\section{Introduction}

The search of Lagrangian models, describing spinning particles,
has a long story. Most popular approach in this direction is
the formulation of the Lagrangian on the space-time,
extended by the anticommuting variables,
which upon quantization provide the system with the nontrivial spin.
This approach is closely related with the supersymmetric field theories
and is essential for the formulation of superparticle systems.

There is another, pure bosonic, approach in which  the Lagrangian
is formulated on the direct product of the
initial space-time by some orbit of Poincar\'e group.
The actions of this sort are, in fact, the particle counterparts
of  Born-Infeld type systems. This approach seems to be interesting
due its visible relation with orbit method of Kirillov-Konstant-Souriau
\cite{souriau}. Most developed investigation of such systems has been
presented in Refs. \cite{tomsk}.
Notice that the  bosonic approach is the only correct
in  $(2+1)$-dimensional systems, due to anyonic nature of planar particle.

The bosonic approach  admits an aesthetically attractive modification,
where  the  additional bosonic variables are encoded
in the dependence of the Lagrangian on higher-order derivatives.
These  systems seem to be interesting not only for their clear geometrical
meaning. The  higher-derivative parts in their
Lagrangians can be generated by some
field-theoretical mechanism, e.g. arising as quantum corrections.
Investigation of such sort of particle systems
became popular after remarkable work of
Polyakov \cite{polac}, where he  show, that evaluation of
the effective action of $CP^1$ model minimally coupled to the
 Chern-Simons field for the charged solitonic excitation
results in the action
\begin{equation}
{\cal S}_{eff} =\int (m + {\pi\over 2\theta}K_2)|d{\bf x}|,
\label{polac}\end{equation}
where $K_2$ is  world-line's torsion and $\theta$ is field coupling
 strength.

Later it was found, that this system describes  anyonic  analog of
Majorana field equations \cite{3d}.
Due to further  studies it became a part of physical folklore
that  relativistic systems can get nontrivial spin,
 if one adds to the Lagrangians the  higher-derivatives terms
(more precisely, the terms, depending on
 reparametrization invariants (extrinsic  curvatures) of world-line).
Some significant observations were done in connection with this
 subject, particularly, in the  description of three- and four- dimensional
particle with Majorana spectrum \cite{maj},
 $4d$ massless particles \cite{pl}.
The relation of the mentioned massless particle  model
 with $W-$ algebras has also been established \cite{rr}.

However, such  systems were not  studied completely
 even in the four- dimensional space-time,
where only the first and second extrinsic curvatures  were
considered. \\
In this note we attempt to fill this gap and
consider the simplest four-dimensional analog of (\ref{polac})
given by the action \cite{pc}
\begin{equation}
{\cal S} =\int(c_0+cK_3)d{\tilde s},\quad
d{\tilde s}=|d{\bf x}|\equiv sd\tau\neq 0
\label{0}\end{equation}
where $\tau$ is an arbitrary evolution parameter
and $K_3$ is  the  torsion (highest curvature) of a worldline in $4d$ space.

We will show that this system  possesses interesting properties
which make it drastically different from other
 four- and three- dimensional massive particle systems depending
on extrinsic curvatures:
\begin{itemize}
\item it  has a zero spin;
\item it possesses, in addition to reparametrization invariance, the
 extra gauge degrees of freedom: its classical trajectories are
restricted by the condition
$$
 \frac{K^2_2K_3}{K^2_1}=|\alpha|, \quad \alpha={c_0}/{c}
$$
while the mass spectrum is described
the conformal
mechanics with the energy ${\cal E}$,
$$ dq\wedge dp, \quad {\cal H}=\frac{p^2}{2}\pm\frac{\alpha^2}{2q^2},
\quad
-2{\cal E}/\alpha^2= \left\{
\begin{array}{cc}
M^2/c^2_0\mp 1,&
 {\rm if}\;\;\alpha < 0,\\
\pm M^2/c^2_0 \pm 1,&
 {\rm if}\;\;\alpha >0
\end{array}\right. ,
$$
where
$q=K_1/K_2$, and  $M$ denotes the mass of the system.\\
When  $\alpha<0$, the upper sign corresponds to the
time-like trajectories, while other solutions corresponds
to the space-like ones.

\item  When $\alpha<-1/2$, the solutions
with space-like trajectories  possess  a {\it discrete} spectrum with
 massive and tachionic sectors,
while the solutions with time-like trajectories possess continuous
spectrum  containing massive, massless, and tachionic solutions.
\end{itemize}
The paper is arranged as follows:

In {\it Section 2} we construct the Hamiltonian system corresponding
to the model under consideration (\ref{0}).
We show that the system possesses an extended gauge invariance and
a zero spin.
The  geometry  of its classical trajectories and quantum spectrum
(in the Euclidean space) are  defined by the one-dimensional
conformal mechanics (with a repulsive potential).

In {\it Section 3}  we reformulate
 the Hamiltonian system constructed in Section 2 in the Minkowski space
     and consider the properties of its mass spectrum.

\setcounter{equation}{0}
\section{Hamiltonian  formulation}
In this section we give
the Hamiltonian formulation of the model (\ref{0}).

 Recall
that the extrinsic curvatures $K_a$ of a (non-null) curve in
four-dimensional
space can be defined via the Frenet equations for the moving frame
${\bf e}_a$ :
\begin{eqnarray}
&{\dot{\bf x}}=s{\bf e}_1,\quad {\dot{\bf e}}_a=k_a^{\;b}{\bf e}_b,
\quad{\bf e}_a{\bf e}_b=\eta_{ab},\quad a,b,c=1,2,3,4; &\label{f}\\
& k_{a}^{\; c}\eta_{cb}=
\left(
\begin{array}{cccc}
0   &k_1 &0   &0\\
-k_1&0   &k_2 &0\\
0   &-k_2&0   &k_3\\
0   &0   &-k_3&0
\end{array}\right),\quad k_{a-1}=sK_{a-1},& \label{mat}
\end{eqnarray}
 where   ${\bf x}$  are  coordinates of four-dimensional space,
and ${\bf e}_a$ are  elements of the moving frame.\\
While $K_1, K_2$ are positive quantities, the sign of the
highest curvature $K_{3}$ (torsion) is not  uniquely defined
by the Frenet equations \cite{postnicov}.
Without loss of generality we assume  below that  $K_3>0$.

 In the  Euclidean space $\eta_{ab}=\delta_{ab}$, so the
 Frenet equations read
\begin{equation}
 {\dot{\bf e}}_a=k_a{\bf e}_{a+1}-k_{a-1}{\bf e}_{a-1},\quad
{\bf e}_{0}={\bf e}_5\equiv 0.
\label{ffe}\end{equation}

One can transform the Frenet equations in the Euclidean space
 into those in the Minkowski space,
performing
 the  following transition
\begin{equation}
({\bf e}_{\underline a},\; k_{\underline a},\;
k_{{\underline a}-1}, s)\to
(i{\bf e}_{\underline a},\; ik_{\underline a},\;
ik_{{\underline a}-1},(-i)^{\delta_{1{\underline a}}}s)
\label{tr}\end{equation}
 for some index  ${\underline a}$.   \\
Indeed, this transformation preserves the form of the
matrix in (\ref{mat}),
while  the element ${\bf e}_{\underline a}$ becomes time-like:
${\bf e}^2_{\underline a}=-1$.
So, we can give our basic derivations for the Euclidean case,
 reformulating them for the Minkowski space
  for a final analysis.

  It follows  from  (\ref{ffe})
that
\begin{equation}
s=\sqrt{{\dot{\bf x}}^2},\quad
k_a={\dot{\bf e}}_a{\bf e}_{a+1}=\sqrt{{\dot{\bf e}_a}^2-k^2_{a-1}}.
\label{cf}\end{equation}
Thus, the  Lagrangian appearing in the action (\ref{0})
in the {\it Euclidean} space
can be  replaced   by  the following
one
\begin{equation}
 L=c_0s+c\sqrt{{\dot{\bf e}}_3^2-k^2_2}+{\bf p}({\dot{\bf x}}-s{\bf e}_1)+
\sum_{i}{\bf p}_{i-1}({\dot{\bf e}}_{i-1}-k_{i-1}{\bf e}_{i}+
k_{i-2}{\bf e}_{i-2})-\sum_{i,j}d_{ij}({\bf e}_i{\bf e}_j-\delta_{ij}),
\label{l}\end{equation}
where
${\bf x}, {\bf p},{\bf e}_i,{\bf p}_{i-1}, s, k_i, d_{ij}$
are independent variables,
$i,j=1,2,3$.\\

Now we can perform the Legendre transformation for
this Lagrangian
 (referring for
details to \cite{tmp}).\\
The variables  ${\bf p}_{i-1}$ represent the momenta
conjugated to
 ${\bf e}_{i-1}$,
whereas the momenta conjugated to $(s, k_{i-1}, d_{ij})$
lead to the trivial constraints
\begin{equation}
 p^s\approx 0,\quad p^{i-1}\approx 0,\quad p^{ij}\approx 0.
\label{erunda}\end{equation}
Setting  $k_3\neq 0$  we find that the momentum
conjugated to ${\bf e}_3$ is of the form
\begin{equation}
{\bf p}_3=c\left({\dot{\bf e}}^2_3-k^2_{2}\right)^{-1/2}
{\bf{\dot e}}_3.
\label{pN}\end{equation}
Taking into account  (\ref{cf}), we get the  constraints
\begin{equation}
{\bf p}_3{\bf e}_{3}\approx 0,\quad
{\bf p}_3{\bf e}_{1}\approx 0,
\quad{\bf p}^2_3- ({\bf p}_3{\bf e}_{2})^2-c^2\approx 0.
\label{PhiNN0}\end{equation}
Then the construction of primary
Hamiltonian system becomes straightforward.

To simplify  the resulting system, one can stabilize
   trivial primary constraints
(\ref{erunda}) and exclude
them  from our considerations, which makes
the variables $s, k_{i-1}, d_{ij}$  lagrangian multipliers.
 Without
loss of generality  one can
also impose the gauge conditions (see for a details \cite{tmp})
$${\bf p}_3{\bf e}_2\approx 0,\quad
 {\bf p}_{2}{\bf e}_{2}\approx 0, \quad
{\bf p}_{2}{\bf e}_{1}\approx 0. $$

After these manipulations we get
the   Hamiltonian system
\begin{equation}
\begin{array}{c}
\omega=d{\bf p}\wedge d{\bf x}+
\sum_{i} d{\bf p}_i\wedge d{\bf e}_i, \\
\;\;\\
{\cal H}=s({\bf p}{\bf e}_1 -c_0)+\sum_{i} k_{i-1}\phi_{i-1.i}+
{k_3\over 2c}({\bf p}^2_3-c^2)+\sum_{i,j}d_{ij}
({\bf e}_i{\bf e}_j-\delta_{ij}),
\end{array}
\label{ss}
\end{equation}
with  primary constraints
\begin{eqnarray}
&{\bf p}_i{\bf e}_{j}\approx 0,\quad
i\geq j &\label{primary}\\
&{\bf e}_i{\bf e}_j-\delta_{ij}\approx 0, &
\label{u}\\
&{\bf p}{\bf e}_1 -c_0\approx 0,&\label{phi0}\\
&\phi_{i-1.i}\equiv{\bf p}_{i-1}{\bf e}_{i}-
{\bf p}_{i}{\bf e}_{i-1} \approx 0,
\quad{\bf p}^2_3-c^2\approx 0,&\label{phi}
\end{eqnarray}
where variables $s,k_i, d_{ij}$ play the role of
Lagrangian multipliers.\\

Let us stabilize the primary constraints (\ref{primary})-(\ref{phi}).\\
Stabilization of  (\ref{primary}) gives the following fixation
of the Lagrangian multipliers $d_{ij}$:
\begin{equation}
2d_{i.j}=k_3c\delta_{i.3}\delta_{j.3}-sc_0\delta_{1.i}\delta_{1.j},
\label{d}\end{equation}
so that  the equations of motion read
\begin{equation}
\begin{array}{c}
 \dot{\bf x}={\bf e}_1,\\
{\dot{\bf e}}_1=k_1{\bf e}_2,\\
 {\dot{\bf e}}_2=-k_1{\bf e}_1+k_2{\bf e}_3,\\
 {\dot{\bf e}}_3=-k_2{\bf e}_2+k_3{\bf p}_3/c\\
 {\dot{\bf p}}_3=-k_3c{\bf e}_3-k_2{\bf p}_2,\\
{\dot{\bf p}}_2=-k_1{\bf p}_1+k_2{\bf p}_3,\\
{\dot{\bf p}}_1=-s{\bf p}+k_1{\bf p}_2+sc_0{\bf e}_1, \\
{\dot{\bf p}}=0.
\end{array}\label{hem}\end{equation}
Stabilizing the remaining primary constraints,
 we get the following first-stage secondary constraints
\begin{equation}
{\bf p}{\bf e}_2\approx 0,
\quad{\bf p}_{1}{\bf e}_{3}\approx 0,
\quad {\bf p}_3{\bf p}_2\approx 0.
\label{secondary}\end{equation}
From the  ortogonality of (${\bf e}_i, {\bf p}_3/c$) to ${\bf p}_2$,
which follows  from (\ref{phi}),(\ref{primary}),(\ref{secondary})
 we conclude that
\begin{equation}
{\bf p}_2\approx 0.
\label{secondary1}\end{equation}
Stabilizing the first and second  constraints from (\ref{secondary})
and  the constraint (\ref{secondary1}), we get
 \begin{eqnarray}
&k_2=qk_1, \quad k_3=sc_0/cq^2, &\label{multiplyers}\\
& {\bf p}_1=q{\bf p}_3,&\label{tertiary}
\end{eqnarray}
 where
\begin{equation}
1/q\equiv{{\bf p}{\bf e}_3\over c_0}\neq 0.
\end{equation}
Then we get
\begin{equation}
 {\bf p}=c_0{\bf e}_1+c_0{\bf e}_3/q+
p{\bf p}_3,\label{p}
\end{equation}
where
\begin{equation}
 p\equiv {\bf p}{\bf p}_3/c^2.\label{pi}
 \end{equation}
Consistency of the equations of motion (\ref{hem}) with the Frenet equations
implies that
\begin{equation}
{\bf e}_a=\{{\bf e}_i,\;{\bf e}_4\equiv{\bf p}_3/c\}:
\quad {\bf e}_a{\bf e}_b= \delta_{ab}.\label{on}\end{equation}
Therefore, the initial Hamiltonian system can be formulated purely in
terms of the moving frame ${\bf e}_a$, coordinates ${\bf x}$, and momentum
${\bf p}$, while the relations (\ref{on}), (\ref{p})
play the role of constraints.

Taking into account the  constraint (\ref{p})),
we get the
expressions for the rotation generators:
\begin{equation}
{\bf J}={\bf p}\vee{\bf x}+\sum_{i}{\bf p}_i\vee{\bf e}_i=
{\bf p}\vee ({\bf x}-q{\bf p}_3/c_0)\label{J}\end{equation}
and the Casimirs
\begin{equation}
{\bf p}^2=(cp)^2+c^2_0(1+1/q^2),\quad
{\bf W}^2\equiv ({\bf p}\vee {\bf J})^2=0.
\end{equation}
Thus, {\it the system possesses a  zero spin
despite the dependence of the initial Lagrangian
on higher derivatives.}\\

The expression for rotation generators (\ref{J}) hints us to introduce the
``effective" coordinate
\begin{equation}{\bf X}\equiv {\bf x}-q{\bf p}_3/c_0\;:
\quad {\bf{\dot X}}=s{\bf p}/c_0,\quad {\bf p}=const
\label{X}\end{equation}
whose evolution equations look similar to the convenient relation
 between velocity and momentum of a massive particle
(note, that in spite of these relations
the mass of the system is not equal to $c_0$).

The reduction of the initial  Hamiltonian system (\ref{ss})
  by the constraints
 (\ref{on}), (\ref{p}), (\ref{tertiary}) leads to
 the following unconstrained system
\begin{equation}
 \omega^{red}= d{\bf p}\wedge d{\bf X}+\frac{c^2}{c_0}
{dp\wedge dq},\quad
{\cal H}^{red}=\frac{sc^2}{c_0}\left(
 p^2/2+\frac{c^2_0}{2c^2q^2}-\frac{{\bf p}^2- c^2_0}{2c^2}\right).
\label{fd}\end{equation}
The external curvatures of the system are related with the modular
``coordinate"  $q$ as follows
\begin{equation}
 \frac{K_2}{K_1}=q, \quad K_3=\frac{c_0}{cq^2},\quad\Rightarrow\quad
\frac{K^2_2K_3}{K^2_1}=\frac{c_0}{c}.
\label{Kq}\end{equation}
So, {\it  the system under consideration
possesses extended gauge invariance
 since the trajectories
 with the same ratio $K_{1}/K_{2}$ are gauge equivalent.}\\

Reducing the system (\ref{fd}) by ${\bf p}$, to exclude
the  trivial part of  dynamics (\ref{X}),
we get that the
evolution of the parameter $q$ is described by the textbook
mechanical system \cite{ll}
\begin{equation}
dp\wedge dq,\quad
{\cal H}_0=p^2/2+\frac{c^2}{2c^2_0q^2},\quad {\cal E}=
\frac{{\bf p}^2-c^2_0}{2c^2}.
\label{cm}\end{equation}
This  system  can be immediately integrated  at the classical
level
$$ q^2=\frac{({\bf p}^2-c^2_0){\tilde s}^2}{2c^2}+
\frac{2}{{\bf p}^2-c^2_0},\quad{\bf p}^2>c^2_0,$$
as well as at the quantum one.

Thus,  adding of the torsion term increases the absolute value of momenta
 of the system with respect to the torsionless system,
${\bf p}^2\geq c^2_0$.
The system (\ref{cm}) is known in literature as a conformal mechanics,
 due to the symmetry of its  action under conformal  transformations
generated  by
\begin{equation}
{\cal H}_0=p^2/2+\frac{c^2}{2c^2_0q^2},
\quad{\cal D}=p q,\quad  {\cal K}={q^2}/{2}\;.
\end{equation}
The ``energy"  spectrum is  continuous and has the lowest bound ${\cal E}=0$
which is not normalizable.
In \cite{nc} the conformal symmetry of this system
has been used for solving the problem of the ground state.

Recently this mechanism has been found to be adequate
to the problem of motion of charged  particle in the field
of  a charged black hole \cite{k}. Due to this
 observation the interest to the conformal mechanics
and its supergeneralizations \cite{pashnev},
 is renewed in the context of
study of probe  $D0$-brane dynamics in the external $D$-brane  field
(see e.g.\cite{branes} and refs therein).
Conformal mechanics arises in our model  a different context:
it defines  local gauge symmetry of the  particle system.

\setcounter{equation}{0}
\section{Transition to Minkowski space}

In the previous Section  we constructed the constrained Hamiltonian system
corresponding to the action (\ref{0}) in the Euclidean space.
We found that the
evolution and  the geometry of this system are described in terms of  the
non-relativistic  conformal mechanics with repulsive potential.
The purpose of this Section is to consider the relativistic aspects of
this system, i.e. to investigate the model (\ref{0}) in the
Minkowski space.
As we have mentioned in the beginning of Section 2,
 we can
use the results concerning the Euclidean case,
since the  Frenet formulae in the Euclidean space  can be transformed
 into the ones in
Minkowski space with the use of transition (\ref{tr}),
 where ${\underline a}$ denotes
an element of moving frame which becomes time-like, i.e.
${\bf e}^2_{\underline a}=-1$.
Correspondingly,
for the transition to {\it time-like} trajectories we have to choose
 ${\underline a}=1$, while for the transition to
 {\it space-like} trajectories
  we have to choose ${\underline a}\neq 1$.
 We do not consider here systems with {\it light-like} trajectories.

For convenience we will use the notation
\begin{equation}
{\bf p}^2\equiv-M^2,\quad \alpha=\frac{c_0}{c},
\end{equation}
so that $M^2>,=,< 0$  corresponds to
the massive, massless, and tachionic sectors of the system, respectively.

To reformulate the system under consideration  in
 the Minkowski space we have to supply  the transition
 (\ref{tr})  with appropriate
 transformations of characteristic constants $c_0, c$ and momenta
${\bf p}_i$ which  preserve the form of
the initial action (\ref{0})
 and the initial Hamiltonian system, namely,
\begin{equation}
\begin{array}{cccc}
{\underline a}=1:&({\bf e}_1,{\bf p}_1, k_1, s, c_0)&\to &
( i{\bf e}_1, -i{\bf p}_1, ik_1,-is, ic_0);\\
{\underline a}=2:&({\bf e}_2,{\bf p}_2, k_2, k_1)&\to &
( i{\bf e}_2, -i{\bf p}_2, ik_2, ik_1),\\
{\underline a}=3:&
({\bf e}_3,{\bf p}_3, k_3, k_2, c)&\to &
( i{\bf e}_3, -i{\bf p}_3, ik_3, ik_2, -ic),\\
 {\underline a}=4:&
(c, k_3)&\to &( -ic,  ik_3).
\end{array}
\label{htr}\end{equation}
Indeed, it is easy to see that it is only the
ortonormality condition
${\bf e}_a{\bf e}_b=(-1)^{\delta_{{\underline a}a}}\delta_{ab}$
that is changes in (\ref{ss}) under this transformation.
Consequently, the reduced system (\ref{cm}) with  appropriately
changed parameters describes the effective Hamiltonian system corresponding
to the action (\ref{0}) in the Minkowski space.

The transition  (\ref{htr}) induces the following
transformation of the parameters and coordinates
of the reduced system (\ref{cm})
\begin{equation}
(p,q,\alpha, c_0, s)\to
\left\{
\begin{array}{cc}
(p, iq, i\alpha, ic_0, -is)&{\rm if } \;{\underline a}=1\\
(p,q,\alpha, c_0, s)& {\rm if } \;{\underline a}=2\\
(ip, -iq, i\alpha, c_0, s ) &{\rm if } \;{\underline a}=3\\
(-p, q, i\alpha, c_0, s ) &{\rm if } \;{\underline a}=4.
\end{array}\right.
\end{equation}
Thus,  the reduced system  reads
\begin{equation}
dp\wedge dq,\quad
\epsilon_{{\underline a}}s\left(
\frac{p^2}{2} +
\frac{\epsilon_{{\underline a}}\alpha^2}{2q^2}
-{\cal E}_{\underline a}\right)\approx 0,
\end{equation}
 where
$$
{\cal E}_{{\underline a}}=
-(-1)^{\delta_{4{\underline a}}}\frac{\alpha^2}{2}
({M^2}/{c^2_0}+(-1)^{\delta_{1{\underline a}}}),\quad
\epsilon_{{\underline a}}=\left\{
\begin{array}{cc}
1,& {\rm if }\;\;{\underline a}=1,2\\
-1,& {\rm if }\;\;{\underline a}=3,4.
\end{array}\right.
$$
The
expressions for curvatures (\ref{Kq}) read
\begin{equation}
 \frac{K_2}{K_1}=(-1)^{\underline a}q, \quad
K_3=(-1)^{\underline a}\frac{\alpha}{q^2},\quad\Rightarrow\quad
\frac{K^2_2K_3}{K^2_1}=(-1)^{\underline a}\alpha .
\label{Kqm}\end{equation}

Due to  positivity of curvatures $K_i$, we conclude that
\begin{itemize}
 \item  ${\underline a}=1,3$, $\quad q<0$,\hspace{0.5cm} if\hspace{0.5cm}
 $\alpha<0$,
\item
${\underline a}=2,4$, $\quad 0<q$,\hspace{0.5cm} if\hspace{0.5cm} $\alpha>0$.
 \end{itemize}
Thus,  in both cases the system possesses the solutions described
by conformal    mechanics with a repulsive (${\underline a}=1,2$) and an
attractive (${\underline a}=3,4$) potentials.
The classical solutions of these  systems are of the
form
$$
q^2=\left\{
\begin{array}{cc}
{\cal E}_{\underline a}{\tilde s}^2+(-1)^{[{\underline a}]}
{\alpha^2}/{{\cal E}_{\underline a}},&{\rm if} \;\; {\cal E}\neq 0,\\
2\alpha {\tilde s},&{\rm if}\;\; {\cal E}=0,\;\;
{\underline a}=3,4
\end{array}\right.
$$
When the potential
is attractive (${\underline a}=3,4$),
the parameter ${\tilde s}$ is defined on the domain
$$
{\tilde s}\in\left\{
\begin{array}{cc}
] -|\alpha/{\cal E}|,\;\;|\alpha/{\cal E}|[, & {\rm if}\; {\cal E}<0,\\
]|\alpha/{\cal E}| ,\;\;\infty [,& {\rm if} \;{\cal E}>0.
\end{array}\right.
$$
While  the  quantum  spectrum of the mechanics  with a repulsive potential
is continuous, the spectrum of  conformal mechanics with an attractive
potential
can be both  continuous and discrete.
 The discrete spectrum corresponds to the
strongly attractive potential ($|\alpha|>1/2$)
and has an infinite number of energy levels \cite{pr}.\\

We first consider the systems with repulsive potential
\begin{equation}
dp\wedge dq,\quad
{\cal H}_0=\frac{p^2}{2} +
\frac{\alpha^2}{2q^2},\quad \frac{2{\cal E}}{\alpha^2}=\left\{
\begin{array}{cc}
(1-{M^2}/{c^2_0}),& {\rm if}\;\; \alpha<0,\;\;{\underline a}=0\\
-(1+{M^2}/{c^2_0}),&{\rm if}\;\; \alpha>0,\;\;{\underline a}=1
\end{array}\right.
\end{equation}
Notice that while in the first case trajectories are {\it time-like},
those are {\it space-like}  in the second case. \\
Since the energy ${\cal E}$ is positive both in classical and quantum cases,
we get the restrictions on
the admissible value of the mass
\begin{itemize}
\item $M^2<c^2_0$,$\;$ for $\alpha<0$ and  time-like trajectories
(${\underline a}=1$),
\item  $M^2<-c^2_0$,$\;$ for $\alpha>0$, ${\underline a}=2$.
\end{itemize}
Thus,  in the first case there are massive, massless, and tachionic
states, while in the second case the system is tachionic.\\

Now we consider the systems with an attractive potential. In this case
all trajectories are space-like and are defined by the mechanics
\begin{equation}
dp\wedge dq,\quad
{\cal H}_0=\frac{p^2}{2} -
\frac{\alpha^2}{2q^2},\quad \frac{2{\cal E}}{\alpha^2}=
\left\{
\begin{array}{cc}
-(1+{M^2}/{c^2_0})&
{\rm if}\;\; \alpha<0,\; {\underline a}=3\\
(1+{M^2}/{c^2_0})&
{\rm if} \;\;\alpha>0,\; {\underline a}=4
\end{array}\right.
\end{equation}
The spectrum of conformal mechanics with an attractive potential is
continuous    for  ${\cal E}>0$ and for  ${\cal E}\leq 0, \alpha^2<1/4$
(see \cite{ll}).

If the potential is ``strongly attractive" $\alpha^2>1/4$,
and  ${\cal E}<0$,
the system has a dicrete spectrum with an
infinite number of bound states  given
by the expression \cite{pr}
$$
{\cal E}_n=-\hbar^2 B^2\exp{\frac{-2\pi n}{\sqrt{\alpha^2-1/4}}},\quad
n=0,\pm1,\pm2,\ldots,
$$
where $B$ is an undefined "phase" factor.   \\
So,
\begin{equation}
M^2_n/c^2_0=-1+
(-1)^{{\rm sgn}\alpha}\left(\frac{B\hbar}{\alpha}\right)^2
\exp{\frac{-2\pi n}{\sqrt{\alpha^2-1/4}}},\quad |\alpha|>1/2.
\end{equation}
Therefore,  for $\alpha>0$ the  discrete branch corresponds to pure
tachionic states, while for $\alpha<0$ it contains massive, massless,
and tachionic  sectors. \\

{\large Acknowledgments.} I am  thankful to M.Plyushchay,
who  suggested  to investigate this  model and gave numerous advices
 and C.Sochichiu for valuable discussions and useful comments.

This  work has been partially supported by
grants INTAS-RFBR No.95-0829, INTAS-96-538 and INTAS-93-127-ext.


\section{INTRODUCTION}

In today's standard inflationary scenario the amplified vacuum
fluctuations
of the inflaton field provide the seeds for large-scale structure
formation
\cite{linde90,padmanabhan93,brandenberger92/liddle93}. It is thus of
interest to understand the effect of these quantum
fluctuations on the universe dynamics and the implications they may have
on cosmological observations.

In most inflationary models (exponential inflation), the geometry of
spacetime during the period of inflation can be reasonably well described
by
de Sitter spacetime. This spacetime also seems favored by quantum
cosmology since most
accepted ``initial condition'' proposals, which are either quantum
``tunneling from nothing'' \cite{vilenkin84/86} or ``no-boundary''
condition \cite{hartle83}, both predict it.

It has been recently pointed out that fluctuations in the stress-energy
tensor of quantum fields may be important for some states in curved
spacetimes or even flat spacetimes with non-trivial topology \cite{kuo93}.
Hu and Phillips, for instance, have computed the energy density
fluctuations
of quantum states in spatially closed Friedmann-Robertson-Walker models
and have
shown that these could be as important as the energy density itself \cite
{hu97}. If so, these fluctuations may induce relevant back-reaction
effects
on the gravitational field (the spacetime geometry).

In this paper we compute the fluctuations of the stress-energy
tensor for a scalar field in de Sitter spacetime. We consider a massless
minimally coupled field in the Euclidean vacuum state, which is a de
Sitter invariant state. The motivation for considering the massless
minimal coupling case
is the fact that it mimics the behavior of gravitons in a curved
background
as well as that of the perturbations of the inflaton field in usual
inflationary models. As for the state, the reasons why we chose the
Euclidean vacuum are twofold. On the one hand, the high degree of symmetry
makes the computations simpler. On the other hand, there are at least two
serious physical motivations. First, it naturally arises in exponential
inflation models and for a massive field it has been shown to be the state
to which any other state tends asymptotically in a de Sitter background
\cite{hawking83}; and second, it is selected by the most popular boundary
conditions in quantum cosmological models \cite{halliwell89}.

Here we compute the two-point correlation for spacelike separated points
of the
stress-energy tensor and find that these fluctuations are important.
Therefore the back reaction of these fluctuations on the spacetime
geometry
could be relevant, and will be the subject of further work within the
context of stochastic semiclassical gravity and Einstein-Langevin
equation \cite{martin98}.
Related to this fact it is worth mentioning that Abramo, Brandenberger and
Mukhanov \cite{abramo97a,abramo97b} have shown that the second order
contribution
to back reaction of inflaton and gravitational perturbations during the
inflationary period can be important even below the ``self-reproduction''
scale.
Although the spirit of our approach to the back-reaction problem is
slightly
different, we believe a partial connection with their work may exist.
These issues will also be addressed in future investigations.

The plan of the paper is the following. In section \ref{sec2} we give a
very brief review of de Sitter spacetime, its properties and the solution
of the Klein-Gordon equation in such a background. In section \ref{sec3},
following the proposal made in \cite{garriga93}, we discuss the invariant
vacuum state for a massless minimally coupled field that we are going to
use. In section \ref{sec4} we deal with the fluctuations of the
stress-energy tensor of the scalar field in de Sitter background and
compute the noise kernel (i.e. the two-point correlations), which
characterizes these fluctuations, for spacelike separated points. We
discuss our results in the final section.
Throughout the paper we will use the $(+,+,+)$ sign convention of Misner,
Thorne and Wheeler's book \cite{misner71}.

\section{DE SITTER SPACETIME} \label{sec2}

In this section we give a brief summary of some useful properties and
definitions related to de Sitter spacetime. For a more detailed
exposition, the reader is referred to \cite{hawking73/mottola85,allen85}.

Four dimensional de Sitter space-time has positive constant curvature and
is thus maximally symmetric. Its ten dimensional group of isometries is
$O(4,1)$. It can be represented as a hyperboloid embedded in a five
dimensional spacetime with Minkowskian metric $\eta _{AB}$, so that
$\eta_{AB}\xi ^A(x)\xi ^B(x)=H^{-2}$, where the Hubble constant $H$ is
related to the scalar curvature $R=12H^2$ and $\xi ^A(x)$ is the position
vector in the
five-dimensional Minkowskian embedding spacetime (with $A,B=0,\ldots ,4$)
corresponding to the point $x$ of de Sitter. One can define the biscalar
\begin{equation}
\label{00}Z(x,y)\equiv H^2\eta _{AB}\xi ^A(x)\xi ^B(y) ,
\end{equation}
$Z>1$ for timelike separated points, $Z=1$ for points connected by null
geodesics and $Z<1$ for spacelike separated points. It is worth
emphasizing that there is no geodesic connecting two points with
$Z<-1$. For points which are geodesically connected an alternative
expression to (\ref{00}) is \cite{allen85}:
\begin{equation}
\label{0}Z(x,y)=\cos \sqrt{\frac{R\,\sigma (x,y)}6},
\end{equation}
where $\sigma (x,y)\equiv \frac 12 s^2(x,y)$ with $s(x,y)$ being the
geodesic distance
between the two points. We will use the closed coordinates $(\eta ,\chi
,\theta ,\varphi )$, which cover the whole de Sitter spacetime. The line
element reads:
\begin{equation}
\label{01}ds^2=(H\sin \eta )^{-2}\left( -d\eta ^2+d\chi ^2+\sin ^2\chi
\left( d\theta ^2+\sin ^2\theta \,d\varphi ^2\right) \right),
\end{equation}
where $\eta \in (0,\pi )$ and $(\chi ,\theta ,\varphi )$ are the usual
hyperspherical parametrization of the $S^3$ spatial surfaces, which are
invariant under the $O(4)$ subgroup of isometries.

A scalar field of mass $m$ satisfies the Klein-Gordon equation
$\left( \Box -m^2-\xi R\right) \phi (x)=0$, where $\xi$ is a
dimensionless constant which determines the coupling of the field to the
spacetime curvature. We look for a set of mode solutions that can be
written as $u_{klm}(x)=H\sin \eta\,X_k(\eta )\,Y_{klm}(\chi ,\theta
,\varphi)$, where $Y_{klm}$ (with $k=0,1,2,\ldots ;l=0,1,\ldots ,k-1$ and
$m=-l,\ldots ,l$) are the $S^3$ spherical harmonics obeying
$\Delta^{(3)}Y_{klm}=-k(k+2)Y_{klm}$, which constitute a $(k+1)^2$
dimensional representation of $O(4)$. Substituting into the Klein-Gordon
equation, $X_k(\eta )$ satisfy the following equation:
\begin{equation}
\label{04}\frac{d^2X_k}{d\eta ^2}+\left\{ (k+1)^2+(H\sin \eta )^{-2}\left[
m^2+\left( \xi -\frac 16\right) R\right] \right\} X_k=0.
\end{equation}
The general solution to this equation is
\begin{equation}
\label{05}X_k(\eta )=(\sin \eta )^{-\frac 12}\left[ A_kP_{k+\frac 12%
}^\lambda (-\cos \eta )+B_kQ_{k+\frac 12}^\lambda (-\cos \eta )\right],
\end{equation}
where $\lambda \equiv \sqrt{\frac 94-\frac{12}R(m^2+\xi R)}$, the
coefficients $A_k$ and $B_k$ satisfy the normalization condition
$A_kB_k^{*}-A_k^{*}B_k=i\Gamma (k+\frac 32%
-\lambda )/\Gamma (k+\frac 32+\lambda )$ and $P_{k+\frac 12}^\lambda
(z)$ and $Q_{k+\frac 12}^\lambda (z)$ are associated Legendre functions of
the first
and second kind respectively. Given such a complete set of orthonormal
solutions, one can expand the field operator as $\hat{\phi }%
(x)=\sum_{klm}\left( \hat{a}_{klm}u_{klm}(x)+\hat{a}%
_{klm}^{+}u_{klm}^{*}(x)\right) $ and build the associated Fock space of
states in the usual way. If we require the vacuum state to be $O(4,1)$
invariant and of Hadamard type, $A_k$ and $B_k^{}$ are uniquely determined
\cite{allen87,chernikov68/schomblond76}:
\begin{eqnarray}
\label{07}
A_k&=&\frac \pi 4\frac{\Gamma (k+\frac 32-\lambda )}{\Gamma (k+\frac 32
+\lambda)}, \\
B_k&=&-\frac{2i}\pi A_k .
\end{eqnarray}
The corresponding state is the so-called Euclidean vacuum, also known as
Bunch-Davies vacuum.

\section{DISCUSSION ABOUT THE CHOSEN STATE} \label{sec3}

The most natural state to choose is the Euclidean vacuum, which is
invariant
under the de Sitter group of isometries and is a Hadamard state. This is
the
one selected in quantum cosmology when Hartle-Hawking no-boundary proposal
or Vilenkin's tunneling wave function are taken as initial conditions
\cite
{halliwell89}. In addition, when the field is massive, the field state
asymptotically tends to it \cite{hawking83}.

Both the Green's functions and the renormalized expectation value of the
stress-energy tensor have been computed for the Euclidean vacuum in
several
works \cite{bunch78,birrell84}:
\begin{equation}
\label{08}G^{\left( 1\right) }(x,y)\equiv \left\langle 0\right| \{
\hat{\phi }(x),\hat{\phi }(y)\} \left| 0\right\rangle
=\frac{%
2H^2}{(4\pi )^2}\,\Gamma\!\left( \frac 32-\lambda \right) \Gamma\!\left(
\frac 32+\lambda \right) F\!\left( \frac 32-\lambda ,\frac 32+\lambda
,2;\frac{1+Z(x,y)}2\right),
\end{equation}
\begin{eqnarray}
\label{09}
\left\langle
\hat{T}_{\mu \nu }(x)\right\rangle _{ren}=-\frac{g_{\mu \nu}(x)}{64\pi^2}
\left\{ m^2\left[ m^2+\left( \xi -\frac 16\right) R\right] \left[ \psi\!
\left( \frac 32-\lambda \right) +\psi\! \left( \frac 32+\lambda \right)
+\ln
\frac R{12m^2}\right] \right. \nonumber \\
\left. -m^2\left( \xi -\frac 16\right) R-%
\frac 1{18}m^2R-\frac 12\left( \xi -\frac 16\right)
^2R^2+\frac{R^2}{2160}%
\right\},
\end{eqnarray}
where $F(a,b,c;z)$ is a hypergeometric function and $\psi (z)\equiv
(d/dz)\,\ln \Gamma (z)$.

From (\ref{08}) one can see that the symmetrized two-point function $%
G^{\left( 1\right) }(x,y)$ has an infrared divergence (as $m^2\rightarrow
0$%
\/) in the minimal coupling case. On the other hand, the expectation value
of the stress-energy tensor remains finite, but is ambiguous, i.e., it
depends on the way the limit $m^2\rightarrow 0$, $\xi \rightarrow 0$ is
taken. In fact, Allen \cite{allen85} proved that in the massless minimally
coupled case there can be no Fock space built on a de Sitter invariant
vacuum. Allen and
Folacci suggested that in this case one should consider instead $O(4)$
invariant states \cite{allen87}. On the other hand Garriga and Kirsten
pointed out that such a peculiar behavior comes from the zero mode, which
corresponds to a constant solution of the Klein-Gordon equation that
appears
when $k=0$ in (\ref{04}) \cite{garriga93}. They also showed that a special
treatment of the zero mode
seems to give an invariant vacuum. The Hilbert space of states is then the
tensor product of the zero mode part, which is equivalent to the Hilbert
space for a one dimensional non-relativistic free particle, times the Fock
space corresponding to the non-zero modes (the whole space is not a Fock
space, in agreement with Allen's result). The field operator is written
\begin{equation}
\label{001}\hat{\phi }(x)=\sum_{%
{{klm} \atop {k\neq 0}}
}\left( \hat{a}_{klm}u_{klm}(x)+\hat{a}_{klm}^{+}u_{klm}^{*}(x)%
\right) +\frac H{\sqrt{2}\pi }\left( \hat{Q\,}\frac
12+\hat{P}\left(
\eta -\frac 12\sin 2\eta -\frac \pi 2\right) \right) ,
\end{equation}
where $\hat{Q}$ and $\hat{P}$ are the position and momentum
operators of the free particle Hilbert space and $\hat{a}_{klm}^{+}$
and $\hat{a}_{klm}$ are the creation and annihilation operators of the
Fock space associated to all the modes with $k\neq 0$.
Note that the two functions that multiply the operators $\hat{Q}$ and
$\hat{P}$, when divided by $\sin \eta$, are solutions of equation
(\ref{04}) for $k=0$ (with $m=\xi=0$). In fact, the constant
function multiplying the operator $\hat{Q}$ is precisely the aforesaid
zero mode. These operators satisfy the usual commutation relations:
\begin{equation}
\label{002}
\begin{array}{c}
\left[ \hat{a}_{klm},\hat{a}_{k^{\prime }l^{\prime }m^{\prime }}\right]
=0,\qquad \left[ \hat{a}_{klm},\hat{a}_{k^{\prime }l^{\prime}
m^{\prime }}^{+}\right] =\delta _{kk^{\prime }}\delta _{ll^{\prime}}
\delta_{mm^{\prime }},\qquad [ \hat{Q},\hat{P} ] =i, \\
\protect{} [ \hat{a}_{klm},\hat{Q} ] =0,\qquad [ \hat{a}_{klm},\hat{P} ]
=0 .
\end{array}
\end{equation}
The vacuum is defined as follows:
\begin{equation}
\label{003}
\begin{array}{c}
\hat{P}\left| 0\right\rangle =0 ,\\ \hat{a}_{klm}\left|
0\right\rangle =0\;,\,k\neq 0 .
\end{array}
\end{equation}
As a consequence of the first condition, this vacuum is not normalizable,
so strictly speaking it is not an
element of the Hilbert space but just a limit. However, as it has been
pointed out by Garriga and Kirsten, this fact is not so
strange: it also happens for the ground state of a non-relativistic free
particle. These authors also computed the renormalized expectation
value of the stress-energy tensor for this state:
\begin{equation}
\label{2}\left\langle \hat{T}_{\mu \nu }(x)\right\rangle
_{ren}=\frac{119}{138240\pi ^2}\,R^2\,g_{\mu \nu }(x)=\frac{119}{960\pi
^2}\,H^4\,g_{\mu \nu }(x).
\end{equation}
Two aspects should be stressed. First, the expression is de Sitter
invariant. Second, the energy density is lower than that of the $O(4)$
invariant
states introduced by Allen an Folacci, the stress-energy tensor of which
is not de Sitter invariant. It is also remarkable that the expectation
value for other states of observables such as the dispersion of the
smeared field or the stress-energy tensor tend
asymptotically ($\eta \rightarrow \pi $ or equivalently $t\rightarrow
\infty $, where $t$ is the proper time for a comoving observer) to the
same value as that of the invariant vacuum \cite{garriga93}.

\section{COMPUTATION OF THE STRESS-ENERGY TENSOR FLUCTUATIONS}
\label{sec4}

The quantity that characterizes the stress tensor fluctuations and which
determines, via the Einstein-Langevin equation, the fluctuations in the
spacetime geometry \cite{martin98} is the noise kernel, defined by the
symmetrized two-point correlation function $\frac 12\left\langle
\{\hat{t}_{\mu \nu}(x),\hat{t}_{\rho \sigma }(y)\} \right\rangle = \Re
\left\langle
\hat{t}_{\mu \nu}(x)\hat{t}_{\rho \sigma }(y) \right\rangle$,
where $\hat{t}_{\mu \nu}
(x)\equiv \hat{T}_{\mu \nu }(x)-\left\langle \hat{T}_{\mu \nu
}(x)\right\rangle $ and $\hat{T}_{\mu \nu }(x)\equiv\lim_{x^{\prime
}\rightarrow x}{\cal D}_{\mu \nu}\,(x,x^{\prime})\left( \hat{\phi
}(x)\hat{\phi
}(x^{\prime })\right) $ with ${\cal D}_{\mu \nu }(x,x^{\prime })\equiv
\left(\nabla_\mu ^x\nabla _\nu ^{x^{\prime}}-\frac 12g_{\mu \nu }(x)\nabla
_\alpha ^x\nabla ^{x^{\prime }\alpha }\right)$ (we have chosen
point-splitting regularization for convenience). It is easy to see that
$\left\langle
\hat{t}_{\mu \nu}(x)\hat{t}_{\rho \sigma }(y) \right\rangle$
is equivalent to $\left\langle \hat{T}_{\mu \nu }(x)\hat{T}_{\rho
\sigma }(y)\right\rangle -\left\langle \hat{T}_{\mu \nu
}(x)\right\rangle \left\langle \hat{T}_{\rho \sigma }(y)\right\rangle$.
Note that this expression is finite in the following sense: one can
compute it suitably regularized, then the potentially divergent terms
cancel and one can remove the regularization (letting $x^{\prime
}\rightarrow x$ and $y^{\prime }\rightarrow y$).
\begin{eqnarray}
\label{3}
\left\langle
\hat{T}_{\mu \nu }(x)\hat{T}_{\rho \sigma }(y)\right\rangle
-\left\langle \hat{T}_{\mu \nu }(x)\right\rangle \left\langle
\hat{T}_{\rho \sigma }(y)\right\rangle =\lim_{{x^{\prime }\rightarrow
x}\atop{y^{\prime }\rightarrow y}}{\cal D}_{\mu \nu }(x,x^{\prime}){\cal
D}_{\rho\sigma}(y,y^{\prime})&&\left( \left\langle \hat{\phi}(x)\hat{\phi}
(x^{\prime })\hat{\phi }(y)\hat{\phi }(y^{\prime })\right\rangle \right.
\nonumber \\
&&\left. -\left\langle \hat{\phi }(x)\hat{\phi }(x^{\prime
})\right\rangle \left\langle \hat{\phi }(y)\hat{\phi }(y^{\prime
})\right\rangle \right).
\end{eqnarray}
The expression inside the parenthesis is computed in the appendix, where
we find that it is finite as we pointed out above, and equals
$G^{+}(x,y)G^{+}(x^{\prime},y^{\prime})+G^{+}(x,y^{\prime})G^{+}
(x^{\prime},y)$. The final result after removing the regularization is:
\begin{eqnarray}
\label{6}
\left\langle
\hat{t}_{\mu \nu }(x)\hat{t}_{\rho \sigma }(y)\right\rangle &=&
\Bigl[
 \nabla _\rho ^y\nabla _\mu ^xG^{+}(x,y)\nabla _\sigma ^y\nabla _\nu
^xG^{+}(x,y)+\nabla _\sigma ^y\nabla _\mu ^xG^{+}(x,y)\nabla _\rho
^y\nabla
_\nu ^xG^{+}(x,y) \nonumber \\
&& -g_{\mu \nu }(x)\nabla _\rho
^y\nabla _\alpha ^xG^{+}(x,y)\nabla _\sigma ^y\nabla ^{x\alpha
}G^{+}(x,y)-g_{\rho \sigma }(y)\nabla _\alpha ^y\nabla _\mu
^xG^{+}(x,y)\nabla ^{y\alpha }\nabla _\nu ^xG^{+}(x,y) \nonumber \\
&& +\frac 12g_{\mu \nu }(x)g_{\rho \sigma }(y)\nabla _\beta ^y\nabla
_\alpha ^xG^{+}(x,y)\nabla ^{y\beta }\nabla ^{x\alpha }G^{+}(x,y)\Bigr].
\end{eqnarray}

Thus, we need to determine the Wightman function $G^{+}(x,y)\equiv
\left\langle 0\right| \hat{\phi }(x) \hat{\phi }(y)\left|
0\right\rangle $ for our chosen state. If we consider the Euclidean vacuum
for a massive minimally coupled field and take the massless limit, the
Wightman function diverges. As already explained, this is connected to
the impossibility of having a Fock space built on a de Sitter invariant
vacuum in the massless case. To apply Garriga and Kirsten's special
treatment of the zero mode, we have to consider separately the
contribution
from the zero mode and that from all the rest. To compute the latter, one
can use the expression for the massive case, with the mass acting as a
regulator of the infrared divergence, subtract the contribution from the
zero mode, which contains all the infrared divergences, and then remove
the regulator, i.e., take the massless limit.

When $x$ and $y$ are spacelike separated, $G^{+}(x,y)=\frac 12\left(
G^{\left( 1\right) }(x,y)+G(x,y)\right) =\frac 12 G^{\left( 1\right)
}(x,y)= \Re G^{+}(x,y)$
since the commutator $G(x,y)\equiv \left\langle 0\right| [ \hat{%
\phi }(x),\hat{\phi }(y)] \left| 0\right\rangle $ vanishes for
causally disconnected points. In this case the noise kernel coincides with
expression (\ref{6}). We can also take advantage of the results in refs.
\cite{garriga93,allen87} to write:
\begin{equation}
\label{7}G^{\left( 1\right) }(x,y)=G_{NZM}^{\left( 1\right)
}(x,y)+\frac{H^2%
}{8\pi ^2}\left\langle 0\right| \hat{Q}^2\left| 0\right\rangle ,
\end{equation}
where
\begin{equation}
\label{8}G_{NZM}^{\left( 1\right) }(x,y)\equiv\frac R{48\pi ^2}\left[
\frac 1{1-Z(x,y)}-\ln \left( 1-Z(x,y)\right) -\ln (4\sin \eta _x\sin \eta
_y)-\sin^2\eta _x-\sin ^2\eta _y\right] .
\end{equation}
Several remarks are in order. $G_{NZM}^{\left( 1\right) }(x,y)$
corresponds
to the non-zero mode contribution whereas the contribution from the zero
mode reduces to $H^2/8\pi ^2 \,\left\langle 0\right| \hat{Q}%
^2\left| 0\right\rangle $ since $\hat{P}\left| 0\right\rangle =0$.
This
term is actually divergent but is independent of $x$ and $y$, so that it
will give no contribution to (\ref{6}). Furthermore, the term $\ln
(2\sin \eta _x)+\ln (2\sin \eta _y)+\sin ^2\eta _x+\sin ^2\eta _y$
will not contribute either, because in (\ref{6}) there are always
derivatives with respect to both $x$ and $y$ acting on $G^{+}(x,y)$.
Consequently, we only need to take into account the following part of the
two-point function $G^{(1)}(x,y)$:
\begin{equation}
\label{9}{\cal G}(x,y)\equiv \frac R{48\pi ^2}\left[ \frac 1{1-Z(x,y)}-\ln
\left( 1-Z(x,y)\right) \right] .
\end{equation}
If we consider spacelike separated points which are geodesically
connected,
we can use expression (\ref{0}) to derive the following results:
\begin{eqnarray}
\label{10}
\nabla _\mu ^x\sigma (x,y)=s(x,y)s_\mu (x), \\
\label{11}
\nabla _\rho ^y\nabla _\mu ^x\sigma (x,y)=s_\rho (y)s_\mu (x),
\end{eqnarray}
where $s^\mu (x)$ is the unit vector tangent at point $x$ to the geodesic
joining $x$ and $y$. The derivatives of $Z(x,y)$ are then
\begin{eqnarray}
\label{12}
\nabla _\mu ^xZ(x,y)=
\sqrt{\frac R{12}\left( 1-Z^2(x,y)\right) }\;s_\mu (x), \\ \nabla _\rho
^y\nabla _\mu ^xZ(x,y)=\frac R{12}\left( 1-Z^2(x,y)\right) s_\rho (y)
s_\mu (x).
\end{eqnarray}
Thus, after a few algebraic manipulations, we have
\begin{equation}
\label{13}
\nabla _\rho ^y\nabla _\mu ^xG^{+}(x,y)=
\nabla _\rho ^y\nabla _\mu ^x{\cal G}(x,y)=\frac{H^4}{4\pi ^2}
\frac{1+4Z(x,y)}{\left( 1-Z(x,y)\right) ^2}\,s_\rho (y)s_\mu (x) .
\end{equation}
Substituting this result into (\ref{6}), we get our final result for the
noise kernel corresponding to spacelike separated points:
\begin{eqnarray}
\label{14}
\left\langle
\hat{t}_{\mu \nu }(x)\hat{t}_{\rho \sigma }(y)\right\rangle
=\frac{H^8}{16\pi ^4}\left( \frac{1+4Z(x,y)}{\left( 1-Z(x,y)\right)
^2}\right)
^2 && \bigl[ 2 s_\mu (x)s_\nu (x)\,s_\rho (y)\,s_\sigma (y)+g_{\mu \nu
}(x)\,s_\rho (y)\,s_\sigma (y) \nonumber \\
&&  +g_{\rho \sigma }(y)s_\mu
(x)s_\nu (x)+\frac 12g_{\mu \nu }(x)g_{\rho \sigma }(y)\bigr] .
\end{eqnarray}

\section{CONCLUSION}

Comparing (\ref{14}) with the ``square'' of (\ref{2}) one realizes that
the
contribution from the stress-energy fluctuations is at least as important
as that coming from the expectation value. Note that in order to compare
both
expressions it is convenient to consider the metric and the tangent vector
components as referred to an orthonormal base.
Of course, given that $H^4 << m_p^2 H^2$, this stochastic back-reaction
source is still much smaller than the dominant term which
drives inflation (usually coming from the potential of the inflaton scalar
field) and which is responsible for the near de Sitter geometry of the
spacetime. It seems interesting to point out that the
expectation value (\ref{2}) yields a negative value for the energy density
because of two facts. First, Ford and collaborators (see for instance ref.
\cite{kuo93}) have suggested that one may expect important stress-energy
fluctuations especially in those cases where the energy density is
negative. Second, this is in agreement with some of the results found in
\cite{abramo97b}.

It is only the contribution to the variations of the potential coming from
fluctuations of the inflaton field which are usually considered when
addressing the generation of large-scale gravitational inhomogeneities.
Expressions of the sort $\left\langle \phi ^2(x)\right\rangle
=\frac{H^3}{4\pi ^2}\Delta t$ or related ones \cite{ford82/linde82},
where proper infrared (related to initial conditions) and ultraviolet
cut-offs (only scales larger than the horizon are considered:
$\lambda_{phys}>H^{-1}$) have been imposed, are of frequent use in the
literature \cite{linde90,starobinsky82}.
These fluctuations for the smeared inflaton field can be interpreted as if
the smeared field were undergoing a sort of ``Brownian motion''
\cite{vilenkin83}. Such an
interpretation still applies to Garriga and Kirsten's vacuum as they show
when analyzing the dispersion of the smeared field \cite{garriga93}.
The fluctuations associated to a given cosmic scale arise from the
stochastic contribution, when leaving the horizon, from the modes of the
inflaton field perturbations with a wavelength corresponding to that
scale. The usual treatment just deals with first order variations of the
potential, which are assumed to provide the dominant contribution, and
does not take into account the back-reaction contribution from the kinetic
terms of the quantum perturbations of the inflaton field \cite{linde90}.
The latter can be modeled, at least partially, by a free massless
minimally coupled quantum scalar field. This is precisely the case we
considered in this paper. Furthermore, as mentioned above, we have found
that the contribution from the fluctuations of the stress-energy tensor
can be as important as the expectation value itself. Within the
inflationary context, this contribution to back reaction becomes
especially relevant above the ``self-reproduction'' scale and could have
important implications for the stochastic inflation approach, where this
effect is never taken into account.

One can try to extract information about large-scale fluctuations from
correlations
for spacelike separated points which are beyond the horizon distance \cite
{hawking82}. Unfortunately, for such points $Z(x,y)<-1$ and there is no
geodesic connecting them, thus one cannot directly use our results, which
were obtained using (\ref{10}) and (\ref{11}), which in turn rely on
the particular expression (\ref{0}) for geodesically connected points. If
one is interested in the $Z(x,y)<-1$ case, then the general expression
(\ref{00}) should be used. The relative importance of the fluctuations
that we have found for geodesically connected points encourages us to
pursue this research further.

\section*{ACKNOWLEDGMENTS}

It is a pleasure to thank Esteban Calzetta, Larry Ford, Jaume Garriga,
Bei-Lok Hu, Rosario Mart\'\i n and Xavier Montes for useful comments and
stimulating discussions. This work has been partially supported by the
CICYT
Research Project number AEN95-0590, and the European Project number
CI1-CT94-0004. A.R. also acknowledges support of a grant from the
Generalitat de Catalunya.


\section{Introduction} \label{sec:Intro}
Some of the most remarkable predictions of quantum field theory arise from
zero point fluctuations of quantum states. One of the best known examples of
this is the Unruh effect \cite{Fulling:73,Davies:75,Unruh:76}, in
which an accelerating detector measures the zero point fluctuations of the
inertial (Minkowski) vacuum state and finds they have a thermal spectrum.
Another is the Casimir effect \cite{Casimir:48}, in which the walls (or
boundaries) of a box experience a net force due to the difference between the
zero point fluctuations of the states inside the box and out.

It seems reasonable, then, to expect accelerating boundaries to produce
interesting effects, and indeed they do. Following studies of the Casimir
effect between moving mirrors in 1+1 dimensions by Moore \cite{Moore:70},
DeWitt pointed out that the single moving mirror problem would be interesting
and could be solved exactly in (1+1)D\cite{DeWitt:74} . Fulling
\cite{Fulling:73} and DeWitt\cite{DeWitt:75} have shown that a {\em uniformly}
accelerating mirror will indeed alter the quantum state in the vicinity of the
mirror. However, a far more interesting result was obtained by Davies and
Fulling\cite{FullingDavies:76,DaviesFulling:77} for a mirror
experiencing {\em non-uniform} acceleration in (1+1)D. Such a mirror actually
emits fluxes of quantum radiation, as though the mirror were knocking zero
point quanta out of the vacuum and off to infinity. More precisely, they found
that a mirror with 2-velocity $u^\mu = d x^\mu/d\tau$ ($u\cdot u = -1$)
and acceleration $a^\mu$ emits a flux
\begin{equation}
   \frac{dE}{d\tau} = - \langle T_{\mu\nu}\rangle u^\mu n^\nu =
   -\frac{\hbar}{12\pi} \frac{d}{d\tau}~(a\cdot n),
\label{eq:1Dflux}
\end{equation}
in the direction of a unit spatial vector $n^\nu$ orthogonal to $u^\mu$. This
holds for either choice (``left'' or ``right'') of $n^\mu$.  Thus, a mirror
whose acceleration is increasing (algebraically) toward the right will emit
a stream of negative energy to the right and a numerically equal positive
stream to the left.

The implications of this result are intriguing. Davies\cite{Davies:72} and
Ford\cite{Ford:78} first raised the possibility that the negative-energy flux
from a moving mirror could be used to cool a hot body and thus violate the
second law of thermodynamics in a quantum context, and this paradox was
further discussed by Deutsch, Ottewill and Sciama\cite{Deutschetal:82}.
Limitations on the extent of such violations in flat spacetime (``quantum
interest'') have been formulated by Ford and Roman\cite{FordRoman:99} and
others (\cite{Flanagan:97}, \cite{Pretorius:99}, {\em etc.}).

More recently, Anderson\cite{Anderson:94} has used this result in the context
of the Geroch gedankenexperiment\cite{Bekenstein:81}. In this experiment, a
box with mirrored walls is filled with radiation and lowered adiabatically
toward a black hole. Unruh and Wald\cite{UnruhWald:82} have shown that such a
box is subject to a buoyancy force and will eventually reach a floating point
above the black hole. Anderson has examined this further and shown that the
ground state inside the box is the Boulware state, whose energy (which becomes
increasingly negative as the box descends) is fed by Davies-Fulling fluxes
from the reflecting walls. This accounts for the buoyancy felt by the box.

The volume of literature on moving mirrors is impressive, but it bears noting
that all the results mentioned above are obtained in 1+1
dimensions. Indeed, if one includes the result for moving mirrors in curved
space-times obtained by Ottewill and Takagi\cite{OttewillTakagi:88} with those
reviewed above, the (1+1)D theory of moving mirrors can be considered
essentially complete. This is due largely to the conformal properties of
quantum field equations in (1+1)D, which allow boundaries, and even
space-time itself, to be flattened, thereby enabling one to obtain results for
complicated geometries from those for much simpler geometries.

This is not the case in 3+1 dimensions, where only partial results are
available. The case of constant acceleration has been solved for both
plane\cite{CandelasDeutsch:77,CandelasDeutsch:78} and
spherical\cite{FrolovSerebriany:79} mirror geometries. Ford and
Vilenkin\cite{FordVilenkin:82} have extended the plane mirror result to
include non-constant acceleration for the case when the acceleration and its
derivatives are small. More recently, Hadasz {\em et al.}\cite{HadaszEtc:98}
have considered arbitrary (radial) motion of a spherical
mirror, but have restricted their attention to the ``S-wave approximation''
where only spherically symmetric modes are considered. Because of this
restriction, their result can be related to the 1+1 dimensional results of
Davies and Fulling\cite{FullingDavies:76,DaviesFulling:77}.

Consideration of quasi-stationary processes (e.g. slow descent of a mirror in
a strong gravitational field) requires knowledge of the flux emitted by a
mirror whose acceleration is changing slowly, though it may be large. Our
objective in this paper is to derive an interesting and relatively simple
result of this type in 3+1 dimensions. The central tool is the use of a
Green's function perturbation technique.  Evaluating the perturbation is much
more manageable if the Green's functions for the unperturbed problem are
available in closed form. This is actually the case for a uniformly
accelerated spherical mirror, as shown by Frolov and
Serebriany\cite{FrolovSerebriany:79}. The mirror's history is then a
three-dimensional pseudo-sphere of radius $b$, say, and the unperturbed
problem is just the Minkowski-signature analogue of finding the
four-dimensional electrostatic potential of a point charge in the presence of
an earthed conducting 3-sphere of radius $b$. This is easily solved by the
method of images.

Our objective in this paper is to solve the perturbed Frolov-Serebriany
problem, {\em i.e.} to examine the effect of small spherically symmetric
non-uniformities in the acceleration of a spherical mirror.

It is well to stress at the outset that the solution for a plane mirror cannot
be derived from ours by a straightforward limiting process. The single
parameter $b$, whose reciprocal gives the unperturbed acceleration, also gives
the minimum radius attained by the mirror as seen from its center. Thus, the
plane limit $b\rightarrow\infty$ is inseparable from small acceleration.
(There are reasons to expect the planar case to be considerably more
complicated. Formally, the Wightman Green's function is now an infinite sum of
McDonald (Bessel) functions. Geometrically, any light ray reflected
non-orthogonally of a uniformly accelerated plane mirror will re-encounter the
mirror an infinite number of times; in the spherical case there is just one
encounter.)

Also, we concern ourselves only with calculating the outward flux, which we
expect to be the most interesting stress-energy component. In fact, it turns
out to be somewhat more interesting than one might expect. We find that it
has a remarkable property; although it could, in principle, depend on
the entire retarded history of the mirror, to first order in the mirror
perturbation it depends only the behavior of the mirror at the most recent
retarded time, {\em i.e.} it is local.

This article is organized as follows: in Section \ref{sec:UniAcc} we review
the Frolov-Serebriany result for a mirror expanding with uniform acceleration.
In Section \ref{sec:Perturb} we present our Green's function perturbation
scheme, and in Section \ref{sec:Eval} we use it to evaluate the corrections to
the Frolov-Serebriany Green's function, with some of the more cumbersome
details relegated to Appendix \ref{sec:AppA}. Section \ref{sec:Flux} is
concerned with calculating the quantum flux from these perturbations, with
details again left to Appendices (\ref{sec:AppB} and \ref{sec:AppC}).
Finally, in Section \ref{sec:Conclusion} we offer some concluding remarks.

\section{Uniformly accelerating spherical mirror}\label{sec:UniAcc}
In the case where the mirror's acceleration is uniform, the Green's functions
for the massless fields can be obtained in simple closed form by the method of
images, as noted by Frolov and Serebriany\cite{FrolovSerebriany:79}. In this
Section we shall briefly review these results.

Consider first the static potential due to a point charge $q'$ in Euclidean
4-space at a distance $R'$ from the center of an earthed conducting 3-sphere
of radius $b$. The Dirichlet boundary condition can be reproduced by
introducing a co-radial image charge $q''=-(b/R')^2q'$ at radius $R''=b^2/R'$.

In the Lorentzian analogue of this problem, we are concerned with Green's
functions for the wave equation $\Box \varphi = 0$ in Minkowski space-time,
with Dirichlet boundary conditions on the pseudo-sphere ({\em i.e.} the
time-like hyperboloid of one sheet) $R=b$, where now
\begin{equation}
   R^2 \equiv \eta_{\mu\nu}x^\mu x^\nu=x^2+y^2+z^2-t^2=\rho^2+z^2-t^2,
\label{eq:rad}
\end{equation}
in a self-evident notation.

The pseudo-sphere $R=b$ represents the history of a spherical mirror of radius
$b$ (constant as measured in its instantaneous rest frame), whose center is
fixed at the spatial origin $x=y=z=0$ and which moves with uniform
acceleration $a=b^{-1}$.

The image construction gives for the retarded Green's function
$G_{ret}(x,x')$, satisfying
\begin{equation}
   \Box G_{ret}(x,x')=-\delta^4(x,x'),
\label{eq:reteq}
\end{equation}
the expression
\begin{equation}
   G_{ret}(x,x')=\frac{1}{2\pi}~\theta(t-t')\left\{\delta[(xx')^2]-\left(
      \frac{b}{R'}\right)^2\delta[(xx'')^2]\right\}.
\label{eq:Gretconst}
\end{equation}
Here $\theta$ is the unit step (Heaviside) function, $\delta$ is the Dirac
distribution, $(xx')^2$ is the squared Minkowski interval
\begin{equation}
   (xx')^2\equiv\eta_{\mu\nu}(x^\mu-{x'}^\mu)(x^\nu-{x'}^\nu),
\label{eq:Minkint}
\end{equation}
and the image source is located at
\begin{equation}
   {x''}^\mu = \left(\frac{b}{R'}\right)^2{x'}^\mu.
\label{eq:imagepoint}
\end{equation}

Similarly, the Wightman function
\begin{equation}
   W(x,x')=\langle0|\varphi(x)\varphi(x')|0\rangle
\label{eq:Wdef}
\end{equation}
for a massless scalar field takes the form
\begin{equation}
   W(x,x')=\frac{1}{4\pi^2}\left\{\frac{1}{(xx')^2+i(t-t')\epsilon} -\left(
      \frac{b}{R'}\right)^2\frac{1}{(xx'')^2+i(t-t'')\epsilon}\right\},
\label{eq:Wconst}
\end{equation}
with $\epsilon \rightarrow +0$.

\section{Nearly uniform acceleration: perturbing the boundary}
\label{sec:Perturb}
The corresponding Green's functions for a spherical mirror whose acceleration
is slightly non-uniform can be derived from the preceding results by
superposing the effect of a small perturbation on the history of the mirror,
{\em i.e.}, the time-like 3-space $\Sigma$ on which Dirichlet boundary
conditions are imposed.

Consider generally the problem of solving
\begin{equation}
   \Box \Phi = 0, \hspace{1cm} \Phi=0 ~~\mbox{on}~~ \Sigma.
\label{eq:BVP}
\end{equation}
Suppose that $\Sigma$ is a small perturbation of a simpler time-like 3-space
$\Sigma_0$, obtained by displacing $\Sigma_0$ a distance $\delta n(x)$ along
its outward normal $n$, and that we know the solution $\Phi_0$ of the problem
\begin{equation}
   \Box \Phi_0 = 0, \hspace{1cm} \Phi=0 ~~\mbox{on}~~ \Sigma_0,
\label{eq:unpertBVP}
\end{equation}
with the same initial boundary conditions.

Then (\ref{eq:BVP}) can be reformulated as a problem with boundary conditions
specified on the unperturbed boundary $\Sigma_0$:
\begin{equation}
   \Box \Phi = 0, \hspace{1cm} \Phi=-\frac{\partial\Phi_0}{\partial n}~ \delta n(x)
      ~~\mbox{ on }~~ \Sigma_0.
\label{eq:pertBVP}
\end{equation}

The causal solution for the perturbation $\delta \Phi \equiv \Phi - \Phi_0$
follows from Green's identity:
\begin{equation}
   \delta \Phi(x') = \int_{\Sigma_0} d\Sigma~\frac{\partial G_{ret}(x',x)}{\partial n}
      ~\delta n(x)~\frac{\partial}{\partial n} \Phi_0(x),
\label{eq:fieldpert}
\end{equation}
where $G_{ret}$ is the retarded Green's function for the unperturbed boundary
value problem (\ref{eq:unpertBVP}) and $x$ is in $\Sigma_0$. Equation
(\ref{eq:fieldpert}) defines a linear operation applied to $\Phi_0$, which we
shall write for brevity as
\begin{equation}
   \delta \Phi(x') = L_{\delta n}(x')~\Phi_0(\cdot).
\label{eq:pertop}
\end{equation}

To obtain the effect of the perturbation on the Wightman function
(\ref{eq:Wdef}), we note that it can be written as a mode sum
\begin{equation}
   W(x,x') = \int \frac{d^3 k}{(2\pi)^3} ~f_{\boldmath{\mbox{${\scriptstyle k}$}}} (x) ~\overline{f_{\boldmath{\mbox{${\scriptstyle k}$}}}}(x'),
\label{eq:Wsum}
\end{equation}
where $\{ f_{\boldmath{\mbox{${\scriptstyle k}$}}}(x)\}$ is a complete set of initially positive frequency
solutions of (\ref{eq:BVP}) and the bar denotes complex conjugation. Applying
(\ref{eq:pertop}) to each mode separately and summing the results yields
\begin{equation}
   \delta W(x,x') = L_{\delta n}(x)W_0(\cdot,x')+L_{\delta n}(x')W_0(x,\cdot),
\label{eq:modepert}
\end{equation}
where $W_0$ denotes the unperturbed Wightman function given by
(\ref{eq:Wconst}).

\section{Wightman function for nearly uniform acceleration}\label{sec:Eval}
Evaluation of the expression (\ref{eq:modepert}) for $\delta W$ in the case
where the mirror's acceleration departs slightly from uniformity is somewhat
lengthy but straightforward. Here, we outline the results of this calculation
reserving the technical details for Appendix \ref{sec:AppA}.

It will be assumed that the mirror remains spherical as viewed by an observer
at its center $x=y=z=0$. Then its history is
\begin{equation}
   R=b(1+f(t)), \hspace{1cm} |f(t)| \ll 1
\label{eq:boundpert}
\end{equation}
in terms of this observer's time $t$, where $f(t)$ is arbitrary but small.
The corresponding advanced and retarded times are written
\begin{equation}
   v=t+r,\hspace{1cm}u=t-r,\hspace{1cm}r=\sqrt{x^2+y^2+z^2}.
\label{eq:uvdef}
\end{equation}

The radial energy flux measured by a stationary observer outside the mirror is
\begin{equation}
   F= T_{uu}-T_{vv}.
\label{eq:statflux}
\end{equation}
Each of these terms can be dealt with by a similar procedure. Let us consider
$T_{uu}$; it is evident from (\ref{eq:Wdef}) that its expectation value at an
event $p_1$ outside the mirror is
\begin{equation}
   \langle T_{uu}(p_1)\rangle = \left[ \partial_{u_1} \partial_{u_2} W(p_1,p_2)
      \right]_{p_2=p_1}
\label{eq:fluxdef}
\end{equation}
for a minimally coupled massless scalar field, and a similar but more
complicated expression for conformal coupling (see (\ref{eq:quantSETconf})
below). Note that regularization and symmetrization are not needed to evaluate
this component of the stress-energy.

We can now present the results for $\delta W$.  The spherical symmetry of
(\ref{eq:boundpert}) allows us to take both $p_1$ and $p_2$ in the $z-t$ plane
($x=y=0$).  Calculation shows (see Appendix \ref{sec:AppA}) that the first
term of (\ref{eq:modepert}) contributes
\begin{equation}
   \frac{8\pi^2\zeta^2}{(R_1^2-b^2)(R_2^2-b^2)} L_{\delta n}(p_1)W(\cdot,p_2)
      = \frac{z_1z_2}{\zeta} \int_{-\infty}^{t_1^*} dt \frac{f(t)}{(t-t_0)^3}
      -\frac{z_1z_1^*}{(R_1^2-b^2)(t_1^*-t_0)^2}
\label{eq:Wterm1}
\end{equation}
where $\zeta$ and $t_0$ are defined by
\begin{equation}
   \zeta=z_1t_2-z_2t_1, ~~~~~\zeta t_0 = \frac{1}{2}(z_1-z_2)(z_1z_2-b^2)
      +(z_1t_2^2-z_2t_1^2),
\label{eq:zetadef}
\end{equation}
and $p_i^*$ (with coordinates $x_i^*,t_i^*$) is the event nearest to $p_i$
($i=1,2$) at which the past light-cone of $p_i$ intersects the (unperturbed)
mirror (see Fig. 1).
\begin{figure}[hbtp]
\begin{center}
\leavevmode \epsfbox{2d_intersect.eps}
\end{center}
\caption{The $z-t$ plane. The mirror profile is the hyperbola. $p_1$ and $p_2$
are the two events at which the Wightman function is to be evaluated and $v_0$
is their common advanced time.  $p_1^*$ and $p_2^*$ are the intersections of
their past light-cones with the mirror in the $z-t$ plane.
\label{fig:1}}
\end{figure}

The contribution of the second term in (\ref{eq:modepert}) is obtained by
interchanging $p_1$ and $p_2$ in (\ref{eq:Wterm1}). Because $\zeta$ is an odd
function of $p_1,~p_2$ ($t_0$ is even), the term involving the integral
changes sign. Thus, the sum of the two contributions,
\begin{equation}
   -\frac{8\pi^2\zeta^2}{(R_1^2-b^2)(R_2^2-b^2)}\delta W(p_1,p_2) =
      \frac{z_1z_2}{\zeta} \int_{t_1^*}^{t_2^*} dt \frac{f(t)}{(t-t_0)^3}
      +\sum_{i=1}^2\frac{z_iz_i^*}{(R_i^2-b^2)(t_i^*-t_0)^2}f(t_i^*),
\label{eq:deltaW}
\end{equation}
depends only on the mirror's history between the retarded times $t_1^*$ and
$t_2^*$. Particle and anti-particle modes interfere destructively in the case
of a spherical mirror to eliminate the effects of the past history and
produce (when we take the coincidence limit $p_2 \rightarrow p_1$) a
purely local expression for the stress-energy.

To perform the partial derivatives $\partial_{u_1}$ and $\partial_{u_2}$ in the
expression (\ref{eq:fluxdef}) for~~$T_{uu}$, we require mutual independence of
the $u$-coordinates of the events $p_1$ and $p_2$. But these are the only
coordinates which need be independent. To evaluate $T_{uu}$, it suffices to
consider $\delta W(p_1,p_2)$ in a partial pre-coincidence limit $v_1=v_2$, as
in Fig. \ref{fig:1}.

These remarks in principle apply, {\em mutatis mutandis}, also to the
evaluation of~~$T_{vv}$, but there is a complication. In the pre-coincidence
limit $u_2\rightarrow u_1$, the points $p_1^*$ and $p_2^*$ tend to coincidence
with each other and with the point on the mirror having coordinate $t_0$, so
that the integrand in (\ref{eq:deltaW}) becomes infinite while the interval of
integration shrinks to zero. The evaluation of $T_{vv}$ is discussed further in
Appendix \ref{sec:AppC}. Here, we merely note that $T_{vv}$ makes no
radiative contribution (proportional to $r^{-2}$) to the flux
(\ref{eq:fluxdef}). This follows at once from the identity
\begin{equation}
   \partial_u\left(r^2 \partial_u \left(r^2T_{vv}\right)\right) =  \partial_v
      \left( r^2 \partial_v \left( r^2 T_{uu}\right)\right),
\label{eq:consEq}
\end{equation}
which is a consequence of the conservation of $T_{\mu \nu}$ and the vanishing
of the trace $T^\alpha_\alpha$ for a conformal scalar field in flat space. In
(\ref{eq:consEq}), the derivative $\partial_v$ increases the fall-off with
distance, but this does not hold for $\partial_u$, which can operate on the
retarded displacement $f(u)$ in (\ref{eq:boundpert}). Thus, $T_{vv}$ falls off
more strongly than $T_{uu}$. The detailed calculation (Appendix
\ref{sec:AppC}) shows that $T_{vv}\sim r^{-6}$ as $r\rightarrow\infty$.

\section{Flux from a non-uniformly accelerating mirror}\label{sec:Flux}
The expectation value of the stress-energy is derivable from the partial
derivatives of the Wightman function $W(x,x')=W_0+\delta W$ in the coincidence
limit, with the unperturbed part $W_0$ given by (\ref{eq:Wconst}) and the
perturbation $\delta W$ by (\ref{eq:deltaW}). The unperturbed part becomes
singular in the limit $x' \rightarrow x$, but is easily regularized by
subtracting the value of $W_0$ in free space without the mirror, {\em i.e.},
the first term of (\ref{eq:Wconst}), leaving the second (image) term as the
sole contribution to $(W_0)_{reg}$. The perturbation $\delta W$ is regular in
the coincidence limit.

For a massless scalar field, two different stress tensors are commonly
considered: (a) the minimal stress-energy, given classically by
\begin{equation}
   \left(T_{\mu \nu}\right)_{min} = \varphi_{,\mu} \varphi_{,\nu} -
      \frac{1}{2} g_{\mu \nu} (\nabla \varphi)^2
\label{eq:classSETmin}
\end{equation}
and quantum mechanically by
\begin{equation}
   \langle T_{\mu \nu}(x)\rangle_{min} = \left[(\partial_{\mu}\partial_{\nu'}-\frac{1}{2}
      g_{\mu\nu}~\partial^{\alpha}\partial_{\alpha'})W_{reg}(x,x')\right]_{sym~;~x'=x},
\label{eq:quantSETmin}
\end{equation}
in which ``sym'' indicates symmetrization in (x,x') and in the partial
derivatives; (b) the conformal (trace-free) stress-energy, defined classically
by
\begin{equation}
   \left(T_{\mu \nu}\right)_{\mathit conf} = \frac{2}{3}\varphi_{,\mu}
      \varphi_{,\nu}- \frac{1}{3}\varphi\varphi_{,\mu\nu}-
      \frac{1}{6}g_{\mu \nu} (\nabla \varphi)^2
\label{eq:classSETconf}
\end{equation}
and quantum mechanically by
\begin{equation}
   \langle T_{\mu \nu}(x)\rangle_{\mathit conf}=\frac{1}{3}\left[
      (2\partial_{\mu}\partial_{\nu'}- -\partial_{\mu'}\partial_{\nu'}
      -\frac{1}{2} g_{\mu\nu}~\partial^{\alpha}\partial_{\alpha'})
      W_{reg}(x,x')\right]_{sym~;~x'=x},
\label{eq:quantSETconf}
\end{equation}

We begin by reviewing the Frolov-Serebriany\cite{FrolovSerebriany:79} results
for {\em uniform} acceleration. Differentiating the regularized form of
(\ref{eq:Wconst}), we easily find
\begin{eqnarray}
   \langle T^{(0)}_{\mu \nu}(x)\rangle_{min} &=& - \frac{b^2}{\pi^2(R^2-b^2)^4}
      \left(x^{\mu}x^{\nu}-\frac{1}{2}g^{\mu\nu}R^2\right),
\label{eq:constSETmin}\\
      \langle T^{(0)}_{\mu \nu}(x)\rangle_{\mathit conf} &=& 0.
\label{eq:constSETconf}
\end{eqnarray}
This last result is quite remarkable, because the conformal stress is not
likely to vanish for a spherical mirror at rest (it certainly does not for
electromagnetic fields \cite{Boyer:68,Davies:72,MiltonEtc:78}).
It appears that the effects of uniform acceleration exactly cancel the static
Casimir stresses.

The effects of non-uniform acceleration are more complicated. We shall simply
quote the result for the conformal radial out-flux $\langle T_{uu}
\rangle_{\mathit conf}$ at a point $(r,t)$ outside the mirror, leaving to
Appendix \ref{sec:AppB} an outline of the derivation:
\begin{equation}
   \langle T_{uu}(t,r) \rangle_{\mathit conf} = - \frac{1}{1440 \pi^2}
      \frac{1}{rv} \left\{\frac{q}{m^3n}\frac{d^2\alpha}{d\chi^2} -
      \frac{4}{m^3n^2r} (npr-q^2)\frac{d \alpha}{d \chi} +
      \frac{2 s^2}{mn^3r^2}\left[q(\alpha+f)+p\frac{df}{d\chi}\right]\right\}.
\label{eq:SETpert}
\end{equation}
The notation is as follows: the advanced and retarded times, $v$ and $u$ are
defined as in (\ref{eq:uvdef}), and we write
\begin{eqnarray}
   m&=&-\frac{u}{2},~~~n~=~\frac{1}{2v}(R^2-b^2)~=~-\frac{1}{2}(u+b^2/v),
\nonumber\\
   p&=&\frac{1}{8}(u^2-b^2),~~~q~=~\frac{1}{8}(u^2+b^2),~~~
      s~=~-\frac{1}{2}(v+b^2/v);
\label{eq:notdefs}
\end{eqnarray}
$\chi$ is the pseudo-angle along the (unperturbed) mirror trajectory ({\em
i.e.}, $\tau=b\chi$ is the mirror's proper time). Equation
(\ref{eq:boundpert}), giving the trajectory of the perturbed mirror, is now
written $R=b(1+f(\chi))$, and
\begin{equation}
   \alpha = f''(\chi) - f(\chi)
\label{eq:alphadef}
\end{equation}
is a measure of the non-uniformity of the acceleration $A$, which is given by
\begin{equation}
   A = (1+\alpha)/b.
\label{eq:acceldef}
\end{equation}
In (\ref{eq:SETpert}), $\chi$ refers to the pseudo-angle at the point $p^*$,
which is the nearest retarded point to (r,t), as in Fig. \ref{fig:2}.
The corresponding expression for the canonical flux is too long to reproduce
here, and is also deferred to Appendix B.

\begin{figure}[hbtp]
\begin{center}
\leavevmode \epsfbox{3d_intersect.eps}
\end{center}
\caption{Intersection of the history of a uniformly accelerating sphere
(hyperboloidal cylinder) and the past null cone of an exterior point $p$. The
intersection (which lies entirely within the shaded plane) is represented by
the bold curve. The nearest retarded point to $p$ on the mirror's worldsheet,
$p^*$, is the only point whose perturbation contributes to the flux at $p$.
This figure has been dimensionally reduced; each point represents a circle.
\label{fig:2}}
\end{figure}

The limit $b \rightarrow \infty$ corresponds to a slowly accelerating, nearly
plane mirror. This was the case studied by Ford and
Vilenkin\cite{FordVilenkin:82}. It is straightforward to show that in this
limit our result (\ref{eq:SETpert}) for the conformal flux (and also our
result for the canonical flux) reduces to the expressions they give.

To obtain a more intuitive grasp of the physical meaning of the complex
expression (\ref{eq:SETpert}), we can evaluate the flux $F$ radiated at
retarded time $u=-b$ ~~-~~ {\em i.e.} when the mirror is near its minimum
radius $r_M \approx b$ ~~-~~ as measured by a stationary observer at radius
$r\gg b$ and the same retarded time. Using (\ref{eq:statflux}), taking the
appropriate limit of (\ref{eq:SETpert}), and noting that $T_{vv}$ does not
contribute to the flux to leading order (as discussed at the end of Section
\ref{sec:Eval}) we find
\begin{equation}
   F=-\frac{\hbar}{720 \pi^2}\left(\frac{R_0}{r}\right)^2\left\{A
      \frac{d^2 A}{d \tau^2}+2A^3\left(A-\frac{1}{R_0}\right)\right\},
\label{eq:flux}
\end{equation}
where $R_0=b(1+f(0))$ is the proper radius of the mirror at the time of
emission $\chi = 0$, and we have restored Planck's constant to display the
correct dimensionality. We recall that this perturbative result is correct
to linear order in deviations from uniform acceleration $(A-1/R)$, $dA/d\tau$,
$d^2A/d\tau^2$, but the acceleration $A$ itself is arbitrary.

\section{Concluding Remarks}\label{sec:Conclusion}
Our chief interest in this paper has been in the quantum flux radiated by the
mirror and we have explicitly computed only those components of the
stress-energy tensor from which it arises. However, the remaining components
can be derived straightforwardly (though with some labour) from our expression
(\ref{eq:deltaW}) with the methods of Appendix \ref{sec:AppB}. It would be
useful to have these to round out the picture.

The relevant Green's functions for an unperturbed, uniformly accelerating
spherical mirror have a simple closed form, and this has enormously simplified
our perturbative calculation. This simplification is bought at a price: we are
limited to spherical mirrors whose acceleration $A$ and proper radius $R$ are
nearly reciprocals. We cannot decouple the plane limit $R\rightarrow \infty$
from the limit of small acceleration, and cannot disentangle curvature
(Casimir) effects from the effects of acceleration.

It is evident that much remains to be done before we can claim anything
approaching a comprehensive understanding of the quantum dynamics of
three-dimensional mirrors.
\acknowledgements{We are indebted to Valeri Frolov and Tom Roman for
discussions and to the latter for calling our attention to the prior work of
Ford and Vilenkin \cite{FordVilenkin:82} on slowly accelerating plane mirrors.

This research was supported by the Canadian Institute for Advanced Research
and by NSERC of Canada.}