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\def\CA{{\cal A}}
\def\CB{{\cal B}}
\def\CC{{\cal C}}
\def\CD{{\cal D}}
\def\CE{{\cal E}}
\def\CF{{\cal F}}
\def\CG{{\cal G}}
\def\CH{{\cal H}}
\def\CI{{\cal I}}
\def\CK{{\cal K}}
\def\CL{{\cal L}}
\def\CM{{\cal M}}
\def\CN{{\cal N}}
\def\CO{{\cal O}}
\def\CP{{\cal P}}
\def\CQ{{\cal Q}}
\def\CR{{\cal R}}
\def\CS{{\cal S}}
\def\CT{{\cal T}}
\def\CW{{\cal W}}
\def\CX{{\cal X}}
\def\CY{{\cal Y}}
\def\CZ{{\cal Z}}
\def\sabsatz{\par\smallskip\noindent}
\def\mabsatz{\par\medskip\noindent}
\def\babsatz{\par\vskip 1cm\noindent}
\def\tabsatz{\topinsert\vskip 1.5cm\endinsert\noindent}
\def\newline{\hfil\break\noindent}
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\def\Ret{\Re e\,}
\def\Imt{\Im m\,}
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              \hbox{\vrule width.4pt height4pt\kern4pt\vrule width.4pt}
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\def\tr{{\rm tr}\,}
\def\supp{{\rm supp}\;}
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\def\Grad{{\rm Grad}\,}
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\def\Div{{\rm Div}\,}
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\def\B1{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
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\def\locnet{\{\CA(O),\CA,\alpha,\BR^d\}}
\def\ad{{\rm ad}\,} 
\headline={\hfill{\fivepoint HJB--- Mar/94}}
%
\tabsatz
\centerline{\hbf Theory of Local Observables and KMS--Condition}
\mabsatz
\centerline{Dedicated to {\caps Bert Schroer} at the occasion of his
60th birthday}
\vskip 0.4cm
\centerline{\caps H.J. Borchers}
\mabsatz
\centerline{Institut f\"ur Theoretische Physik} 
\centerline{Universit\"at G\"ottingen}
\centerline{Bunsenstrasse 9, D 37073 G\"ottingen}
\babsatz
\vskip 0.8cm
%
{\narrower \sabsatz
{\sbf Abstract:}\newline{\srm
We will look at C$^*$--dynamical systems with the translations of $\BR^d$
as dynamical group and will define KMS--functionals for $\beta$ taking
values in $\BC^d$. The properties of such functionals will be investigated.
In particular if a KMS--functional exists for a complex $\beta$ then
it will be shown that there exists a KMS--state for the real part of $\beta$.
If the system is a theory of local observables then it will be shown that 
locality condition implies that there do not exist $\beta$--KMS--functionals 
such that $\beta$ is spacelike or has a spacelike real part.}\sabsatz}
\mabsatz
%
{\bbf 1. Introduction:}
\mabsatz

It is generally believed that the thermodynamical functions should be
analytic in the temperature except for points of phasetransitions.
In most textbooks non--analyticity is taken as definition for phasetransition.
In this setting the surface of coexistence becomes cuts in some 
analytic domain.
The detailed structure of these cuts is used to classify 
phasetransitions.

In quantum statistical mechanics the temperature--states are defined
by the KMS--condition [HHW67]. Therefore, it is not obvious that
temperature--states should have analyticity properties in the temperature
or rather in the inverse temperature $\beta$. If we allow
arbitrary C$^*$--dynamical systems then one cannot expect any analyticity
in the inverse temperature $\beta$. This follows from the investigations
of O. Bratteli and others [Bra82]. % refs
They have shown that to any given closed subset of the
positive real axis there exists a C$^*$--dynamical system which has 
$\beta$--KMS.--states only for $\beta$'s belonging to this given set.
Therefore, one can only hope for an acceptable behaviour in the
temperature if we are dealing with reasonable dynamics. From the
experience with quantum field theory we know that locality and 
spectrum condition seem to be good for defining acceptable dynamics. 

If we use the theory of local observables then
we have to ask for the direction of the time axis. 
Starting from the Gibbs approach to the temperature--states, the time
direction is fixed because one considers boxes at rest. But in 
any other approach,
starting from the infinite system, there is no convincing argument for a 
distinguished time--direction. This phenomenon is known from the kinetic
theory of gasses. Therefore, one should allow any direction inside the forward
light cone as a time--direction. Indeed, looking at the free field, 
P. Junglas [Jun85] %refs
was able to show that in this example KMS--states exist for all $\beta$
belonging to the forward and backward light cone. 

Unfortunately it is unknown whether or not every local net allows 
KMS--states for all temperatures. One only knows that the nuclearity
condition of Buchholz and Wichmann [BuWi86] %refs
implies the existence of KMS--states for all temperatures [BuJu89]. %refs
These states are locally normal with respect to the vacuum state.

An investigation of complex temperatures will be 
started in this note. The first step is the generalization of 
$\beta$--KMS--states to 
complex values of the temperature. In the complex  it is not 
reasonable anymore to
stay with the concept of states, we rather have to introduce the notation of
KMS--functionals. The problem of analyticity will be discussed in a later 
investigation. Having analyticity in mind might serve as a justification
for looking at KMS--functionals (and not states) when dealing
with complex temperatures.

The second aspect of this note is the introduction of C$^*$--dynamical
systems where the translation group of $\BR^d$ is acting as a group of 
automorphisms. In such a situation we allow the temperature -- rather
the inverse temperature -- to be elements of $\BC^d$.

In the next section we shall define the KMS--functionals for arbitrary 
complex	values of the inverse temperature and shall look at the properties of 
these functionals.
In the third section we shall turn to the theory of local observables and
shall derive some consequences of the locality condition. The main outcome
of the third section is the result that the locality condition
does not allow spacelike temperatures.

In reacent years there have been other investigations also using the 
locality condition especially for the so--called
relativistic KMS--condition [BrBu92,93,94].  % refs
Our interest in this note is the functional
dependence on the temperature and therefore this investigation is
complementary to those of the relativistic KMS--condition. 
But hopefully the connection
between the two subjects will be established one day. 

For the rest of this section we collect some mathematical tools.\newline
The theory of KMS--states is closely related to the 
Tomita--Takesaki theory [Tom67], [Tak70]. % refs
Since we need some results from this we collect its main aspects
for the convenience of the reader.\newline
Let $\CH$ be a Hilbert space and $\CM$ be a von Neumann algebra 
acting on this space with commutant $\CM'$. A vector $\Omega$ is cyclic 
and separating for $\CM$ if $\CM\Omega$ and $\CM'\Omega$ are dense in $\CH$.
If these conditions are fulfilled then a modular operator $\Delta$ and a 
modular conjugation $J$	is associated
to the pair $(\CM,\Omega)$
such that
\newline
(i) $\Delta$ is self-adjoint, positive, invertible and fulfills
$$\Delta\Omega=\Omega,\qquad J\Omega=\Omega,$$
(ii) the unitary group $\Delta^{\imath t}$ defines a group of automorphisms
of $\CM$
$$\sigma_t(\CM):=\ad \Delta^{-\imath t}\CM =\CM \qquad \forall t\in\BR,$$
(iii) the operator $J$ is a conjugation i.e. $J$ is antilinear and
$J^2=1$ and commutes with $\Delta^{\imath t}$, i.e.
$$\ad J\Delta =\Delta^{-1},$$
$J$ maps $\CM$ onto its commutant
$$\ad J\CM  =\CM',$$
(iv) for every $A\in\CM$ the vector $A\Omega$ belongs to the domain
of $\Delta^{\frac12}$.\newline
The operators $S:=J\Delta^{\frac12}$ and $S^*=J\Delta^{-\frac12}$
have the property
$$\eqalign{
SA\Omega=A^*\Omega&\qquad \forall A\in \CM,\cr
S^*A'\Omega=A^{'*}\Omega&\qquad\forall A'\in\CM'.\cr}$$
(v) The function
$$f(t):=(\Omega,B\sigma_t(A)\Omega),\qquad A,B\in \CM$$
has an analytic continuation into the strip $S(0,1)=\{z;0<\Imt z <1\}$.
At the upper boundary it takes the values
$$f(x+\imath)=(\Omega,\sigma_t(A)B\Omega).$$
This relation is called the KMS--boundary--condition.\newline
(vi) Let $\CM$ be a von Neumann algebra with cyclic and separating vector 
$\Omega$
and let $\ad U(t)$ define a group of automosphisms of $\CM$ and assume
$U(t)\Omega=\Omega$ and
$(\Omega,BU(t)A\Omega)$ fulfill the KMS--boundary condition for $\beta=1$
then $U(t)$ is the modular group of $\CM$.

For the proof see the exposition of Takesaki [Tak70] or textbooks as
for instance Bratteli and Robinson [BrRo79] or Kadison and Ringrose [KaRi89].
% refs

We often have to use the polar decomposition of functionals. If $\phi \in
\CA^*$ then there exists a positive linear functional $\omega$ and a partial
isometry $V\in\CA^{**}$ with
$$\phi(A)=\omega(V^*A).$$
The partial isometry $V$ is unique if we require that $V^*V=E$ is the
support--projection of $\omega$. The positive functional $\omega$ is
faithful on the von Neumann algebra $\CA^{**}_E$.
%
\babsatz
{\bbf 2. C$^*$--dynamical systems and complex temperatures}
\mabsatz

We start the investigation by looking at a C$^*$--dynamical system where
the dynamics are given by the translation group of $\BR^d$. We assume the 
following
\mabsatz
{\bf 2.1 Notation:}
\mabsatz
1. $\{\CA,\alpha,\BR^d\}$ denotes a C$^*$--dynamical system where\sabsatz
\item{a)} $\CA$ is a C$^*$--algebra,
\item{b)} $\alpha: \;\BR^d\longrightarrow \;Aut(\CA)$, i.e., $\alpha_a$ is 
a representation
of the translations of $\BR^d$ as automorphisms of $\CA$,
\item{c)} the representation $\alpha_a$ is strongly continuous. This means
for $A\in\CA$ the function $a\to\alpha_a(A)$ is a continuous function with
values in $\CA$ equipped with the norm topology.
\sabsatz
2. $\CA_0$ denotes those elements in $\CA$ such that
$\alpha_a(A)$ has an extension as entire analytic function
for $a\in\BC^d$. $\CA_0$ is norm-dense in $\CA$.
\sabsatz
3. For $A\in\CA_0$ one has the relation
$(\alpha_{\overline z}(A))^*=\alpha_z(A^*)$ for all $z\in
\BC^d$. This is due to the fact that both sides are analytic in 
$z\in \BC^d$ and coincide for real $z$.
\sabsatz
4. We denote by $\CA_{00}$ those elements in $\CA_0$ for which $\alpha_a(A)$
is an entire analytic function of exponential growth. $A\in\CA_{00}$ if
$\alpha_a(A)$ is the Fourier-transform of an operator-valued distribution
with compact support. $\CA_{00}$ is  norm-dense in $\CA_0$ and hence
also in $\CA$.
\mabsatz

Next we want to introduce the temperature--functionals. Here we follow
the definition of Haag, Hugenholtz, and Winnink [HHW67] % refs
by means of the KMS boundary condition. We will only use
the subset of analytic elements in the algebra.
Since we allow the temperature to be in $\BC^d$ we look at these functionals
as a vector--bundle over $\BC^d$.
\mabsatz
{\bf 2.2 Definition:}\sabsatz{\it
Let $\{\CA,\alpha,\BR^d\}$ be a C$^*$--dynamical system fulfilling the 
assumptions 2.1.
For $z\in \BC^d$ we set
$$F_z=\{\varphi\in\CA^*;\varphi(B\alpha_{\imath z}(A)-AB)=0 \;\forall\;
A,B\in\CA_0\}.$$
We call $F_z$ the KMS--fibre over $z$.} 
\mabsatz

First we collect some obvious results, which holds for every KMS--fibre.
Notice that every KMS--fibre contains at least the element zero.
\mabsatz
{\bf 2.3 Lemma:}\sabsatz{\it
Let $\{\CA,\alpha,\BR^d\}$ be a C$^*$--dynamical system fulfilling the 
assumptions 2.1, then every KMS--fibre has the properties:\sabsatz
\item{$(i)$} The set $F_z$ is weakly closed.
\item{$(ii)$} The sets $F_z$ are invariant under all automorphisms
commuting with the translation, in particular with the translations
themselves. This means 
$$\varphi\in F_z\;{\rm implies}\;\alpha_a\varphi\in F_z,\quad 
\forall a\in\BR^d.$$
\item{$(iii)$} If $\varphi\in F_z$ then $\varphi^*\in F_{\overline z}$.
\item{$(iv)$} If $\varphi\in F_z$, $z=\xi+\imath\eta$ and $A,B\in \CA$ 
then the function $f(x)=\varphi(B\alpha_{x\xi}(A))$
has an analytic continuation in the variable $\zeta=x+\imath y$ 
into the strip
$S(0,1):=\{\zeta;0<\Imt \zeta<1\}$. In this strip $f(\zeta)$ is bounded by
$$|f(\zeta)|\leq \norm A\norm B\norm\varphi,$$
and $f(\zeta)$ has continuous boundary values. 
We find the relation
$$f(x+\imath)=\varphi(\alpha_{x\xi+\eta}(A)B)$$
at the upper boundary.} 
\mabsatz

{\sl Proof\/}: (i) $F_z$ is defined as the annihilator of family of
elements. Hence it is the intersection of weakly closed linear subspaces.
\newline
(ii) is due to the fact that $\alpha_a$ is a commutative group.
This implies that the set of elements, annihilated by $F_z$, is invariant
under translations. \newline
(iii) Recall the definition of $\varphi^*$ 
by the formula $\varphi^*(A)=\varphi(A^*)$. This implies by the fact that
$\alpha_x$ is a $*$--automorphism:
$$\varphi^*(B\alpha_{\imath\overline z}(A)-AB)=
\varphi(\alpha_{-\imath z}(A^*)B^*-B^*A^*).$$ 
Replacing $A^*=\alpha_{\imath z}(A')$ we obtain 
$$\varphi^*(B\alpha_{\imath \overline z}(A)-AB)=
\varphi(A'B^*-B^*\alpha_{\imath z}(A'))=0.$$
(iv) For $A,B\subset \CA_0$
we know that $f_{A.B}(\zeta):=\varphi(B\alpha_{\imath \zeta\xi}(A))$ is 
analytic in $\zeta$. Since $\varphi\in F_z$ we find
$$\varphi(B\alpha_{(x+\imath)\xi}(A))=\varphi(B\alpha_{\imath z}(
\alpha_{x\xi+\eta}(A))=\varphi(\alpha_{x\xi+\eta}(A)B).$$
This implies that the function $f_{A,B}(\zeta)$ is bounded in the
strip $S(0,1)$ with the estimate
$$|f_{A,B}(\zeta)|\leq\norm\varphi \norm A \norm B,\quad {\rm for}
\quad \zeta\in S(0,1).$$ 
If we take $A,B\in\CA$ and choose sequences $A_n,\;B_n\subset \CA_0$
converging in norm to $A$ and $B$ respectively, then $f_{A_n,B_n}(\zeta)$
is a sequence of bounded analytic functions on $S(0,1)$ which converge
on the boundary of the strip. The limit--function
is continuous on the boundary because the translations act continuously. 
Hence we obtain the convergence of the analytic functions.
$\hfill\Box$
\mabsatz

Next we want to look at the invariance properties of KMS--functionals.
It is known that $\beta$--KMS--states are invariant under 
"time" translations. For complex temperatures one finds the following
results:
\mabsatz
{\bf 2.4 Proposition:}\sabsatz{\it
Assume $\varphi\in F_z$ and let $z=\xi+\imath\eta$ then we obtain:
$$\eqalign{
\alpha_{\lambda\xi}\varphi&=\varphi\quad \forall\;\lambda\in \BR\cr
\alpha_{n\eta}\varphi&=\varphi\quad n=0,\pm 1,\pm 2,.... \cr}$$
The first equation shows that $\alpha_{\lambda\xi}\varphi$ is
constant while $\alpha_{\lambda\eta}\varphi$ is	periodic.}
\mabsatz

{\sl Proof\/}:
Choosing $B=1$ we obtain $\varphi(\alpha_{\imath z}(A))=\varphi(A)$ 
which implies\newline
$\varphi(\alpha_{n\imath z}(A))=\varphi(A)$. Replacing $A$ by 
$\alpha_{n\eta}A$ we obtain from the last equation 
$$|\varphi(\alpha_{\imath n\xi}(A))|=|\varphi(\alpha_{n\eta}(A))|
\leq \norm{\varphi}\norm{A}.$$ 
Now choosing $A\in\CA_{00}$ and replacing
$A$ by $\alpha_{x\xi}(A)$ we see that
the function $\varphi(\alpha_{\zeta\xi}(A))\quad \zeta\in \BC$ is bounded 
and hence $\varphi(\alpha_{\zeta\xi}(A))$ does not depend on $\zeta$.
As $\CA_{00}$ is norm dense in $\CA$ we obtain the first equation. 
Using this result we obtain the second equation from the relation
$\alpha_{\imath nz}\varphi=\varphi$.$\hfill\Box$
\mabsatz

The next Lemma collects	results concerning the polar decomposition of states.
The first part is of general nature and the second and third part 
is valid for KMS--functionals only. 
\mabsatz
{\bf 2.5 Lemma}\sabsatz{\it
A. Let $\CA$ be a $C^*$-algebra, $\alpha\in Aut(\CA)$ and $\varphi\in \CA^*$.
Assume $\omega=|\varphi|$ and $\alpha \varphi=\varphi$. Then
\sabsatz
\item{a.} $\alpha\omega=\omega.$
\item{b.} If $\varphi(A)=\omega(V^*A)$ with $V^*V$ the support--projection
of $\omega$ then $\alpha(V^*V)=V^*V$.
\item{c.} If $\omega=\omega_{\psi}$ then one has $\alpha(V)\psi=
V\psi$. 
\sabsatz
B. The KMS--boundary condition
$$\varphi(B\alpha_{\imath z}(A))=\varphi(AB)$$
can be extended to all elements belonging to $\CA^{**}$. This means
for $A,B\in\CA^{**}$ the function $\varphi(B\alpha_{x\xi}(A))$ in $x$
is the boundary--value of an analytic function holomorphic in the
strip $0<\Imt x<1$. It has continuous boundary--values at $\Imt x=1$
and satisfies
$$\varphi(B\alpha_{(x+\imath)\xi}(A))=\varphi(\alpha_{x\xi+\eta}(A)B).$$
\sabsatz
C. Assume $\varphi\in F_z$ and $\varphi(A)=\omega(V^*A)$ where
$E=V^*V$ is the support--projection of $\omega$ then one has:
\sabsatz
\item{d.} $EV=VE=V\in \CA^{**}_E$.
\item{e.} $\alpha_a(V)=V$ for $a=\lambda\xi$ or $a=\eta$.
\mabsatz}
 
{\sl Proof\/}:
If $\varphi=0$ then the statement is trivial. Now assume $\norm{\varphi}=1$
and let $\varphi(A)=\omega(V^*A)$ be the polar decomposition of
$\varphi$ in $\CA^{**}$ where $V^*V$ is the support--projection of 
$\omega	> 0$. With this decomposition we obtain 
$\omega(V^*\alpha(V))=\varphi(\alpha(V))=
\varphi(V)=\omega(V^*V)$. This implies the following:
If we write $\omega=\omega_{\psi}$ we conclude from the equation 
$(V\psi,V\psi)=(V\psi,\alpha(V)\psi)$ that $\alpha(V)\psi=V\psi$
because $\norm{V\psi}=1$ and $\norm{\alpha(V)\psi}\leq 1$. This is
statement A.c. Now
we obtain $\norm{\alpha(V)\psi}^2=\omega(\alpha(V^*V))=\norm{V\psi}^2
=\omega(V^*V)$. If we denote
$\alpha(V^*V)=F$ then we obtain $(\psi,F\psi)=(\psi,\psi)$ which
implies $F\psi=\psi$ and we get $F\geq E$
since $E$ is the support--projection of $\psi$.
Replacing $\alpha$ by $\alpha^{-1}$ we obtain the
invariance of $E=V^*V$. Hence we have shown A.b.
Combining A.b.,c. we find
$\omega(\alpha(A))=\omega(V^*V\alpha(A))=\omega(\alpha(V^*VA))=
(\alpha(V)\psi,\alpha(V\pi(A))\psi)=(V\psi,\alpha(V\pi(A))\psi)=
\varphi(\alpha(VA))=\varphi(VA)=(V\psi,V\pi(A)\psi)=\omega(A)$. This shows
part A. of the lemma. 

We know from Proposition 2.4 that $\omega$ is invariant under
translations in the direction of $\xi$. Hence there exists a continuous 
representation $U(x)$ with
$U(x)\pi(A)\psi=\pi(\alpha_{x\xi}(A))\psi$ in the representation space
$\CH_\omega$. This implies that the two equations 
$$\eqalign{
(\psi,V^*\pi(B)U(x)\pi(A)\psi)&=\varphi(B\alpha_{x\xi}(A))\cr
(\psi,V^* \pi(\alpha_{-\eta}(A))U(-x)\pi(B)\psi)&=\varphi(\alpha_{-\eta}(A)
\alpha_{-x\xi}(B))\cr}$$
can be extended to all of $\CA^{**}$ as continuous functions. 
This follows from Kaplansky's density theorem and from the continuity of 
$U(x)$.	We know that the approximating functions are connected by bounded
analytic functions. This implies by Vitali's theorem the convergence
of the analytic functions, so that also the limiting functions satisfy the
KMS--boundary condition.

We know from A.b that $\alpha_z(E)=E$. This implies by the KMS--condition
for $(\CA^{**}_E,\psi)$	and the invariance of $\varphi$ the relation
$\varphi(EA)=\varphi(E\alpha_{\imath z}(A))=\varphi(AE)=\varphi(A)$. Using 
the representation of $\varphi$ we obtain: $\omega(V^*EA)=\omega(V^*A)$ or if
$\omega=\omega_{\psi}$ we find the relation
$(EV-V)\psi=0$ by inserting $A=(EV-V)$. Applying $\pi(\CA)'$  
to this equation we get $EVE-VE=0$.
By construction we also have $VE=V$ and therefore  $V\in\CA^{**}_E$.
This is the first statement of C. From $\alpha(E)=E$ we obtain 
$\alpha(V)\in \CA^{**}_E$. The invariance of $V$
follows from $\alpha_a(V)\psi=V\psi$ since $\psi$ is separating for 
$\CA^{**}_E$. $\hfill\Box$
\mabsatz

>From this last lemma we can draw important conclusions for the
representation defined by the positive functional $\omega$.
\mabsatz
{\bf 2.6 Proposition:} \sabsatz{\it
Let $\CA$ fulfill the conditions 2.1 and assume $\varphi\in F_z$ with
$\norm{\varphi}=1$. Let $\omega=|\varphi|$ and $\omega=\omega_\psi$. 
Then the cyclic vector
$\psi$ is also separating for the representation $\pi$.}
\mabsatz

{\sl Proof\/}: Let us denote the central support of $E$ by $F$. We show
that the assumption $E\neq F$ leads to a contradiction. Using the polar
decomposition of $\varphi$ and the KMS--boundary condition we obtain:
$$\eqalign{
\omega(B\alpha_{\imath\xi}(A))&=\omega(V^*VB\alpha_{\imath\xi}(A))=
\varphi(VB\alpha_{\imath\xi}(A))\cr
&=\varphi(\alpha_{\eta}(A) VB)=\omega(V^*\alpha_{\eta}(A)VB)\cr}$$
>From this equation we see the following: If $E\neq F$ then there
exists an element $B\in \CA^{**}$ with $0\neq B=EB=B(F-E)$. Moreover, we know
that $\pi(\alpha_{\imath \xi}(\CA_0))\psi$ is dense in the representation
space. Hence there exists an element $A\in\CA_0$ with 
$\omega(B\alpha_{\imath \xi}(A))\neq 0$. This is the contradiction
because $\omega(. B)=0$. $\hfill\Box$

Using the Lemma 2.5 and the last proposition we can say more about 
the structure of elements in the fibre $F_z$.
\mabsatz
{\bf 2.7 Proposition:} \sabsatz{\it
Assume $\varphi\in F_z,\; z=\xi+\imath \eta$ and 
$\varphi(A)=\omega(V^*A)$ with
$E=V^*V$ the support--projection of $\omega$.
Define for $A\in \CA^{**}_E$ the automorphism $\beta(A)=V^*AV$, then
one has on $\CA^{**}_E=\pi(\CA)''$:
\sabsatz
\item{$(i)$} $\beta\omega=\omega$.
\item{$(ii)$} $\beta$ commutes with $\alpha_{\eta}$ and $\alpha_{r\xi}$ for
$r\in\BR$.
\item{$(iii)$} $\beta^{-1}=\alpha_{\eta}$.
\mabsatz}

{\sl Proof\/}: First we obtain by means of the polar decomposition of 
functionals:
$$\eqalign{
\omega(B\alpha_{\imath\xi}(A))&=\omega(V^*VB\alpha_{\imath\xi}(A))=
\varphi(VB\alpha_{\imath\xi}(A))\cr
&=\varphi(\alpha_{\eta}(A) VB)=\omega(V^*\alpha_{\eta}(A)VB).\cr}$$
Next notice that Lemma 2.3 implies $\omega(.V)\in F_{\overline z}$
if $\omega(V^*.)\in F_z$ and hence we find together
with 2.5.e
$$\eqalign{
\omega(B\alpha_{\imath\xi}(A))&=
\omega(B\alpha_{\imath\xi}(A)V^*V)= \omega(B\alpha_{\imath\xi}(AV^*)V)\cr
&=\omega(\alpha_{-\eta}(AV^*)BV)=\omega(\alpha_{-\eta}(A)V^*BV).\cr}$$
Hence we obtain by comparing the two representations of
$\omega(B\alpha_{\imath\xi}(A))$:
$$\omega(\beta(\alpha_{\eta}(A))B)=\omega(\alpha_{-\eta}(A)\beta(B)).
\eqno(*)$$
Choosing $A=1$ we find
$$\omega(B)=\omega(\beta(B))$$
which is the first statement. In order to show the second statement 
we use the
invariance property of $\omega$ and of $V$ established in 2.5.a. and e.
respectively. Together with $(*)$ we obtain
$$\omega(\beta(\alpha_{\eta}(A))B)=\omega(\alpha_{-\eta}(A)\beta(B))=
\omega(\beta^{-1}(\alpha_{-\eta}(A))B).$$
Since $\omega$ is separating for $\CA^{**}_E$ we find $\beta\alpha_{\eta}
=\beta^{-1}\alpha_{-\eta}$.

We know from 2.5.A.a. and 2.7.i. that $\omega$ is invariant under 
$\beta\circ\alpha_{\eta}$.
Let $\omega=\omega_{\psi}$ and define the unitary operator $U$ by
$$UA\psi=\beta(\alpha_{\eta}(A))\psi.$$
As in the proof of (i) we have
$$\omega(B\alpha_{\imath\xi}(A))=\omega(\beta(\alpha_{\eta}(A))B).$$
This means (For simplicity we write $A$ instead of $\pi(A)$.):
$$(B^*\psi,\alpha_{\imath\xi}(A)\psi)=
(\beta(\alpha_{\eta}(A))^*\psi,B\psi),$$
and if we introduce the operator $S$ for the pair $(\pi(\CA)'',\psi)$ we see
from the last equation
$$\alpha_{\imath\xi}(A)\psi\in\CD(S^*)$$
and hence we obtain
$$S^*\alpha_{\imath\xi}(A)\psi=\beta(\alpha_{\eta}(A))^*\psi=
S\beta(\alpha_{\eta}(A))\psi$$
This implies that both sides belong to the domain of $S$ and hence
we get
$$\Delta^{-1}\alpha_{\imath\xi}(A)\psi=\beta\circ\alpha_{\eta}(A)\psi.$$
Here $\Delta$ is the modular operator of the pair $(\pi(\CA)'',\psi)$.
Multiplying both sides with $\alpha_{\imath\xi}(A)\psi$ and using the 
commuting property of $\alpha_{\lambda\xi}$ with $\beta\alpha_{\eta}$ 
we find with $A\in\CA_0$ and with Lemma 2.5.e.:
$$\eqalign{
(\alpha_{\imath\xi}(A)\psi,\Delta^{-1}\alpha_{\imath\xi}(A)\psi)&=
(\alpha_{\imath\xi}(A)\psi,\beta(\alpha_{\eta}(A))\psi)\cr
&=(\alpha_{\imath\xi/2}(A)\psi,\beta(\alpha_{\eta}(\alpha_{\imath\xi/2}(A)))
\psi)\cr
&=(\alpha_{\imath\xi/2}(A)\psi,U\alpha_{\imath\xi/2}(A)\psi).\cr}$$
Since the lefthand side is positive and since 
$\{\alpha_{\imath\xi/2}(A)\psi\}$ is dense in $\{\pi(A)\psi\}$ we obtain 
$U\geq 0$ which implies $U=1$.$\hfill\Box$\par
\mabsatz

>From this proposition we obtain the following result concerning 
$F_{\xi+\imath\eta}$ and $F_\xi$:
\mabsatz
{\bf 2.8 Theorem:} \sabsatz{\it
Assume $\varphi\in F_z$ and $\varphi(A)=\omega(V^*A)$ with
$z=\xi+\imath \eta$ then:\sabsatz
$(i)$
$$\omega\in F_{\xi}.$$
$(ii)$ The fibre $F_{\xi}$ is generated by its positive 
elements.\newline
$(iii)$	Assume $\omega\in F_{\xi}$ is a state and $\eta\in\BR^d$. If\sabsatz
\item{$(\alpha)$} $\alpha_{\eta}\omega=\omega$, and
\item{$(\beta)$} if $E$ is the support of $\omega$ and if $\alpha_{\eta}$ is
inner for the algebra $\CA^{**}_E$. \newline
Then with $z=\xi-\imath\eta$ there exists an element $\varphi\in F_z$
with $|\varphi|=\omega$.}
\mabsatz

{\sl Proof\/}: We know that $\alpha_{\eta}$ is inner for the algebra 
$\pi(\CA)''$. Therefore, with the same calculation used in the
proof of 2.7.(i) we obtain
$$\omega(B\alpha_{\imath\xi}(A))=\omega(\beta(\alpha_{\eta}(A))B)=
\omega(AB).$$
Hence $\omega\in F_{\xi}$. \newline
(ii) If $\varphi\in F_{\xi}$ then by Lemma 2.3 also $\varphi^*\in 
F_{\xi}$. So $F_{\xi}$ is generated by its self-adjoint 
elements. Next assume $\varphi(.)=\omega(U.)$ with $U^2$ the 
support--projection of $\omega$. Since $\varphi$ and 
$\omega$ both belong to
$F_{\xi}$ this holds also for $2\varphi^+=\omega+\varphi$ and for 
$2\varphi^-=\omega-\varphi$. \newline (iii)
Since $\alpha_{\eta}$ is inner in $\CA^{**}_E$ exists $V\in\CA^{**}_E$
with $\alpha_{\eta}(A)=V^*AV$ on $\CA^{**}_F$. Hence we obtain
$$\omega(V^*B\alpha_{\eta}\circ\alpha_{\imath\xi}(A))=\omega(\alpha_{\eta}(A)
V^*B)=\omega(V^*AB).$$
Here we have used that $VV^*=E$ is the identity of the representation
$\pi_\omega$.
This shows $\omega(V^*.)$ belongs to $F_z$. $\hfill\Box$
\mabsatz

We end this section with a remark on possible intersection of two
fibres.
\mabsatz
{\bf 2.9 Lemma:}
\sabsatz{\it
Assume we have $\varphi\in F_{z_1}\cap F_{z_2}$ then  
$\alpha_{\imath(z_1-z_2)}$ restricted to $\CA^{**}_E$
is trivial. In this expression $E$ denotes the support--projection of
$|\varphi|$.}
\mabsatz

{\sl Proof\/}: We obtain 
$$\varphi(B\alpha_{\imath(z_1-z_2)}(A))=\varphi(\alpha_{-\imath z_2}(A)B)=
\varphi(A\alpha_{\imath z_2}(B))=\varphi(BA).$$
${B^*V\psi}$ is dense in the representation space of $\omega$
since $\varphi(.)=\omega(V^*.)$ and since $V$ is unitary in $\CA^{**}_E$
This implies $U(z_1-z_2)$ is trivial when restricted to $\CA^{**}_E$,
where $U$ is defined by the equation $U(z_1-z_2)\pi(A)\psi=
\pi(\alpha_{(z_1-z_2)}A)\psi$.
$\hfill\Box$
\newpage
{\bbf 3. KMS-condition and local nets}
\babsatz
In this section we will assume in addition to 2.1
that we are dealing with a theory of local observables.
We introduce the following 
\mabsatz
{\bf 3.1 Notation:}
\sabsatz
A theory of local observables will be denoted by $\locnet$. The symbols
have the following meaning:\newline
(i) A $C^*$-algebra $\CA(O)$ fulfilling isotony	is associated with
every bounded open region $O$ in the Minkowski space $\BR^d$. \newline
(ii) $\CA$ denotes the $C^*$-algebra generated by all $\CA(O)$.\newline
(iii) The local net fulfills locality, i.e. if $O_1$ and $O_2$ 
are spacelike separated then the two algebras $\CA(O_1)$ and $\CA(O_2)$
commute.\newline
(iv) $\alpha$ is a representation of $\BR^d$ in Aut($\CA$) with the
property
$$\alpha_a\CA(O)=\CA(O+a)\quad a\in \BR^d.$$
(v) We use the following Minkowski scalar product: $x^2=x_0^2-(\vec x)^2$.

The aim of this section is to show that there do not exist "spacelike"
temperatures in the theory of local observables. By 
Theorem 2.8 this implies also that there is no complex temperature 
with spacelike real part. In other words: The result known from the free 
fields, that
the possible values of $\imath\beta$ belong to the forward or backward
tube, is a general feature of local quantum field theories.
We will show
\mabsatz
{\bf 3.2 Theorem:}
\sabsatz{\it
Let $\locnet$ be as described above. Assume $\xi$ is a spacelike vector.
If $\omega\in F_{\xi}$ is a state and if $\{\pi,\CH,\Omega\}$
is the G.N.S.-representation of $\omega$ then
$$\CA /\ker \pi \quad{\rm is\ an\ abelian\ algebra.}$$
This means in particular: If $\CA$ is simple then $F_{\xi}=\{0\}$.}
\mabsatz

For the preparation of the proof of the theorem we remark first:
\mabsatz
{\bf 3.3 Lemma:}\sabsatz{\it
Let $\locnet$ be as described above. Assume $O$ is a bounded open
region centered at the origin. Let $\xi$ be a spacelike vector.  
If $\omega\in F_{\xi}$ is a state and $A,B\in\CA(O)$ then 
$$f(z)=\omega(B\alpha_{z\xi}(A))$$
is a periodic function with period $\imath$. If $a>0$ is the smallest
number such that
$$O+x\xi\subset O'\quad {\rm for}\quad |x|>a$$
then $f(z)$ is analytic except for the cuts
$$[-a,a]+\imath n,\qquad n=0,\pm 1,\pm 2,...$$
and $f(z)$ is bounded everywhere:
$$|f(z)|\leq \max\{\sqrt{\omega(BB^*)\omega(A^*A)},
\sqrt{\omega(B^*B)\omega(AA^*)}\}.$$ }
\mabsatz

{\sl Proof\/}:
Since $\omega\in F_{\xi}$ we obtain from the KMS-condition
$$\omega(B\alpha_{(x+\imath)\xi}(A))=\omega(\alpha_{x\xi}(A)B).$$
If $|x|>a$ then $\alpha_{x\xi}(A)$ and $B$ commute. For $|x|>a$ these 
two relations imply 
$f(x)=f(x+\imath)$ and hence $f(z)$ is periodic in the imaginary 
direction with period $\imath$. Since $f(z)$ is analytic in the strip
$0<\Imt z<1$ we see that $f(z)$ is analytic except for the cuts. Since
$f(z)$ is bounded everywhere it must take its maximum at the cuts.
By the Cauchy-Schwarz inequality we obtain for $\Imt z=+0$:
$$|f(z)|\leq \sqrt{\omega(BB^*)\omega(A^*A)},$$
and for $\Imt z= 1-0$:
$$|f(z)|\leq \sqrt{\omega(B^*B)\omega(AA^*)}.$$
This shows the lemma.$\hfill\Box$
\mabsatz

In order to obtain more properties of this state we will compute
its Fourier-transform. We obtain
\mabsatz
{\bf 3.4 Lemma:}
\sabsatz{\it 
With 
$$\CF f(\epsilon)=\int \omega(B\alpha_{x\xi}(A))e^{-\imath\epsilon x}dx$$
we obtain
$$\CF f(\epsilon)=\cases{g(\epsilon)\frac1{1-e^{-\epsilon}}
\quad{\rm for}\quad \epsilon>0 \cr
-g(\epsilon)\frac{e^{\epsilon}}{1-e^{\epsilon}}\quad{\rm for} \quad 
\epsilon<0 \cr}=g(\epsilon)\frac1{1-e^{-\epsilon}}.\eqno(1)$$
The function $g(\epsilon)$ is the Fourier-transform of the commutator
$$g(\epsilon)= \int\limits_{-a}^a \omega([B\alpha_{x\xi}(A)])
e^{-\imath\epsilon x}dx\eqno(2)$$
and hence it is an entire analytic function. At the origin of $\CF f(\epsilon)$
there will appear
an additional $\delta$-function contribution describing the cluster
property of $f(x)$.}
\mabsatz

{\sl Proof\/}:
For $\epsilon>0$ we can deform the path of integration into the lower
half plane and remain with the integrals around the cuts. So we obtain
in this situation
$$\eqalign{
\CF f(\epsilon)&=\sum_{n=0}^{-\infty} \int\limits_{-a+n\imath}^{a+n\imath}
\omega([B\alpha_{x\xi}(A)])e^{-\imath \epsilon z}dz \cr
&=\int\limits_{-a}^a \omega([B\alpha_{x\xi}(A)])e^{-\imath \epsilon x}dx
\frac1{1-e^{-\epsilon}}.\cr}$$
If $\epsilon<0$ we can deform the contour of integration into the
upper half plane. In this situation two things are changing, namely the sum
goes only from 1 to $\infty$ and the integration around the cuts changes
the direction, which accounts for the sign in the formula. Since the 
function $f(x)$ is the Fourier-transform of a measure, the only additional
contribution at the origin can be a $\delta$-function. But in the following
its exact coefficient is not of interest.$\hfill\Box$
\mabsatz

Next we want to compute the Fourier-transform of the commutator, i.e.
we want to look at the function $g(\epsilon)$. But first let us introduce
some notations. Let $\omega\in F_{\xi}$ be a state.
Let $(\pi,U(x\xi),\psi)$ be the G.N.S.--representation given by
$\omega$, then we can write 
$$U(x\xi)=\int\exp\{\imath \epsilon x\}\,dE(\epsilon)$$ 
where $E(\epsilon)$ is the spectral resolution of the one parametric
group $U(x\xi)$. From the KMS--condition follows that
$U(x\xi)$ is also the
modular group for $\pi(\CA)''$. We obtain
\mabsatz
{\bf 3.5 Lemma:}
\sabsatz{\it
In the situation described above we have with
$$  (\psi,\pi(B)U(x\xi)\pi(A)\psi)=\int 
(\psi,\pi(B)dE(\epsilon)\pi(A)\psi)e^{\imath x\epsilon}$$
the equation
$$\omega([B,\alpha_{x\xi}(A)])=\int 
(\psi,\pi(B)dE(\epsilon)\pi(A)\psi)(1-e^{-\epsilon})e^{\imath x\epsilon}
\eqno(3)$$
where the expression on the right hand side is well defined because of
the KMS--condition.}
\mabsatz

{\sl Proof\/}:
With $S=J\Delta^{1/2}$ we obtain
$$\eqalign{
\omega([B,\alpha_{x\xi}(A)])&= (\pi(B^*)\psi,U(x\xi)\pi(A)\psi)
-(\pi(A^*)\psi,U(-x\xi)\pi(B)\psi)\cr
=&(\pi(B^*)\psi,U(x\xi)\pi(A)\psi)
-(S\pi(A)\psi,U(-x\xi)S\pi(B^*)\psi)\cr
=&(\pi(B^*)\psi,U(x\xi)(1-\Delta)\pi(A)\psi)\cr}$$
The last expression is well defined since $\pi(A)\psi$ and
$\pi(B^*)\psi$ are both in the domain of definition of $\Delta^{1/2}$.
Taking now the inverse Fourier-transform we obtain the stated result.
$\hfill\Box$

Since $(1-e^{-\epsilon})$ vanishes at the origin the 
contribution coming from the invariant state drops out.
Details are described in the next result.
\mabsatz
{\bf 3.6 Lemma:}
\sabsatz{\it
Let $E_0$ be the projection onto the invariant vectors of $U(x\xi)$.
Define
$$E'(\epsilon)=\cases{E(\epsilon)\quad{\rm for}\quad \epsilon<0\cr
E(\epsilon)-E_0 \quad{\rm for}\quad \epsilon\geq 0\cr}$$
then
$$g(\epsilon)\frac{d\epsilon}{1-e^{-\epsilon}}=(\psi,\pi(B)dE'(\epsilon)\pi(A)\psi)$$
is an entire analytic function of exponential type $a$.}
\mabsatz

{\sl Proof\/}:
>From (2) and (3) we obtain
$$g(\epsilon){d\epsilon}=(\psi,\pi(B)dE(\epsilon)\pi(A)\psi)
(1-e^{-\epsilon}).\eqno(3')$$
>From (2) we know that $g(\epsilon)$ is an entire analytic
function of exponential type $a$. Eqn. (3) implies $g(0)=0$. Hence
$g(\epsilon)(1-e^{-\epsilon})^{-1}$ is again an	entire analytic
function of exponential type $a$. If we want to divide (3') by
$(1-e^{-\epsilon})$ we must remember that this factor has annihilated
the $\delta(\epsilon)$ contribution, i.e. at zero we only obtain the 
continuous part of the spectral family. This is the statement of
the lemma. $\hfill\Box$
\mabsatz

{\sl Proof of the Theorem\/}:
The function $\{f(z)-(\psi,\pi(B)E_0\pi(A)\psi)\}{d\epsilon}$ is
the Fourier-transform of
$$(\psi,\pi(B)dE'(\epsilon)\pi(A)\psi).$$
By Lemma 3.6 this expression is entire analytic and of 
exponential type $a$. Hence its Fourier-transform vanishes for $|x|>a$. 
Since $f(z)$ is the boundary value of an analytic function we get
the vanishing everywhere. This implies
that $f(z)$ is constant and coincides with 
$(\psi,\pi(B)E_0\pi(A)\psi)$. Next notice that
$\cup \CA(O)$ is dense in $\CA$. This implies that $f(z)$ is constant
for arbitrary $A,B\in\CA$. Therefore, the translations in the direction
$\xi$ are represented trivially. 
Together with the locality--condition follows
that $\CA/\ker\pi$ is abelian.$\hfill\Box$ 
\mabsatz

This result means that there exist no spacelike temperatures
and by Theorem 2.8 also no complex temperatures with spacelike
real part. Therefore, $\imath$ times temperatures can only
belong to the forward--  or to the backward tube. 

The KMS--functionals in the forward tube are connected with the 
vacuum state and those in the backward tube are related to the plenum
(ceiling) state.
\babsatz
{\bbf References}
\def\ref{\par\vskip 10pt \noindent \hangafter=1
\hangindent 22.76pt}
\parskip 5pt
{\baselineskip=3ex\eightpoint\smallskip
%\font\eightit=cmti8
%\font\eightbf=cmbx8
\def\it{\eightit}
\def\bf{\eightbf}
%
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\bye