\documentstyle[12pt,twoside]{article}
\pagestyle{myheadings}
\markboth{ }{ }
\def\greaterthansquiggle{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}}
\def\lessthansquiggle{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\beqa}{\begin{eqnarray}}
\newcommand{\eeqa}{\end{eqnarray}}
\newcommand{\beqan}{\begin{eqnarray*}}
\newcommand{\eeqan}{\end{eqnarray*}}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}
\newcommand{\no}{\nonumber}
\newcommand{\bob}{\hspace{0.2em}\rule{0.5em}{0.06em}\rule{0.06em}{0.5em}\hspace{0.2em}}
\newcommand{\grts}{\greaterthansquiggle}
\newcommand{\lets}{\lessthansquiggle}
\def\dddot{\raisebox{1.2ex}{$\textstyle .\hspace{-.12ex}.\hspace{-.12ex}.$}\hspace{-1.5ex}}
\def\Dddot{\raisebox{1.8ex}{$\textstyle .\hspace{-.12ex}.\hspace{-.12ex}.$}\hspace{-1.8ex}}
\newcommand{\Un}{\underline}
\newcommand{\ol}{\overline}
\newcommand{\ra}{\rightarrow}
\newcommand{\Ra}{\Rightarrow}
\newcommand{\ve}{\varepsilon}
\newcommand{\vp}{\varphi}
\newcommand{\vt}{\vartheta}
\newcommand{\dg}{\dagger}
\newcommand{\wt}{\widetilde}
\newcommand{\wh}{\widehat}
\newcommand{\br}{\breve}
\newcommand{\A}{{\cal A}}
\newcommand{\B}{{\cal B}}
\newcommand{\C}{{\cal C}}
\newcommand{\D}{{\cal D}}
\newcommand{\E}{{\cal E}}
\newcommand{\F}{{\cal F}}
\newcommand{\G}{{\cal G}}
\newcommand{\Ha}{{\cal H}}
\newcommand{\K}{{\cal K}}
\newcommand{\cL}{{\cal L}}
\newcommand{\M}{{\cal M}}
\newcommand{\N}{{\cal N}}
\newcommand{\cO}{{\cal O}}
\newcommand{\cP}{{\cal P}}
\newcommand{\Q}{{\cal Q}}
\newcommand{\R}{{\cal R}}
\newcommand{\cS}{{\cal S}}
\newcommand{\T}{{\cal T}}
\newcommand{\U}{{\cal U}}
\newcommand{\V}{{\cal V}}
\newcommand{\W}{{\cal W}}
\newcommand{\X}{{\cal X}}
\newcommand{\Y}{{\cal Y}}
\newcommand{\Z}{{\cal Z}}
\newcommand{\st}{\stackrel}
\newcommand{\dfrac}{\displaystyle \frac}
\newcommand{\dint}{\displaystyle \int}
\newcommand{\dsum}{\displaystyle \sum}
\newcommand{\dprod}{\displaystyle \prod}
\newcommand{\dmax}{\displaystyle \max}
\newcommand{\dmin}{\displaystyle \min}
\newcommand{\dlim}{\displaystyle \lim}
\def\QED{\\ {\hspace*{\fill}{\vrule height 1.8ex width 1.8ex }\quad}
\vskip 0pt plus20pt}
\newcommand{\hy}{${\cal H}\! \! \! \! \circ $}
\newcommand{\h}[2]{#1\dotfill\ #2\\}
\newcommand{\tab}[3]{\parbox{2cm}{#1} #2 \dotfill\ #3\\}
\def\nz{\ifmmode {I\hskip -3pt N} \else {\hbox {$I\hskip -3pt N$}}\fi}
\def\zz{\ifmmode {Z\hskip -4.8pt Z} \else
{\hbox {$Z\hskip -4.8pt Z$}}\fi}
\def\qz{\ifmmode {Q\hskip -5.0pt\vrule height6.0pt depth 0pt
\hskip 6pt} \else {\hbox
{$Q\hskip -5.0pt\vrule height6.0pt depth 0pt\hskip 6pt$}}\fi}
\def\rz{\ifmmode {I\hskip -3pt R} \else {\hbox {$I\hskip -3pt R$}}\fi}
\def\cz{\ifmmode {C\hskip -4.8pt\vrule height5.8pt\hskip 6.3pt} \else
{\hbox {$C\hskip -4.8pt\vrule height5.8pt\hskip 6.3pt$}}\fi}
\newtheorem{theorem}{Theorem}
\newtheorem{definition}{Definition}
\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary}
\newtheorem{prp}{Proposition}
\def\lint{\int\limits}
\voffset=-24pt
\textheight=22cm
\textwidth=15.9cm
\oddsidemargin 0.0in
\evensidemargin 0.0in
\normalsize
\sloppy
\frenchspacing
\raggedbottom
\begin{document}
\bibliographystyle{plain}
\begin{titlepage}
\begin{flushright}
UWThPh-1995-17
\end{flushright}
\vspace{2cm}
\begin{center}
{\Large \bf How Hot Is the de Sitter Space? }\\[50pt]
H. Narnhofer, I. Peter, W. Thirring \\
Institut f\"ur Theoretische Physik \\
Universit\"at Wien
\vfill
{\bf Abstract} \\
\end{center}
We show that the unique invariant locally Minkowskian state
of quantum fields in de Sitter space $M$ has for an observer
moving along with a Killing vector field a temperature
$\frac{1}{2\pi} \sqrt{\frac{1}{R^2} + |a|^2}$ where $R$ is the radius
of $M$ and $a$ his acceleration. States with another temperature
cannot be locally Minkowskian all over $M$.
\vfill
\begin{center}
To appear in the International Journal of Modern Physics B, Memorial Issue
to commemorate the scientific activity of Hiroomi Umezawa.
\end{center}
\vfill
\end{titlepage}
\section{Introduction}
The Unruh effect [1], namely that the vacuum looks to an accelerated observer
like a thermal state is astonishing since it combines the completely
different area of geometry with that of statistical mechanics where the
thermal states are consequences of molecular chaos. It was pointed
out by Bekenstein [2], Hawking and Gibbons [3] that in general
relativity some
geometrical configurations produce states of a certain temperature.
Sewell [4] observed the strong relation of the Hawking radiation to the
Unruh effect and its mathematically precise formulation as a
Bisognano--Wichmann theorem [5,6].
It has been emphasized by Figari, H\"oegh-Krohn and Nappi [17] and recently
by Bros and Moschella [7] that also in de
Sitter space with radius $R$ the most natural state --- the one which
is invariant under the full de Sitter group and that has the best
analyticity properties and thus corresponds to the vacuum state over
Minkowski space --- locally satisfies a KMS condition with the
natural temperature $1/2 \pi R$ (if $\hbar = c = k = 1$). In this note
we want to refine this result by discussing what an accelerated and a
geodesic observer experiences in the course of time in a de Sitter space
and compare it with a accelerated and non--accelerated observer in
Minkowski space.
There is no doubt that in non--relativistic quantum mechanics the
precise definition of the temperature $T = \beta^{-1}$ is given by the
KMS condition. There it is linked to the time evolution $\tau_t$ of
the algebra $\A$ of observables by requiring for the temperature state
$\omega$
\beq
\omega(a \tau_t b) = \omega(b \tau_{-t + i\beta} a) \qquad
\forall \; a,b \in \A.
\eeq
In relativity there is no preferred time and in a curved space $M$ time
may not even be globally definable. In a pseudo--Riemannian space any
timelike vector field $X$ can be used to define a time locally and
if we normalize it to $\langle X|X \rangle = 1$ the parameter $t$ of
its flow measures the physical proper time of the lines of flow
$\tau^*_t : M \ni z \ra z(t)$. However the corresponding transformation
of a quantum field $\Phi$,
\beq
\tau_t \Phi(z) = \Phi(\tau^*_{-t}(z))
\eeq
will in general not be an automorphism of $\A$ even when restricted to
a neighbourhood $\Lambda$ of $z$. The best one can hope for in the
general case is that there exists a shrinking sequence
$\Lambda^{(n)}$, $\bigcap_n \Lambda^{(n)} = z$ and an automorphism
$\tau_t$ of $\A_{\Lambda^{(1)}}$ such that
$\tau_t \A_{\Lambda^{(n)}} \subset \A_{\bar \Lambda_t^{(n)}}$
for some $\bar \Lambda_t^{(n)}$ such that
$\bigcap_n \bar \Lambda_t^{(n)} = z(t)$. This minimal requirement can
be met in relativistic quantum field theory by a geometrical transformation
(2). However for those theories the algebraic structure of $\A$ and the
metric structure of $M$ are so tightly bound together that (2) will
give an automorphism of $\A$ only if $\tau^*$ is an isometry and thus
if $X$ is a Killing vector field. Fortunately in the case we are
interested in, namely maximally symmetric spaces, there are plenty of
Killing vector fields and even for each geodesic $z(t)$ there is a
Killing vector field $X$ which has $z(t)$ as a flow line. Thus the time
defined by $X$ includes the proper time experienced by one freely
falling observer. However our Killing vector fields will not be
geodesic so that its flow lines except this special one describe
accelerated motion. Futhermore $\langle X|X \rangle \neq$ constant
so that the flow parameter $t$ does not describe the proper time of the
accelerated observer and has to be renormalized because the physical
temperature is related to the physical time. With these precautions
we find the following.
For each geodesic trajectory in de Sitter space there is a unique Killing
vector field $X$ such that this trajectory is a flow line of $X$. In
Minkowski space this Killing vector field is a translation and is
geodesic everywhere. In de Sitter space it is not, in fact it is
timelike only in a certain region (a ``wedge''). Thus the other flow lines
have a certain acceleration $a$ and an observer on them experiences a
temperature $\frac{1}{2\pi} \sqrt{\frac{1}{R^2} + |a^2|}$. Only the
geodesic observer has $a = 0$ and feels the Figari, H\"oegh-Krohn, Nappi,
Gibbons--Hawking--Bros--Moschella $1/2\pi R$. The result of Minkowski
space where a geodesic observer feels temperature 0 is reached in the
limit $R \ra \infty$. Our result can be interpreted as an Unruh effect
in the ambient 5--dimensional Minkowski space. There even the geodesics
of de Sitter space have an acceleration $a_5$ and generally
$|a_5^2| = \sqrt{\frac{1}{R^2} + |a|^2}$. Thus an observer feels like moving
in a 5--dimensional space though the fifth dimension has otherwise no
physical reality. Since the de Sitter universe is homogeneous and
isotropic the temperature of this background depends only on the
acceleration of the observer and not on his (or her) position or
velocity. In particular there is neither red shift nor Doppler shift
in contradistinction to our background radiation.
Another important difference to Minkowski space is that whereas the latter
supports KMS states which are globally regular for any temperature in
de Sitter space states with another temperature than the natural one
become irregular on the horizon. Since the horizon is observer dependent
we have the remarkable situation that if the globally regular structure
of the geometry is to be respected by the state it dictates what the
temperature has to be.
Since the verification of these claims draws on results from various
branches of mathematics at the risk of boring some experts we first
collect these facts for the convenience of the reader. Our results are
more or less direct consequences of known facts, see f.i. [6], [19],
[20].
\section{Vector fields on a (pseudo--) Riemannian manifold $M$}
We shall denote vector fields by $G,K,X,Y,Z$ and their scalar products
by $\langle X|Y \rangle$, etc. The Lie (resp. the covariant) derivative
in direction of $X$ is denoted by $L_X$ (resp. $D_X$) and they are
related by $D_X Y = D_Y X + L_X Y$. A Killing vector field $K$ is
characterized by
$L_K \langle Y|Z\rangle = \langle L_K Y|Z\rangle + \langle Y|L_K Z\rangle$
$\forall \; Y,Z$
and a geodesic vector field $G$ by $D_G G = 0$. On the contrary,
$D_X \langle Y|Z\rangle = \langle D_X Y|Z \rangle + \langle Y|D_X Z\rangle$,
$L_X Y = - L_Y X$ and therefore $L_X X = 0$ holds $\forall \; X,Y,Z$.
For $K =$ Killing denote by $H_c$, $c \in {\bf R}$ the submanifold
$\{ x \in M: \langle K|K\rangle (x) = c\}$ so that its tangent space
$T(H_c)$ is spanned by the $\{ X: L_X \langle K|K\rangle = 0\}$.
\paragraph{Proposition I}
\begin{enumerate}
\item[(i)] $X \in T(H_c) \Longleftrightarrow \langle X|D_K K\rangle = 0$
\item[(ii)] $K_1,K_2 = \mbox{Killing } \Ra \langle K_1 | D_{K_1} K_2
\rangle = 0$ (in particular $K \in T(H_c)$)
\item[(iii)] If $K_1,K_2 =$ Killing and $L_{K_1} K_2 = \lambda K_2$,
$\lambda \in {\bf R}$ then $\langle D_{K_1} K_1|K_2\rangle = \lambda
\langle K_1 | K_2\rangle$
\item[(iv)] $G =$ geodesic, $\langle G|G\rangle \neq 0
\Longleftrightarrow \dfrac{G}{|\langle G|G\rangle|^{1/2}} =$ geodesic
\item[(v)] If $K =$ Killing then $K =$ geodesic $\Longleftrightarrow
\langle K|K\rangle =$ const.
\item[(vi)] If $K = f G$ on a $K$--invariant submanifold $h$ for some
function $f$ then $\left. D_K K\right|_h = \left. \kappa K\right|_h$,
$\kappa =$ const. on flowlines of $K$.
\end{enumerate}
\paragraph{Remarks}
\begin{enumerate}
\item $K \in T(H_c)$ means that $\langle K|K \rangle$ remains constant
along the flow lines of $K$. $H_c$ may have the dimension of $M$, then
$K$ is geodesic on $H_c$ or its dimension may be less by 1 and $K$ is not
geodesic.
\item The situation (vi) arises in a space with Minkowski signature
$(1,-1,\ldots,-1)$ of the metric if $N = \{x : K(x) = 0\} \subset H_0$
is spacelike and $\dim M = \dim H_0 +1 = \dim N + 2$. Then $\forall \; x
\in N \; \exists$ exactly 2 vectors $T_x(H_0) \ni G_{1,2}(x) \in
T_x(N)^\perp$ which are lightlike and future--directed. Denote by
$G_{1,2}$ the geodesic vector fields whose flowlines through $x$ go
in direction $G_{1,2}(x)$. Since the Killing vector field $K$
preserves the properties which characterize these geodesics it
shifts them into themselves. This means $K = f_1 G_1$ or $f_2 G_2$ on
$H_0$.
\item (iii) means that $K_{1,2}$ give a realization of the Anosov group [8]
and $\lambda$ is the Lyapunov exponent. If $D_{K_1} K_1 = \kappa K_1$
(which according to (vi) happens only on $H_0$) and
$\langle K_1|K_2\rangle \neq 0$ then $\lambda$ equals the ``surface
gravity $\kappa$ of the Killing horizon $H_0$''.
\end{enumerate}
\paragraph{Proof}
\begin{enumerate}
\item[(i)]
\beqan
L_X \langle K|K\rangle &=& 0 = D_X \langle K|K\rangle = 2 \langle D_X K|K
\rangle = 2 \langle D_K X|K\rangle + 2 \langle L_X K|K\rangle \\
&=& 2(D_K - L_K) \langle X|K\rangle - 2 \langle X|D_K K\rangle =
- 2 \langle X|D_K K\rangle.
\eeqan
\item[(ii)]
$$
\langle K_1|D_{K_1} K_2 \rangle = \langle K_1|D_{K_2} K_1\rangle -
\langle K_1|L_{K_2} K_1\rangle = \frac{1}{2} (D_{K_2} - L_{K_2})
\langle K_1|K_1\rangle = 0
$$
and
$$
0 = \langle K|D_K K \rangle = \frac{1}{2} D_K \langle K|K \rangle =
\frac{1}{2} L_K \langle K|K\rangle.
$$
\item[(iii)] According to (ii)
$$
\langle D_{K_1} K_1|K_2\rangle = D_{K_1} \langle K_1|K_2\rangle =
L_{K_1} \langle K_1|K_2\rangle = \langle K_1|L_{K_1}K_2\rangle =
\lambda \langle K_1|K_2\rangle.
$$
\item[(iv)]
$$
D_G \frac{G}{|\langle G|G\rangle|^{1/2}} =
\frac{D_G G}{|\langle G|G\rangle|^{1/2}} -
G \frac{\langle D_G G|G\rangle}{|\langle G|G\rangle|^{3/2}} = 0.
$$
\item[(v)] $\forall \; Y$ we have
$$
D_K \langle K|Y\rangle = \langle D_K K|Y\rangle + \langle K|D_K Y\rangle
= \langle D_K K|Y\rangle + \langle K|D_Y K\rangle + \langle K|L_K Y
\rangle
$$
but also $= L_K \langle K|Y\rangle = \langle K|L_K Y\rangle$. Thus
$$
\langle D_K K|Y\rangle = - \frac{1}{2} D_Y \langle K|K\rangle =
- \frac{1}{2} L_Y \langle K|K\rangle
$$
and therefore $D_K K = 0 \Leftrightarrow \langle K|K\rangle =$ const.
\item[(vi)] Since $h$ is $K$--invariant we get with $f D_G = D_{f G}$
after restriction to $h$
\beqan
0 &=& L_K (K - fG) = -L_K(f) G - f L_K G = -L_K(f) G - f D_K G + f D_G K \\
&=& - \frac{1}{f} L_K(f)K - f^2 D_G G + D_K K
= -L_K (\ln f) K + D_K K \Ra D_K K = \kappa K
\eeqan
with
$$
\kappa = L_K(\ln f).
$$
Taking again $L_K$ of $D_K K = \kappa K$ and observe $L_K D_K =
D_{L_K K} + D_K L_K$, we see $L_K \kappa = 0$.
\end{enumerate}
To a timelike Killing vector field $X$ one can intrinsically associate
an acceleration vector field
\beq
a = \langle X|X\rangle^{-1} D_X X.
\eeq
The motivation is the following. To any vector field corresponds a flow
$$
\frac{dz^i(t)}{dt} = X^i(z(t)) \quad \mbox{and thus} \quad
\frac{d^2z^i}{dt^2} = X^i{}_{,k} \frac{dz^k}{dt}.
$$
Now in Riemann normal coordinates where the connection vanishes at
$z(t)$ we have
$$
(D_X X)^i(z(t)) = X^i{}_{,k} X^k(z(t)), \quad \mbox{thus} \quad
\frac{d^2z^i(t)}{dt^2} = (D_X X)^i(z(t)).
$$
Now mind that the flow parameter is not the proper time $s$ of an
observer on this trajectory. $s$ is normalized by
$\langle dz/ds|dz/ds\rangle = 1$, that is to say
$(ds/dt)^2 = \langle X|X\rangle$. If $X$ is Killing then according to (ii)
this is constant along the flow lines and thus
$$
\frac{D_X X}{\langle X|X\rangle} = \left(\frac{dt}{ds}\right)^2
\frac{d^2z}{dt^2}
$$
has the significance of the acceleration felt physically. Furthermore
$$
0 = L_X \langle X|X\rangle = D_X \langle X|X\rangle = 2 \langle D_X X|X
\rangle,
$$
thus $D_X X$ is spacelike if $X$ is timelike if the metric is Minkowskian.
\section{Maximally symmetric spaces}
A maximally symmetric space $M$ of dimension $m$ is locally isomorphic
to the submanifold
$\{x_i \in {\bf R}^{m+1},x_i x_k \eta^{ik} = \pm R^2, \eta^{ik} = 1$
for $i = k = 0,1,\ldots,n-1$, $= -1$ for $i = k = n,n+1,\ldots,m$ and
zero otherwise$\} \subset {\bf R}^{m+1}$. The metric of $M$ is $\eta^{ik}$
restricted to $M$. The $SO(n,m+1-n)$ group generated in ${\bf R}^{m+1}$
by the vector fields $L_{ik} = x_i \partial_k - x_k \partial_i$,
$k > i = 0 \ldots m$ leaves $M$ and $\eta$ invariant so that one can
speak of the restriction of $L_{ik}$ to $M$ and there they define
$m(m+1)/2$ Killing vector fields.
\paragraph{Proposition II} In the de Sitter space ($n = 1$, $m = 4$) there
are no timelike Killing vector fields which are also geodesic.
\paragraph{Proof:} Since the Killing vector fields are a linear space
any element can be written $K = v_{ik} x_i \partial_k$,
$v_{ik} = - v_{ki} \in {\bf R}$. According to Prop. I, (v) we only have
to see whether $\langle K|K\rangle = - x_i v_{ik} \eta_{km} v_{mn} x_n$
can be a constant $> 0$ on $M$. This happens iff $v \eta v = c \eta$,
$c \in {\bf R}^+$, which implies $(\det v)^2 = c^5 > 0$. But since $v$
is a real antisymmetric $5 \times 5$ matrix it has one eigenvalue 0
and hence $\det v = 0$.
\paragraph{Remark} This is not a general feature of curved maximally
symmetric spaces since on $S^3$ ($n = 0$, $m = 3$) the matrix
$$
v = \left( \ba{rcrc} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\
0 & 0 & -1 & 0 \ea \right),
$$
$v^2 = -1$ generates a $K$ which is geodesic.
In the de Sitter case the $K$ which are in some region timelike are of
the form $L_{0i}$, $i = 1, \ldots, 4$ which is timelike only in the two wedges
$|x_i| > |x_0|$. In one, say $H^+_0 \equiv \{x_i > |x_0|\}$ time flows
upwards, in $H^-_0 \equiv \{x_i < - |x_0|\}$ it flows downwards.
This makes one--particle states
unstable in theories with interaction [16]. A particle may tunnel
through to the other side of $M$ where it appears with the opposite
energy which causes some instability.
The intersections of $M$ with planes
going through the origin of $M$ are the geodesics. The Killing vector
field which leaves this plane invariant has this geodesic as flow line.
In contradistinction to the Killing vector fields there are globally
defined timelike geodesic vector fields. They are not Killing and to these
observers the universe appears to be first contracting and then
expanding. (The corresponding Hamiltonian is time--dependent [9].)
The wedge is the union of the future light cones emerging from the points
of the geodesic intersected with the union of its past light cones. This
set is the same for all timelike curves in the wedge. Thus one can say
that the wedge is the part of the de Sitter space with which an observer
can communicate. This means it consists of the points where one can
receive a message from the observer and an immediate reply can still
reach him.
The covariant derivative $D_X$ in $M$ and the one $\bar D_X$ in the
ambient ${\bf R}^{m+1}$ are related via the second fundamental form $S$,
\beq
\bar D_X Y = D_X Y + S(X,Y).
\eeq
Since $M$ has constant curvature and all sectional curvatures are the
same Gauss' famous ``Theorema egregium'' tells us that for $M$ we simply
have
$$
S(X,X) = \frac{\langle X|X\rangle}{R} \nu
$$
where $\nu$ is the unit vector field $\perp T(M)$. Thus the acceleration
$a_4$ of $X$ in $M$ and $a_5$ in ${\bf R}^5$ are related by
$$
a_5 = a_4 + \frac{\nu}{R}
$$
or since $a_4$ and $\nu$ are spacelike and $a_4 \perp \nu$
$$
|\langle a_5|a_5\rangle|^{1/2} = \sqrt{|\langle a_4|a_4\rangle | +
\frac{1}{R^2}}.
$$
Since the ambient space is flat we have $D_X X = X^k{}_{,\ell} X^\ell
\partial_k$. For the Killing vector field $L_{01} = x_0 \partial_1 -
x_1 \partial_0$ we find $\bar D_X X = x_0 \partial_0 - x_1 \partial_1$ and
$\langle \bar D_X X | \bar D_X X\rangle = - \langle X|X\rangle$ on $M$.
Thus
\beq
|a_5^2| = \frac{1}{\langle X|X \rangle}.
\eeq
Furthermore the Laplace--Beltrami operator ${}\Box{}$ on $M$ is the
angular part of the one on the flat ambient ${\bf R}^5$ [10].
Finally a time flow given by a $L_{0i}$,
$i = 1,2,3,4$ gives a realization of the Anosov group (Prop. I, (iii))
if the transversal shift is identified with $L_{0i} - L_{ij}$,
$j = 1,\ldots,4$. Since on the Killing horizon for $K = L_{01}$ we
have $D_K K = \kappa K$ we have here ``surface gravity''
$\kappa =$ Lyapunov exponent.
\section{KMS states}
A faithful state $\omega$ over an algebra $\A$ with a continuous
automorphism group $\tau_t$ is said to be $\tau$--KMS if
$\omega(b \tau_t a)$ is analytic for $0 < \mbox{Im } t < \beta$,
continuous on the boundary and $\omega(ab) = \omega(b \tau_{i\beta}(a))$
$\forall \; a,b \in \A$.
\paragraph{Proposition III}
\begin{enumerate}
\item[(i)] If $\omega$ is $\tau$ and $\sigma$--KMS then $\tau_t = \sigma_t$
$\forall \; t$.
\item[(ii)] If $\omega$ is $\tau$--KMS and invariant under some
automorphism $\sigma$, $\omega \circ \sigma = \omega$, then
$[\tau,\sigma] = 0$.
\end{enumerate}
\paragraph{Proof}
\begin{enumerate}
\item[i)]
$$
F(t) := \omega(\tau_t(a) \sigma_t(b)) = \omega(\sigma_t(b) \tau_{t+i\beta}
(a)) = \omega(\tau_{t + i\beta}(a) \sigma_{t+i\beta}(b) = F(t + i\beta)
$$
for all $a,b$ in $\A$. For a dense set $\wt \A$ $F(t)$ is entire and can be
continued periodically in Im~$(t)$ and since it is bounded in the strip
it is bounded in all of {\bf C}. But for an analytic function
$|F(t)| < M$ $\forall \; t \in {\bf C}$ implies $F(t) =$ const. By the
same argument $\omega$ is invariant under $\sigma$ such that
$\omega(\sigma_t^{-1} \tau_t(a)b) = \omega(ab)$ $\forall\; t \in {\bf R}$,
$a,b \in \wt \A$. Since $\omega$ is faithful this implies
$\sigma_t^{-1} \tau_t = id$.
\item[(ii)] Consider $\bar \tau_t = \sigma^{-1} \tau_t \sigma$.
$$
\omega(b \bar \tau_{i\beta} a) = \omega(b \sigma^{-1} \tau_{i\beta} \sigma a) =
\omega(\sigma(b) \tau_{i\beta} \sigma a) = \omega(\sigma a \cdot \sigma b)
= \omega(ab).
$$
Thus $\omega$ is also $\bar \tau$--KMS and because of (i) this implies
$\bar \tau = \tau$ or $\sigma \tau = \tau \sigma$.
\end{enumerate}
\paragraph{Conclusion:} Because of (ii) a state invariant under the full
de Sitter group $SO(1,4)$ cannot be KMS for any 1--parameter subgroup
$\tau_t$ since $SO(1,4)$ has trivial center and $\tau_t$ would have to
commute with all elements of the invariance group.
This conclusion is not so sad since we have no global time automorphism
either. The only candidates are given by Killing vector fields $L_{0i}$
which are timelike only in the wedges $H^\pm_0$.
The $L_{0i}$ gives an automorphism $\tau_t$ of the subalgebra
$\A_{H^\pm_0}$ but the other $L_{kj}$ do not leave $H^\pm_0$
invariant and thus do not create automorphisms of $\A_{H^\pm_0}$.
Hence there is no contradiction in the restriction
$\left. \omega \right|_{\A_{H^\pm_0}}$ being $\tau_t$--KMS. Because of
(i) it cannot be KMS for another automorphism and here it is not the state
but the subalgebra $\A_{H^\pm_0}$ which singles out a distinguished
time.
If $\A$ contains operators such that their commutator (or anticommutator)
is a multiple of the identity (as for free bosons or fermions) then
this fixes their correlation function in a KMS state $\omega_\beta$ with
an arbitrary temperature $\beta^{-1}$. By taking the Fourier transform of
(1) with respect to $t$ we deduce [11]
\beq
\omega_\beta(ab) = \int_{-\infty}^\infty \frac{d\nu dt}{2\pi} e^{i\nu t}
\frac{e^{\beta \nu}}{e^{\beta \nu} -1} \omega_\beta ([\tau_t(a),b]) =
\int_{-\infty}^\infty \frac{d\nu dt}{2\pi} e^{i\nu t}
\frac{1}{e^{-\beta \nu} +1} \omega_\beta ([ \tau_t(a),b]_+)
\eeq
and verify
\beqan
\omega_\beta([a,b]) &=& \int_{-\infty}^\infty \frac{d \nu dt}{2\pi}
\cos \nu t \left( \frac{e^{\beta\nu}}{e^{\beta\nu} -1} +
\frac{e^{-\beta\nu}}{e^{-\beta\nu} - 1}\right)
\omega_\beta([\tau_t(a),b]) \\
&=& \int_{- \infty}^\infty dt \delta(t) \omega_\beta([\tau_t(a),b]).
\eeqan
Relation (7) holds under the assumption that such a state $\omega_\beta$
exists and will in fact be used to show that the de Sitter space
supports no other than the natural temperature if the state is to be
locally regular on all of $M$.
\section{Quantum fields in de Sitter space}
We shall concentrate on real free scalar fields $\Phi$ since they show
already the relevant features. (For spinors things work the same way,
see [12]. Since our main argument is based on the properties of the de
Sitter group it should carry over to the interacting case but these
theories have not yet been constructed.)
If we use as coordinates the Euclidean coordinates of
${\bf R}^5$ and $dx$ denotes the invariant measure
$$
\int d^5 x \delta(x^2_0 - x^2_1 - x^2_2 - x^2_3 - x^2_4 + R^2)
$$
supported on $M$, then $\A$ is generated linearly by the Weyl operators
\beq
W(f) = \exp [i\int dx f(x) \Phi(x)], \qquad f \in {\bf C}^\infty_0
({\bf R}^5) \mbox{ real}.
\eeq
The algebraic properties are characterized by a symplectic form
$\sigma$,
\beq
W(f) W(g) = e^{-i \sigma(f,g)/2} W(f + g)
\eeq
where $\sigma(f,g) = \int dx dx' f(x) \Delta(x-x') g(x')$ and $\Delta$
is the real odd function
\beq
[\Phi(x), \Phi(x')] = i \Delta(x - x').
\eeq
If the de Sitter group is to be realized by geometrical automorphisms
$\Delta$ can depend only on the invariant distance $(x - x')^2 =
- 2R^2 - 2xx'$. Thus $\Delta$ admits a type of Lehmann--K\"allen
representation
\beq
\Delta(x) = i \int_0^\infty da \rho(a) \int d^5k e^{ikx} \delta(k^2-a)
\ve(k_0).
\eeq
If $\Phi$ were to obey a Klein--Gordon equation $(\Box - m^2) \Phi(x) = 0$
on $M$, $m$ would be related to the degree of homogenity of $\rho$
as the radial part of $\Box^5$ gives on a function of radial degree of
homogeneity $\nu$ an additional $\nu(\nu + 3)$. Since our argument
is independent of the mass we shall not pursue this further and work
with a generalized free field.
A quasifree state $\omega$ over $\A$ is characterized by [13]
\beq
\omega(W(f)) = e^{- \sigma(f,Jf)/4}.
\eeq
for some operator $J$ such that
$$
\sigma(f,Jg) = \int dx dx' f(x) \Delta^{(1)}(x,x') g(x')
$$
where the real symmetric function $\Delta^{(1)}(x, x')$ is
$$
\omega([\Phi(x),\Phi(x')]_+) = \Delta^{(1)}(x, x').
$$
If $J^2 = -1$ one can in the usual way introduce creation and
annihilation operators and represent $\A$ irreducibly in a Fock
space so that $\omega$ is pure.
If $\omega$ is to be invariant then $\Delta^{(1)}(x, x')$ should
depend only on $(x-x')^2$ and to meet reality and evenness
$\Delta^{(1)}$ and
positivity of $\omega$, $J$ must be in $k$--space multiplication by
$c i \ve(k_0)$, $c \in {\bf R}^+$, $\ve(x) = x/|x|$.
If $c = 1$ we have $J^2 = -1$
and $\omega$ is pure. Furthermore in this case
\beq
\Delta^{(1)}(x) = \int da \rho(a) \int d^5k e^{ikx} \delta(k^2 -a)
\eeq
such that
\beq
\omega(\Phi(x) \Phi(x')) = \int da \rho(a) \int d^5k e^{ik(x-x')}
\delta(k^2 - a) \Theta(k_0), \qquad
\Theta = \frac{1 + \ve}{2}
\eeq
satisfies for small $x - x'$, ($x,x'$ near $(0,0,0,0,R)$) by suitable
normalization of $\int da \rho(a)$ the principal of local
definiteness, i.e. it tends to $\lim_{\ve \downarrow 0} (x-x'-i\ve)^{-2}
(2\pi)^{-2}$ where $\ve$ is in the upper lightcone. There are other
invariant states [14] but these requirements single out the
``Euclidean vacuum''.
For this state the work of Bisognano and Wichmann has been carried over
to $M$ by Bros and Moschella and they showed that $\left. \omega\right|_W$
is $\tau$--KMS if $\tau_t$ is generated by $L_{01}$ and the wedge
$H^+_0 = \{ x \in M : x_1 > |x_0|\}$ and $\beta = 2\pi$.
The argument boils down to the following. $L_{01}$ generates
$\tau^*_{i\alpha}(x) = (x_0 c + ix_1s,x_1 c + ix_0s,x_\perp)$ with
$\ba{c} c \\ s \ea = \ba{c} \cos \\ \sin \ea \alpha$ and
$x_\perp = x_{2,3,4}$. Since
$\omega(\Phi(x') \tau_{i\alpha}(\Phi(x))) = \Delta^+((x' -
\tau^*_{i\alpha}(x))^2)$ and the singularity of $\Delta^+$ sits at the
origin we have to see whether we can continue $\alpha$ from 0 to $2\pi$
without that $(x' - \tau^*_{i\alpha}(x))^2$ touches 0. By invariance we
may assume in the wedge $x_0 = 0$ then
$$
(x' - \tau^*_{i\alpha}(x))^2 = (x'_0 - ix_1 s)^2 - (x'_1 -x_1 c)^2
- (x'_\perp - x_\perp)^2
$$
and
\beqan
\mbox{Re }(x' - \tau^*_{i\alpha}(x))^2 &=& x'_0{}^2 - x'_1{}^2 - x_1^2
+ 2x'_1 x_1 c - (x'_\perp - x_\perp)^2 \\
\mbox{Im }(x' - \tau^*_{i\alpha}(x))^2 &=& - 2x'_0 x_1 s.
\eeqan
Since $x_1 > 0$ in $H^+_0 \mbox{ Im} = 0$ only if $s = 0 \Leftrightarrow
\alpha = \pi$ or $x'_0 = 0$.
In both cases Re~$< 0$ and thus in $H^+_0$ we can move $\alpha$ from 0 to
$2\pi$ without encountering the singularity. When we approach $2\pi$
the only difference is that now the real axis approached from the
other side. This corresponds exactly to the difference between
$\Delta^+(x-x')$ and $\Delta^+(x'-x)$ and hence
$$
\omega(\Phi(x'), \tau_{2\pi i} \Phi(x)) = \omega(\Phi(x) \cdot \Phi(x'))
$$
which is just the KMS condition we have been looking for. It holds
$\forall \; x,x' \in H^+_0$ and the only difference to Bisognano--Wichmann
is that in $M$ $\tau^*_t$ also generates one geodesic line and thus
also a freely falling observer sees a temperature.
\paragraph{Remarks}
\begin{enumerate}
\item The wedge $H^+_0$ cannot be invariant under another automorphism
of the de Sitter group not commuting with $L_{01}$. This would
contradict Prop. II, (ii) since $\omega$ is invariant under all of
them. Thus the apparent contradiction can be avoided only if these
transformations do not create automorphisms of $\A_{H^+_0}$.
\item One might ask whether there are states with a different temperature
for $\A_{H^+_0}$. In fact (7) gives an explicit representation for states
for arbitrary $\beta$. However the analysis of [11] carries directly over
and shows that at the edge of the horizon where $x_0,x_1 \ra 0$ these
states do not satisfy the principle of local definiteness. This requires
that in the small distance limit the propagator has the same singularity
as in Minkowski space. As we have seen in Sect. 5 the commutator is by
construction the same as in Minkowski space and what happens is that the
expectation value of the anticommutator has in front a factor $1/2\pi\beta$
instead of $1/(2\pi)^2$. The same effect with $\beta < 2\pi$
occurs if one goes in Minkowski space by a Bogoliubov transformation
into a wrong vacuum. For $\beta > 2\pi$ positivity of the state gets
lost since $\Delta^{(1)}$ becomes too small compared with $\Delta$.
This means that for $T$ larger than the ``natural'' temperature there
are infinitely many negative energy quanta in the tangent space of the
edge of the wedge and smaller temperatures do not exist at all.
To see how this
happens here one can directly take over the analysis of [11] where with
the notation of their Sect. 3 we have when projected from ${\bf R}^5$
to $M$
\beqan
\Delta(\tau_1 + \tau,x_1|\tau_2,x_2) &=& \int d^5k \delta(k^2-m^2) \ve(k_0)
e^{ik_u (u_1 e^\tau -u_2) + ik_v(v_1 e^{-\tau}-v_2) + ik^\perp(x_1^\perp
- x_2^\perp)} \cdot \\
&& \cdot \delta(u_1 v_1 + x_1^{\perp 2} - R^2)
\delta(u_2 v_2 + x_2^{\perp 2} - R^2).
\eeqan
The only change is that $x^\perp$ has an extra dimension. This is
however fixed by the $\delta$--functions and the main point is not
affected. Thus only for $\beta = 2\pi$ the state can be extended
beyond the horizon without violating local definiteness.
\item The KMS property is not the only equilibrium feature of $\omega$.
Because of the Anosov property mentioned in Sect. 3 and the invariance
of $\omega$ we know from general arguments that we get sensitive
dependence on initial conditions and exponential decay in $t$ of the
correlation functions [8].
\item A light source which keeps shining and never exhausts itself seems
strange and one might wonder whether it could not be used to construct a
perpetuum mobile. That this is not so shows another characterization of
KMS states, the passivity [18]. It states that by an external perturbation
depending periodically on time one can only invest energy into the system
but never extract any from it.
\end{enumerate}
Everything which has been said about the wedge $H^+_0$ can also be said about
subalgebras supported in $H^+_c = \{x \in H_c, x_1 > 0\}$. $H^+_c$ is
invariant under the flow of $L_{01}$ and therefore the associated
automorphism leaves these subalgebras invariant. Thus the wedge algebra
has plenty of invariant subalgebras, its Anosov property excludes only
finite-dimensional invariant subalgebras. We shall now show that because of
Haag duality they all have the same strong closure and their consideration
does not give an essentially new information.
\paragraph{Definition (5.1)} Let $\A$ be an algebra localized in a space-time
region $\Delta$ and denote by $\A;$ its commutant and by $\A^\perp$ the
strong closure of the algebra localized in the causal complement of
$\Delta$. We have the inclusions
\begin{enumerate}
\item[(i)] $\A'' \supset \A$, equality holds if $\A$ is strongly closed
(von Neumann's theorem).
\item[(ii)] $\A^{\perp\perp} \supset \A$, equality holds if $\Delta$ is
causally closed.
\item[(iii)] $\A' \supset \A^\perp$ by causality, equality is called Haag
duality.
\item[(iv)] $\B' \supset \A'$, $\B^\perp \supset \A^\perp$ for any
subalgebra $\B$ of $\A$.
\end{enumerate}
\paragraph{Proposition (5.2)}
\begin{enumerate}
\item[(i)] Let $\A$ be causally closed. Then for Haag duality
$\A'' = \A^{\perp\perp} = \A^\perp{}' = \A'{}^\perp = \A$ is necessary
and $\A'' = \A^\perp{}'$ is sufficient.
\item[(ii)] If $\B$ is Haag dual and $\A^\perp = \B^\perp$, then $\B$ is
strongly dense in $\A$.
\item[(iii)] If in addition $\B \subset \A$, then $\A$ is also Haag
dual.
\end{enumerate}
\paragraph{Proof:}
\begin{enumerate}
\item[(i)] Necessity: From (5.1) follow the inclusions
$\A'{}^\perp \subset \A^{\perp\perp} \supset \A \subset \A''
\subset \A^\perp{}'$. If $\A$ is causally (and therefore strongly)
closed the inclusions in the middle are equalities and so are the ones
at the ends if Haag dualoty holds.
Sufficiency: $\A' = \A''' = \A^\perp{}'' = \A^\perp$ since $\A'$ and
$\A^\perp$ are strongly closed.
\item[(ii)] $\B'' = \B^\perp{}' = \A^\perp{}' \supset \A''$.
\item[(iii)] $\A' \supset \A^\perp = \B^\perp = \B' \supset \A'$.
\end{enumerate}
\paragraph{Corollary.} For the quasifree bosons we are considering Haag
duality holds and the causal complements of all $H^+_c$ is $H^-_0$.
Thus all $\A_{H^+_c}$, $c > 0$, are strongly dense in $\A_{H^+_0}$.
\section{Conclusions}
Our first postulate is that the physical temperature is linked to the
physical time. After all one measures the temperature of astrophysical
objects by measuring their frequency spectrum.
Furthermore we work with a given metric which is supposed to measure the
physical time. This postulate has been extended by Connes and Rovelli [15]
to the generally covariant situation of quantum gravity and our more
modest application illustrates some of the points made there.
Secondly we argued that
the physical timeflow has to be represented by an automorphism group of
the observables. To call the parameter of transformations which do not
preserve the structure of the laws of nature the time seems illusory
since this will not be what is measured by a clock. If these
transformations are to be geometrical they have to be flows of Killing
vector fields. In the case of de Sitter space there are as many Killing
vector fields as in Minkowski space and there a time automorphism
exists at
least locally though not globally. If there were no Killing vector
fields one would have to withdraw to the infinitesimal level by some
scaling limit. This is presumably the way how the background radiation
in our universe has to be interpreted. Also in de Sitter space $M$ one
could project a KMS state with respect to the $x_0$--shift of the
5--dimensional Minkowski field theory onto the field theory on $M$.
This state will be faithful and therefore defines a global modular
automorphism which will not be geometric. The corresponding geometrical
generator of the $x_0$--shift is $x_0$--dependent and thus does not
generate a group of unitaries. However to the extent that the universe in
some coordinate system is only adiabatically expanding one may then define by
some scaling limit a local temperature in the tangent space of each
point. This temperature will then be different at different points and may
show red shift and Doppler shift. The algebraic
structure of free quantum fields is fixed by the geometrical setting
but the multitude of states requires a selection principle. We adopt the
principle of local definiteness which requires that at the infinitesimal
level the state looks like a Minkowski vacuum. This singles out one among
the invariant states and then our claims follow immediately. For each
Killing vector field $K = L_{0i}$ the state restricted to the algebra
in the wedge where $K$ is timelike is KMS for $\beta = 2\pi$. Since
this is measured in the parameter $t$ of the flow and not the proper
time $s$ we have to use from Sect. 2 that along a trajectory
$(s/t)^2 = \langle K|K\rangle$ and thus
$\beta_{\rm phys} = 1/T = 2\pi \langle K|K\rangle^{1/2}$. Now we
recognize with Sect. 3 $\langle K|K\rangle^{-1/2}$ as the 5--dimensional
acceleration $|a_5|$ and when expressed in the intrinsic acceleration
$a_4$ we come to our conclusion
$$
T = \frac{|a_5|}{2\pi} = \frac{1}{2\pi} \sqrt{\frac{1}{R^2} + |a^2_4|}.
$$
Though Minkowski and de Sitter space belong both to the class of
maximally symmetric spaces the curvature changes physics quite a bit.
Therefore we shall conclude by confronting the various properties of the
two spaces how they are reflected by free quantum fields.
$$
\begin{tabular}{p{7cm}p{7cm}}
$\qquad \qquad$ {\bf Minkowski} & $\qquad \qquad \qquad$ {\bf de Sitter} \\[5pt]
\multicolumn{2}{c}{\bf Vector fields} \\[5pt]
There are globally timelike geodesic vector fields which are also Killing.
& There are globally timelike geodesic vector fields but they are not
Killing. There are global Killing vector fields but they are only in some
regions timelike. Each has only a single flow line which is geodesic. \\[5pt]
\multicolumn{2}{c}{\bf States} \\[5pt]
There is one invariant locally definite state. For each timelike global Killing
vector field there are for all temperatures KMS states which are everywhere
locally definite. For all geodesic obserserves the state of zero temperature
is the distinguished invariant state.
& There is one invariant locally definite state. Each Killing vector field
which is locally timelike has one geodesic observer. For him the
distinguished invariant state has a temperature $1/2\pi R$. There is no
state which is everywhere locally definite which would have another
temperature for him.
\end{tabular}
$$
\section*{Acknowledgements}
We are grateful to J. Bros and D. Buchholz for stimulating discussions
and to H. Rumpf and H. Urbantke for valuable remarks.
%\newpage
\begin{thebibliography}{99}
\bibitem{1} W.G. Unruh, Phys. Rev. {\bf D14} (1976), 870.
\bibitem{2} J. Bekenstein, Phys. Rev. {\bf D7} (1973), 2333--2346.
\bibitem{3} S.W. Hawking, G.W. Gibbons, Phys. Rev. {\bf D15(10)} (1977),
2738--2791.
\bibitem{4} G.L. Sewell, Ann. Phys. (N.Y.) {\bf 141} (1982), 201--224.
\bibitem{5} J. Bisognano, E. Wichmann, J. Math, Phys. {\bf 16(4)} (1975),
985--1007; J. Math. Phys. {\bf 17(4)} (1976), 303--321.
\bibitem{6} B.S. Kay, R.M. Wald, Phys. Rep. {\bf 207} (1991), 49--136.
\bibitem{7} J. Bros, U. Moschella, Saclay preprint (1995).
\bibitem{8} G.G. Emch, H. Narnhofer, G.L. Sewell, W. Thirring, J. Math.
Phys. {\bf 35}/11 (1994), 5582.
\bibitem{9} W. Thirring, Acta Phys. Austr. Suppl. IV (1967) 269.
\bibitem{10} T. Wyrozumski, Class. Quantum Grav. {\bf 5} (1988),
1607--1613.
\bibitem{11} R. Haag, H. Narnhofer, U. Stein, Commun. Math. Phys. {\bf 94}
(1984), 219--238.
\bibitem{12} I. Peter, Thesis, University of Vienna (1995).
\bibitem{13} O. Bratteli, D. Robinson, Vol. II, Springer, New York, 1981.
\bibitem{14} B. Allen, Phys. Rev. {\bf D32} (1985), 3136.
\bibitem{15} A. Connes, C. Rovelli, Class. Quantum Gravity {\bf 11}
(1994), 2899.
\bibitem{16} O. Nachtmann, Dynamische Stabilit\"at im de Sitter Raum,
Sitzungsber. d. \"Osterr. Akademie d. Wiss., Mathem.--naturw. Klasse,
Abt. II, 176 Bd., 8. bis 10. Heft, 1967.
\bibitem{17} R. Figari, R. H\"oegh-Krohn, C.R. Nappi, Commun. Math.
Phys. {\bf 44}, 265 (1975).
\bibitem{18} W. Pusz, S.L. Woronowicz, Commun. Math. Phys. {\bf 58},
273 (1978).
\bibitem{19} S.J. Summers, R. Verch, Modular Inclusion, the Hawking
Temperature and Quantum Field Theory in Curved Space-Time,
preprint UFIFT, University of Florida.
\bibitem{20} R. Wald, ``Quantum Field Theory in Curved Spacetime and
Black Hole Thermodynamics'', Univ. of Chicago Press, 1994.
\end{thebibliography}
\end{document}