%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentstyle[prl,aps,twocolumn]{revtex}
%
%%%%%%%%%%%%%%%%%%%%%%%%% LAYOUT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%% Newcommands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\AdS}{\mbox{{AdS}$_n\,$}}
\newcommand{\RR}{{\bf R}}
\newcommand{\CC}{{\bf C}}
\newcommand{\NN}{{\bf N}}
\newcommand{\AdSG}{\mbox{SO$_0(2,n-1)$}}
\newcommand{\AdSGG}{\mbox{SO$(2,n-1)$}}
\newcommand{\CA}{{\cal A}}
\newcommand{\CH}{{\cal H}}
\newcommand{\CO}{{\cal O}}
\newcommand{\CW}{{\cal W}}
\newcommand{\rva}{{| 0 \rangle}}
\newcommand{\lva}{{\langle 0 |}}
\newcommand{\rphi}{{| \,\phi \rangle}}
\newcommand{\lphi}{{\langle \phi |}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Titlepage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\draft
\title{Hawking--Unruh Temperature and Einstein Causality
in Anti--de Sitter Space--Time}
%
\author{Detlev Buchholz}
\address{Institut f\"ur Theoretische Physik,
Universit\"at G\"ottingen, D--37073 G\"ottingen, Germany}
\author{Martin Florig and Stephen J.\ Summers}
\address{Department of Mathematics, University of Florida, Gainesville
FL 32611, USA}
%
\date{May, 1999}
%
\maketitle
%
%%%%%%%%%%%%%%% TEXT DES ARTIKELS %%%%%%%%%%%%%%%%%%%%
%
\begin{abstract}
If the vacuum is passive for uniformly accelerated observers in anti--de
Sitter space--time ({\em i.e.} cannot be used by them to operate a
{\em perpetuum mobile}), they will (a) register a unique value of the
Hawking--Unruh temperature and (b) find that
products of observables which are
localized in complementary wedge--shaped regions necessarily
commute with each other in the vacuum state.
Thus the stability properties of the vacuum induce a
``geodesic causal structure'' on this space--time
which seems incompatible with true interaction.
\end{abstract}
\pacs{PACS numbers: 04.62.+v, 11.10.Cd, 11.30.Er}
\narrowtext
\noindent Quantum field theory in anti--de Sitter space--time
(AdS) has recently
received considerable attention \cite{Spires}. It seems therefore
worthwhile to clarify in a model independent setting
the universal properties of such theories
which are implied by generally accepted and physically meaningful
constraints. We report here on the results of an
investigation of this question \cite{BuFlSu} which
brought to light an intimate
relation between stability properties of the vacuum, the value of the
Hawking--Unruh temperature registered by uniformly accelerated
observers, and locality properties of their respective observables.
As will be explained, these results hold in
any theory which allows one to specify the spacetime localization
properties of observables, and they suggest that quantum field theories
on AdS cannot describe true interaction.
We consider here AdS of any dimension $n \geq 3$. It can conveniently be
described in terms of Cartesian coordinates in the ambient space
$\RR^{n+1}$ as the quadric
$$
\AdS = \{x\in\RR^{n+1} :
x^2 \equiv x_0^2 - x_1^2 - \dots - x_{n-1}^2 + x_n^2
= R^2 \}
$$
with metric $g = \mbox{diag}(1,-1,\dots,-1,1)$ in diagonal form.
Its isometry group is O$(2,n-1)$ whose identity component will be
denoted by \AdSG.
Any quantum field theory on AdS has to determine the following
structure, which is indispensible for its physical interpretation:
\noindent (i) \ A Hilbert space $\CH$, describing the physical states,
and a continuous unitary representation $U$ of
the symmetry group \AdSG, acting on $\CH$ \cite{restriction}.
\noindent (ii) \ A set of operators which
describe the observables of the theory. Since these operators can
be added and multiplied, they generate an algebra $\CA$ which
will be called the algebra of observables \cite{observables}.
It is a fundamental feature of quantum field theory and its modern
ramifications that the spacetime localization properties of
observables enter into the formalism in an essential way.
One can tell whether an observable is localized about
a spacetime point, a string, loop, disk etc. Thus, given any
region $\CO \subset \AdS$, one can identify
those observables which are localized within that region
and define a corresponding subalgebra $\CA (\CO) \subset \CA$
which is generated by them. Since there exist no fewer observables
if the region becomes larger, one obviously has
$$
\CA (\CO_1) \subset \CA (\CO_2) \quad \mbox{if} \quad \CO_1 \subset
\CO_2.
$$
The localization of observables has
to be compatible with the action of the
spacetime symmetry group. If an observable $A$
is localized in the region $\CO$, say, its image
$U(\lambda) A U(\lambda)^{-1}$ under the action of
$\lambda \in \AdSG$
should be localized in the transformed region $\lambda \CO$.
The transformation properties of the
observables under the action of the
unitaries $U(\lambda)$ may be quite complicated, but
fortunately this detailed information is not needed here. All that matters
are the geometrical aspects of the transformations, which can be
summarized as follows.
\noindent (iii) For each region $\CO\subset \AdS$
and $\lambda \in \AdSG$
one has the equality (in the set--theoretic sense)
$$
U(\lambda) \CA (\CO) U(\lambda)^{-1} = \CA (\lambda \CO).
$$
We emphasize that we do not postulate here
from the outset local commutation relations of
the observables. For, in contrast to
the case of globally hyperbolic space--times,
the principle of Einstein causality does
not provide any clues
as to which observables on AdS should commute.
Instead, we will derive such commutation relations from
stability properties of the vacuum.
Since this stability aspect sheds also new
light on the phenomenon of the Hawking--Unruh
temperature, we explain this point in
somewhat more detail.
We begin by describing the observers in AdS which are of
interest here. Let $x_O \in \AdS$ be any point and let
$\lambda (t)$, $t \in \RR$, be any one--parameter subgroup of
$\AdSG$ such that $t \rightarrow \lambda (t) x_O$ is an
orthochronous curve \cite{past}. We interpret this
curve as the worldline of some observer
(assumed to be male for concreteness). Depending on the
initial data chosen, this observer will be geodesic or
experience some constant acceleration. Points in the
neighborhood of $x_O$ will in general also give rise
to orthochronous curves under the action of the
chosen subgroup of $\AdSG$, and we denote by $\CW$ the
connected neighborhood of $x_O$ in \AdS consisting of all
such curves. The region $\CW$, which can be all of AdS
or a subset of it, is that part of the space--time where
the observer can perform measurements. His
observables are described by the self--adjoint elements of
$\CA(\CW)$ and his dynamics is given by
$e^{\, itM} \equiv U(\lambda (t)) $ with generator (``Hamiltonian'')
$M$ \cite{parametrization}.
Let us discuss next what this observer can say about the
properties of the vacuum, described by the
unit vector $\rva \in \CH$. First, he should find that the
vacuum is in a stationary state with respect to his dynamics, {\em i.e.}
$$
M \, \rva = 0.
$$
Second, he can check whether the vacuum
is suitable to operate a {\it perpetuum mobile\/} of
the second kind. To this end he
would perform all kinds of operations in $\CW$ which
can be described by perturbations of the generator $M$ in
that region. Time--dependent perturbation theory
then tells us that the state of the vacuum changes
under the influence of such perturbations and is
described by a vector $V \rva$ for some unitary operator
$V \in \CA (\CW)$ after the perturbation has been turned off.
Since this challenge of the second law ought to fail, the
observer should find that, no matter what he does, the
energy of the final state is no less than that of
the initial vacuum, {\em i.e.} he cannot extract energy from the vacuum.
In formula form,
$$
\lva \, V^* M V \, \rva \geq \lva \, M \, \rva = 0
$$
for all {\it unitary\/} operators $V \in \CA (\CW)$. This
property of a state --- not being able to perform work in
a cyclic process --- is called passivity \cite{PuWo}.
Finally, the observer can test with the help of time
averages of his observables (yielding order parameters)
whether the vacuum is a mixture of different phases. Since the
vacuum is the most elementary system, the answer should be
negative, {\em i.e.} all order parameters should have sharp values.
This is the content of the relation (mixing property)
$$
\lim_{T \rightarrow \infty}
\frac{\mbox{\footnotesize $1$}}{\mbox{\footnotesize $T$}}
\int_0^T \! dt \, \lva A(t) B \rva =
\lva A \rva \, \lva B \rva \, ,
$$
for $A,B \in \CA(\CW)$, where
$A(t) \equiv e^{\, itM } A e^{\, -itM }$.
These basic features of the vacuum can be summarized
as follows.
\noindent (iv) The vacuum $\rva$ is, for all geodesic and all uniformly
accelerated observers, a stationary, passive and mixing state.
Let us now turn to the analysis of the implications
of these assumptions.
Here we profit from a deep result of Pusz and Woronowicz
for arbitrary quantum dynamical systems~\cite{PuWo}. In the present
context this result says that the vacuum vector $\rva$ is,
as a consequence of its passivity and mixing properties,
either a ground state for the dynamics
$M$ of the above observer,
or it satisfies, for some {\it a priori\/} unknown $\beta \geq 0$,
the Kubo--Martin--Schwinger (KMS) condition, which can
be presented in the form \cite{forms}
$$
\lva B e^{\, -\beta M} A^* \rva = \lva A^* B \rva ,
$$
for all $A,B \in \CA(\CW)$. In the latter case the
respective observer would interpret the vacuum
as a thermal equilibrium state at temperature $\beta^{-1}$.
It is convenient here to proceed to another formulation of the
above result. Making use of the fact that
$\lva A^* B \rva = \overline{\lva B^* A \rva}$ and that
the set of vectors $\CA ( \CW) \, \rva$ is dense in $\CH$
\cite{BuFlSu}, the above relation allows one to define an anti--unitary
operator $J$ on $\CH$ by setting~\cite{modular}
$$
J A \rva \equiv e^{\, -(\beta/2) M } A^* \rva \ \
\mbox{for} \ \ A \in \CA(\CW).
$$
Our main task will be to determine
the specific properties of this operator.
In order to simplify the necessary computations,
we consider the particular choice of region
$$
\CW = \{ x \in \AdS : x_1 > |x_0|\, , x_n > 0 \}
$$
on which the one--parameter subgroup of
boosts $\lambda_{0 1} (t)$, $t \in \RR$,
in the $0$--$1$--plane acts in an
orthochronous manner. For any $x_O \in \CW$, the curve
$t \rightarrow \lambda_{0 1} (t) x_O$ is the worldline of some
observer, as described above.
We shall determine the temperature felt by this
observer. Since the generator $M_{0 1}$ of
his dynamics is transfomed into $- M_{0 1}$
by the adjoint action of the rotation
$e^{\, i\pi M_{1 2}}$ in the $1$--$2$--plane,
it cannot be a positive operator. Hence $\rva$ is not a
ground state for this observer; it must therefore
satisfy the KMS--condition for some $\beta \geq 0$. So what is the
value of $\beta$ ?
To answer this question, we apply the methods in
\cite{BoBu}. We pick a region $\CO$ such that
$\lambda_{0 2} (s) \, \CO \subset \CW$
for the boosts $\lambda_{0 2} (s)$ in the
$0$--$2$--plane with sufficiently small parameter $s$.
By the group law in \AdSG \, we have
$$ e^{\, itM_{0 1}} e^{\, isM_{0 2}} =
e^{\, is(\cosh(t)M_{0 2} + \sinh(t) M_{1 2})} e^{\, itM_{0 1}}.
$$
Taking into account that the boost $e^{\, isM_{0 2}}$ is the dynamics
of some other observer, we infer from (iv)
that $\rva$ is invariant under its action.
Thus we get for any vector $\rphi \in \CH$ and
operator $A^* \in \CA(\CO)$
\begin{eqnarray*}
\lefteqn{ \lphi \, e^{\, itM_{0 1}}
\, e^{\, isM_{0 2}} A^* e^{\, -isM_{0 2}} \rva}
\\ & & = \lphi \,
e^{\, is(\cosh(t)M_{0 2} + \sinh(t) M_{1 2})}\, e^{\, itM_{0 1}}
A^* \rva.
\end{eqnarray*}
At this point we are in the position to combine the KMS--condition with
a basic result in the representation theory
of Lie groups, due to Nelson \cite{Ne}.
In the present situation this result can
be stated as follows: There exists a dense set of vectors
$\rphi \in \CH$ such that the corresponding vector functions
$$
u,v \rightarrow e^{\, i (u M_{0 2} + v M_{1 2})} \, \rphi
$$
are analytic in a fixed neighborhood
of the origin of $\CC$.
Returning to the analysis of the above equality,
we note that
according to condition (iii) the operator
$e^{\, isM_{0 2}} A^* e^{\, -isM_{0 2}}$
appearing in the matrix element on the left hand side is an element of
$\CA(\lambda_{0 2} (s)\CO) \subset \CA(\CW)$. So, in view of
the KMS--condition, we can continue this term analytically in $t$ to
$i\beta/2$. In the expression on the right hand side
there appears a product of two unitary operators
which both depend on $t$.
If we choose for $\rphi$ a Nelson vector, as described above, the
first operator, standing next to $\lphi$, can be analytically
continued in $t$ to $i\beta/2$, provided $s$ is sufficiently
small. The second operator, standing next to $ A^* \rva$,
can likewise be continued to $i\beta/2$
by the KMS--condition. So we arrive at
\begin{eqnarray*}
\lefteqn{ \lphi \, e^{\, -(\beta/2)M_{0 1}}
\, e^{\, isM_{0 2}} A^* e^{\, -isM_{0 2}} \rva}
\\ & & = \lphi \,
e^{\, is(\cos(\beta/2) M_{0 1} + i \sin(\beta/2) M_{1 2})}
e^{\, -(\beta/2)M_{0 1}} A^* \rva.
\end{eqnarray*}
Bearing in mind the definition of
$J$ and the invariance of the vacuum under
the boosts $e^{\, -isM_{0 2}}$, we can bring this
relation into the more transparent form
$$
\lphi \, J
e^{\, isM_{0 2}} A \rva = \lphi \,
e^{\, is(\cos(\beta/2) M_{0 2} + i \sin(\beta/2) M_{1 2})} J A \rva.
$$
As this holds for the dense sets of vectors
$\rphi$ and $A \rva$, respectively, we conclude that
$$
J e^{\, isM_{0 2}} =
e^{\, is(\cos(\beta/2) M_{0 2} + i \sin(\beta/2) M_{1 2})} J.
$$
The operator on the left hand side is anti--unitary, so the
same must be true for the operator on the right hand side. This
is only possible if $\beta$ is an {\it integer\/} multiple of $2 \pi$,
for otherwise the operator appearing in the
exponential function is not skew-adjoint. By a more
refined functional analytic argument one can restrict $\beta$
even further and show that its only possible value is
$\beta = 2 \pi$ \cite{BuFlSu}.
Proceeding to the proper
time scale of the observer \cite{parametrization},
we conclude that he is exposed to the
Hawking--Unruh temperature
$(1/2\pi)((\dot{\lambda}_{0 1}(0)\, x_O)^2){}^{-1/2}$,
in accordance with the value found in
model computations \cite{DeLe}.
By similar arguments one can compute the temperature
felt by other observers, such as
geodesic ones, for whom it is zero \cite{BuFlSu}.
Thus we have established the following general fact:
{\it Each geodesic or uniformly accelerated observer
testing the vacuum state in AdS finds
a universal value of the Hawking--Unruh temperature which depends only
on his particular orbit.}
This result is entirely a
consequence of the passivity of the vacuum. Hence
this state is the only one which is passive for all observers.
Having computed the value of $\beta$, let us return now to the
analysis of $J$. Plugging $\beta = 2 \pi$ into the above equality
for $J$, we see that for small $s$
$$
J e^{\, isM_{0 2}} =
e^{\, - is M_{0 2}} J.
$$
This relation can be extended to
arbitrary $s$ by iteration, if one
decomposes $e^{\, isM_{0 2}}$ into an $m$--fold product
$(e^{\, i(s/m)M_{0 2}})^m$ for sufficiently large $m$.
In a similar manner one
can determine the intertwining properties of $J$
with other one--parameter subgroups of
\AdSG, and thereby with all unitaries $U(\lambda)$.
We skip these computations and only state the result
\cite{BuFlSu}:
$$J U(\lambda) = U(\theta \lambda \theta) J, $$
where $\theta$ is the reflection which changes the sign of the
$0$--$1$--coordinates of the points in \AdS. This result also
allows us to show explicitly
that $J^2 = 1$ \cite{modular}: For the intertwining
relation and the anti--unitarity of $J$ imply
$$
J e^{\, - (\beta/2) M_{0 1}} = e^{\, (\beta/2) M_{0 1}} J,
$$
hence we have, in view of the defining equation for $J$,
\begin{eqnarray*}
J^2 A \Omega & = & J e^{ \, - (\beta/2) M_{0 1}} A^* \Omega =
e^{\, (\beta/2) M_{0 1}} J A^* \Omega \\ & = &
e^{\, (\beta/2) M_{0 1}} e^{\, - (\beta/2) M_{0 1}} A^{**} \Omega =
A \Omega ,
\end{eqnarray*}
for all $A \in \CA(\CW)$. Summing up, we have established the
following analogue of the TCP--theorem:
{\it The unitary representation $U$ of \AdSG \, extends to a
representation of \AdSGG, in which the reflection $\theta$
is implemented by the anti--unitary operator $J$.}
This result is a purely group--theoretic statement
which does not say anything about the adjoint action
of $J$ on the observables. In order to gain insight
into the nature of this latter action, we consider the observables
which are localized in the region
$$
\CW^{\prime} \equiv
\{x\in\AdS : - x_1 > | x_0 |, x_n > 0\}.
$$
Since the regions $\CW$ and $\CW^{\prime}$ are obtained by
intersecting opposite wedge--shaped regions in the ambient
space with \AdS, we call them {opposite wedges}.
By a rotation in the $1$--$2$--plane about the angle $\pi$,
the region $\CW$ is transformed into $\CW^{\prime}$, and
{\it vice versa}. Thus, if $A \in \CA(\CW)$ one has
$A^{\prime} \equiv e^{\, i \pi M_{1 2}} A
e^{\, -i \pi M_{1 2}}\in \CA(\CW^{\prime})$, cf.\ (iii).
By the intertwining relation for $J$, we have
$J e^{\, i \pi M_{1 2}} = e^{\, -i \pi M_{1 2}} J =
e^{\, i \pi M_{1 2}} J$, where we used the fact that
$e^{\, i 2 \pi M_{1 2}} =1$
\cite{restriction}. Finally, the group structure implies that
$e^{\, i \pi M_{1 2}} e^{\, - (\beta/2) M_{0 1}} =
e^{\, (\beta/2) M_{0 1}} e^{\, i \pi M_{1 2}}$. Combing these
facts, we can compute
\begin{eqnarray*}
J A^{\prime} \Omega & = & J e^{\, i \pi M_{1 2}} A \Omega
= e^{\, i \pi M_{1 2}} J A \Omega \\ & = &
e^{\, i \pi M_{1 2}} e^{\, - (\beta/2) M_{0 1}} A^* \Omega =
e^{\, (\beta/2) M_{0 1}} e^{\, i \pi M_{1 2}} A^* \Omega \\ & = &
e^{\, (\beta/2) M_{0 1}} A^{\prime *} \Omega.
\end{eqnarray*}
Picking any $A^{\prime} \in \CA(\CW^{\prime})$,
$B \in \CA(\CW)$, we obtain with the help of this result
\begin{eqnarray*}
\lefteqn{\lva A^{\prime *} B \rva = \overline{\lva B^{*} A^{\prime}\rva}
= \lva B^{*} J^* J A^{\prime}\rva} \\ & & =
\lva B^{* *} e^{\, -(\beta/2) M_{0 1}}
e^{\, (\beta/2) M_{0 1}} A^{\prime *} \rva
= \lva B A^{\prime *} \rva.
\end{eqnarray*}
Thus we find that the elements of $\CA(\CW)$ and
$\CA(\CW^{\prime})$ commute in the vacuum state.
It is apparent that the same statement holds also
for the algebras corresponding to all transformed opposite
wedges $\lambda \CW$ and $\lambda \CW^{\prime}$. This
commutativity, in the vacuum state, of observables which are
localized in certain complementary regions is sometimes called
{\it weak locality}. So we can state:
{\it Observables which are localized in opposite wedges of AdS
are weakly local with respect to each other.}
This establishes the asserted general relation between passivity
properties of the vacuum and commensurability properties of
observables. In theories
where the basic fields satisfy c--number commutation relations,
it follows from this result that observables
localized in opposite wedges commute also in the
usual (operator) sense, in accord with the structure found in
concrete models \cite{Fr}.
Before we turn to the discussion of our results, let us
mention another property of $J$ which holds in all theories complying with
the following stronger locality condition:\\[1pt]
(v) Observables which are localized in opposite wedges
of AdS commute with each other.
This situation prevails in the models of current interest \cite{Rehren}.
It implies that the operator $J$ implements
the geometric action of the reflection $\theta$ also on the observables,
$$
J \CA (\CW) J^* = \CA (\theta \CW).
$$
As a matter of fact, the validity of this relation is equivalent to the
strong locality condition. The proof of this result relies
on methods of Tomita--Takesaki theory and will be given elsewhere
\cite{BuFlSu}. There it will also be shown that mild constraints on
the mulitplicities of the eigenvalues $2 \pi \NN_0$ of the generator $M_{0 n}$
(existence of the partition function $\mbox{Tr} \, e^{\, - \beta
M_{0 n}}$) imply that
observables in opposite wedges are statistically (causally)
independent \cite{BuWi}.
We have established in this letter a tight relation between stability
properties of the vacuum and commensurabilty properties of observables on
AdS. Such relations exist also in other space--times, but
they are of particular interest in the case of AdS. For they
reveal that
observables in opposite wedges $\CW$ and $\CW^{\prime}$ necessarily
commute with each other, either weakly or strongly.
This result
appears to be very strange at first sight, since any point in $\CW$ can be
connected with any other point in $\CW^{\prime}$ by some timelike
curve. Hence measurements at these points should affect each other
and not be commensurable (let alone be statistically independent).
After a moment's reflection, however, one sees that
the regions $\CW$ and $\CW^{\prime}$, although causally connected in the
usual sense, cannot be connected by timelike geodesics.
So the following picture emerges, which explains the unexpected
commensurability properties of the observables: the perturbations
which are produced by observers in the region $\CW$ move, after having
left the region, along geodesics and therefore do not reach $\CW^{\prime}$.
In noninteracting theories this picture is reasonable, and free field
models show that it is also consistent. But it seems to be in conflict
with the patterns of interaction processes.
Imagine an experimenter who produces unstable
particles in the region $\CW$. By preparing carefully the initial
momenta of the particles, he can determine to some extent
the spacetime point on their respective orbits
where they will decay. As the decay products no longer move
along the original geodesics, some can reach the region
$\CW^{\prime}$ and perturb measurements there. But also
the quantum effects of measurements would produce
these unstable particles with a certain probability. It seems
therefore impossible that all observables in $\CW$ and $\CW^{\prime}$
are commensurable in this case. So the inevitable ``geodesic''
causal commutation relations of observables in AdS seem to exclude
true interaction. The only way out of this clash between
geometry and interaction seems to be to proceed
from AdS to its covering space.
\bigskip
The authors profitted from a
correspondence with Jacques Bros. One of them (DB)
would like to thank the University of Florida for hospitality and
financial support and the Deutsche Forschungsgemeinschaft DFG
for a travel grant.
\begin{thebibliography}{99}
%\vspace*{-30pt}
\bibitem{Spires} We refrain from giving references and refer the
interested reader to the HEP database, where
a comprehensive list of articles on this topic
may be retrieved.
\bibitem{BuFlSu} D. Buchholz, M. Florig and S.J. Summers,
(unpublished)
\bibitem{restriction} It is sufficient here to
consider the subspace of ``bosonic'' states, so we do not
need to proceed to the covering group of the space--time
symmetry group.
\bibitem{observables}
We work in the framework of
von Neumann algebras without dwelling on
this point any further. For more information on the
mathematical background, cf.\\[1pt]
R.\ Haag, {\it Local Quantum Physics}, Springer 1992.
\bibitem{past} It is meaningless to talk about past and future
in AdS, but one can still assign a time direction to causal curves.
\bibitem{parametrization} We choose a fixed parametrization of
the pertinent subgroups of \AdSG. The proper time of the
observer is obtained by rescaling $t$ with
$((\dot{\lambda}(0) \, x_O)^2){}^{1/2}$.
\bibitem{PuWo} W.\ Pusz and S.L.\ Woronowicz, Commun.\ Math.\ Phys.\
{\bf 58}, 273 (1978).
\bibitem{forms} Since $e^{\, - \beta M}$ is in general an
unbounded operator, this relation is to be understood in the sense of
quadratic forms.
\bibitem{modular} The operator $J$ is called
modular conjugation in Tomita--Takesaki theory.
Some of our explicit computations could be abbreviated by
using general results from this theory.
\bibitem{Ne} E.\ Nelson, Ann.\ Math.\ {\bf 70} 572 (1959).
\bibitem{BoBu} H.J.\ Borchers and D.\ Buchholz, Ann.\
Inst.\ H.\ Poincar\'e {\bf A 70} 23 (1999).
\bibitem{DeLe} S.\ Deser and O.\ Levin, Class.
Quant. Grav. {\bf 14} L163 (1997).
\bibitem{Fr} C.\ Fronsdal, Phys.\ Rev.\ {\bf D 10} 589 (1974).
\bibitem{BuWi} D.\ Buchholz and E.H.\ Wichmann, Commun.\ Math.\ Phys.\
{\bf 106} 321 (1986).
\bibitem{Rehren} K.H.\ Rehren, {\it Algebraic Holography},
G\"ottingen preprint (1999), hep-th/9905nnn
\end{thebibliography}
\end{document}
\end
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%