll \omega_{0,k}^{2}.\label{cond1} 
\eeqa
On the other hand, the condition
for the quadratic term of (\ref{eom2}) to be neglected 
($2v\sqrt{2 \beta'} |t-t_{*}| \gg v^{2}(t-t_{*})^{2}$) is
\beqa
|t-t_{*}|\ll \frac{8\beta'}{v}.\label{cond2}
\eeqa
If we choose $t_{1}-t_{*}=-8\beta'/10 v$ to satisfy (\ref{cond2})
(please note the sign),
the condition to satisfy (\ref{cond1}) is written as
\beqa
v\ll 2\beta'.\label{cond3}
\eeqa
which is compatible with the condition to regard $c_{0}$ and $w$
as only massive fields.
In this section we consider such a system.
One may worry that for any set of $\beta$ and $v$,
there are tachyonic modes which do not satisfy (\ref{cond3})
because $2\beta'=2\beta-k^{2}$, and, tachyons with large $k$
break the condition.
In fact, in such a set-up, 
$v$ is so slow that the tachyonic fluctuation $\tilde{c}$
blows up enough 
immediately, before the tachyon modes with large $k$  
``wake up''. So, the above point 
is not practically a problem.



The initial fluctuation for $z_{0}=0.5 l_{s}$ is
\beqa
\Delta_{k}(0)=\frac{1}{\sqrt{T_{p}\omega_{0,k}}}\label{yuragi2}
\eeqa
where $\omega_{0,k}=\sqrt{(z_{0})^{2}-2\beta'}$.
Substituting the basis functions (\ref{basis1}) and (\ref{yuragi2})
for the formula (\ref{formaldelta}),
we have
\beqa
\Delta_{k}(t=t_{1})^{2}= \frac{1}{T_{p}\omega_{1}}\{1 
+O(\frac{v}{2\beta})\}\label{omegac}
\eeqa 
where $\omega_{1}\equiv \omega_{0,k}(t=t_{1})$.
We note that the leading term of (\ref{omegac}) represents 
spreading behavior of $\Psi$ coherently as $\omega^{2}$ 
decreases,
and the next leading terms correspond to its oscillating behavior around 
the leading width;
the wave function tends to spread on its own
as in the free particle case, but it cannot always 
spread coherently with the shape of the potential, so, it oscillates. 
We use only the leading term of (\ref{omegac}) below,
neglecting the other terms of the order $O(\frac{v}{2\beta})$.
Someone may say it's a waste of effort because
it is easy to infer the leading term.
The analysis, however, enables us to 
clarify the definite range of parameters 
appropriate for the approximation.

We use Airy functions
$Ai(x)$ and $Bi(x)$ for $t\ge t_{1}$ as basis functions,
whose asymptotic behaviors for large $x$ are
\beqa
Ai(x)\sim\frac{1}{2\sqrt{\pi}x^{1/4}}e^{-\frac{2}{3}x^{3/2}}
\nonumber\\
Bi(x)\sim\frac{1}{2\sqrt{\pi}x^{1/4}}e^{\frac{2}{3}x^{3/2}}.\label{asym1}
\eeqa
Using them and (\ref{omegac}),
we obtain $\Delta_{k}(t)^{2}$ as
\beqa
\Delta_{k}(t)^{2}&=&\frac{1}{T_{p}}[  \frac{\pi^{2}}{\omega_{1}}
\{Bi'(\tau_{1})Ai(\tau)-Ai'(\tau)Bi(\tau)\}^{2}\nonumber\\
& &+\frac{\omega_{1}\pi^{2}}{(8v^{2}\beta')^{1/3}}
\{Bi(\tau_{1})Ai(\tau)-Ai(\tau_{1})Bi(\tau)\}^{2}
],\label{movingck}
\eeqa
where $Ai'(x)=\frac{dAi}{dx}$, $\tau\equiv \alpha^{1/3}_{k} (t-t_{*}) 
=\{8v^{2}(\beta-k^{2}/2)\}^{1/3}( t-t_{*})$ and 
$\tau_{1}\equiv \alpha^{1/3}_{k} (t_{1}-t_{*})$.
We denote here a typical time scale of the decay as 
$T^{(II)}\equiv 1/(8v^{2}\beta)^{1/6}$.
Taking into account (\ref{asym1}), 
one can derive its asymptotic form 
for relatively large $t-t_{*}$.
A rough one is $\Delta_{k}(t)\sim 
\exp [ \frac{2}{3} (8v^{2}\beta')^{1/4}(t-t_{*})^{3/2} ]
/\sqrt{T_{p}}$. 

Using (\ref{typfldef}) and (\ref{movingck}),
the the typical amplitude of $|\tilde{c}(x^{\mu})|$
is evaluated as 
\beqa
|\tilde{c}(x^{\mu})|_{{\rm typical}}^{2}\equiv B(t)^{2} f_{0}(x)^{2} 
=\frac{\Omega_{p-2} }{2 (2\pi)^{p-1}}
\int_{0}^{\sqrt{2\beta}} k^{p-2}dk \Delta_{k}(t)^{2}
\sqrt{\frac{2\beta}{\pi}}e^{-2\beta x^{2}}.\label{movingcfinal}
\eeqa
As in the case (I), the blow-up of the tachyonic fluctuation
is localized around $x=0$ with the  width $\delta=1/\beta$.
The behavior of the 
time dependent part of $|\tilde{c}(x^{\mu})|_{{\rm typical}}$ 
($B(t)$ over $\sqrt{g_{s}}$) 
in the specific case of D-4-branes with 
$2\beta=10^{-2}$ and $v=10^{-4}$ is plotted in Fig.9
where $T^{(II)}\simeq 36.84 l_{s}$.  
\begin{figure}[h]
\begin{center}
\includegraphics[width=8cm]{f9}
\caption{$\log_{10}(B(t)/\sqrt{g_{s}})$ vs. 
$(t-t_{0})/T^{(II)}$ for D-4-branes; $2\beta=10^{-2}$ and $v=10^{-4}$.}
\end{center}
\end{figure}

The factor that characterizes the behavior of $B(t)$ most can
be extracted  from (\ref{movingcfinal}) as
$\exp [ \frac{2}{3} (8v^{2}\beta)^{1/4}(t-t_{0})^{3/2} ]$.
The coefficient of $(t-t_{0})^{3/2}$ strongly depends on 
$v$ as well as $\beta$, so, we can see that
if $v$ increases, the blow-up behavior of the tachyon becomes
much more radical. 
The physical interpretation of this consequence is  as follows:
After the distance $z$ becomes shorter than $z_{0}$,
the tachyonic modes start to ``wake up'' in turn and blow up 
one after another. The larger $v$ gets, the earlier  
the tachyonic modes with some momenta ``wake up'' to blow up.
So, we can say the blow up behavior is very sensitive to
the velocity around the critical distance.
%as in ref.\cite{}    

A consistency check can be done 
by comparing the behavior of
(\ref{movingcfinal}) with the amplitude of $\tilde{c}(x_{\mu})$  
inferred on the basis of the classical field equation
$d^{2}\tilde{c}/dt^{2}\simeq \alpha_{0} (t-t_{0})\tilde{c}$.
%(which is written as $d^{2}\tilde{c}/d\tau^{2}$)
Its solutions are of course Airy functions $Ai$ and $Bi$
as a function of $\tau_{0}\equiv\alpha^{1/3}(t-t_{0})$.
Its asymptotic form is inferred as 
$e^{2\tau_{0}^{\frac{3}{2}}/3}\equiv
\exp [ \gamma_{0}^{(II)} (t-t_{0})^{\frac{3}{2}}]$ (though
the prefactor cannot be estimated by the classical argument),
which agrees with the roughest estimation presented above. 
The momentum correction to the coefficient $\gamma_{0}$ can be made
by defining $\gamma_{{\rm eff}}^{(II)}\equiv d\ln(B(t))/d(t-t_{0})^{3/2}$,
whose behavior for $t\ge t_{0}$ with 
$2\beta=10^{-2}$ and $v=10^{-4}$ is drown in Fig.10.
\begin{figure}[h]
\begin{center}
\includegraphics[width=8cm]{f10}
\caption{$\gamma_{{\rm eff}}^{(II)}/\gamma_{0}^{(II)}$ 
vs. $(t-t_{0})/T^{(II)}$ for D-4-branes}
\end{center}
\end{figure}
The typical momentum which the tachyon modes possess,
defined in (\ref{momav}) in the previous section, 
can also be estimated. 
For example, for a specific value of parameter 
$2\beta=10^{-2}$ and $v=10^{-4}$,
$<k^{2}>/k^{2}_{{\rm max}}$ is given in Fig.11.
\begin{figure}[h]
\begin{center}
\includegraphics[width=8cm]{f11}
\caption{$<k^{2}>/k^{2}_{{\rm max}}$ vs. $(t-t_{0})/T^{(II)}$ 
for D-4-branes.}
\end{center}
\end{figure}
Based on that, the typical wavelength of the tachyon modes in this case is
about the order of
$\lambda_{{\rm typical}}\sim \lambda_{{\rm min}}/\sqrt{0.03}
\sim 400 l_{s}$ for $k^{2}=0.03 k^{2}_{{\rm max}}$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Time-evolution of the D-p-brane's shape, 
pair-creations of strings, and their physical
implications in brane-world scenarios}


In this subsection we first discuss the behavior of the 
shape of the branes after recombination 
and a large number of pair-creations of fundamental 
strings for the case (II).
Then, we consider physical consequences of the above two behaviors
and finally speculate its implication for the braneworld scenarios.

The essential difference of this case from the case (I) is that 
in the case (II), there are two scalar fields which have non-trivial 
VEV's: $X_{p+1}^{(0)}$ and $X_{9}^{(0)}$.
Since the two after tachyon condensation are not commutative,
it is probable that one cannot determine 
definite positions of the branes in both of the
dimensions $x_{p+1}$ and $x_{9}$ simultaneously.
In this subsection we assume that if we diagonalize $X_{p+1}^{(0)}$,
regardless of  $X_{9}^{(0)}$,
we can interpret each of the diagonal elements 
as the position of branes in the $x_{p+1}$  direction. 
We will discuss the problem related to  the non-commutativity and 
uncertainty in the final section.

Using (\ref{defctilder}), we rewrite the result in terms of 
the field $d$ as $|d(x_{\mu})|_{{\rm typical}} = 
\frac{|\tilde{c}|_{{\rm typical}} }{\sqrt{2}}=
\frac{B(t)f_{0}(x)}{\sqrt{2}}$,
which is essentially the same as (\ref{dbehavior}) 
in the case (I) with $A(t)$ replaced by $B(t)$,
and so is the diagonalized matrix $X_{p+1}^{(0)}$.
Thus, based on the above assumption,
we can proceed the discussions parallel to the ones done in the case (I): 
The shape of each brane projected on the $x_{p}x_{p+1}$-plane
is almost hyperbola at the first stage,
but then, deforms into the one with multiple (three) extremes
due to localization of tachyon condensation.
The only thing we have to check is that the  
terms newly added to the action, including higher order ones,
do not interfere the branes to have
the multiple extremes, i.e. $B(t)$ to reach the critical value
$(\pi\beta/2)^{1/4}$ given in (\ref{wkbcond1}).
Within the WKB approximation, the behavior is inevitable;
$z(t)$ only plays a role of mass, and $w$ and $c_{0}$ are still 
massive fields, strongly suppressed by $g_{s}^{1/2}$.
As for the higher order terms, the most dominant term
is again (\ref{ss4}) while  the second order terms are also the 
same order as in the case (I). Therefore, we conclude that
also in the case (II), the branes comes to have multiple extremes
if the shape of the branes can be defined and is adequate to discuss.  

As for the another interesting phenomanon, a large number of 
pair-creations of fundamental strings connecting the recombined branes,
it certainly happens also in the case (II), in the same way as
discussed in subsection 2.3. This is because the
non-commutativity of $X_{p+1}$ and $X_{9}$ arising in this case
does not affect the blow-up of $F_{0p}$;
In either gauge where $X_{p+1}$ or $X_{9}$ is diagonalized, 
only the off-diagonal elements of $F_{0p}$ blow up.

One of the physical consequences of the above two phenomena is
that {\it the branes (strongly) 
dissipate their energy into radiations of
gravitational and (RR and NSNS) gauge fields}
(though Yang-Mills theory does not include 
the effect of dissipation).
The reason for the radiations to occur is as follows:
acceleration of the D-p-branes means acceleration of a large mass and 
electric charges of RR (p+1)-form gauge field.
Thus, such a radical acceleration causes 
strong radiations of gravitational and RR gauge fields.
In addition, pair-created strings have electric charges of
the NSNS 2-form gauge field $B_{\mu\nu}$.
If the number of the charges varies, so does the value of the field,  
which is propagated as its radiation. Thus, it is probable that
vast number of their pair-creations give rise to (relatively strong) 
radiation of NSNS gauge field. 

Another physical consequence, based on our analysis,
is that {\it the branes have a stronger tendency
to dissipate their energy} than we had expected on the basis of 
our naive intuition.
The reason is as follows;
suppose that, based on our naive intuition,  
the shape of the branes (on $x_{p}x_{p+1}$-plane)
would develop, keeping the shape to be like a hyperbola
as the condensation goes on. 
Then, dissipation of the energy of course would happen,
but in the case the energy distribution would be rather
``well-balanced", not extremely partial.
Thus, dissipation of the energy would be
 expected to be rather mild, not so radical.
In addition, when the branes are wrapped 
around the compact space, the shape of the brane 
would oscillate some times due to the effect of compact spaces
while radiating its energy, and would rest wrapping around some 
supersymmetric cycle.    
However, our analysis reveals that in
the true case the energy released by tachyon condensation
is concentrated around the point $x=0$,
and the part of the branes around $x=0$ is 
accelerated more than the other parts,
resulting in the shape with multiple extremes.
This shape symbolizes concentration of the energy. 
Though Yang-Mills theory does not include 
the effect of dissipation  (due to the radiations or the coupling to
some other fields), its effect certainly exists and should be 
stronger when the energy is localized in a smaller region
or when only some smaller part of branes is accelerated in a 
concentrated way. 
Even if the shape of the brane cannot be defined in this case,
or the term ``shape'' is not appropriate for the setting,
localization of  tachyon condensation and that of the released energy 
will happen.
Therefore, we conclude that also in the case (II),
the brane system have a relatively 
stronger tendency to dissipate its energy than naively expected.   
  
If one apply this setting to the braneworld scenarios of inflation,
the property may imply that
the tachyonic preheating era tends to be rather short, as in 
ref.\cite{linde1}, although the cause of 
dissipation in this case is different from that in 
ref.\cite{linde1}.\footnote{
The dissipation in ref.\cite{linde1} is due to the coupling to 
the other scalar fields, while that in our case is due to radiations
because of the local acceleration.}
We believe this work is of value in that we 
extract the physical property or the tendency of the whole process 
of recombination from only low order perturbative analysis.  



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Discussion}
\setcounter{equation}{0}
\setcounter{footnote}{0} 

Finally, we discuss the problem of non-commutativity, 
uncertainty and diagonalization procedure (gauge choice) 
of VEV's of adjoint scalar fields 
$X_{p+1}^{(0)}$ and $X_{9}^{(0)} $ which appear in the case (II).
 
The commutation relation of the two is
\beqa
[X_{p+1}^{(0)},X_{9}^{(0)}] = z(t) \left(
\begin{array}{cc}
0 & -d^{*} \\
d & 0 
\end{array}
\right). 
\label{comm}
\eeqa
It is plausible for the relation to give an uncertainty relation
\beqa
\Delta x_{p+1} \Delta x_{9} \ge |z||d|.\label{uncer}
\eeqa
Let us assume that this uncertainty relation holds.
Then, the right hand side (r.h.s.) of (\ref{uncer}) is a function of 
time, so, the uncertainty is time-dependent in this case;
when the tachyon still does not condensate,
r.h.s. of (\ref{uncer}) vanishes and positions
of the branes in both directions can be determined
with arbitrary accuracy,
while the product of the accuracies of positions in two directions
cannot be less than the r.h.s.
when the tachyon starts to condensate.
Then, if one focuses on the positions in $x_{p+1}$ direction
and do not discuss or determine those in $x_{9}$ direction,
it seems possible to discuss the shapes of branes projected on 
$x_{p}x_{p+1}$-plane
with some high accuracy. 
Thus, we argue that the discussion in section 3
holds.
 
As for the ``shape'' (or profile) in  $x_{9}$ direction, 
suppose one diagonalizes $X_{p+1}$ and describes their positions
in the direction with the accuracy of the order $|d|$,
i.e. the accuracy one can determine whether the branes are 
combined or not. 
Then, according to (\ref{uncer}),
one is able to tell the positions in $x_{9}$ direction
only with the accuracy of order $|z|$, which means that we cannot
say anything about the branes' shapes in $x_{9}$ direction.

One may notice that there seems a bit problematic thing: 
When one takes the gauge which diagonalizes  $X_{9}$ instead,  
no evidence of recombination process can be observed.
It may be possible to attribute this problem to
the  gauge choice: On the one hand there is a gauge where 
geometric information of the branes is easy to extract 
(in this case the gauge $X_{p+1}$ is diagonalized; we denote 
this as $X_{p+1}$-diagonal gauge). On the other hand, 
there is a gauge where
the behavior of fluctuations like tachyon condensation
is easy to describe (in this case the gauge $X_{9}$ is diagonalized).
 
In addition, we would like to speculate that
diagonal elements of not diagonalized scalar fields  
might have some geometric meaning.
In the $X_{p+1}$-diagonal gauge, the VEV of $X_{9}$ is written as
\beqa
X_{9}^{(0)\  {\rm diag}} = U^{-1}X_{9}^{(0)}U =
\frac{z(t)}{2\sqrt{(\beta x)^{2}+|d|^{2}}}
\left(
\begin{array}{cc}
 \beta x & -d^{*} \\
-d & -\beta x
\end{array}
\right). 
\label{x9}
\eeqa
The diagonal elements represent two kink-like shapes (profiles)
along x-direction with a width $|d|/\beta$ with 
both ends approaching $\pm z(t)/2$, which fits our expectation
for the branes' shapes projected on $x_{p}x_{9}$-plane.  
It may be a merely coincidence, and the statement
that the diagonal elements describe the shapes of the branes,
may be false against (\ref{uncer}),
but there might be a possibility that it has some geometric meaning.
Let us remember the solution of dielectric (spherical) 
D2/D0 bound states
given by Myers in ref.\cite{myers}. We concentrate
the N=2 (two D0) case. The solution is
\beqa
\begin{array}{ccc}
X_{1}= \frac{R}{\sqrt{3}}
\left(
\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}
\right),
 &
X_{2}= \frac{R}{\sqrt{3}}
\left(
\begin{array}{cc}
0 & -i \\
i & 0
\end{array}
\right),&
X_{3}= \frac{R}{\sqrt{3}}
\left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right)
\end{array}
\eeqa
where $ R$ is the extent of the spherical 
D-2-brane to which two D-0-branes are bound.
If one focuses on $x_{3}$ direction, 
the two D-0-branes are considered to locate in the north 
and south pole of the sphere in the $x_{3}$ direction. 
Then, one cannot determine the positions of two D0's
in $x_{1}$ and $x_{2}$ direction, but the zeros 
in diagonal elements of them may be interpreted as 
center of mass positions or expectation values of their positions.
Analogous to this case, one might be able to
find out some geometrical meaning
in the diagonal elements of $X_{9}^{(0)\  {\rm diag}} $, 
though there is  also a possibility that 
they have no sense. 
Anyway, the above point, related with non-commutativity, gauge
choice and uncertainty, 
is interesting and to be explored in the future. 

\vskip 3ex
\noindent
{\large\bf Acknowledgement}
\vskip 3ex

%
I would especially like to thank Hidehiko Shimada for many fruitful
discussions, encouragement and much adequate advice 
in proceeding this work.
I also thank to Prof. K. Hashimoto and S. Nagaoka for useful 
communication and discussion.
I also like to thank Dr. Taro Tani for many useful discussion 
and encouragement, and Prof. M. Kato for useful comments (questions),
directing my attention to some aspects I was not considering. 
This work is supported by Japanese Society for the Promotion of 
Science under Post-doctorial Research Program (No.13-05035). 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix


\section{The time-dependent width for a Gaussian wave function
in terms of basis functions}
\setcounter{equation}{0}

%\setcounter{section}{5}

In this appendix we first review derivation of the propagator (kernel) 
$K$ of the harmonic oscillator with a time-dependent frequency,
and then the fact that
an initially Gaussian wave function keeps to be
Gaussian within WKB approximation,
and finally present the formula for
the time-dependent width of the Gaussian in terms of
basis functions. See for more detail ref.\cite{fey}.

If one find two independent solutions $A(t)$ and $B(t)$ (which 
we denote as basis functions) to the ``WKB-approximated 
equation of motion
\beqa
\frac{d^{2} x(t)}{dt^{2}}= -\omega(t)^{2} x(t),
\label{eoma}
\eeqa
we can immediately write essential part of the propagator\cite{fey}
in terms of $A(t)$ and $B(t)$ up to WKB approximation.
The solution representing the  classical path of a motion 
from $(x_{0},t_{0})$ to $(x_{1},t_{1})$ is
\beqa
x(t)&=&\frac{1}{B_{1}A_{0}-A_{1}B_{0}}[\{ B_{1}A(t)-A_{1}B(t)\} x_{0}
+\{ -B_{0}A(t)+A_{0}B(t)\} x_{1}]\\
&\equiv & h_{0}(t) x_{0}+ h_{1}(t) x_{1}\label{classicalsol}
\eeqa
where we abbreviate $A(t_{1})$ as $A_{1}$, etc.
Substituting (\ref{classicalsol}) for the action 
$S=\int_{t_{0}}^{t_{1}} dt \ m(\dot{x}^{2}-\omega^{2}x^{2})/2$, 
and exponentiating 
the one times $i$, we have the propagator $K(x_{1},t_{1};x_{0},t_{0})=
F e^{iS_{{\rm cl}}}$ where $S_{{\rm cl}}$ is quadratic in $x$ as
\beqa
S_{{\rm cl}}=\frac{m}{2}\sum_{i=0,1}x_{i}x_{j} \alpha_{ij}(t_{1};t_{0})
\eeqa
where   
\beqa
\alpha_{ij}(t_{1};t_{0})\equiv \int_{t_{0}}^{t_{1}} dt 
[\frac{dh_{i}}{dt} \frac{dh_{j}}{dt} -\omega(t)^{2} h_{i}h_{j} ].
\eeqa
$F$ is a time-dependent normalization but not dependent on $x_{i}$,
which can be determined 
in several ways. 
(In this paper we determine the factor $F$ so that the propagator 
in each case $K$
satisfies the Schrodinger equation.)
Since the wave function for $t=t_{1}$ is given by
$\Psi(x_{1},t_{1})=\int dx_{0} K(x_{1},t_{1};x_{0},t_{0})
\Psi(x_{0},t_{0})$,
the absolute value of
an initially Gaussian wave function remains to be Gaussian,
since the above is only a Gaussian integral.
The behavior of the system  is represented by the 
time-dependent width of the Gaussian
obtained  in terms of the basis functions as
\beqa
\Delta(t)^{2}=\Delta(t_{0})^{2}
\frac{\alpha_{00}(t;t_{0})^{2}}{\alpha_{01}(t;t_{0})^{2}}
+\frac{1}{\Delta^{2}(t_{0})^{2}(\alpha_{01})^{2}}.\label{formaldelta}
\eeqa 
Thus, finding the basis functions leads directly to the 
time-evolution behavior of the width and the system up to WKB
approximation as far as the the initial wave function is Gaussian.
We note that normalizations of
$A(t)$ and $B(t)$ do not appear in
(\ref{classicalsol}) and hence also not in the final result.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end {document}


