text:
\section{ Perturbations and their Back-reaction on the
Microsuperspace Sector}
\setcounter{equation}{0}
As we mentioned in the Introduction, the one-plus-one $\lambda \phi^4$ field
theory has minisuperspace sectors just as gravitation does.  The
minisuperspace closest in concept to the cosmological minisuperspace
is the space-independent mode of $\phi$, $\phi_0 (t)$.  This
minisuperspace has the property that if it is plugged into the
action (1.2)
it gives a reduced action
\begin{equation}
S = L\int [{{1}\over {2}} \dot \phi_0^2 - {{\mu^2}\over {2}}
\phi_0^2 - \lambda \phi_0^4 ]dt,
\end{equation}
where we have assumed that the $t$ = const. slices have an $S^1$ topology
with $z = \pm
L/2$ identified.  When the action is varied with respect to $\phi_0$
it gives the correct equation of motion for $\phi_0$,
\begin{equation}
 \ddot \phi _0 + \mu^2 \phi_0 + 4\lambda \phi_0^3 = 0.
\end{equation}
Of course, the Hamiltonian form of (2.2),
\begin{equation}
 S = L\int [\pi_0 \dot \phi_0 - ({{1}\over {2}} \pi^2_0 +
{{\mu^2}\over {2}} \phi_0^2 + \lambda \phi_0^4 )] dt
\end{equation}
also gives the correct equations of motion for $\pi_0$ and $\phi_0$.

As we also mentioned in the Introduction, minisuperspace quantization of
the $\phi_0$ mode consists of quantizing the theory given by
(2.3) with $\phi_0$ and $\pi_0$, the configuration variable and its
conjugate momentum as the only variables of the problem, and this
quantization can be taken as a simplified toy model of quantum
cosmology.  We will not attempt at this point to quantize this system
but will give an approximate solution as part of the more general
problem discussed below.

As a model for studying when the quantization of a gravitational
quantum minisuperspace gives useful information, we will extend our
$\lambda \phi^4$ minisuperspace in a way similar to that used by
Halliwell and Hawking\cite{H-Hawk} to study the volume of superspace near the
$k = +1$ FRW minisuperspace.  They used a truncated action which in
our case can be constructed by first expanding the full $\phi (z,t)$
in a real Fourier series of the form of (1.3),
where for convenience we will set all of the $\phi^{(s)}_n (t)$ equal
to zero since the final result will not be qualitatively different if
these terms are kept.  If we put this form of $\phi$ into (1.2) we find
\begin{eqnarray}
S &=& L \int [ {{1}\over {2}} \dot \phi_0^2 + {{1}\over {4}} \sum_n
\dot \phi_n^2 - {{\mu^2}\over {2}} \phi_0^2 - \lambda \phi_0^4 -
{{1}\over {4}} \sum_n \phi_n^2 \left ({{2\pi n}\over {L}} \right )^2
- {{\mu^2}\over {4}} \sum_n \phi_n^2 -\nonumber \\
& &-
\lambda \phi_0 \sum_{n,m,\ell} A_{nm\ell} \phi_n \phi_m \phi_{\ell}
- 3\lambda \phi_0^2 \sum_n \phi_n^2 - \lambda \sum_{n,m,\ell ,k}
B_{mn\ell k} \phi_n \phi_m \phi_{\ell} \phi_k ]dt,
\end{eqnarray}
where $A_{mn\ell}$ and $B_{mn\ell k}$ are constants given by the
integrations over products of $\cos (2n\pi z/L)$.  The truncated
action assumes that the $\phi_n$ are small enough that the cubic and
quartic terms in the $\phi_n$ are negligible, while the quadratic
terms are large enough to affect the behavior of the $\phi_0$. In
this case the action reduces to
\begin{equation}
 S = L \int [{{1}\over {2}} \dot \phi_0^2 + {{1}\over {4}}\sum_n
\dot \phi_n^2 - {{\mu^2}\over {2}} \phi_0^2 - \lambda \phi_0^4 -
{{\mu^2}\over {4}} \sum_n \phi_n^2 - {{1}\over {4}} \sum_n \phi_n^2 \left
({{2\pi n}\over {L}} \right )^2 - 3 \lambda \phi_0^2 \sum_n \phi_n^2
]dt.
\end{equation}
We will not attempt here to justify the exclusion of the quartic and
cubic terms, but will instead discuss the meaning of the classical
equations of motion derived from (2.5).  The equation for $\phi_0$ is
\begin{equation}
\ddot \phi_0 + \mu^2 \phi_0 + 4\lambda \phi_0^3 + 6\lambda \phi_0
\sum_n \phi_n^2 = 0,
\end{equation}
while the equations for the $\phi_n$ are
\begin{equation}
\ddot \phi_n + \mu^2 \phi_n + \left ({{2\pi n}\over {L}} \right
)^2 \phi_n + 12 \lambda \phi_0^2 \phi_n = 0.
\end{equation}
Up to the order we have kept the $\phi_n$ these are exact.
These exact equations can be interpreted as simple perturbation
equations for the $\phi_n$ and (2.6) as an equation for $\phi_0$ that
includes the first non-zero term of the back reaction of the $\phi_n$
on the $\phi_0$.  However, this interpretation only makes sense if we
assume that the amplitudes $\phi_n$ are more or less randomly
distributed, and are not correlated in such a way as to represent a
large concentration of field $\phi$ at some point $z = z_0$.  Perhaps
the best way to qualify this idea is to call it a ``cosmological''
paradigm.  The usual picture of the universe is that it is made up of
matter condensations that are locally inhomogeneous, but that these
condensations are spread throughout space in such a way that on the
average the matter density is homogeneous.  It is this averaged
density that drives the spatially homogeneous gravitational mode.  Of
course, any gravitational field can be broken up into harmonic modes,
and the homogeneous mode will be affected by the inhomogeneous modes,
but if, say, the universe is half empty, the homogeneous mode cannot
reasonably be interpreted as a homogeneous cosmological background
driven by the averaged matter density.

In order to study the meaning of quantum minisuperspace solutions in
the context of the present theory, we want to first solve the quantum
problem in the microsuperspace sector where the $\phi_n$ are set
equal to zero, and in this case the action is (2.3) and corresponds
to a one-dimensional anharmonic oscillator.  We are not assuming that
either $\mu^2$ or $\lambda$ is necessarily positive, so there are four
possible microsuperspace quantum systems, Case I ($\mu^2 >0, \lambda >
0$), Case II ($\mu^2 < 0, \lambda > 0$), Case III ($\mu^2 > 0, \lambda <
0$), and Case IV ($\mu^2 < 0, \lambda < 0$).  We can quantize the
problem given by (2.3) by realizing the operator $\hat \pi_0$ as $-i
\partial / \partial \phi_0$ operating on a state function $\Psi
(\phi_0, t)$, which leads to the Schr\"odinger equation
\begin{equation}
-{{1}\over {2}} {{\partial^2 \Psi}\over {\partial \phi^2_0}} +
{{\mu^2}\over {2}} \phi_0^2 \Psi + \lambda \phi_0^4 \Psi = i
{{\partial \Psi}\over {\partial t}}.
\end{equation}
We would like to compare the solutions of this equation with
solutions for the full superspace of all possible modes, ($\phi_0,
\phi_n$) in order to see if the presence of the higher modes,
$\phi_n$, affects the behavior of the $\phi_0$ mode in such a way as
to cause drastic changes in its quantum dynamics.  However,
as we said in the Introduction, in order
to avoid renormalization problems we will work with a simpler system
similar to that used by Kucha\v r and Ryan \cite{Kuchar}, where we will imbed
the $\phi_0$ (microsuperspace) mode in an extended minisuperspace
where only one of the $\phi_n$ is non-zero .
Since in the truncated action the $\phi_n$ modes do not interact
among themselves, this ansatz is consistent.

We will now need the Hamiltonian form of (2.5).
In order to simplify our notation, we will define $x \equiv \sqrt{L}
\phi_0$, $y \equiv \sqrt{L/2}\phi_n$, and $p_x$ and $p_y$ their
conjugate momenta.  With these definitions the
final action for $x$ and $y$ becomes
\begin{equation}
 S = \int [p_x \dot x + p_y \dot y - \{ {{1}\over {2}}p_x^2 +
{{1}\over {2}}p_y^2 + {{\mu^2}\over {2}} x^2 + {{\lambda}\over {L}}
x^4 + [{{\mu^2}\over {2}} +
{{1}\over {2}}\left({{2\pi n}\over {L}} \right)^2]y^2 +
6{{\lambda}\over {L}}x^2 y^2 \}]dt.
\end{equation}
In this case the minisuperspace state function $\Psi (x, y, t)$ obeys
the Schr\"odinger equation
\begin{equation}
-{{1}\over {2}} {{\partial^2 \Psi}\over {\partial x^2}} - {{1}\over {2}}
{{\partial^2 \Psi}\over {\partial y^2}} + {{\mu^2}\over {2}} x^2
\Psi + {{\lambda}\over {L}} x^4 \Psi +{{1}\over {2}}\left [ \mu^2 +
\left ( {{2\pi n}\over {L}} \right )^2 \right ] y^2 \Psi +
6{{\lambda}\over {L}} x^2 y^2 \Psi = i {{\partial \Psi}\over
{\partial t}}.
\end{equation}
Of course, the interaction term $6(\lambda /L) x^2 y^2 \Psi$ gives
the only influence of the $\phi_n$ mode on the $\phi_0$ mode.
To simplify our notation further, we will define $m^2 \equiv \mu^2
+ (2n\pi /L )^2$.  In the
next section we will find approximate solutions to (2.10), first a
microsuperspace solution where $y$ and $\partial^2 \Psi / \partial
y^2$ are put equal to zero and second a minisuperspace solution to the full
equation. The solution to the full (2.10) will be made as close as
possible to a coherent state centered around $y = 0$, and we will
investigate its behavior relative to the microsuperspace solution.
\def\tilde {\cal S}{\tilde {\cal S}}
\section{Quantum
Solutions for the Microsuperspace and Minisuperspace Models}
\setcounter{equation}{0}
In this section we will first write down the solution to the Schr\"odinger
equation for the microsuperspace model and subsequently for the full
minisuperspace model for Case I, i.e, ${\mu}^2 >0, \lambda >0$.
Since the quantum problem for even the
microsuperspace sector is not exactly solvable, we will
use ${\lambda \over L}$ as a perturbation parameter and obtain a perturbative
solution to first order in ${\lambda \over L}$.
However, since ultimately we are interested
in comparing the quantum dynamics of the truncated model with that of
the full model, we will look for time dependent
solutions that are analogous to
coherent states for these models,
rather than the stationary solutions that are
found in the usual applications of quantum perturbation theory .

Let us start by considering the Schr\"odinger equation for the
microsuperspace model. This can be written down from (2.8):
\begin{equation}
i {\partial\Psi\over \partial t}
= {-{1\over 2}}{{\partial}^2\Psi\over {\partial x^2}}
+ \left({1\over 2}\mu^2 x^2 + {\lambda\over L} x^4 \right)\Psi .
\end{equation}
We make the following ansatz for the wave function
\begin{equation}
{\Psi}_{micro} = e^{-{\cal S}}  ,
\end{equation}
where ${\cal S} = {\cal S}_0 + {\lambda\over L } {\cal S}_1$.
We will now substitute the above
ansatz in the Schr\"odinger equation (3.1) and
retain terms only up to linear order in ${\lambda \over L}$.  As far as we
know, this is the first attempt to apply perturbation theory to the
exponent ${\cal S}$ of $\Psi$.  Other attempts have directly
perturbed the function $\Psi$, and correspond to expanding
$\exp(-{\lambda\over L} {\cal S}_1)$ as $1 - ({\lambda\over L}) {\cal S}_1$.

Now, to lowest order we
obtain an equation for ${\cal S}_0$ given by,
\begin{equation}
-i {\partial {\cal S}_0\over \partial t}
= -{1\over 2}{\left({\partial {\cal S
}_0\over \partial x }\right)}^2
{+{1\over 2}}{{\partial}^2 {\cal S}_0\over {\partial x^2}}
+ {1\over 2}\mu^2 x^2     ,
\end{equation}
and to $O({\lambda \over L})$ we obtain
\begin{equation}
-i {\partial {\cal S}_1\over \partial t}
= - \left({\partial {\cal S}_0\over \partial x }\right)
\left({\partial {\cal S}_1\over \partial x }\right)
{+{1\over 2}}{{\partial}^2 {\cal S}_1\over {\partial x^2}}
+ x^4  .
\end{equation}
Let us concentrate on the zeroth order solution, i.e, ${\cal S}_0$ first. We
should mention that since this essentially only involves the quantum
solution to the harmonic oscillator problem, it is exactly solvable,
and the solutions are well known. In particular, the coherent state
solution is known, and we can write it down directly. However, we will
go through it in some detail merely to illustrate our technique of
obtaining it.
Separating ${\cal S}_0$ into real and imaginary parts as
${\cal S}_0 = {{\cal S}_0}^R + i{{\cal S}_0}^I$
, the problem now reduces to solving the real and imaginary parts of
eqn. (3.3 ), which are given by
\begin{equation}
- {\partial{{\cal S}_0}^R\over \partial t}
= - \left({\partial {{\cal S}_0}^R\over \partial x }\right)
\left({\partial{{\cal S}_0}^I\over \partial x }\right)
+{1\over 2}{{\partial}^2 {{\cal S}_0}^I\over {\partial x^2}}
\end{equation}
and
\begin{equation}
 {\partial{{\cal S}_0}^I\over \partial t}
= -{1\over 2}{\left({\partial {{\cal S}_0}^R\over \partial x }\right)}^2
+{1\over 2}{\left({\partial {{\cal S}_0}^I\over \partial x }\right)}^2
  {+{1\over 2}}{{\partial}^2 {{\cal S}_0}^R\over {\partial x^2}}
+ {1\over 2}\mu^2 x^2
\end{equation}
respectively.
Let us now specialize our ansatz to the following form ( clearly
suggested by the known coherent state solution):
\begin{equation}
{\cal S}_0 = {\mu\over 2}{\left[x - g(t)\right]}^2 + iP(t)x ,
\end{equation}
where $P$ and $g$ are real functions of time. Substituting the
real and imaginary parts of ${\cal S
}_0$ from (3.7) into eqns. (3.5) and
(3.6)
respectively, we obtain the equations satisfied by these functions
by equating equal powers of $x$.
These are given by:
\begin{equation}
\ddot{g}(t) + {\mu}^2 g(t) = 0
\end{equation}
and
\begin{equation}
P(t) = -\dot{g} .
\end{equation}
Eqn. (3.8) is simply the equation for a classical harmonic oscillator
with coordinate $g$ and the solution we choose is
\begin{equation}
g(t) = x_0 \cos \mu t ,
\end{equation}
where $x_0$ is a real constant determined by initial conditions. Then
$P(t)$ can be determined through eqn. (3.9 ) easily. We will not write
down the solution for $P(t)$ explicitly, since it corresponds to a
phase in the wave function, and we are ultimately interested in the
probability density $|\Psi(x)|^2$. Therefore, what we are after is
really the quantity $e^{-2{\cal S}^R}$, and as a result we will avoid
calculating the imaginary parts of ${\cal S}$ explicitly throughout the
rest of this paper. It is evident from eqn. (3.9) and (3.10) that
we have recovered the usual coherent state solution for the
wavefunction to the lowest order, and it is a Gaussian peaked around
the classical trajectory given by (3.10).  The rest of the solution
will represent a perturbation on this coherent state.

Thus
\begin{equation}
{{\cal S}_0}^R = {\mu\over 2}{\left(x - x_0 \cos \mu t\right)}^2  .
\end{equation}

Let  us continue on to the $O({\lambda \over L})$ part.
The equations for the real and imaginary parts are given by
\begin{equation}
- {\partial{{\cal S}_1}^R\over \partial t}
= - \left({\partial {{\cal S}_0}^R\over \partial x }\right)
\left({\partial{{\cal S}_1}^I\over \partial x }\right)
- \left({\partial {{\cal S}_0}^I\over \partial x }\right)
\left({\partial{{\cal S}_1}^R\over \partial x }\right)
+{1\over 2}{{\partial}^2 {{\cal S}_1}^I\over {\partial x^2}}
\end{equation}
and
\begin{equation}
 {\partial{{\cal S}_1}^I\over \partial t}
= - \left({\partial {{\cal S}_0}^R\over \partial x }\right)
\left({\partial{{\cal S}_1}^R\over \partial x }\right)
+ \left({\partial {{\cal S}_0}^I\over \partial x }\right)
\left({\partial{{\cal S}_1}^I\over \partial x }\right)
+{1\over 2}{{\partial}^2 {{\cal S}_1}^R\over {\partial x^2}} + x^4 .
\end{equation}
Guessing from the form of the lowest order solution, we specialize
our ansatz further to
\begin{eqnarray}
{{\cal S}_1}^R &=& \alpha x^4 + \beta x^3 + \gamma x^2 + \sigma x +
\omega ,\nonumber \\
{{\cal S}_1}^I &=& B x^3 + C x^2 + D x + E .
\end{eqnarray}
We follow exactly the same strategy as before to solve for the
unknown functions. We first substitute the ansatz (3.14) in eqns (3.12)
and (3.13) and then obtain equations satisfied by these functions by
equating equal powers of $x$. In this case of course, execution of
this strategy is much more complicated since we obtain coupled
equations for the functions. However, as shown as in  Appendix A,
these do indeed form a consistent set and can be successively
solved such that each satisfies a harmonic oscillator equation
driven by a source provided by a combination of the other functions
that have already been solved for.

Here we will  write down the final solution for only the real part
${{\cal S}_1}^R$. The coefficients $\alpha, \beta, \gamma, \sigma , \omega$
in (3.14) are given as follows
\begin{eqnarray}
\alpha &=& {1\over 4\mu} ,\nonumber \\
\beta  &=& {x_1}^3\cos{3\mu t} + {x_0\over 2\mu}\cos{\mu t} ,\nonumber \\
\gamma &=& {3\over 4{\mu}^2} + C_1 \cos{2\mu t}
- {3\over 2}{x_0}^2{x_1}^3 \cos{4\mu t} + {3\over 2}{x_0}^2 t \sin
{2\mu t} ,\nonumber\\
\sigma &=& d_1 \cos {\mu t}
-{3\over 2\mu}\left[\left({1\over 2}{x_0}^3 + {x_1}^3
\right) - {x_0}C_1 \right]\cos {3\mu t} , \nonumber \\
& & + {3\over 4}{x_0}^2{x_1}^3
\cos {5\mu t}
 + {3x_0\over \mu}t \sin{\mu t} -
{3\over 2}{x_0}^3 t\sin{3 \mu t} ,\nonumber\\
\omega &=& -{1\over 4\mu}g^4 - {3\over 4{\mu}^2}g^2 - {3g\beta\over
2\mu}
 -g^3 \beta - g\sigma - {\gamma\over 2\mu} ,
\end{eqnarray}
where $x_1 , C_1 $ and $d_1$ are arbitrary constants.
As stated before, we do not need the imaginary part, but as shown in
the Appendix, since the equations are coupled we do need to solve
for $B, C,$ and $D$ to obtain all the coefficients of the real part.
Thus from equations (3.2) , (3.11) and (3.15) we have the complete
unnormalized wavefunction for the microsuperspace sector to
$O({\lambda \over L})$ in the exponent.

At this point we would like to comment on the so called secular
terms of the form $t\sin{2\mu t}$ appearing in the solution. Such
terms are known to appear in the straightforward application
of perturbation theory to solve the classical anharmonic oscillator
problem \cite{mscale}. In that context they are known to be pathological,
because it is known that the full solution must have a behavior
bounded in time and clearly the secular terms  have the
incorrect behavior since they grow with time. This turns out to
be an artifact of a naive application of perturbation theory, and the
correct perturbative solution without secular terms can be obtained
by using more sophisticated methods such as the method of multiple
scales \cite{mscale}. We believe that the ``quantum secular terms" in our
problem have the same origin, since the center of our almost
coherent state approximately obeys the classical equation of motion.
Thus we can probably get a solution free of secular terms by
an application the more sophisticated perturbation techniques adapted
to quantum mechanics  but we will not pursue this further in this paper.
We will therefore confine our analysis to short time scales before
the secular terms become dominant.

Now let us proceed to the full problem, i.e, to  solve the
Schr\"odinger equation for the minisuperspace model, corresponding to
the action given in eqn.(2.9) , where now we have an additional
coordinate $y$ that couples to the microsuperspace sector.
The Schr\"odinger equation in this case is given by the full eqn.
(2.10).
We follow exactly the same steps as before, with the ansatz
\begin{equation}
{\Psi}_{mini} = e^{- \left({\tilde {\cal S}}_0 + {\lambda\over L} {\tilde {\cal S}}_1
\right)}.
\end{equation}
We notice that the lowest order problem is that of two uncoupled
harmonic oscillators with masses $\mu$ and $m$ corresponding to
the coordinates $x$ and $y$ respectively. So, we see from our
experience with the microsuperspace case that
$e^{-{\tilde {\cal S}_0}}$
can  be written down directly as
a product of two independent coherent state solutions for the
individual oscillators with ${\tilde {\cal S}}_0$ given by:
\begin{equation}
{\tilde {\cal S}}_0  = {\mu\over 2}{(x - x_0 \cos {\mu t})}^2
+  {m\over 2}{(y - y_0 \cos {m t})}^2  + iPx + i\bar{P}y ,
\end{equation}
where $y_0$ is a constant to be fixed by initial conditions and
$\bar{P}$ is a known function of time satisfying an equation
identical to (3.9) in the $y$ coordinate.

The real and imaginary parts of the $O({\lambda \over L})$ equation obtained by
substituting the ansatz in the Schr\"odinger equation are given as
follows
\begin{eqnarray}
 {\partial{{\tilde {\cal S}}_1}^I\over \partial t}
&=& - \mu (x - g)
\left({\partial{{\tilde {\cal S}}_1}^R\over \partial x }\right)
- \dot{g}
\left({\partial{{\tilde {\cal S}}_1}^I\over \partial x }\right)
+{1\over 2}{{\partial}^2 {{\tilde {\cal S}}_1}^R\over {\partial x^2}} + x^4 \nonumber \\
& & - m (y - f)
\left({\partial{{\tilde {\cal S}}_1}^R\over \partial y }\right)
- \dot{f}
\left({\partial{{\tilde {\cal S}}_1}^I\over \partial y }\right)
+{1\over 2}{{\partial}^2 {{\tilde {\cal S}}_1}^R\over {\partial y^2}} + 6x^2y^2
\end{eqnarray}
\noindent and
\begin{eqnarray}
- {\partial{{\tilde {\cal S}}_1}^R\over \partial t}
&=& - \mu (x -g)
\left({\partial{{\tilde {\cal S}}_1}^I\over \partial x }\right)
+ \dot{g}
\left({\partial{{\tilde {\cal S}}_1}^R\over \partial x }\right)
+{1\over 2}{{\partial}^2 {{\tilde {\cal S}}_1}^I\over {\partial x^2}}\nonumber \\
& & - m (y - f)
\left({\partial{{\tilde {\cal S}}_1}^I\over \partial y }\right)
+ \dot{f}
\left({\partial{{\tilde {\cal S}}_1}^R\over \partial y }\right)
+{1\over 2}{{\partial}^2 {{\tilde {\cal S}}_1}^I\over {\partial y^2}} ,
\end{eqnarray}
where $f = y_0\cos {m t}$ .

\noindent  We then specialize the ansatz to the following
\begin{eqnarray}
{{\tilde {\cal S}}_1 }^R &=& {\tilde \alpha}x^4 + {\tilde \beta}x^3 +{\tilde
\gamma}x^2 + {\tilde \sigma}x + \delta x^2 y^2 + {\tilde \omega}\nonumber \\
& & + \epsilon x y^2 + \kappa y^2 + \nu x^2 y + \theta xy + \xi y
\end{eqnarray}
and
\begin{eqnarray}
{{\tilde {\cal S}}_1 }^I &=& {\tilde B}x^3 + {\tilde C}x^2 + {\tilde D}x +
{\tilde E} + F x^2 y \nonumber \\
& &+G x y^2 + L y^2 + M xy + Ny .
\end{eqnarray}
The explicit solutions for these quantities  are
written down in gory detail in
Appendix A for the sake of completeness.
However, we are only interested in comparing the
two solutions under the conditions that the initial conditions are
such that the microsuperspace and minisuperspace solutions start out as
close as possible. Therefore here we will present the solution to the
full problem under these conditions, with  a large number
of integration constants set to zero. In fact, we will set $y_0 = 0$
as well, which means the lowest order solution will be initially
centered around $y = 0$, which is necessary to have the solutions
to the lowest order start out as close as possible. Thus
in this case we have,
\begin{eqnarray}
{\tilde \alpha} & =& \alpha ,\nonumber \\
{\tilde \beta}  & = & \beta ,\nonumber \\
{\tilde \gamma} & = & \gamma + {3\over 2\mu(\mu + m)} ,\nonumber \\
{\tilde \sigma} & = & \sigma + {3x_0\over m}t \sin \mu t ,\nonumber \\
\delta          & = & {3\over \mu + m} ,\nonumber \\
\epsilon        & = & {3\mu x_0\over m(m +\mu)}\cos \mu t , \nonumber \\
\kappa          & = & {3\over 2m(\mu + m )} - {3{\mu}^2{x_0}^2\over
2m({\mu}^2 - m^2)} \cos 2 \mu t ,
\end{eqnarray}
and $\nu , \theta$ and $\xi$ vanish.
We have plotted the $|\Psi_{micro}|^2$ and
$|\Psi_{mini}|^2\Big|_{y=0}$  for a succession of time intervals up
$t= 2\pi /\mu$, after which the secular terms
become dominant and the results
cannot be trusted. The wave functions in this section are
presented in their unnormalized form. It can be easily shown that
they are normalizable to $O({\lambda \over L})$. However, in the plots in

Figs. (1), (2), and (3) we have used a rough normalization of dividing
the probability density by the $x$ independent factor in the
unnormalized wave function. This seems to serve our purpose, since
we are more interested in tracking the motion of the center of the
wave packet. One can in principle numerically calculate a
normalization factor, but since such a
normalization can only be valid to order ${\lambda \over L}$, and, of course,
cannot be trusted beyond the timescale where the
secular terms start to dominate, we decided against this.
Notice that the behavior of the wave packets in the microsuperspace
and minisuperspace cases is at least qualitatively similar, in that
both are Gaussian-like packets that oscillate about $x = 0$ at roughly the
same frequency.
These results will be used in the next section to try to develop
a set of criteria for a ``good" microsuperspace.

\setcounter{equation}{0}
\section { Criteria for a Good Microsuperspace }
As we showed in the previous section, for  Case I microsuperspaces
the minisuperspace solution which begins as a Gaussian-like peak
around $y = 0$ and some value $x_0$ of $x$, and whose velocity is
initially zero, follows essentially the trajectory of the
microsuperspace (where $y$ and $p_y$ are set equal to zero before
quantization).  We have allowed our packet to evolve up to a point
where the secular terms begin to grow too large and the shape and
motion of the packet can no longer be trusted.

It is obvious in this case that the microsuperspace gives us at least
a qualitative picture of the behavior of the full minisuperspace wave
packet.  However, at least technically, the form and motion of the $y
= 0$ slice of the full packet is not the same as that of the
microsuperspace packet.  We can solve for the position of the peak of
$\mid \Psi \mid^2$ of the two packets by solving $\partial {{\cal
S}}^R/
\partial x = 0 $ in the microsuperspace case and $\partial {\cal S}^R/\partial
x \mid_{y=0} = 0$ in the minisuperspace case.  It is only reasonable
to solve these equations to first order in $\lambda /L$ because
${\cal S}$
is only valid to this order.  The peak of $\mid \Psi \mid^2$ will be
at $x_m = x_0 \cos \mu t + (\lambda /L) x_{1m}^{(0)}$ for the
microsuperspace and $x_m = x_0 \cos \mu t + (\lambda /L)
x_{1m}^{(1)}$ for the minisuperspace.  The $x_{1m}$ are
\begin{eqnarray}
 x^{(0)}_{1m} &=& {{1}\over {\mu}}[ -{{x_0^3}\over {8\mu}}\cos 3\mu t
+ \left( {{15}\over {8\mu}} x_0^3 + {{3x_0}\over {2\mu^2}} \right )
\cos \mu t  + \left ( {{3x_0^2}\over {2}} + {{3x_0}\over {\mu}} \right
)t \sin
\mu t ],  \\
x^{(1)}_{1m} &=& x^{(0)}_{1m} + {{3x_0}\over {\mu (\mu + m)}} \cos
\mu t + {{3x_0}\over {m}} t \sin \mu t.
\end{eqnarray}
The changes in (4.2) are a slight shift in the zero-order amplitude,
$x_0 \cos \mu t$, and a change in the secular term $t \sin \mu t$.
Notice that for large $m$ (high mode number $n$ or small $L$) that
these corrections will be small.
As we saw from the graphs in Sec 3, these changes do not affect the
qualitative behavior of the wave packet, but it is obvious that they
do change the detailed behavior of the system.  Whether one says that
the microsuperspace system is a good model of the minisuperspace
system depends on exactly what questions one asks about the system.
However, given the coincidence of the graphs of the motion of the wave
packet in Sec. 3, we can reasonably say that
for Case I the microsuperspace is
``good'' in that it adequately represents the qualitative
behavior of the larger
minisuperspace.

For the other cases we will have to discuss in more detail the
meaning of a good minisuperspace.  In Case I the potential term in
$xy$-space has the form of a bowl (Figure 4),
and it is not surprising that
a wave packet exists that moves along the $y = 0$ line and is
confined to that line by the rising potential on either side of it.
The bowl shape also means that if the wave packet is given a small
initial motion in the $y$-direction it will (as can be seen in
Appendix A) only oscillate around the $y =0$ line while
following essentially the trajectory given in (4.2).  This leads us
to another question about the meaning of minisuperspace quantization.
We have based our ideas up to this point on the question of whether a
wave packet follows the minisuperspace (in our case microsuperspace)
trajectory when it ``feels'' the surrounding areas of superspace. The
new question is whether the trajectory of the wave packet is stable
with respect to small changes in the initial conditions that set it
up.  It is not entirely clear that such instability is a drawback,
since in the case of relativistic cosmology the initial conditions
are not ours to decide. However, too much dependence on initial conditions
to make a minisuperspace state a good description of the quantum
physics of the system is probably best avoided.  Of course, we cannot
escape all
dependence on initial conditions.  There are
examples, such as Bianchi type I cosmological minisuperspaces, where
the three-curvature of $t =$ const. slices that serves as a potential
term is zero, which means that trajectories are marginally stable and
any small changes in initial conditions in a wave packet will cause
the packet to move slowly away from any given minisuperspace
trajectory.  However, it is probably best to be suspicious of all
cases where the minisuperspace trajectory is, in the classical
regime, unstable with respect to small changes in initial conditions.
We will call such cases ``classically unstable'', and also use
``classically stable'' in the obvious way. A gravitational
minisuperspace example of a system that is unstable with respect to
slight changes in initial conditions is given in Ref. \cite{GRR2}.

Given these caveats we can now look at the cases other than Case I of
the $\lambda \phi ^4$ theory.  Case II ($\mu^2 < 0$, $\lambda >0$)
  corresponds to a
one-dimensional form of the usual ``Mexican hat'' potential of
inflationary theory.  If we consider the minisuperspace we must
divide Case II into two subcases, II$_a$ where $\mu^2 < 0$, $\lambda
> 0$, $m^2 > 0$, and II$_b$ where $\mu^2 < 0$, $\lambda > 0$, $m^2 <
0$. Of course, $m^2$ is greater or less than zero if $-\mid \mu
\mid^2 + (2n\pi /L)^2$ is greater or less than zero.
For II$_a$ Fig. (5a) shows the potential in $x$ and $y$.  Here the
$y=0$ line is always a minimum with respect to $y$ of the potential.
This means that the wave packet will be confined by the rising
potential
in the $y$-direction, and independent of the complicated motion in the
$x$-direction, we can expect that the packet will have the same
stable behavior as in Case I, so again II$_a$ will imply a ``good''
microsuperspace.  Case II$_b$ is less straightforward.  Figure (5b)
shows the potential there.  For small $x$ and $y$ the potential term
is similar to the ``crown'' of the Mexican hat potential, falling off
in all directions from $x = y = 0$.  As we move away from this point
along the $y = 0$ line we arrive at minima at $x = \pm (\mid \mu
\mid)/2)
\sqrt{L/\lambda}$, then the potential rises sharply as the $(\lambda
/L) x^4$ term takes over.  In the $y$-direction the $y = 0$ line is a
maximum of the potential at $x = 0$, and remains so until $\mid x\mid$
reaches
$(1/\sqrt{3} ) [1 - (2n\pi/\mid \mu \mid L )^2]^{1/2} (\mid \mu \mid /2)
\sqrt
{L/\lambda}$. So, between this value of $\mid x \mid$ and
$\mid x \mid  = (\mid \mu \mid /2)
\sqrt{L/\lambda}$ the potential in the $y$-direction has a relative
minimum at $y = 0$.  This implies that for Case II$_b$, small
oscillations of the wave packet around the $x$-minima have almost
exactly the same character as similar oscillations in Case I, so for
initial conditions that imply these oscillations, the microsuperspace
does give a reasonable approximation to the motion of the full
minisuperspace.  However, wave packets that begin near $x = 0$, or
those which oscillate around the $\mid x \mid = (\mid \mu \mid /2)
\sqrt {L/\lambda}$ minima whose
amplitudes of oscillation around these
minima are large enough for the wave packet to pass
the $\pm (1/\sqrt{3} ) [1- (2n\pi/\mid \mu \mid L)^2 ]^{1/2}
(\mid \mu \mid /2)
\sqrt{L/\lambda}$ points will behave differently.  As an example we
will consider a wave packet that begins near $x = 0$ and is as close
to $y = 0$, $p_y = 0$ as possible.  For small $x$ and $y$ the
potential is close to an inverted harmonic oscillator, and is
approximately $-{{1}\over {2}} \mu^2 x^2 - {{1}\over {2}} m^2 y^2$.
In this case there exists a wave packet that stays centered around $y
= 0$ while moving toward positive or negative $x$.  Such a wave
packet is discussed in Appendix B.  Here we
encounter the problem discussed above.  If we assume that the
existence of a packet centered around $y = 0$ that remains centered
around $y = 0$ during the evolution of the packet in $x$ implies
that the quantum microsuperspace $y = 0$ describes approximately the
evolution of the minisuperspace packet, then we must say that this
case allows a good microsuperspace.  If, however, we insist that the
microsuperspace must be stable against small changes in initial
conditions, it is obvious that giving the packet a small initial
momentum in the $y$-direction, will cause it to depart wildly from
the microsuperspace, and the $y = 0$ microsuperspace does not give a
good approximate description of the system.

For Case III($\mu^2 > 0, \lambda < 0$),
 $\mu^2 > 0$ implies that $m^2 > 0$.  The $y = 0$
potential has the form of a depression around $x = 0$ that rises to
maxima at $x = \pm (\mu /2) \sqrt{L/\mid \lambda \mid}$, and then
falls off rapidly, allowing quantum tunneling from the depression
into regions of larger $\mid x \mid$.  The potential is shown in
Figs. (6a,b).
Since $m^2 > 0$, for small $x$
the potential rises in the $y$-direction which means that in a small
range of $x$ near $x = 0$ wave packets are confined in $y$, and the
microsuperspace gives an adequate description of the minisuperspace
behavior.  However, as $x$ reaches the values $\pm (1/\sqrt{3}) [ 1 +
(2n\pi /\mu L)^2 ]^{1/2} (\mu / 2) \sqrt {L/ \mid \lambda \mid}$, the
potential goes from having a minimum at $y = 0$ to having a maximum
there.  Notice that this value of $x$ can be smaller or larger than
the maxima of the microsuperspace potential, $x = \pm (\mu /2)
\sqrt{L/\mid \lambda \mid}$(the two possibilities are shown in Figs.
[6a] and [6b], respectively).
If the value lies inside the maxima,
then a wave packet that oscillates around $x = 0$
can reach a region
where the $y = 0$ line is classically unstable if the amplitude of
its oscillation is large enough.  If the value of $x$ where the $y =
0$ line becomes a maximum is outside the maxima of the
microsuperspace potential, then if the amplitude of oscillation of
the wave packet is large enough, it will escape from the depression
and will begin to roll down the slope toward regions of $\mid x \mid
> (1/\sqrt{3}) [ 1 + (2n \pi /\mu L)^2 ]^{1/2} (\mu /2) \sqrt{L /\mid
\lambda \mid}$ into classically unstable regions of the $y = 0$ line.
Notice that in all cases there exists the possibility of quantum
tunneling from the depression out into regions of classical
instability or areas of $x$ where it is possible to roll down into
such regions.  The possibility of tunneling means that if classical
stability is a criterion for the microsuperspace to give an adequate
description of the quantum dynamics of the full system, Case III can
never be considered a good microsuperspace.  The only exception is if
we call the microsuperspace ``good'' if it gives a reasonable
description of the minisuperspace system for a time, and then breaks
down.  In this case the tunneling time could be long, and the wave
packet could stay in the depression for a long enough time to be
useful for the predictions one wants to make.

As in Case II, Case IV ($\mu^2 < 0$, $\lambda < 0$)
gives rise to two sub-cases in the
minisuperspace, Case IV$_a$ where $\mu^2 < 0$, $\lambda < 0$, $m^2
>0$, and Case IV$_b$ where $\mu^2 < 0$, $\lambda < 0$, $m^2 < 0$.  In
both sub-cases, which are shown in Figs. (7a,b), near $x = 0$
the microsuperspace potential can be modeled by that of an
upside-down harmonic oscillator, and the wave packet given by Eq.
(B.12)
will in general roll off $x = 0$ in the direction of larger $\mid x
\mid$.  In the minisuperspace in Case IV$_a$ near $x = 0$ the $y = 0$
line is a minimum, so a $y = 0$ trajectory is classically stable.
However, when $x = \pm (1\sqrt{3}) [(2n\pi/
\mid \mu \mid L)^2 - 1
]^{1/2} (\mid \mu \mid /2 ) \sqrt {L/\mid \lambda \mid}$
the $y = 0$ line
becomes a maximum in $y$, and there is nothing to prevent the wave
packet from moving to these values of $x$, so it will always reach
regions of classical instability.  In Case IV$_b$ near $x = y = 0$
we can always construct a wave packet along the lines of (B.12) that will
stay centered around $y = 0$ while moving along that line.  However,
every point on the $y = 0$ line is classically unstable, so, again, if
classical instability is a criterion for the usefulness of the
microsuperspace, then in Case IV$_b$ it must be assumed that the
microsuperspace description does not give a reasonable approximation
to the full minisuperspace behavior.

In the next section we will try to summarize the status of
the problem of when the
$y = 0$ microsuperspace provides a useful description of the full
minisuperspace behavior and compare the criteria we have developed to
possible gravitational scenarios.
\setcounter{equation}{0}
\section{ Conclusion and Discussion}
We can summarize our conclusions from Section 4 as follows.
The full potential term $({{\mu}^2\over 2})x^2 + \left({\lambda\over
L}\right)x^4 + {m^2\over 2} y^2 + 6 \left({\lambda\over L }\right)
x^2 y^2$ determines whether the microsuperspace gives a good
approximation to the behavior of a minisuperspace wave packet
centered around the $y=0$ sector. In Ref.\cite{Kuchar2}
a gravitational example
of a case where the minisuperspace wave packet had behavior that
diverged wildly from the microsuperspace behavior was given. In all
of the cases of Section 4 a state that remained peaked around
the $y=0$ minisuperspace during its whole evolution in the
$x$-direction existed. The motion of the peak of the minisuperspace
packet did show a minor deviation from the motion of the
microsuperspace packet, so one would argue that none of the
microsuperspaces were truly ``good"
in the sense of faithfully representing the detailed
quantitative behavior of the minisuperspace. However, the microsuperspace
and minisuperspace behaviors were qualitatively similar so that
it is probably best to say that all of the microsuperspaces gave
a reasonable approximation to the true minisuperspace behavior.

The one major difference was in the stability of the packets against
small changes of the $y$-position and $y$-velocity, which may be
taken to be a criterion for the ``badness" of the microsuperspace. If
one does so, then in cases II$_b$ , III, and IV the
microsuperspace cannot be taken to represent the behavior of the
full minisuperspace.

As we stated in the Introduction, here we have considered an
approach different from Ref.\cite{Kuchar}  (henceforth referred to as KR1)
to define a good microsuperspace. However, since we have used
exactly the same model, it is worth making a few comments comparing
the two approaches. In KR1, the idea was so start with the full
Schr\"odinger equation for superspace (minisuperspace in our
language) , expand the wave function in eigenstates of the
minisuperspace Hamiltonian ( the part of the full Hamiltonian that
depends on $y$ and $p_y$) that parametrically depend on $x$, and
find  the conditions under which the full Schr\"odinger equation
could be reduced to a ``projected Schr\"odinger equation " for the
set of ``microsuperspace wave functions" ( the set of
$x, t$ dependent coefficients in the above stated mode expansion)
evolved by the pure microsuperspace  Hamiltonian ($y=0, p_y
= 0$). Expectation values of the microsuperspace operators could
then be obtained  from the density matrix  constructed from the
wave functions.
The criteria that emerge are roughly akin to the Born-Oppenheimer
approximation, i.e, i)the parametrized eigenstates should vary
slowly with $x$ and ii)the $x$ dependent energy eigenvalues should
be small compared with the microsuperspace potential.
There was no attempt to consider different signs of $\lambda$
and $\mu$, so the KR1 model is essentially equivalent to only
Case I.

In our analysis, rather than trying to reduce the Schr\"odinger
equation on the larger space , we compare the two behaviors
on the level of solutions. Our minisuperspace wave function is
a coherent-state-like solution of the full Schr\"odinger equation
while the microsuperspace wave function is a similar solution
to the microsuperspace Schr\"odinger equation $(p_y = 0, y = 0)$,
which has {\it no} information on the $y$ degree of freedom.
So we see that the approaches differ slightly on what is meant
by the microsuperspace wave function. The differences also lie
in that ours is formulated in terms of ``coherent" states for
the full solution, while KR1 relies on a decomposition in terms of
``energy" eigenstates and the criteria are stated in terms of
excitation levels. However, the approaches can be related by
writing the wave function (3.16) in terms of the eigenstates of KR1,
and testing the slow variation criterion, for example. Though the two notions
of the microsuperspace wave function do not quite coincide, we believe
that criterion ii) can be closely related to the criterion that the
difference in the trajectories of the centers of the
$y=0$ minisuperspace wavepacket
and the microsuperspace wavepacket should be small. Our classical
stability
criterion does not seem to have an obvious parallel in KR1.
Overall, it is probably fair to say that while our approach may
involve some loss of generality as opposed to KR1 since it relies
on a specific type of solution, it has the advantage that it
allows one to follow the evolution in greater detail and formulate
more concrete criteria.

The model of Ref.\cite{Sinhu} differed from the
one considered here in that it considered an infinite-dimensional
minisuperspace that was coupled to a curved rather than flat space
and the treatment was as in KR1 based on the analysis of equations
of motion rather than solutions. Apart from the features specifically
tied to curved spacetime, the requirement there of the microsuperspace
potential dominating the backreaction term is loosely analogous to
the criterion of the centers of the two wave packets remaining close.
However, an extension of the treatment of this paper to include
all the infinite number of modes as in \cite{Sinhu} would be a worthwhile
exercise. Of all of the previous attempts at finding criteria that
tell us whether a particular quantum minisuperspace is a good
approximation to a real solution, the present article is most
closely related to \cite{Kuchar2}. There, as here, the motion of a
microsuperspace wave packet was compared directly to that of a
minisuperspace wave packet. There, however, as we have just
mentioned, the behavior of the two packets was extremely divergent,
while here they are qualitatively similar.

Before going on to the case of gravity, we would like to mention
another possible approach that can be tested on quantum mechanical
and field theoretical models. It is clear that if we only
want to know expectation value of operators that are constructed
solely from minisuperspace variables, given the full wave function
, we only require the knowledge of the reduced density matrix
(${\rho}_{red} = Tr_{y} |\Psi><\Psi| )$ to calculate
these objects. A similar idea was discussed in KR1, though it
required more specific assumptions about the expansion of
the wave function in energy eigenstates. The evolution of the
reduced density matrix would be guided by a master equation rather
than a Hamiltonian evolution through the Schr\"odinger equation.
The master equation will contain diffusion and dissipation
terms arising
from the averaged effect of the minisuperspace modes,
 in addition to the pure microsuperspace Liouville operator.
Demanding the smallness of these extra terms will then lead to
criteria for the goodness of the microsuperspace. This approach can
be thought of as elevating the approach of \cite{Sinhu} to the quantum
statistical mechanics level.

The next important question is of course to understand how one
can extend the analysis of this paper to possible gravitational
scenarios. As mentioned in the Introduction, the dynamics in
this case will be dictated by a Wheeler-DeWitt equation (1.1)
rather than a Schr\"odinger equation and we will immediately
be burdened with the host of problems that have posed a barrier
to the construction of a successful theory of quantum gravity
until now. However, without trying to confront the full theory
in all its complexity, one can conceive of  gravitational
minisuperspace models
(not necessarily finite-dimensional) which contain microsuperspaces
embedded in them that are exactly or approximately solvable, as
for example in \cite{Kuchar2}, and apply a similar analysis to them . Two
appropriate candidates appear to be the Gowdy model \cite{Gowdy}
, which has a Bianchi- I microsuperspace embedded
in it, and Halliwell and Hawking's model\cite{H-Hawk} of a closed Robertson
Walk
   er
universe with gravitational perturbations.

In such a picture, as stated before, $gR$  is the appropriate
analog of our potential function which is a determining factor
in the goodness of the microsuperspace. Provided one can define an
appropriate ``time" in such a model, it is clear that while
parameters analogous to $\lambda$ and $\mu$ are fixed by the
given gravitational Hamiltonian (as opposed to being free
parameters that we can vary at will as in this paper), they are
necessarily time dependent. The major and nontrivial task would
then be to understand how to construct criteria analogous to ours for
time dependent coefficients.
Another point that perhaps is worth mentioning is that though
$gR$ is possibly the major factor in deciding on the criteria, the
superspace metric $G_{ijkl}$ will also surely play a role. It
so happens that all the gravitational models that have been analyzed
in this context \cite{Kuchar2} possess a flat superspace metric. It is
therefore important to analyze models that have a curved superspace
metric to study this particular feature.

An interesting point is that it may be possible to formulate
stability criteria for microsuperspace trajectories
in rather general terms without referring to a specific model.
A solution to the Einstein equations is represented by a
driven geodesic in superspace which obeys the equation (for $g_{00} =
-1$, $g_{0i} = 0$)
\begin{equation}
{{d^2 g^{(ij)}}\over {ds^2}} + \Gamma^{(ij)}_{(k\ell )(mn)}
{{dg^{(k\ell )}}\over {ds}} {{dg^{(mn)}}\over {ds}} = {{\delta (\sqrt{g}
R )}\over {\delta g_{(ij)}}},
\end{equation}
where the $\Gamma^{(ij)}_{(k\ell )(mn)}$ are the superspace
Christoffel symbols constructed from $G_{ijk\ell}$,
and $\sqrt{g}R$ is defined above.
Classical stability
of a trajectory in superspace can be studied by considering the
appropriate superspace ``equation of geodesic deviation" for the
trajectory. This type of equation can be used for at least
two purposes. It is possible to find regions of ``classical
instability " of the type found in cases II, III and IV.
It will also serve, by finding examples of classical ``magnetic
mirror" behavior such as those that exist in type IX models, as
an indicator of the existence of quantum ``magnetic mirror"
solutions such as those found in Ref.\cite{Kuchar2}

\vspace{2 cm}
\noindent{\bf Acknowledgement} \\
The research reported here was supported in part by an NSF-CONACYT
grant between the Instituto de Ciencias Nucleares, UNAM and the
University of Utah (P-89).

\renewcommand{\thesection}{A}
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}

\section{Appendix A}
Let us first treat the microsuperspace sector.
Substituting the ansatz (3.14) in equations (3.12) and (3.13)
and equating equal
powers of $x$, we obtain the following coupled equations for the
unknown coefficients. For the imaginary part we get
\begin{eqnarray}
\dot{B} &=& g - 3\mu \beta ,\\
\dot{C} &=& 3 \mu g \beta - 2\mu \gamma - 3 B\dot{g} + {3\over
2\mu} ,\\
\dot{D} &=& 2\mu g \gamma - \sigma \mu - 2\dot{g}C + 3\beta ,\\
\dot{E} &=& \mu g \sigma - D\dot{g} + \gamma ,
\end{eqnarray}
and for the real part,
\begin{eqnarray}
-\dot{\beta} &=& -3 \mu B + {\dot{g}\over \mu} ,\\
-\dot{\gamma} &=& 3 \mu g B - 2\mu C + 3\dot{g}\beta ,\\
-\dot{\sigma} &=& 2 \mu g C - \mu D + 2\dot{g}\gamma +3B ,\\
-\dot{\omega} &=& \mu g D +\dot{g}\sigma + C ,
\end{eqnarray}
and $\alpha = {1\over 4\mu}$.
These equations can be decoupled in the following manner. From (A.5)
one can solve for $B$ in terms of $g$ and $\beta$ and substitute it
into (A.1) to obtain the following differential equation for $\beta$.
\begin{equation}
\ddot{\beta} + 9{\mu}^2 \beta = 4\mu x_0 \cos{\mu t}.
\end{equation}
which is simply an equation for a harmonic oscillator driven by a
known oscillating source, and can be easily solved.
The solution can be written down as
\begin{equation}
\beta = {x_1}^3\cos{3\mu t} + {x_0\over 2\mu}\cos{\mu t} ,
\end{equation}
where $x_1$ is an arbitrary constant. We follow the same procedure
for the rest of the functions.

\noindent Solving for $C$ from (A.6) , substituting in (A.2) and using
the now known solution for $\beta$ we get the following equation for
$\gamma$
\begin{equation}
\ddot{\gamma} + 4 {\mu}^2 \gamma = 18{\mu}^2 x_0 {x_1}^3\cos{4\mu t}
+ 6 \mu {x_0}^2\cos{2\mu t} + 3 .
\end{equation}
To obtain the equation for $\sigma$ one solves for $D$ from (A.7) and
substitutes this in eqn. (A.3). Using the known solutions for $\beta$
and $\gamma$ one obtains the following equation
\begin{eqnarray}
\ddot{\sigma} + {\mu}^2\sigma &=& 9 \mu {x_1}^3 \cos{3\mu t}
-18{\mu}^2{x_0}^2{x_1}^3\cos{5 \mu t} \nonumber \\ &+& 8 {\mu}^2x_0 C_1 \cos{3\mu
t}
+ 12 {\mu}^2 {x_0}^3 t \sin{3 \mu t} + 6 x_0\cos{\mu t} .
\end{eqnarray}
As before, both (A.11 ) and (A.12) are equations for driven harmonic
oscillators and their solutions , which are explicitly written down
in eqn. (3.15) can be obtained by standard methods. The solution
for $\omega$ is trivial, since it can be reduced to quadratures, and
it is easy to see that the right hand side of (A.8) can be written
as a total derivative leading to the solution for $\omega$ given in
(3.15). One can obtain $B, C, D, E$  easily as well, writing them in
terms of the coefficients of the real parts. However, as stated
before in the text, we will not calculate them since they are
unimportant for our considerations.

Now let us treat the minisuperspace sector. Substituting (3.20) and
(3.21) in eqns. (3.18 ) and (3.19) and
equating equal powers of $x$ and $y$, we
obtain the following set of equations for the coefficients of the
imaginary part
\begin{eqnarray}
\dot{\tilde{B}} &=& - 3\tilde{\beta}\mu + 4{\tilde{\alpha}} \mu g , \\
\dot{\tilde{C}} &=& -2\tilde{\gamma}\mu + 3\tilde{\beta}\mu g - 3
\tilde{B}\dot{g}
+ 6\tilde{\alpha} + \nu m f - \dot{ f} F + \delta ,\\
\dot{\tilde{D}} &=& -\tilde{\sigma} \mu + 2\mu\tilde{\gamma} g
- 2\dot{g}\tilde{C}
+ 3{\tilde{\beta}} + m f \theta - M\dot{f} + \epsilon ,\\
\dot{\tilde{E}} &=&
 \sigma \mu g - \dot{g}\tilde{D} + \gamma + mf\xi
-N\dot{f} + \kappa ,\\
\dot{F} &=& -(2 \mu + m )\nu + 2mf\delta , \\
\dot{G} &=&  -(\mu + 2m)\epsilon + 2\mu g \delta , \\
\dot{L} &=& \mu g \epsilon - \dot{g}G + \delta - 2m\kappa ,\\
\dot{M} &=& -  (\mu + m)\theta + 2 \mu g \nu - 2F\dot{g}
+2\epsilon m f - 2\dot{f}G , \\
\dot{N} &=& \theta \mu g - \dot{g}M + \nu - m \xi + 2mf\kappa -
2\dot{f}L .
\end{eqnarray}

For the real part we get
\begin{eqnarray}
-\dot{\tilde{\beta}} &=& -3\tilde{B}\mu + 4\dot{g}\tilde{\alpha} ,\\
-\dot{\tilde{\gamma}} &=& -2\mu \tilde{C} + 3\tilde{B}\mu g +
3\dot{g} \tilde{\beta} + mfF + \dot{f}\nu ,\\
-\dot{\tilde{\sigma}} &=& -\tilde{D}\mu + 2\mu g \tilde{C} +
2\tilde{\gamma}\dot{g} +3\tilde{B} + mfM + \theta\dot{f} + G ,\\
- \dot{\tilde{\omega}} &=& D\mu g + \dot{g}\tilde{\sigma} + \tilde{C}
+\dot{f}\xi + L + mfN ,\\
-\dot{\epsilon} &=& -(\mu + 2m)G + 2\delta\dot{g} ,\\
-\dot{\nu}      &=& -(2\mu + m)F + 2\delta\dot{f} , \\
-\dot{\theta} &=& -M\mu + 2\mu g F + 2\dot{g}\nu - Mm + 2mfG +
2\dot{f}\epsilon ,\\
-\dot{\kappa} &=& \mu gG +\dot{g}\epsilon - 2Lm ,\\
-\dot{\xi}     &=& \mu g M + \dot{g}\theta + F - Nm + 2mfL +
2\kappa\dot{f},
\end{eqnarray}
where $\tilde{\alpha} = \alpha ={1\over 4\mu}$ and $\delta = {3\over
\mu + m}$. We will use the same technique as in the microsuperspace
case to decouple these equations. As before, we solve for an
imaginary-part coefficient in terms of the real-part coefficient and
substitute in the corresponding equation for the time derivative of
the real-part coefficient to obtain a differential equation for it.
We see that the equations for $\tilde{\beta}$ and $\tilde{B}$ are
identical to their microsuperspace counterparts, so that we
conclude $\tilde{\beta} = \beta$. From inspection of the equations
(A.13)
-- (A.30), we see that the pairs that should be used in the above
solve and substitute procedure are the following : $[F,\nu] , [C, \gamma],
[G,\epsilon], [M, \theta], [D,\sigma], [L, \kappa]$ and $[N ,\xi]$.
The process of solving  also proceeds in the above order, so
the solution of a preceding variable can be used to solve for
the subsequent one as before.

\noindent Using the above procedure, the equation satisfied by $\nu$
is given by
\begin{equation}
\ddot{\nu} + {(2\mu + m)}^2 \nu = 12 m y_0 \cos{m t}.
\end{equation}
In all of what follows, we will write down the solution to the forced
harmonic oscillator equations as only the part coming from the
inhomogeneous term, using the freedom in the arbitrary coefficients
to set the homogeneous solution to zero. This is done only for
simplification and it is obvious from the procedure how to
incorporate the homogeneous solution. Bearing this in mind, the
solution to (A.31) can be written as
\begin{equation}
\nu = {3my_0\over \mu(\mu +m)}\cos{m t}.
\end{equation}
Next, the equation for ${\tilde{\gamma}}$ is given by
\begin{equation}
\ddot{\tilde{\gamma}} = 18{\mu}^2 x_0 {x_1}^3\cos{4 \mu t}
+ 6\mu {x_0}^2\cos{2 \mu t} + 6{m^2 {y_0}^2\over \mu}\cos{2 mt}
+ {6\mu\over m +\mu} +3  ,
\end{equation}
and the solution is
\begin{equation}
\tilde{\gamma} = {3\over 4 {\mu}^2}\left( {3\mu +m\over \mu +
m}\right) -{3\over 2}{x_0}^2{x_1}^3\cos{4\mu t} +{3\over 2} {x_0}^2
t\sin{2\mu t} + {3\over 2}{m^2 {y_0}^2\over \mu({\mu}^2 - m^2)}\cos{2
mt}  .
\end{equation}
The equation for $\epsilon$ is given by
\begin{equation}
\ddot{\epsilon} + {(\mu + 2m)}^2\epsilon = 12\mu x_0 \cos{\mu t} ,
\end{equation}
and the solution is
\begin{equation}
\epsilon = {3\mu x_0\over m(m +\mu)}\cos{\mu t} .
\end{equation}
The equation for $\theta$ is given by
\begin{equation}
\ddot{\theta} + {(\mu + m)}^2\theta = 12(m + \mu)x_0 y_0\cos{(\mu +
m)t} ,
\end{equation}
and the solution is
\begin{equation}
\theta = 6x_0 y_0 t \sin{(\mu + m)t}.
\end{equation}
The equation for $\kappa$ is given by
\begin{equation}
\ddot{\kappa} + 4m^2 \kappa = {6m\over \mu + m} +
{6{\mu}^2{x_0}^2\over m} \cos{2\mu t} ,
\end{equation}
and the solution is
\begin{equation}
\kappa = {3\over 2m(\mu + m)} - {3{\mu}^2{x_0}^2\over 2m ({\mu}^2 -
m^2)} \cos{2\mu t} .
\end{equation}
The equation for $\tilde{\sigma}$ is given by
\begin{eqnarray}
\ddot{\tilde{\sigma}} + {\mu}^2 \tilde{\sigma} &=&
6x_0 \left (1 + {\mu\over m} \right) +
3\mu (4{x_1}^3 - {x_0}^3)\cos {3\mu t}
 + 12 {\mu}^2{x_0}^3 t \sin{3
\mu t} \nonumber \\
& - & 18{\mu}^2 {x_0}^3{x_1}^3\cos{5 \mu t}
+ {6 {x_0}{y_0}^2 m (2m - \mu)\over \mu - m}\cos (\mu + 2m)t\nonumber \\
&+& 12{x_0}{y_0}^2 m (\mu + m) t \sin (\mu + 2m)t ,
\end{eqnarray}
and the solution is
\begin{eqnarray}
\tilde{\sigma} &=& -{3\over 2\mu}\left({x_1}^3 + {1\over
2}{x_0}^3\right)\cos{3 \mu t} + {3\over 4}{x_0}^2{x_1}^3\cos{5 \mu t}
-{3{x_0}{y_0}^2\mu\over 2m({\mu}^2 -m^2)}\cos(\mu + 2m)t \nonumber \\
&+& 3x_0 {(m + \mu)\over m\mu}t\sin{\mu t} - {3\over
2}{x_0}^3t\sin{3\mu t} - 3{x_0}{y_0}^2 t \sin (\mu + 2m)t .
\end{eqnarray}
and finally, the equation for $\xi$ is given by
\begin{eqnarray}
\ddot{\xi} + m^2 \xi &=& 6 {y_0}\left[{m(m + 3\mu)\over \mu(\mu +
m)}\right]\cos{mt}
- {6\mu {x_0}^2{y_0}\over (\mu - m)}(2\mu - m)\cos(2\mu +m)t \nonumber \\
&+& 12\mu{x_0}^2{y_0}(\mu + m) t \sin (m + 2 \mu)t ,
\end{eqnarray}
and the solution  is
\begin{equation}
\xi = {3{x_0}^2{y_0}m^2\over 2\mu({\mu}^2 - m^2)}\cos(2\mu + m)t
+ {3y_0 (m + 3\mu)\over \mu (m + \mu)}t\sin {m t}
-3{x_0}^2y_0 t \sin(m + 2\mu)t .
\end{equation}
The solutions that appear in (3.23) are a special case of these with
$y_0 = 0$.  Note that a small $y_0$ implies (from the form of ${\cal
S}_0$) that the peak of the wave packet will oscillate with small amplitude
around the $y = 0$ line.
It can be shown that $\tilde{\omega}$ again can be obtained
from a total derivative, but we do not demonstrate this explicitly
since this acts merely as a normalization factor.

\renewcommand{\thesection}{B}
\renewcommand{\theequation}{B.\arabic{equation}}
\setcounter{equation}{0}

\section{Appendix B}
Here we will demonstrate the quantum solution to the zeroth order
microsuperspace solution for ${\mu}^2 < 0 $, which is essentially
equivalent to a one dimensional upside down harmonic oscillator.
The equations to be solved are again (3.5) and (3.6) with ${\mu}^2$
replaced by $- {|\mu|}^2$. We make the following ansatz for
${\cal S}$
\begin{equation}
{\cal S} = \bar{\alpha}x^2 + \bar{\beta}x + \bar{\gamma}
+ i\bar{A}x^2 + i \bar{B} x  + i \bar{C} .
\end{equation}
Plugging in the above ansatz into the modified equations (3.5) and
(3.6) we obtain the following equations for the unknown coefficients
\begin{eqnarray}
\dot{\bar{\alpha}} &=& 4{\bar{\alpha}}\bar{A} ,\\
\dot{\bar{\beta}}  &=& 2 (\bar{\alpha}\bar{B} + \bar{A}\bar{\beta}) ,
\\
\dot{\bar{\gamma}} &=& \bar{\beta}\bar{B} - \bar{A} ,\\
\dot{\bar{A}} &=& {1\over 2}( 4{\bar{A}}^2 - 4{\bar{\alpha}}^2 -
{|\mu|}^2) ,\\
\dot{\bar{B}} &=& 2(\bar{A}\bar{B} - \bar{\alpha} \bar{\beta} ) ,\\
\dot{\bar{C}} &=& {1\over 2}( {\bar{B}}^2 - {\bar{\beta}}^2 +
2\bar{\alpha}) .
\end{eqnarray}
{}From the above equations we obtain the following equation for
$\bar{A}$
\begin{equation}
\ddot{\bar{A}} - 12\bar{A}{\dot{\bar{A}}} + 16 {\bar{A}}^3 -
4{|\mu|}^2 \bar{A} = 0 ,
\end{equation}
and thus from eqn. (B.2), $\bar{\alpha}$ can be written down as
\begin{equation}
\bar{\alpha} = C_0 \exp\left[{4\int \bar{A} dt}\right]
\end{equation}
where $C_0$ is an arbitrary constant.
The equation for $\bar{B}$ is given by
\begin{equation}
\ddot{\bar{B}} - 4\bar{A}\dot{\bar{B}} + (6{\bar{A}}^2 +
2{\bar{\alpha}}^2  - 2 \dot{\bar{A}})\bar{B} = 0 ,
\end{equation}
where the known solutions for $\bar{A}$ and $\bar{\alpha}$ are
supposed to be inserted where these quantities appear.
Then $\bar{\beta}$ can be determined from
\begin{equation}
\bar{\beta} = {2\bar{A}\bar{B} - \dot{\bar{B}}\over 2\bar{\alpha}}
\end{equation}
using the known solutions of $\bar{A}, \bar{\alpha}$ and $\bar{B}$.
Subsequently, from eqns.(B.4) and (B.7) we see that $\bar{\gamma}$ and
$\bar{C}$ are reduced to quadratures. The normalized wavefunction
can be written as:
\begin{equation}
\Psi(x,t) = {\left({2{\bar{\alpha}}\over \pi}\right)}^{1\over 4}
e^{-{\bar{\gamma}\over 2} -{{{\bar{\beta}}^2\over 4\bar{\alpha}}
-i\bar{C}}}
e^{- (\bar{\alpha} + i\bar{A})x^2 - (\bar{\beta} + i \bar{B})x }.
\end{equation}

Though eqn.(B.12) formally gives a complete solution to the problem,
in practice it is quite difficult to find an exact solution since
eqns. (B.8) and (B.10) are highly nonlinear.
However, a special exact solution
was given by Guth and Pi in Ref.\cite{GuPi}. In our notation this solution
corresponds to
$\bar{B} = \bar{\beta} = 0$ and
\begin{eqnarray}
\bar{\alpha} &=& {|\mu|\sin{2\phi}\over 2\left(\cos{2\phi} +
\cosh(2|\mu|t) \right)} \\
\bar{A} &=& {-|\mu|\sinh(2|\mu|t)\over  2\left(\cos{2\phi} +
\cosh(2|\mu|t) \right)} ,
\end{eqnarray}
where $\phi$ is a real constant of integration related to the width
of the wave packet at $t=0$ such that the wave packet is at its
minimum width at $t=0$. $\bar{C}$ and $\bar{\gamma}$ can be
obtained from the complex identity:
\begin{equation}
{\left[b\cos(\phi - i|\mu|t)\right]}^{-{1\over 2}}
= 2{\bar{\alpha}}^{-{1\over 4}} e^{-{\bar{\gamma}\over 2} - i\bar{C}}
,
\end{equation}
where $b = {\left(|\mu|\sin{2\phi}\right)}^{-{1\over 2}}$
This corresponds to a wave packet whose center remains at the top
of the potential hill at $x=0$ for all times while the spread
grows with time.

\noindent From the wave function (B.12) it can be seen that the center
of our wave packet lies at $x = {\bar{\beta}\over 2\bar{\alpha}} =
<x>$.
We can solve for the motion of the center of
the wave packet quite easily without having to
find the general solution for the wave function.
Using eqns. (B.2) -- (B.7), one can show that
${\bar{\beta}\over 2\bar{\alpha}}$ satisfies the following
equation
\begin{equation}
{\left(   {\bar{\beta}\over 2\bar{\alpha}}\right)}^{\cdot \cdot}
- {|\mu|}^2 {\left(   {\bar{\beta}\over 2\bar{\alpha}}\right)}
= 0 ,
\end{equation}
which has the simple solution
\begin{equation}
{\bar{\beta}\over 2\bar{\alpha}} = x_0 \cosh{|\mu|t}  + {p_0\over
|\mu|}\sinh{|\mu|t}.
\end{equation}
where $x_0$ , $p_0$ are  arbitrary constants which can be interpreted as
the initial position and momentum of the center of the wave packet
respectively.
Eqn. (B.16)
is also merely a statement of Ehrenfest's theorem for this problem.
{}From (B.17) it is clear that (B.13) and (B.14)
gives a special solution with $x_0 =
0$ and $p_0 = 0$,
and from (B.17) it is also evident that for any other initial conditions
the wave packet will rapidly roll down the potential hill.

The above exact solution can also be used to obtain a minisuperspace
solution , i.e, in the two dimensional ($x$- $y$) problem. In the
approximation that we consider a region around the origin such that
$x$ and $y$ are small, for $m^2 <0$,
the full Hamiltonian can be approximated by
one for two decoupled upside down harmonic oscillators since the
coupling terms are of higher than quadratic order. Going through the
the same steps as the above one-dimensional case, one then has a
minisuperspace solution that is a two-dimensional wave packet whose
center remains at $x = y =0$ for all times while its spread grows in
time. Thus the solution will be a product of two Gaussian wave
packets, one in $x$ and the other in $y$. The center of the wave
packet in $y$ will obey an equation identical to (B.16)  with $|\mu|$
replaced by $|m|$.

\newpage

redpajama_set_name: RedPajamaArXiv


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The Marriott Conference Center at the National Center for Employee Development located at 2801 East State Highway 9, Norman, OK, 73071 is currently hiring Cook II.
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chilli sauce. Well, let's take a close look, shall we?
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text: Q: Qt - globally enable touch scrolling I've been using a wacom tablet lately on windows 10 and I do like how Windows natively supports gestural scrolling when you're using a tablet. In trying to implement this in my PyQt apps I was able to get kinetic scrolling working quite nicely with tablet events and a QScroller. However you need to do this for every (scrolling) widget you want to have support for it... So my question is: can this be somehow enabled globally? Perhaps this is a dumb question, but I think it would make sense to have this work for all widgets and scroll bars without reimplementing all over the place and extensive refactoring. Is there a clean way to replace all scroll areas or scroll bars throughout the app?

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text:
Name | Type | Description | Notes
------------ | ------------- | ------------- | -------------
**statement_text** | **string** | The text that will appear on your employee's bank statement when they receive payment | [optional]
**account_name** | **string** | The name of the account | [optional]
**bsb** | **string** | The BSB number of the account | [optional]
**account_number** | **string** | The account number | [optional]
**remainder** | **bool** | If this account is the Remaining bank account | [optional]
**amount** | **double** | Fixed amounts (for example, if an employee wants to have $100 of their salary transferred to one account, and the remaining amount to another) | [optional]

[[Back to Model list]](../README.md#documentation-for-models) [[Back to API list]](../README.md#documentation-for-api-endpoints) [[Back to README]](../README.md)

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text:
Fabrice Begeorgi (* 20. April 1987 in Martigues) ist ein französischer Fußballspieler. Er spielte lange Zeit hauptsächlich auf der Position des linken Außenstürmers, kam jedoch in seiner Zeit bei Werder Bremen meist als linker Außenverteidiger zum Einsatz.

Karriere
Begeorgi begann erst bei seinem Heimatverein FC Martigues mit dem Fußballspielen, bevor er in der Jugend nach Marseille zu Olympique wechselte. Dort kam er dann bereits in der Saison 2005/06 mit 19 Jahren zweimal im UEFA-Pokal zum Einsatz, nachdem die Stammstürmer der ersten Mannschaft ausgefallen waren. Im Jahr darauf wurde er von den Südfranzosen ausgeliehen, um Spielerfahrung zu sammeln, und er ging zum FC Libourne-Saint-Seurin in der 2. Liga. Dort kam er auch regelmäßig zum Einsatz und spielte eine erfolgreiche Saison, woraufhin ihm Marseille einen Profivertrag anbot.

Trotz Vertrag spielte er erst ein weiteres Jahr auf Leihbasis in Liga 2, in dem er aber wegen eines Kreuzbandrisses nur auf wenige Einsätze kam. 2008 schickte man ihn deshalb erneut zu einem anderen Verein, diesmal ins Ausland nach Deutschland zum Zweitligisten TuS Koblenz; der Vertrag wurde jedoch Anfang 2009 aufgelöst. Daraufhin wurde er an Werder Bremen verliehen, wo er hauptsächlich für die zweite Mannschaft vorgesehen war, jedoch auch dem Kader der ersten Mannschaft angehörte. Zu einem Einsatz in der Bundesliga kam er allerdings nicht.

Zur Saison 2009/10 wurde Begeorgis Vertrag in Marseille aufgelöst, er wechselte in die Ligue 2 zum FC Istres. Seit 2010 spielt er beim Ligakonkurrenten AC Ajaccio. Dort war er zunächst Stammspieler und feierte 2011 mit der Mannschaft den Aufstieg in die Ligue 1, kam danach aber nur noch sporadisch zum Einsatz, auch weil er während der Saison 2011/12 bedingt durch einen Kreuzbandriss für eine längere Zeit ausfiel. Anschließend schaffte er nicht den Weg zurück in die erste Elf und entschied sich im September 2013 für einen Wechsel auf Leihbasis zum Drittligisten ES Uzès Pont du Gard. Er kehrte im Sommer 2014 nach Ajaccio zurück, das gerade aus der Ligue 1 abgestiegen war. Die Saison 2014/15 endete lediglich mit dem Klassenverbleib. Anschließend war er ein halbes Jahr ohne Verein, ehe ihn der rumänische Erstligist Petrolul Ploiești verpflichtete. Dort konnte er mit seiner Mannschaft den Abstieg 2016 nicht vermeiden. Im Sommer 2016 wechselte er zum andorranischen Verein UE Engordany. Ein Jahr später schloss er sich dem französischen Viertligisten US Le Pontet an. Seit dem Sommer 2018 spielt er erneut in Andorra, dieses Mal bei UE Sant Julià.

Weblinks

 Einsatzstatistik Frankreich

Einzelnachweise

Fußballspieler (Olympique Marseille)
Fußballspieler (FC Libourne)
Fußballspieler (SC Amiens)
Fußballspieler (TuS Koblenz)
Fußballspieler (Werder Bremen)
Fußballspieler (FC Istres)
Fußballspieler (AC Ajaccio)
Fußballspieler (ES Uzès Pont du Gard)
Fußballspieler (Petrolul Ploiești)
Fußballspieler (UE Engordany)
Franzose
Geboren 1987
Mann

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text:
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This Privacy Policy sets out the basis on which any information and/or data of the Participant, is collected, used and disclosed by Elrond, hereafter referred to as the Company, when accessing or using the Website or the Services. The Company is committed to preserving the privacy of all visitors of the Website and/or Participants and users of the Services. The Policy explains how the Company complies with the General Data Protection Regulation ('GDPR').
By acquiring, holding or using the Services, the Participant is subject to the Terms of Service. Please take time to read the Terms and this Privacy Policy. By proceeding further, you have read and expressly consent to the collection, use, retention, processing, transfer and disclosure of Personal Data under the terms of this Privacy Policy.
Information and/or Personal Data which may be collected, stored, processed, transferred, used and disclosed by Elrond includes but is not limited to information and/or data that can be used on its own or with other information to, directly or indirectly, identify, contact, or locate a single person, or to identify an individual in context.
The following terms "Controller", "Processor", "Data Subject", "Personal Data", "Processing Activity/ies", "Pseudonymisation", "Cross-Border processing of Personal Data", "Supervisory Authority" used in this document shall have the same meaning as in the GDPR.
Other Capitalised terms used in this Privacy Policy shall have the same meaning given to them in the Terms of Service.
Basic Principles Regarding Personal Data Processing
Elrond, as Controller, shall adhere to the GDPR principles as stipulated in Article 5(2) of the GDPR where the "Controller shall be responsible for, and be able to demonstrate, compliance with the principles." The Personal Data must be processed fairly and lawfully; be obtained only for specific and lawful matters; be adequate and up to date; not to be held for any longer than necessary; and be protected in appropriate ways.
Elrond shall be accountable and must disclose information held to the Data Subject when requested and shall not transfer Personal Data outside of the European Economic Area (EEA), unless that country or territory also ensures an "adequate" level of protection. The Processing of Data must be carried out in accordance with the rights of the Data Subjects.
The Company May Collect the Following Information
Information provided by the Participant: when using the Services, subscribing to Elrond's updates, responding to a survey or when filling enquiry forms on the Website and when corresponding by phone, e-mail or other communication services, you may be asked to provide certain information. The information may include Personal Data such as name, address, e-mail address, phone number, financial and/or credit card information, cryptocurrency address, personal description, ID or Passport copy, date of birth, passport number, or other data which can be used for personal identification purposes and required for 'know your customer' that may be required to comply with applicable laws.
Information, collected automatically by Elrond: on each visit to the Website, the Company will automatically collect the following:
Technical information, including the Internet protocol (IP) address used to connect your computer to the Internet, the User login information, browser type and version, time zone setting, browser plug-in types and versions, operating system and platform; information about the visits, including the full Uniform Resource Locators (URL), clickstream to, through and from the Website (including date and time), products viewed or searched for, page response times, download errors, length of visits to certain pages, page interaction information (such as scrolling, clicks, and mouse-overs), methods used to browse away from the page, and any phone number used to call our customer service number.
Information may be collected by cookies and other tracking. Cookies are small data files that are stored on your device when visiting a website, which enable the Company to collect information about your device identifiers, web browsers used to access the Services, pages or features viewed, time spent on pages, mobile app performance and links clicked. Web beacons (or pixel tags) are electronic images that may be used to help deliver cookies, count website visits, understand usage and determine the effectiveness of email marketing campaigns.
Participants are advised that if they wish to deny the use and saving of cookies from this Website on their computer's hard drive they should take necessary steps within their web browsers security settings to block all cookies from this Website.
Information, which is received from third-parties: this includes, but is not limited to Elrond partners, sub-contractors in technical, payment services, advertising networks, search information providers, credit reference agencies, or other third parties who may provide information, for the purpose of fulfilling the Services or to comply with legal requirements. Elrond shall endeavour that these third parties are to provide the same level of data protection and that they shall only carry out their contractual obligations towards Elrond or upon the instructions of the Company and not for any other purposes.
In case of using a location-enabled device: the Company may collect location data or use various means to determine the location.
Elrond may use your Personal Data to supply you with the Services required, to bill you and to contact you when required. Elrond may also analyse the Personal Data provided to help in administering, supporting and improving the business, as per the following ways:
to provide, maintain, deliver and improve the Services and obtaining your views of the Services;
to send periodic emails with products and services;
to follow up with the Participant after correspondence (live chat, email or phone inquiries);
for marketing and retargeting purposes;
to carry out any other purpose for which the information was collected.
If you want to stop receiving information as per the above, you can opt out at any time by contacting us at the email address indicated in this Privacy Policy. At all times, access to the Participant's Personal Data by the Company's own staff and third-party intermediaries, will be restricted to the bare minimum required for the Company to perform our business and the provision of the Services.
Your Personal Data may be processed only if you have unambiguously given your consent or, the processing is necessary for the performance of the Services you request or, for compliance with legal obligations as in the case of Anti Money Laundering and Terrorist Financing Regulations (AML/TF) or, pursuant the public interest or, requested in the exercise of official authority vested in Elrond or a third party with whom the Personal Data is disclosed except where such interest is overridden by the interest to protect the fundamental rights and freedoms that arise from the right of privacy.
As an individual you may exercise your right to access your Personal Data held about you by Elrond by submitting your request in writing via this online request form.
Rectification, deletion
Elrond shall at your request immediately rectify, block or erase your Personal Data that has not been processed according to the GDPR and Data Protection Act or processed unlawfully and where applicable proceed with notifying any third party about the measures undertaken. Although all reasonable efforts will be made to keep your information updated, you are kindly requested to inform us of any change referring to the Personal Data held by Elrond. In any case if you consider that certain information about you is inaccurate, you may request rectification of such data.
Provided that no such notification shall be implemented if it is shown to be impossible or it will entail a disproportionate effort on the part of Elrond.
You have the right to receive, upon request, a copy of the Personal Data that you have provided to Elrond in a structured, commonly used and machine-readable format and to transmit such data to another controller, for free. Elrond shall endeavour to ensure that such requests are processed within one month for free, subject that this request is not excessive and does not affect the rights of other individuals' Personal Data.
Upon request, you have the right to have your Personal Data erased by Elrond. Elrond acting reasonably will take all necessary actions (including technical measures) to inform third-party data Processors to comply with the request unless your Personal Data needs to be retained to comply with legal obligations or court orders.
Data Subject's Support
Elrond shall provide its Users with user support through an online chat with an agent. Username and email address shall be necessary to sign up for online chat. The data collected in this manner shall be processed exclusively for the purpose of providing user support.
International or Cross-border Transfer of Your Data and Your Express Agreement
The transfer of the Participant's data may be necessary for using the Services, to provide updates and to respond to the Participant's inquiries. As the internet operates in a global environment, the use of the internet to collect and process personal data necessarily involves the transmission of data on an international or cross-border basis. By using the Services or by communicating with us, you acknowledge and expressly consent to Elrond's processing and disclosure of your Personal Data in this way. By using the Website, communicating with us, and using our Services, you provide your express consent to Elrond's disclosure of your personal data to third-party intermediaries (if applicable) for the purposes described in this Privacy Policy.
Elrond may share the Participants' data with others as follows:
With partners providing the Services;
When using interactive areas of the Website, like our blog or other online forums, certain information you choose to share may be displayed publicly, such as your username, actions you take and any content you post;
To Courts and Government authorities or bodies if requested to do so under a court order or legal process, or to establish or exercise the Company's legal rights or defend against legal claims;
If Elrond believes that the Participant's actions are inconsistent with the Terms or policies, or to protect the rights, property and safety of Elrond or others;
In connection with, or during negotiations of, any merger, sale of company assets, financing or acquisition of all or a portion of our business by another company;
Between and among the Company; or
With your consent or at your direction.
At any point in time you shall retain all rights to your data.
Elrond protects information collected using physical, technical, and administrative security measures to reduce the risks of disclosure, unlawful processing, accidental loss, destruction, damage and unauthorized access. Should a security breach occur, Elrond will use reasonable endeavours to try to fix it.
Social Sharing and Links to Third Party Websites
When sharing the Website, Services or other Elrond's information on social websites, this will enable the sharing of information with your contacts or the public, depending on the settings you establish with the entity that provides the social sharing feature. Service links to third party websites may be made available to you, and that are not owned or controlled by Elrond. Please be aware that Elrond is not responsible for the privacy practices of such websites. For more information about the purpose and scope of data collection and processing in connection with social sharing features and links to third party websites, Elrond encourages you to visit the privacy policies of the entities that provide these features.
Furthermore, you are advised to conduct yourself appropriately when engaging with Elrond on social media or when sharing the Website, Services or other Elrond's information.
Elrond will retain your Personal Data and that of visitors of the Website for a reasonable period or as long as the law requires. Elrond will retain your data as long as needed for the provision of the Services.
Disposal of Personal Data
When Elrond receives requests to dispose of Personal Data records by Data Subjects, Elrond shall ensure that these requests are handled within a reasonable time frame. Elrond shall keep a record including a log of these requests.
Elrond shall also strive in obtaining adequate disposal mechanisms to ensure no Personal Data is leaked outside of Elrond.
Response to Personal Data Breaches – When Elrond learns of a suspected or actual Personal Data breach, Elrond shall perform an internal investigation and take appropriate remedial measures in a timely manner. Where there is any risk to the rights and freedoms of Data Subjects, Elrond will notify the relevant Supervisory Authorities without undue delay and, when possible, within 72 hours from when it learns of such breach. Data Subjects will be informed when the Personal Data breach is of a "high" level.
By using the Services, or otherwise providing information to Elrond, you consent to the processing and transfer of information in accordance with Romanian law.
You may request Elrond to access your information and Personal Data at any time.
Elrond shall be responsible for and be able to demonstrate compliance with the principles outlined in this Privacy Policy. Elrond may at its own discretion assign or transfer this Privacy Policy and your related information to any person or entity that acquires or is merged with us without restriction.
Any claims, questions, comments and requests regarding this Privacy Policy should be addressed to Elrond's via this online request form.
Elrond may update this Privacy Policy from time to time at its own discretion and in response to changing legal, technical and/or business development. The revised Privacy Policy will take effect immediately upon publication by Elrond and we will replace this page with an updated version. When we update our Privacy Policy, we shall take appropriate action and inform you accordingly by a Privacy Notice consistent with the significant changes we would have made.
However, you acknowledge that it is your sole responsibility to check the "Privacy Policy" page from time to any time so as to be aware of any changes which may have occurred from time to time.
This document was updated on 20 January 2020 and is effective from that date.
You have the right to make a request to access or delete any of your personal data that we might possess. You can make a request via this online request form or by contacting our Data Protection Officer at the address below. You may be required to provide additional information to authenticate your request.
We are very open about privacy and recognize your trust as a great value. If you have any questions about this statement or any privacy issues in our applications or services, feel free to contact our Data Protection Officer via this online request form or by post:
Elrond Network SRL
N. Olahus St., no 5, Centrul de Afaceri, corp B, et.9
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redpajama_set_name: RedPajamaCommonCrawl