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for different temperatures  \hskip 1cm \folio } 
\def\leftheadline{\sevenrm \folio \hskip 1cm C.\ D.\ J\"akel \hfill }
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\vskip 1cm
                  
\noindent
{\twentyrm The Relation between KMS-States}
                  
\vskip .2cm
\noindent
{\twentyrm for Different Temperatures}
                  
\vskip .8cm
                  
\noindent
CHRISTIAN D.\ J\"AKEL

\noindent
{\sevenit  Inst.\ f.\ Theoretische Physik, Universit\"at Wien, 
Boltzmanng.\ 5, A-1090 Wien, Austria}

\noindent
{\sevenit  e-mail: cjaekel@esi.ac.at}
                  
\vskip .3cm     
\noindent {\sevenbf Abstract}. {\sevenrm 
Given a thermal field theory for a certain temperature, 
we present a method to construct the theory at any finite positive temperature provided the 
number of local degrees of freedom is restricted in a physically sensible manner.
Our work is based on a construction invented by Buchholz and Junglas [BJu 89]. 
Starting from the vacuum representation, they
established the existence of thermal equilibrium states (KMS-states) 
for a large class of quantum field theories. The KMS-states were 
constructed as limit points of nets of states which represented strictly localized excitations of 
the vacuum. We adjust their method to the general structure of thermal quantum field theories. 
In a first step we construct states which closely resemble
KMS-states for the new temperature in a 
local region $\scriptstyle \O_\circ \subset \r^4$, but coincide with the given
KMS-state in the space-like complement of a slightly larger region~$\scriptstyle \hat{\O}$.
%%%%%%%
By a weak$^*$-compactness argument there always exists a convergent subsequence of
states as the size of
$\scriptstyle \O_\circ$ and $\scriptstyle \hat{\O}$ tends towards $\scriptstyle \r^4$.
%%%%%%%
Whether or not the limit state of such a subsequence 
is a global KMS-states for the new temperature, 
depends on the surface energy contained 
in the layer in between the boundaries of $\scriptstyle \O_\circ^{(i)}$ and 
$\scriptstyle \hat{\O}^{(i)}$ as $\scriptstyle i \to \infty$. 
This surface energy is controlled by a generalized cluster theorem.} 



\vskip .1cm     
\noindent
{\sevenbf Mathematics Subject Classifications (1991).} {\sevenrm 81T05.}                  

                  
\vskip .8 cm



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\noindent

\Hl{Introduction}

\noindent
Algebraic quantum statistical mechanics starts 
from a $C^*$-algebra $\A$ and a strongly continuous one-parameter group of 
automorphisms $t \mapsto \tau_t$, $t \in \R$ (see e.g.\ [BR][E][R][Se][Th]). 
Such a pair $(\A , \tau)$ is called a $C^*$-dynamical system.
Quantum ergodic theory can be set up in this framework and 
without specifying a state one can, for instance, 
study the mixing properties of such a dynamical system (see
e.g.\ [J\"a 91]). More specific information is available if one distinguishes a certain folium of
states. Before we do so, 
let us briefly outline how a thermal quantum field theory fits into the framework of
algebraic quantum statistical mechanics\footnote{$^\star$}{\sevenrm
Non-relativistic quantum field theories and spin systems fit nicely into the
framework of algebraic quantum statistical mechanics. 
In low dimensions the latter have been worked out in 
great detail (see e.g.\ [BR]). Only recently the benefits of formulating 
thermal quantum field theory (TQFT) in the algebraic framework  
were emphasized in a series of papers~[BJu 86, 89][BB 94][N][J\"a a,b,c,d].}: Assume a QFT  
is specified by a $C^*$-algebra $\A$ together with a
net
%
\# { \O \to \A (\O), \qquad \O \subset \R^4, }
%
of subalgebras associated with open, bounded space-time regions $\O$ in Minkowski space
(as described in the monograph by Haag [H].)
In general, the Hermitian elements of~$\A(\O)$ 
are interpreted as the observables which
can be measured at times and locations in~$\O$ (see also [HK]). 
The time evolution $\tau \colon \R \to Aut(\A)$ will respect the net structure
$\O \to \A(\O)$. If $\tau$ is not a priori strongly continuous, then
we may restrict the net $\O \to \A(\O)$ to a subnet which complies with the continuity 
condition (see e.g.\ [S, Prop.\ 1.18]). 

\Rem{If one prefers to start from  
a more conventional formulation of a QFT in terms of operator-valued distributions, then
the algebra $\A(\O)$ associated with a space-time region~$\O \subset \R^4$ 
may be thought of as being generated by bounded functions of the underlying smeared quantum
fields, currents, etc.\ [BoY]: for instance,
if $\phi(x)$ is any such field and if $f(x)$ is any real test 
function with support in a bounded region~$\O$ of space-time, then the corresponding (unitary)
operator 
%
\# { a := \exp \Bigl( i \int {\rm d}x \, f (x) \phi (x) \Bigr) }
%
would be a typical element of~$\A (\O)$. In this way the quantum fields provide 
a ``coordinate system'' for the algebra $\A$. However, we emphasize that
only the algebraic relations between the elements of $\A$ are of physical significance.}


Equilibrium states are characterized by their time-evolution invariance and stability
against small dynamical (or adiabatic [NT]) perturbations 
of the time-evolution $\tau$ [HKTP]. 
Adding a few technical assumptions such a heuristical
characterisation of an equilibrium state leads to a sharp mathematical criterion [HHW],
named for Kubo [K], Martin and Schwinger~[MS]: 

\Def{A state
$\omega_\beta$ over $\A$ is called a $(\tau , \beta )$-KMS-state for 
some $\beta \in \R \cup \{ \pm \infty \}$, if 
% 
\# { \omega_\beta \bigl( a \tau_{i \beta} (b) \bigr) = \omega_\beta (b a) }
%
for all $a, b$ in a norm dense,
$\tau$-invariant $*$-subalgebra of $\A_\tau \subset \A$. Here  $\A_\tau$
denotes the set of analytic elements for $\tau$.}

\vskip .5cm
\noindent
{\it Remarks:}
\vskip .3cm
\noindent
i.)      There are $C^*$-dynamical 
systems~$(\A, \tau)$, where a KMS-state exists at one and only one 
value~$\beta \in \R$  [BR, 5.3.27].
\vskip .1cm

\noindent
ii.)     The class of 
models of a countable 
number of free, scalar particles proposed by Hagedorn [Ha] provides us with
quantum field theories 
which obey all the Wightman and Haag-Kastler axioms but in which no equilibrium states exist 
above a certain temperature [BJu 86]. 
%
\vskip .1cm

\noindent
iii.)     
One can specify conditions on the 
phase-space properties of a QFT in the vacuum representation, such that KMS-states exist for all
$\beta > 0$ [BJu 89].
%
\vskip .1cm

\noindent
iv.)    Arbitrary KMS-state can be represented 
in a unique manner as a convex superposition of extremal KMS-states.  
If the dynamical system reflects the basic elements of 
physical reality, the one expects for high temperatures
and low densities that the set of KMS-states contains a unique element\footnote{$^\star$}
{\sevenrm For non-relativistic fermions with pair-interaction see [J\"a 95].}
whereas at lower temperature it should 
contain many disjoint extremal elements and their 
convex combinations corresponding to various thermodynamic phases and their possible mixtures.
The symmetry, or lack of symmetry of the extremal
KMS-states is automatically determined by this decomposition. By this mechanism 
spontaneous symmetry breaking 
may occure naturally, if one starts from a unique KMS-state
for some high temperature and then cools down the physical system.


\vskip 1cm

Given a KMS-state $\omega_\beta$ for $(\A, \tau)$
the GNS-representation $(\pi_\beta, \H_\beta, \Omega_\beta)$ provides us with a  
thermal field theory\footnote{$^\dagger$}
{\sevenrm For the Lagrangian formulation of a TQFT
we refer the reader to the books by Kapusta [Ka], Le Bellac~[L] and Umezawa [U], and 
the excellent review article 
by Landsman and van Weert~[LvW]. Recent work in the Wightman framework can be found in 
[BB 92, 95, 96][St].}
%
\# { \O \to {\cal R}_\beta (\O) := \pi_\beta \bigl( \A(\O) \bigr)'' , \qquad \O \in \R^4.}
%
Under fairly general circumstances different 
values of the inverse temperature~$\beta$ lead to unitarily inequivalent 
GNS-representations [T][BR, 5.3.35].
Hence thermal field theories for different temperatures are frequently treated as completely 
disjoint objects even if they refer to the same vacuum theory, i.e., even if
they show identical interactions on the microscopic level. To 
understand the relations between these `disjoint thermal field theories'
seems to be highly desirable; in fact, for understanding of the interrelations between 
the appropriate macroscopic variables (e.g.\ pressure or 
specific heat) which appear in classical thermodynamics  
a possibility to compare unitary inequivalent representations seems necessary.
One simple case is well known:
Assume that~$ \tau$ can be approximated by a net of inner automorphisms 
such that for
$a \in \A$ fixed,  
%
\# { \lim_{\Lambda \to \infty} \| \tau_z (a) - 
{\rm e}^{i z  h_\Lambda} a {\rm e}^{-i z  h_\Lambda} \| = 0 , 
\qquad  h_\Lambda = h_\Lambda^* \in \A ,}
%
uniformly in $z$ on compact subsets of~$\C$. If $ (\A, \tau)$  
has a KMS-state $\omega_{\beta}$ at some~$ \beta \ne 0$,  
then the net of states
%
\# {\omega_\Lambda (a) = { \omega_\beta 
\bigl( {\rm e}^{{1 \over 2}(\beta - \beta')  h_\Lambda} 
a {\rm e}^{{1 \over 2}(\beta - \beta') h_\Lambda} \bigr) 
\over \omega_{\beta} \bigl( {\rm e}^{ (\beta - \beta') 
h_\Lambda} \bigr) },
\qquad a \in \A  ,} 
%
has convergent subsequences and the limit points 
%
$ \omega_\beta (a) := \lim_{i \to \infty} \omega_{\Lambda_i} (a)$,  
$a \in \A $, 
%
are $(\tau, \beta')$-KMS-states 
$ (0 \le \beta' \le \infty)$ [Pe, 8.12.10]. 
But in general, phase transitions may occur while 
we change the temperature. Consequently ``... there is no simple prescription for connecting 
the~$(\tau, \beta)$--KMS-states for different~$\beta$'s'' (c.f.\ [BR, p.78]). 

However, in this article we will provide a prescription, which covers, as
far as relativistic systems are concerned, the physically relevant cases:
Starting form a given thermal field
theory $\O \to {\cal R}_\beta (\O)$, whose number of local degrees of freedom 
is restricted in a physically sensible manner, we 
construct a KMS-state $\omega_{\beta'}$ and a thermal field theory 
%
\# { \O \to {\cal R}_{\beta'}, \qquad \beta' \in \R^+,}
%
for each finite positive temperature $1 / \beta'$.
The method we use is essentially due to Buchholz and Junglas [BJu 89]. 
Although we almost repeat their line of arguments, there are some 
nontrivial deviations due to the mathematical structure one encounters in thermal field theory. 
In a first step we construct product states $\omega_\Lambda$, $\Lambda = (\O_\circ, \hat{\O})$,
which --- up to boundary effects --- closely resemble KMS-states 
for the new temperature $1 / \beta'$ in a local region $\O_\circ \subset \R^4$, but 
coincide with the given KMS-state~$\omega_\beta$
in the space-like complement of a slightly larger region $\hat{\O}$, i.e.,
%
\# {\omega_\Lambda (ab)  = \omega_\Lambda (a) \cdot \omega_\beta (b) \qquad  
\forall a \in \A (\O_\circ), \quad \forall b \in \A(\hat {\O}').}
% 
At this point our
method is semi-constructive. It does not uniquely fix the product states~$\omega_\Lambda$.
Intuitively the choice of a product state $\omega_\Lambda$ corresponds to a 
choice of boundary conditions which decouple the local region $\O_\circ$, 
where the state already resembles an equilibrium state for the
new temperature, from the outside $\hat{\O}'$, i.e., the space-like complement of~$\hat{\O}$. 
Different choices $\omega_\Lambda$, ${\omega_\Lambda}'$ will manifest themself 
in different expectation values
for observables localized in between the two regions $\O_\circ$ and $\hat{\O}'$, 
i.e., we expect
%
\# {\omega_\Lambda  \ne {\omega_\Lambda}'
\Rightarrow \exists a \in \A (\O_\circ' \cap \hat {\O}) \quad \hbox {such \ that} \quad
\omega_\Lambda (a) \ne {\omega_\Lambda}' (a).}
% 
By standard compactness arguments, the
net of states $\Lambda \to \omega_\Lambda$ has convergent subsequences.
Whether or not these subsequences converge to a global KMS-states for the new temperature, 
depends on the surface energy contained in between the two regions
$\O_\circ$ and $\hat{\O}'$ as their size increases. 
Introducing an auxiliary structure, which can be understood as a local
purification, and applying a generalized cluster theorem [J\"a c], we will 
control these surface energies in all thermal
theories which satisfy a certain ``nuclearity condition'' (see e.g.\ [BW][BD'AL 90a,b][BY]
for related work). 
Consequently, we can single out sequences $\Lambda_i = (\O_\circ^{(i)}, \hat{\O}^{(i)})$ 
such that the limit points\footnote
{$^\star$}{\sevenrm We have simplified the notation here. 
In fact, we will have to adjust the relative sizes of a  
triple $\scriptstyle
\Lambda_i = (\O_\circ^{(i)},\O^{(i)}, \hat{\O}^{(i)})$ of space-time regions.} 
%
\# { \omega_{\beta'} (a) := \lim_{i \to \infty} \omega_{\Lambda_i} (a) ,  
\qquad a \in \A,}  
%
are KMS-states for the new inverse temperature $\beta'$,
$ (0 \le \beta' \le \infty)$. Phase transitions are not excluded by our method: 
by choosing different ``boundary conditions''
we may encounter disjoint KMS-states for the new temperature in the thermodynamic limit. 


Loosely speaking, we have found a method to heat up or cool down 
a quantum field theory. Before we proceed,
let us mention a possible application: Consider a quantum field 
theory in which two disjoint 
vacuum states exist while the KMS-state is unique above some critical temperature. 
To start from one vacuum and construct the other, by simply combining 
the method provided by Buchholz and Junglas to heat up the system with our method 
to cool down the system, is a challenging task.

\vskip 1cm

\Hl{A List of Assumptions}

\noindent
Although from the viewpoint of algebraic quantum statistical mechanics it is more natural to 
start from a $C^*$-dynamical system $(\A, \tau)$ and then characterize equilibrium 
states $\omega_\beta$ and thermal representations $\pi_\beta$
with respect to the dynamics, we will assume here that we are given a
thermal field theory $\O \to {\cal R}_\beta (\O)$ acting on some Hilbert space $\H_\beta$. 
How one can reconstruct a $C^*$-dynamical system $(\A, \tau)$ 
from the $W^*$-dynamical system $({\cal R}_\beta, \sigma)$ will be indicated 
at the end of this section ($\sigma$ will be defined in (19)). 

\vskip.5cm
We now provide a list of our assumptions:
\vskip.3cm
 

\noindent
i.) (Thermal QFT). A thermal QFT is specified by a von Neumann algebra ${\cal R}_\beta$,
acting on a Hilbert space~$\H_\beta$, together with a
net
%
\versuch {(Net structure)} { \O \to {\cal R}_\beta (\O), \qquad \O \subset \R^4, }
%
of subalgebras associated with open, bounded space-time regions $\O$ in Minkowski space.
The net $\O \to {\cal R}_\beta (\O)$ satisfies
%
\versuch {(Isotony)} {{\cal R}_\beta (\O_1) \subset {\cal R}_\beta (\O_2), 
\qquad \O_1 \subset \O_2 , }
%
and
%
\versuch {(Locality)} {{\cal R}_\beta (\O_1) \subset {\cal R}_\beta (\O_2)', 
\qquad \O_1' \subset \O_2 .}
%  
As before, $\O'$ denotes the space-like complement of $\O$.
\goodbreak

\vskip .3cm
\noindent
ii.) (Dynamical law). The time-evolution is given by a strongly continuous
one-parameter group of unitaries 
%
\# { t \mapsto {\rm e}^{i H_\beta t} , \qquad t \in \R  .} 
% 
It acts geometrically, i.e.,
%
\# {{\rm e}^{i H_\beta t} {\cal R}_\beta (\O) {\rm e}^{- i H_\beta t}
\subset {\cal R}_\beta (\O + t e ) \qquad \forall t \in \R . }
%
Here $e$ is a unit-vector in the time-direction in a fixed Lorentz-frame. 

\vskip .3cm
\noindent
iii.) (Unique KMS-vector). 
There exists a distinguished vector --- sometimes called the {\sl thermal vacuum vector} ---
$\Omega_\beta$, cyclic and separating for ${\cal R}_\beta$, such that the associated 
vector state $\omega_\beta := (\Omega_\beta \, , \, . \, \, \Omega_\beta)$ 
satisfies the KMS-condition (3). 
Restricting attention to pure phases we assume that $\Omega_\beta$ is the 
unique --- up to a phase --- normalized eigenvector with eigenvalue $\{ 0 \} $ of $H_\beta$. 

\Rem{As a consequence of this assumption, we can associate the following
modular structure with a given thermal QFT: the polar decomposition 
%
$S= J \Delta^{1/2}$
%
of the closeable operator  
%
\# { S_\circ  A \Omega_\beta = A^* \Omega_\beta , \qquad  A \in {\cal R}_\beta ,} 
%
provides a conjugate-linear isometric mapping~$J$
from $\H_\beta$ onto $\H_\beta$ and a positive self-adjoint 
(in general, unbounded, but densely defined and invertible) operator $\Delta$ acting 
on~$\H_\beta$, satisfying the conditions $J^2 = \1 $ and
%
\# { J \Delta^{1/2} A \Omega_\beta = A^* \Omega_\beta 
\qquad
\forall A \in {\cal R}_\beta. }
%
$J$ and $\Delta$ are called the modular objects associated to the pair $({\cal R}_\beta,
\Omega_\beta)$. The modular conjugation $J$ induces an $*$-anti-isomorphism $j \colon 
A \mapsto JA^*J$ between the algebra of quasi-local observables ${\cal R}_\beta$ and its commutant
(Tomita's theorem). The opposite net
%
\# { \O \to j \bigl( {\cal R}_\beta (\O) \big), \qquad \O \subset \R^4, }
%
porovides a perfect mirror image of the net of observables.
The unitary operators $\Delta^{is}$, $s \in \R$, induce a one-parameter group of
$*$-automorphism 
$\sigma \colon s \mapsto \sigma_s$ of ${\cal R}_\beta$, 
%
\# { \sigma_s (A) = \Delta^{is} A \Delta^{-is} , \qquad s \in \R, \quad A \in {\cal R}_\beta .}
%
$\sigma$ is called the modular automorphism. Takesaki has shown that 
$\omega_\beta$ is a $(\sigma, -1)$-KMS-state. Moreover,
$\sigma$ is uniquely determined by this condition. 
We conclude that in a TQFT
the modular automorphism $\sigma$ coincides --- up to a scaling factor --- with
the time-evolution. Consequently, the modular automorphism respects the net structure, i.e.,
%
\# { \sigma_s \bigl( {\cal R}_\beta (\O) \bigr) = {\cal R}_\beta (\O + s \beta \cdot e) 
\qquad \forall s \in \R.}
%
The real parameter $\beta \in \R^+$ appearing (until now $\beta$ was just a dummy index)
in (20) distinguishes a length scale,
called the thermal wave-length. 
In fact, we can turn the argument up side down: given a thermal field theory 
$\O \to {\cal R}_\beta (\O)$,
it is not necessary to provide an explicit expression for the effective Hamiltonian $H_\beta$.
It is already uniquely specified by the pair $({\cal R}_\beta, \Omega_\beta)$:
By Stone's theorem 
there exists a unique self-adjoint generator $H_\beta$, sometimes called the
{\sl effective} Hamiltonian, such that 
%
\# { \Delta = {\rm e}^{- \beta H_\beta} .}
%
Modular theory implies that for $0 \le \beta < \infty$ the operator $H_\beta$
is not semi-bounded but its spectrum is symmetric and
consists typically of the whole real line [A 72] [tBW].} 

% Consequently,  $\omega_\beta$ is an extremal KMS-state and
%
% \# { {\cal R}_\beta \cap {\cal R}_\beta' = \C \cdot \1 , }
%  
% i.e., ${\cal R}_\beta$ is a factor.

\vskip .3cm
\noindent
iv.) (Reeh-Schlieder property). The KMS-vector
$\Omega_\beta$ is cyclic and separating for the local algebras ${\cal R}_\beta (\O)$,
whenever the space-like complement of $\O \subset \R^4$ is not empty. 

\Rem{The Reeh-Schlieder property is a consequence [J\"a e] of 
additivity\footnote{$^\star$}
{\sevenrm The net $\scriptstyle \O \to {\cal R}_\beta (\O)$ is called additive  if
%
$\scriptstyle \cup_i \O_i = \O \Rightarrow \vee_i {\cal R}_\beta (\O_i) 
= {\cal R}_\beta (\O)  $. Here $\scriptstyle \vee_i {\cal R}_\beta (\O_i)$
denotes the von Neumann algebra generated by the algebras 
$\scriptstyle {\cal R}_\beta (\O_i)$.} 
and the relativistic KMS-condition
proposed by Bros and Buchholz [BB 94]. If the KMS-state is locally normal  w.r.t.\ the vacuum 
representation, then the standard KMS-condition 
(together with additivity of the net in the vacuum representation) is also sufficient to 
ensure the Reeh-Schlieder property of the KMS-vector~$\Omega_\beta$~[J].} 

\vskip .3cm
\noindent
v.) (Nuclearity condition). The thermal field theory $ \O \to
{\cal R}_\beta (\O )$ satisfies the following restrictions on its phase-space properties:
The maps $\Theta_{\alpha, {\cal O}}
\colon {\cal R}_\beta (\O) \to \H_\beta $ given by 
%
\# { \Theta_{\alpha, {\cal O}} (A) = \Delta^\alpha A \Omega_\beta , 
\qquad 0 \le \alpha \le 1/2,}
%
are nuclear for $0 < \alpha < 1/ 2$. In addition, the following bound holds for the nuclear norm 
(for $\alpha \searrow 0$ or $\alpha \nearrow 1/2$ and large diameters $r$
of $\O$)
%
\# { \| \Theta_{\alpha, {\cal O}} \|  \le {\rm e}^{cr^d \bigl 
( \alpha^{-m} + (1 / 2 - \alpha)^{-m} \bigr) },}
%
where $c, m, d$ are positive constants. We expect that the constant $d$ in this bound
can be put equal to the dimension of space in realistic theories, but we do not make 
such an assumption here. The constant $m> 0$ may depend on  
the model and the KMS-state.
 
In order to control the thermodynamic limit we have to sharpen this nuclearity condition:
We will 
assume that also 
the map $\Theta^\sharp_{\alpha, {\cal O}} \colon {\cal R}_\beta (\O) \to \H_\beta$
given by
%
\# {\Theta^\sharp_{\alpha, {\cal O}} (A) = {\rm e}^{- \alpha | H_\beta |} \bigl( A 
- (\Omega_\beta \, , \, A \Omega_\beta) \bigr) \Omega_\beta,
\qquad \alpha > 0,} 
%
is nuclear and satisfies (for $\alpha^m$ large in comparison with $ r^d $) 
the following bound on the nuclear norm
%
\# { \| \Theta^\sharp_{\alpha, {\cal O}} \|  \le c'  \cdot r^d \alpha^{-m} .}
%
The bound on the nuclear norm $\| \Theta^\sharp_{\alpha, {\cal O}} \|$  
comes from taking the limit $\alpha^{ m}$ large in comparison with $r^d$ in the expression  
%
\# { {\rm e}^{c  r^d \alpha^{-m}} - 1 ,}
%
where the one is due to the subtraction of the thermal expectation value.

\vskip .3cm
\noindent
vi.) (Regularity from the outside). The net $\O \to {\cal R}_\beta (\O)$ 
is regular from the outside, i.e.,
%
\# { \bigcap_{\hat{\cal O}^{(i)} \supset \O} 
{\cal R}_\beta \bigl( \hat{\O}^{(i)} \bigr) = {\cal R}_\beta (\O) , 
\qquad \hat{\O}^{(i)} \searrow \O. }
%

\Rem{If the KMS-state is locally normal w.r.t.\ the vacuum representation, then
it is sufficient to assume that 
%
\# {\bigcap_{\hat{\cal O}^{(i)} \supset \O} {\cal R} \bigl( \hat{\cal O}^{(i)} \bigr) 
= {\cal R} (\O) , 
\qquad \hat{\O}^{(i)} \searrow \O, }
%
holds in the vacuum representation. For the free field this property was
shown by Araki [A 64].}



In general, the weakly continuous one-parameter group $t \mapsto \sigma_t$ will not be strongly 
continuous. In this case, $({\cal R}_\beta, \sigma)$ will not form a $C^*$-dynamical system.
Nevertheless, by  a suitable smoothening procedure 
we can associate an (abstract) 
$C^*$-dynamical system~$(\A, \tau)$, together with a
net
%
\# { \O \to \A(\O), \qquad \O \subset \R^4, }
%
of subalgebras of $\A$ to the given thermal field theory $\O \to {\cal R}_\beta (\O)$. 
More precisely (once again we refer to [S, 1.18]), there exists 
\vskip .3cm
\halign{ \indent #  \hfil & \vtop { \parindent = 0pt \hsize=34.8em
                            \strut # \strut} 
\cr 
(i)    & a $C^*$-algebra $\A$ and a representation $\pi_\beta \colon
\A \to \B(\H_\beta)$ such that $\pi_\beta (\A)$ is a 
$\sigma$-weakly dense $C^*$-subalgebra of ${\cal R}_\beta$.
%
\cr
(ii)    & a net $\O \to \A(\O)$ of $C^*$-subalgebras of $\A$ 
such that $\pi_\beta \bigl(\A (\O)\bigr)$ is a 
$\sigma$-weakly dense $C^*$-subalgebra of ${\cal R}_\beta(\O)$ for all $\O \subset \R^4$.
%
%
\cr
(iii)    & a strongly continuous automorphism group $t \mapsto \tau_t$ of $\A$  
such that $\pi_\beta \bigl( \tau_t (a) \bigr) =  \sigma_{t / \beta} ( \pi_\beta ( a) \bigr)$
for all $a \in \A$. 
\cr}
\vskip .5cm
\noindent
Moreover, the net $\O \to \A(\O)$ satisfies isotony and locality and $\tau$
respects the local structure of the net $\O \to \A(\O)$, i.e.,
%
\# { \tau_t \bigl( \A (\O) \bigr) = \A (\O + te)
 \qquad \forall t \in \R. }
%
\vskip .5cm

We now introduce subalgebras $\A_p$ of
almost local elements in $\A$ which are analytic with 
respect to the time-translations [BJu 89]. 
For the existence of these subalgebras
it is crucial that the time-evolution~$t \mapsto \tau_t$ is strongly continuous, i.e.,
%
\# { \lim_{t \to 0}  \| \tau_t (a) - a \| = 0
 \qquad \forall a \in \A. 
}
%

\Lm {(Buchholz and Junglas). Let $p \in \N$ be fixed and let $\A_p \subset \A$ be 
the $*$-algebra generated by all
finite sums and products of operators of the form
%
\# {a(f) = \int {\rm d}t \, f(t) \tau_t (a), }
%
where $f$ is any one of the functions 
%
\# {f(t) = const. \, {\rm e}^{- \kappa(t+ w)^{2p}} }
%
(with $\kappa > 0$, $w \in \C$) and $a \in \cup_{\cal O} \A(\O) $ is 
any local operator. It follows that
\vskip .3cm
\halign{ \indent #  \hfil & \vtop { \parindent = 0pt \hsize=34.8em
                            \strut # \strut} 
\cr 
(i)    &  each $b \in \A_p$ is an analytic element with respect to $\tau$, 
i.e., the operator-valued function $t \mapsto \tau_t(b)$ can be 
extended to a holomorphic function on \C ;
%
\cr
(ii)    & each $b \in \A_p$ is almost local in the sense that for any ${\breve r}_i > 0$ 
there is a local 
operator $b_i \in \A(\O_\circ^{(i)})$ such that
%
\# { \| b_i - b \| \leq const. \, {\rm e}^{- \kappa( {\breve r}_i /2)^{2p}}, \qquad \kappa >0,}
%
where the constant $const. > 0$ does not depend on ${\breve r}_i$;
\cr
(iii)    & the algebra $\A_p$ is invariant under $\tau_z$, $z \in \C$, and norm dense in $\A$.
\cr} } 

In Section 6 we will specify conditions on $p$. The new KMS-state $\omega_{\beta'}$
will satisfy the condition (3) for $a, b \in \A_p$. (Note that $\A_p$ is a
norm dense, $\tau$-invariant subalgebra of $\A_\tau$.)  


\vskip 1cm

\Hl{Doubling the Degrees of Freedom}

\noindent
Our first step can be understood as a local purification.  
Consider some $\delta > 0$ and two space-time regions $\O$ and $\hat{\O}$ such that
%
\# { \O + te \subset \hat{\O}  \qquad \forall |t| < \delta .}
%
The split property asserts that
there exists a type I factor ${\cal N}$ such that
%
\# { {\cal R}_\beta (\O) \subset {\cal N} \subset {\cal R}_\beta (\hat{\O}) . }
%

\Rem{In a forthcoming paper [J\"a d] we will show that
the split property for the net of von Neumann algebras $\O \to {\cal R}_\beta (\O)$
follows from the nuclearity condition (22--23). 
If the KMS-state is locally normal w.r.t.\ the vacuum representation, then the
split property for the vacuum representation automatically implies the split property
for the thermal representation.}
  
As a consequence of the split inclusion (36) we find the following

\Lm{Let $\O$ be a bounded space-time region. Then the von Neumann algebra
%
\# {  {\cal M}_\beta (\O) := {\cal R}_\beta (\O) \vee j \bigl( {\cal R}_\beta (\O) \bigr)  }
%
is naturally isomorphic to the tensor product of
${\cal R}_\beta (\O) $ and $ j \bigl( {\cal R}_\beta (\O) \bigr)$. 
I.e., there exists a unitary operator 
$V \colon \H_\beta \to \H_\beta \otimes \H_\beta$ such that
%
\# { V{\cal M}_\beta (\O)V^* = {\cal R}_\beta (\O) \otimes j \bigl( {\cal R}_\beta (\O) \bigr) .}
%
}

\Rem{The algebras ${\cal R}_\beta (\O) $ 
and $ j \bigl( {\cal R}_\beta (\O) \bigr)$ are weakly statistically independent, i.e., 
$0 \ne A \in {\cal R}_\beta (\O) $ 
and $0 \ne B\in j \bigl( {\cal R}_\beta (\O) \bigr)$ implies 
$AB \ne 0$ (Schlieder property) [J\"a b].
In this sense one can speak of a doubling of degrees of freedom. 
The elements of ${\cal M}_\beta (\O)$ will in general not show analyticity properties with respect 
to $\Omega_\beta$. Since the analyticity properties encoded in the KMS-condition
simply reflect the basic stability and passivity properties of an equilibrium state,
it seems that the essence of a thermal field theory gets lost, 
when we `double the degrees of freedom' and consider the net 
%
\# { \O \to {\cal M}_\beta (\O), \qquad \O \subset \R^4, }
%
instead of the net of observables $\O \to {\cal R}_\beta (\O)$. 
But, due to the natural tensor product 
structure of ${\cal M}_\beta (\O)$, we will recover certain 
analyticity properties in the next lemma.}

\Pr{As a consequence of the split property there 
exists a product vector $\Omega_p \in \H_\beta$, cyclic and separating for 
${\cal R}_\beta (\O) \vee {\cal R}_\beta (\hat {\O})' $,
such that
%
\# { (\Omega_p \, , \, A B   \Omega_p) =  (\Omega_\beta \, , \, A \Omega_\beta) 
(\Omega_\beta \, , \, B \Omega_\beta)}
%
for all $A \in {\cal R}_\beta (\O)$ and $B \in {\cal R}_\beta (\hat {\O})'$ [J\"a d].
The product vector $\Omega_p$ can be utilized to define a linear operator $V \colon \H_\beta 
\to \H_\beta  \otimes \H_\beta$ by linear extension of
%
\# { V A B   \Omega_p 
=  A \Omega_\beta \otimes B \Omega_\beta ,}
%
where $A \in {\cal R}_\beta (\O) $ and 
$B \in {\cal R}_\beta (\hat{\O})'$.
The operator $V$ is unitary and, inspecting (41), we find 
%
\# { V {\cal R}_\beta (\O) V^* = {\cal R}_\beta (\O) \otimes \1  
\quad \hbox{and} \quad
 V {\cal R}_\beta (\hat{\O})' V^* = \1 \otimes {\cal R}_\beta (\hat{\O})' . }   
%
The inclusion $j \bigl( {\cal R}_\beta (\O) \bigr) \subset
{\cal R}_\beta (\hat{\O})'$ implies that the von Neumann algebra
%
\# {  {\cal M}_\beta (\O) := {\cal R}_\beta (\O) \vee j \bigl( {\cal R}_\beta (\O) \bigr)  }
%
is naturally isomorphic to the tensor product of
${\cal R}_\beta (\O) $ and $ j \bigl( {\cal R}_\beta (\O) \bigr)$ and the relation (38) is 
a consequence of (42).} 


We introduce an auxiliary one-parameter group of unitary 
operators which respects the 
natural tensor product structure of ${\cal M}_\beta (\O)$:

\Def{A one-parameter group of unitary operators $s \mapsto
\Delta_p^{-is} \colon \H_\beta \to \H_\beta$, $ s \in \R$,  
and an anti-unitary operator $J_p \colon \H_\beta \to \H_\beta$ are given  by linear extension of
%
\# {  \Delta_p^{-is} AB \Omega_p  :=  V^*  \Bigl(  
 \Delta^{-is} A \Omega_\beta \otimes \Delta^{is} B \Omega_\beta \Bigr), \qquad s \in \R,}
%
and, respectively, 
%
\# {  J_p  AB \Omega_p  :=  V^*  \Bigl(  
 J A \Omega_\beta \otimes J B \Omega_\beta \Bigr)}
%
for all  $A \in {\cal R}_\beta (\O)$ and $B \in {\cal R}_\beta (\hat{\O})'$. }

\Rem{By Stone's theorem there exists a unique self-adjoint operator $K_\beta$
such that
%
\# { \Delta_p  = {\rm e}^{- \beta K_\beta}  \qquad \hbox{and} \qquad K_\beta \Omega_p = 0 . }
%
$\Omega_p \in \H_\beta$ is cyclic and separating for 
${\cal R}_\beta (\O) \vee {\cal R}_\beta (\hat {\O})' $. It follows from 
the definition (41) of $V$ and the Reeh-Schlieder property 
of $\Omega_\beta$ that the product vector $\Omega_p$ is also cyclic and 
separating for~${\cal M}_\beta (\O)$.}


\Th{Let us consider a space-time region $\O_\circ$ and some $\delta_\circ > 0$  
such that
%
\# { \O_\circ + te \subset \O  \qquad \forall |t| < \delta_\circ .}
%
For $|s|$ sufficiently small, $\Delta_p^{-is}$ respects the local 
structure of $\M_\beta (\O)$, 
i.e.,
%
\# {\Delta_p^{-is} \M_\beta (\O_\circ) \Delta_p^{is} \subset \M_\beta (\O_\circ + s \beta \cdot e) 
\qquad \forall |s \beta | < \delta_\circ .}
%
Moreover, for $a \in \A(\O_\circ)$ and $|s \beta | < \delta_\circ$
the group of unitaries $s \mapsto \Delta_p^{-is}$ coincides --- up to rescaling ---
with the time-evolution, i.e.,
%
$\Delta_p^{-is} \pi_\beta (a) \Delta_p^{is} 
= \pi_\beta \bigl( \tau_{s \beta} (a) \bigr) $
for $ |s \beta | < \delta_\circ $. 
%
}

\Pr{The inclusion (48) follows from the definition (44)
of $\Delta_p$ and the inclusions 
%
\# {{\cal R_\beta} (\O_\circ) \subset {\cal R_\beta}  (\O_\circ + t e)
\qquad \hbox {and} \qquad 
j \bigl( {\cal R_\beta} (\O_\circ) \bigr) \subset j \bigl(
{\cal R_\beta}  (\O_\circ + t e) \bigr),}
% 
which hold for all $|t | < \delta_\circ$.} 


\Lm{Let $\O$ be bounded. Then
%
\# {  {\cal M}_\beta (\O) \Omega_p \subset   {\cal D} \bigl(\Delta_p^{ \alpha} \bigr)
\qquad \forall 0 \le \alpha \le 1 / 2 }
%
and $J_p \Delta_p^{1/2} M \Omega_p = M^* \Omega_p $
%
for all  $M \in {\cal M}_\beta (\O)$.}

\Pr{By definition,
%
%
$J_p^2 = \1 $, $J_p \Omega_p = \Omega_p$ and
%
%
\& {  J_p \Delta_p^{1/2} AB \Omega_p 
& =  V^*  \Bigl(  
A^* \Omega_\beta \otimes  B^* \Omega_\beta \Bigr)
\cr
& = A^* B^* \Omega_p
= (AB)^* \Omega_p }
%
for all  $A \in {\cal R}_\beta (\O)$ and 
$B \in j \bigl( {\cal R}_\beta (\O) \bigr)$. 
Since 
%
\# {  p^\alpha  \le 
\max (1 , p) < 1 + p \qquad \forall 0 \le \alpha \le 1  }
%
and $p > 0$, the spectral resolution of the positive operator $\Delta_p^{1/2}$ 
implies
%
\# {  {\cal M}_\beta (\O) \Omega_p \subset   {\cal D} \bigl(\Delta_p^{ \alpha} \bigr)
\qquad \forall 0 \le \alpha \le 1 / 2.}
%
}

Nevertheless, $J_p$ and $\Delta_p$ are {\sl not}
the modular objects associated to the pair $\bigl( {\cal R}_\beta (\O) \vee 
j \bigl( {\cal R}_\beta (\O) \bigr), \Omega_p \bigr)$.  


\Th{Let $\O_\circ$ and $\O$ denote two space-time regions 
as specified in (47). The inclusion of von Neumann algebras 
${\cal M}_\beta (\O_\circ) \subset {\cal M}_\beta (\O)$ is  a standard\footnote{$^\star$}{\sevenrm
A split inclusion $\scriptstyle {\cal A} \subset {\cal B}$ is called standard, if there exists
a vector $\scriptstyle \Omega$ which is cyclic for $\scriptstyle {\cal A}' \wedge {\cal B}$ 
as well as for $\scriptstyle {\cal A}$ and $\scriptstyle {\cal B}$.} split
inclusion and there exists a unitary operator $W \colon \H_\beta \to \H_\beta \otimes \H_\beta$ 
such that
%
\# { W {\cal M}_\beta (\O_\circ) W^* = {\cal M}_\beta (\O_\circ) \otimes \1 
\quad \hbox{and} \quad
 W {\cal M}_\beta (\O )' W^* = \1 \otimes {\cal M}_\beta (\O)' .}   
%
}



\Pr{From the split inclusions
%
\# { {\cal R}_\beta (\O_\circ) \subset {\cal N}_\circ \subset {\cal R}_\beta (\O ) 
\quad \hbox{and} \quad
j \bigl( {\cal R}_\beta (\O_\circ) \bigr) \subset 
j \bigl( {\cal N}_\circ \bigr) \subset 
j \bigl( {\cal R}_\beta (\O ) \bigr)   }
%
we infer that there exists a type I factor, namely ${\cal N}_\circ \vee 
j \bigl( {\cal N}_\circ \bigr)$, 
such that
%
\# { {\cal M}_\beta (\O_\circ) \subset {\cal N}_\circ \vee j \bigl( {\cal N}_\circ \bigr)
\subset {\cal M}_\beta (\O)  .}
%
All infinite type I factors with infinite commutant on 
the separable Hilbert space
$\H_\beta$ are unitarily
equivalent to $\B(\H_\beta) \otimes \1  \, $ [KR, Chapter 9.3]. 
Thus there exists a unitary operator $W \colon \H_\beta \to \H_\beta \otimes \H_\beta$ such that 
%
\# { {\cal N}_\circ \vee j \bigl( {\cal N}_\circ \bigr) 
= W^* \bigl( \B(\H_\beta)  \otimes \1  \, \bigr) W .}
% 
Now consider $\omega_\beta ( \, . \, ) := (\Omega_\beta \, , \, .\, \Omega_\beta)$ 
and $\omega_p ( \, . \, ) := (\Omega_p \, , \, .\, \Omega_p)$
as two normal states over 
${\cal M}_\beta (\O_\circ)$ and ${\cal M}_\beta (\O)'$, 
respectively. Set
%
\# { \phi_p (C) := (\omega_\beta \otimes \omega_p) (W C W^*) \qquad \forall
C \in {\cal M}_\beta (\O_\circ) \vee {\cal M}_\beta (\O)'.}
%
$\phi_p$ is normal and
satisfies 
%
\# { \phi_p (M N) = \omega_\beta (M) \cdot \omega_p (N)  }
%
for all $M \in {\cal M}_\beta (\O_\circ)$ and $N \in {\cal M}_\beta (\O)'$.
In the presence of a separating vector each normal 
state is a vector state [KR, 7.2.3]. In fact, there exists a unique 
vector $\eta$ in the natural positive 
cone ${\cal P}^\natural \bigl({\cal M}_\beta (\O_\circ) \vee {\cal M}_\beta (\O)', \Omega_\beta
\bigr) $
%
such that 
%
\# { (\eta \, , \, MN \eta) = \phi_p (M N) =   (\Omega_\beta \, , \, M \Omega_\beta) 
(\Omega_p \, , \, N \Omega_p)  }
%
for all $M \in {\cal M}_\beta (\O_\circ)$ and $N \in {\cal M}_\beta (\O)'$ 
[BR, 2.5.31].  Thus the unitary operator $W \colon \H_\beta \to 
\H_\beta \otimes \H_\beta$ can now be specified by linear extension of
%
\# { W M N \eta  =  M \Omega_\beta \otimes N \Omega_p  }
%
for all $M \in {\cal M}_\beta (\O_\circ)$ and $N \in {\cal M}_\beta (\O)'$.
Consequently,
%
\# { W {\cal M}_\beta (\O_\circ) W^* = {\cal M}_\beta (\O_\circ) \otimes \1 
\quad \hbox{and} \quad
 W {\cal M}_\beta (\O )' W^* = \1 \otimes {\cal M}_\beta (\O)' .}   
%
$\Omega_\beta$ is cyclic and separating for  ${\cal M}_\beta (\O_\circ)$ and
$\Omega_p$ is cyclic and separating for 
${\cal M}_\beta ( \O  )'$. Thus $\Omega_\beta \otimes
\Omega_p$ is cyclic and separating for  ${\cal M}_\beta (\O_\circ)
\otimes {\cal M}_\beta ( \O  )' $ and the split inclusion (56) is standard [DL].
}


\vskip 1cm


\Hl{Localized Excitations of a KMS-state}

\noindent
Taking the auxiliary structure developed in the previous section into account, 
we can now adapt the method of Buchholz and Junglas 
to a thermal representation. 

\Def{Let $\O_\circ , \O$ and $ \hat {\O}$ denote three space-time regions 
such that --- for some $\delta_\circ$, $\delta >0$ ---
%
\# {\O_\circ + te \subset \O  \qquad \forall |t| < \delta_\circ
\qquad
\hbox{and}
\qquad
\O + te \subset \hat {\O}  \qquad \forall |t| < \delta .}
%
The Hilbert space $\H_\Lambda  \subset \H_\beta$,
$\Lambda := (\O_\circ , \O , \hat {\O})$, of localized excitations of the 
KMS-state $\omega_\beta$ is given by 
%
\# {\H_\Lambda  
:= \overline {  {\cal M}_\beta ( \O_\circ )   \eta  }.}
%
The projection onto $\H_\Lambda$ is denoted by $E_\Lambda$.}

\noindent{\sl Notation.} Here
${\cal M}_\beta ( \O_\circ )$ denotes the von Neumann algebra generated by
${\cal R}_\beta ( \O_\circ )$ and $j \bigl( {\cal R}_\beta ( \O_\circ ) \bigr)$ and
$\eta \in \H_\beta$ denotes the unique\footnote{$^\dagger$}
{\sevenrm Fixing the product vector with respect to some natural positive 
cone is mathematically convenient,
but not necessary. In fact, we expect that  
different `boundary conditions' are realized by different choices of $\scriptstyle \eta$.
In the thermodynamic limit the different choices of boundaries might lead to different phases.}
product vector in the natural positive 
cone ${\cal P}^\natural \bigl({\cal M}_\beta (\O_\circ) \vee {\cal M}_\beta (\O)', \Omega_\beta
\bigr) $
satisfying
%
\# { (\eta \, , \, MN \eta) =   (\Omega_\beta \, , \, M \Omega_\beta) 
(\Omega_p \, , \, N \Omega_p)  }
%
for all $M \in {\cal M}_\beta (\O_\circ)$ and $N \in {\cal M}_\beta (\O)'$. 
As before, $\Omega_p$ denotes the unique product vector in the natural positive 
cone ${\cal P}^\natural \bigl({\cal R}_\beta (\O) \vee {\cal R}_\beta (\hat {\O})', \Omega_\beta
\bigr) $
satisfying
%
\# { (\Omega_p \, , \, A B   \Omega_p) =  (\Omega_\beta \, , \, A \Omega_\beta) 
(\Omega_\beta \, , \, B \Omega_\beta) }
%
for all $A \in {\cal R}_\beta (\O)$ and $B \in {\cal R}_\beta (\hat {\O})'$.
\vskip .5cm
 
Recall that $W$ is unitary and fulfills
%
\# { W M N W^*  =  M
\otimes N,}
%
for $M \in {\cal M}_\beta (\O_\circ)$ and $N \in {\cal M}_\beta (\O)'$.
Using the isometry $W$, we can write
%
\# { \H_\Lambda
 =  W^*  \overline { {\cal M}_\beta (\O_\circ)  \Omega_{\beta} 
\otimes  \Omega_p}
=  W^* ( \H_{\beta} 
\otimes  \Omega_p ) .}
%
and $E_\Lambda  =
W^*  ( \1 \otimes P_{\Omega_p}) W $, where $P_{\Omega_p} \in \B (\H_\beta)$ denotes the 
projection onto $ \C \cdot \Omega_p$.


\Prop{Given a triple $\Lambda := ( \O_\circ, \O, \hat {\O})$ of space-time regions
as specified in (63), we find:
\vskip .3cm
\halign{ \indent #  \hfil & \vtop { \parindent = 0pt \hsize=34.8em
                            \strut # \strut} 
\cr 
(i)    & $\H_\Lambda$ is invariant under the application of elements from 
       ${\cal M}_\beta (\O_\circ)$, i.e.,
%
\# { {\cal M}_\beta (\O_\circ) \H_\Lambda  = \H_\Lambda .}
%
\vskip -.2cm
%
\cr
(ii)    & The vectors of $ \H_\Lambda$ induce product states on 
${\cal M}_\beta (\O_\circ) \vee \M_\beta (\O)'$: if $\Psi \in \H_\Lambda$, then  
%
\# {(\Psi \, , \, MN \Psi) = (\Psi\, , \, M \Psi) (\Omega_p \, , \, N \Omega_p)}
%
for all  
$M \in {\cal M}_\beta (\O_\circ)$ and $N \in \M_\beta (\O)'$.
%
%
\cr
(iii)    & The vectors states associated with $ \H_\Lambda$ represent strictly localized
excitations of the KMS-state, i.e., they coincide with $\omega_\beta$
in the space-like complement of $\hat {\O}$:  if $\Psi \in \H_\Lambda$, then  
%
\# {(\Psi \, , \, \pi_\beta (a) \Psi) = \omega_\beta (a) 
\qquad \forall a \in \A^c (\hat{\O}) .}
%
Here $\A^c (\hat{\O})  $ denotes the algebra  generated by the set
$\{ a \in \A : [a , b ] = 0 \, \, \forall b \in \A(\hat{\O}) \}$
and not the commutant of $\pi_\beta \bigl( \A(\hat{\O})  \bigr)$ in $\B(\H_\beta)$. 
%
%
\cr
(iv)    & $\H_\Lambda$ is complete in the following sense: to every normal state $\phi$
on ${\cal M}_\beta (\O_\circ)$ there exists a $\Phi \in \H_\Lambda$ such that
%
\# {(\Phi\, , \, M \Phi) = \phi(M)  }
%
for all $M \in {\cal M}_\beta (\O_\circ)$.
\cr} } 

\Pr {We simply adapt the proof of the corresponding result 
by Buchholz and Junglas to our situation:
\vskip .2cm
\noindent
\halign{ # \hfil & \vtop { \parindent =0pt \hsize=36,6em
                            \strut # \strut} \cr 
(i)  & follows from the definition,
%
\cr
(ii) &
follows from (64) and (65).
\cr
(iii) &
follows from (64), (65) and (66).
\cr
(iv) 
& 
Since ${\cal M}_\beta (\O_\circ)$ has a cyclic and separating vector, there exists a vector 
$ \tilde {\Phi} \in \H_\beta$ which induces the given normal state
$\phi$ on  ${\cal M}_\beta (\O_\circ) $.
By definition, $\Phi := W^* ( \tilde {\Phi} 
\otimes  \Omega_p )  \in \H_\Lambda $ satisfies (iv).
\cr}}




In order to show that the restriction of the operator $\Delta_p^{\alpha}$
to the sub-space $\H_\Lambda$ is of trace class
for $ 0 < \alpha < 1  / 2$ we need the following  

\Lm{Assume the nuclearity conditions (22--25) hold for the algebra ${\cal R}_\beta (\O)$.
It follows that the maps  $\vartheta_{\alpha, {\cal O}} \colon {\cal M}_\beta (\O)  \to \H_\beta$,
%
\# { M  \mapsto 
\Delta_p^{\alpha} M  \Omega_p ,  \qquad 0 < \alpha < 1/2, }
%
and $\vartheta^\sharp_{\alpha, {\cal O}} \colon {\cal M}_\beta (\O)  \to \H_\beta$,
%
\# { M  \mapsto {\rm e}^{- \alpha | K_\beta |} \bigl( M - (\Omega_p\, , \, M \Omega_p) \bigr)
\Omega_p ,  \qquad \alpha > 0, }
%
are nuclear. 
%   I.e.,  
%   there exists a sequence of vectors $\Phi_i \in \H_\beta$ and of linear functionals 
%   $\phi_i \in {\cal M}_\beta^* (\O)$ such that 
%   $\sum_i \| \phi_i \| \, \| \Phi_i \| < \infty$ and 
%
%   $\vartheta_{\alpha, {\cal O}} (M) = \sum_i  \phi_i (M) \cdot \Phi_i$,  for all
%   $M \in {\cal M}_\beta (\O)$. 
%
Moreover, the nuclear norm of $\vartheta_{\alpha, {\cal O}}$ can be estimated by  
%
\# { \|  \vartheta_{\alpha, {\cal O}}  \|   \le 
{\rm e}^{2 c r^d \bigl 
( \alpha^{-m} + (1 / 2 - \alpha)^{-m} \bigr) } ,
\qquad \quad c, m, d > 0, }
%
where $r$ denotes the diameter of $\O$ and $c, m, d$ 
are the constants appearing in the 
nuclearity condition (23) for ${\cal R}_\beta (\O)$.}

\Pr{Let $A \in {\cal R}_\beta (\O)$ and $B \in j \bigl( {\cal R}_\beta (\O) \bigr)$. 
By definition,
%
\# { \vartheta_{\alpha, {\cal O}} (AB)   
=  V^*  \Bigl( \Delta^\alpha A \Omega_\beta  \otimes \Delta^{-\alpha} B \Omega_\beta \Bigr) .}
%
The maps
%
$ A \mapsto \Delta^\alpha  A  \Omega_\beta$
%
and
%
$B \mapsto \Delta^{ - \alpha } B \Omega_\beta$
are nuclear for $0 < \alpha < 1 /2$. The tensor product of two nuclear maps is again a
nuclear map and the norm is bounded by the product of the nuclear norms [P].}



\Prop {Let $\Lambda( \O_\circ , \O, \hat {\O} )$ be a triple of space-time regions
as specified in (63). For $0 < \alpha < 1  / 2$
the operator $\Delta_p^{\alpha} E_\Lambda $,
acting on the Hilbert space $\H_\beta$, is of trace-class, and 
%
\# {\| \Delta_p^{\alpha} E_\Lambda \|_1  
\le
{\rm e}^{2 c r^d \bigl 
( \alpha^{-m} + (1 / 2 - \alpha)^{-m} \bigr) } ,
\qquad \quad c, m, d > 0, }
%
where $r$ denotes the diameter of $\O$ and $c, m, d$ 
are the constants appearing in the 
nuclearity condition (23) for ${\cal R}_\beta (\O)$.} 

\Pr{The proof of this proposition is more or less identical to the one given by
Buchholz and Junglas [BJu 89] in the vacuum case. 
\vskip .1cm
\noindent
i.) The first step is to construct a convenient orthonormal basis of $\H_\Lambda$. 
Let $\{ \Psi_i \}_{i \in \n} $ be an orthonormal basis of $\H_\beta$ with
$\Psi_1 = \Omega_\beta$. Set
%
\# { U_{i,j} : = W^*  (M_{i,j} \otimes \1 ) W  ,}
%
where $M_{i,j} \in \B ( \H_\beta )$ are matrix units given by
%
\# { M_{i,j} \Psi : = (\Psi_j, \Psi) \Psi_i
\qquad \forall \Psi \in \H_\beta. }
%
Since $W^*  \bigl( \B ( \H_\beta ) \otimes \1  \bigr) W =
{\cal N}_\circ \vee j \bigl( {\cal N}_\circ \bigr)$, we infer from (78) that 
$U_{i,j} \in {\cal N}_\circ \vee j \bigl( {\cal N}_\circ \bigr) $.
Furthermore, we find
%
%
\# { U_{i,j}^* = U_{j,i}  , 
\qquad
U_{i,j}  U_{k,l} = \delta_{j,k} U_{i,l}  , 
\qquad
\slim_{ N \to \infty } \sum_{i = 1 }^N  U_{i,i} = \1 \, .}
%
Set
%
\# {  U_{i,1} \eta = W^*  (\Psi_i \otimes \Omega_p) .} 
%
$\{   U_{i,1} \eta \}_{i \in \n}$ is the desired orthonormal basis of $\H_\Lambda $.
Introducing an isometry $I \in {\cal N}_\circ \vee j \bigl( {\cal N}_\circ \bigr)$ by
%
\# {  N \eta = I N \Omega_p \qquad  \forall N \in \bigl( {\cal N}_\circ 
\vee j ( {\cal N}_\circ) \bigr)' ,} 
%
we can represent this orthonormal basis by vectors
%
\# { \Phi_i := U_{i,1} \eta  = U_{i,1} I \Omega_p }, 
%
where $U_{i,1} I \in {\cal N}_\circ \vee j \bigl( {\cal N}_\circ \bigr)
\subset {\cal M}_\beta (\O)$. It follows that
%
\# { \Phi_i \in {\cal D} \bigl( \Delta_p^{\alpha} \bigr) \qquad \forall 0 < \alpha < 1 /2,
\quad \forall i \in \N.}
%
Especially, $\eta =: \Phi_1 \in {\cal D} \bigl( \Delta_p^{\alpha} \bigr)$ 
for all $0 < \alpha < 1 /2$.
\vskip .2cm 
\noindent
ii.) By polar decomposition of the closeable operator $\Delta_p^{\alpha} E_\Lambda$ we get
%
\# { \Delta_p^{\alpha} E_\Lambda
= F \cdot | \Delta_p^{\alpha} E_\Lambda | , }
%
where $F$ is a partial isometry with range in $\H_\Lambda$.
Introducing a set of linear functionals $\phi_i$ (which can be chosen to be continuous with
respect to the ultra-weakly topology induced by ${\cal M}_\beta (\O)$ [BD'AF 90b])
and vectors $\Phi_i \in \H_\beta$ corresponding to the 
nuclear map  $\vartheta_{\alpha, {\cal O} }$
we obtain
%
\& { \Tr \, | \Delta_p^{\alpha} E_\Lambda |
&= \sum_i ( U_{i,1} I  \Omega_p \, , \, 
F^*  \Delta_p^{\alpha}     
U_{i,1} I \Omega_p ) 
\cr
&= \sum_i \bigl( U_{i,1} I  \Omega_p \, , \,  F^*  
\vartheta_{\alpha, {\cal O} } ( U_{i,1} I ) \bigr) 
\cr
&= \sum_i \sum_n     \phi_n ( U_{i,1} I ) 
\cdot  ( U_{i,1} I \Omega_p \, , \,  F^* \Phi_n)  
\cr
&\le \sum_i \sum_n   | \phi_n ( U_{i,1} I ) |
\cdot   \|    U_{1,i}  F^*  \Phi_n  \| .}
%
Buchholz and Junglas  have shown the following inequality [BJu 89]:
%
\# {
\sum _i  | \psi ( U_{i,1} ) | \cdot
\| U_{1,i} \Psi \|
\le  \| \psi \| \,  \|  \Psi \|, 
}
%
which holds for any vector $\Psi \in \H_\beta$ and any ultra-weakly continuous linear
functional~$\psi$ on~${\cal M}_\beta (\O)$. Consequently,
%
\# { \Tr \, | \Delta_p^{\alpha} E_\Lambda | \le  \sum_n   \| \phi_n \| \,  \| \Phi_n  \| .}
%
Taking the infimum with respect to all decompositions of the respective nuclear maps
we find
%
\# { \Tr \,  | \Delta_p^{\alpha} E_\Lambda |
\le   \| \vartheta_{\alpha, {\cal O} } \| . }
%
}


\vskip 1cm


\Hl{Local KMS-states}

\noindent
Proposition 4.3 will allow us
to define ``local quasi-Gibbs'' states, which are {\sl local} $(\tau, \beta')$-KMS-states
for the temperature $ T' = 1/ \beta'$ in
the interior of $\O_\circ$ and $(\tau, \beta)$-KMS-states
for the temperature $1/\beta$ outside of 
$\hat {\O}$. Before we do so, 
we give a precise meaning to the statement
that a local excitation $\omega_\Lambda$ of a KMS-state $\omega_\beta$
satisfies a {\sl local KMS-condition} for a new positive inverse
temperature~$\beta'$ in some bounded region 
$\O_\circ$. Note that any $\beta' > 0$ can be decomposed
into some $0 < \alpha < 1/2$ and some (minimal) $n \in \N$ such that $\beta' = \alpha n \beta$.


\Def{Let $\beta' > 0$ and let $n \in \N$ be the smallest natural number such that 
%
\# {n \alpha \beta = \beta' \qquad \hbox{for \  some \ $\alpha$}, \quad 0 \le \alpha \le 1 / 2.}
%
A state $\omega_\Lambda $, which describes a local excitation of a KMS-state $\omega_\beta $,
satisfies the {\sl local KMS-condition} at inverse temperature $\beta'$
in some bounded space-time region $\O_\circ \subset \R^4$ iff for any 
subregion $\O_{\circ \circ} \subset \O_\circ$
whose closure is contained in the interior of $\O_\circ$ there exists a $\delta_{\circ \circ} >0$ 
and a 
function $F_{a,b}$ for every pair 
of operators $a$, $b \in \A (\O_{\circ \circ})$ such that
\vskip .2cm
\halign{ #  \hfil & \vtop { \parindent =0pt \hsize=36,6em
                            \strut # \strut} \cr 
(i)  & $F_{a,b}$ is defined on 
%
\& { {\cal G}_{n , \alpha} & := \{z \in \C : 0 < \Im z < n \alpha \beta \}
\setminus \cr
& \qquad \setminus \{ z \in \C : | \Re z | \ge \delta_{\circ \circ}, \Im z =  k \alpha \beta, \, 
k = 1, \ldots, n-1 \}. }
% 
\vskip -.2cm
\cr
(ii)    & $F_{a,b}$ is bounded and analytic in its domain of definition. 
\cr
(iii)   &  $F_{a,b}$ is continuous for 
           $ \Im z  \searrow k \alpha \beta$ and $\Im z  \nearrow k \alpha \beta$, 
$k = 1, \ldots, n-1$. 
\cr
(iv)   &  $F_{a,b}$ is continuous at the boundary for 
$ \Im z  \searrow 0 $ and $\Im z  \nearrow n \alpha \beta$. 
\cr
(v)    & The respective boundary values are 
%
\# { F_{a,b} (t) = \omega_\Lambda \bigl( a \tau_t(b) \bigr)
\quad \hbox{and} \quad F_{a,b} (t + i n \alpha \beta) = \omega_\Lambda \bigl( \tau_t(b) a \bigr)  
\quad \hbox{\rm for} \qquad |t| < \delta_{\circ \circ}.}
%
\cr}
}

\vskip -.8cm

\Rem{To heat up the system locally is quite simple: For $\beta' < \beta / 2$ we find $n= 1$, i.e.,
no cuts appear in
${\cal G}_{1 , \alpha} = \{z \in \C : 0 < \Im z <  \alpha \beta \}$. The real problem is to cool 
down the system locally. One needs at least $n$ cuts, where
$n$ is the minimal natural number such that $\beta' = n \alpha \beta$, $0 \le \alpha \le 1/2$.
Whether or not it is useful to operate  with more cuts then necessary is unknown to us.} 



\Prop{Let $\Lambda:= ( \O_\circ , \O, \hat {\O} )$ be a triple of space-time regions,
as specified in~(63). Let $n\in \N$ be the minimal natural
number such that $\beta' = n \alpha \beta$, $0 \le \alpha \le 1/2$.
Set --- for $ n $ and $ \alpha $ fixed ---
%
\# { \rho_\Lambda = {  \bigl( E_\Lambda \Delta_p^{\alpha} E_\Lambda \bigr)^n   
\over \Tr \, \bigl( \Delta_p^{\alpha} E_\Lambda \bigr)^n }
\qquad \hbox {and} \qquad \omega_\Lambda  (a) 
=  \Tr \, \rho_\Lambda \pi_\beta (a)  
\quad  \forall a \in \A .}
%
Then $\rho_\Lambda$ is a density matrix, i.e., $\rho_\Lambda> 0$ and $\Tr \, \rho_\Lambda =1$,
%
and the following statements hold true:
\vskip .3cm
\halign{ #  \hfil & \vtop { \parindent =0pt \hsize=36,6em
                            \strut # \strut} \cr 
(i)  & The states $\omega_\Lambda$ are product states which coincide with
the given KMS-state $\omega_\beta$ outside of $\hat{\O}$, i.e.,
%
\# { \omega_\Lambda  ( a b') 
= \omega_\Lambda  (a)  \, 
\omega_\beta (b') }
%
for all $a \in \A(\O_\circ)$, $b' \in \A^c (\hat {\O})$. As before,
$\A^c (\hat{\O})  $ denotes the algebra  generated by the set
$\{ a \in \A : [a , b ] = 0 \, \, \forall b \in \A(\hat{\O}) \}$. 
\cr
(ii)   &  The states $\omega_\Lambda$ are local $(\tau, n \alpha \beta)$-KMS-states 
for the space-time region $\O_\circ$.  
\cr}  
}

\Pr{(i) Let $a \in \A(\O_\circ)$ and $b' \in \A^c (\hat {\O})$. Since
$E_\Lambda \in {\cal M}_\beta (\O_\circ)' \subset \pi_\beta \bigl( \A( \O_\circ ) \bigr)'$, 
it follows that $[ E_\Lambda , \pi_\beta (a) ] = 0$.
Moreover, $E_\Lambda = W^* (\1 \otimes P_{\Omega_p}) W$
implies
%
\# { E_\Lambda \pi_\beta(b') E_\Lambda =
\omega_\beta ( b') E_\Lambda \qquad \forall b' \in \A^c (\hat{\O}) .}
%
Using the cyclicity of the trace  
we find
%
\& { \omega_\Lambda  ( ab')
&= { \Tr \bigl( E_\Lambda \Delta_p^{\alpha} E_\Lambda \bigr)^n \, 
\pi_\beta (a)  E_\Lambda \pi_\beta (b') E_\Lambda
\over \Tr \bigl( \Delta_p^{\alpha} E_\Lambda \bigr)^n }
\cr
&=\omega_\Lambda  ( a )  \omega_\beta(b') .}
\vskip .1cm
\noindent
(ii) Consider the case $n = 2$. Let $\delta_{\circ \circ} > 0$ and 
$\O_{\circ \circ}$ be an open space-time region
such that $\O_{\circ \circ} + t e \subset \O_{\circ}$ for $|t| < \delta_{\circ \circ}$.
Let $a, b \in \A(\O_{\circ \circ})$. By assumption,   
%
$a \tau_t (b) \in \A (\O_\circ)$ for  $ |t | < \delta_{\circ \circ} $. 
%
Set
% 
\# {F_{a,b}^{(1)} (z ) := 
{ \Tr \, \pi_\beta (a) E_\Lambda    
\Delta_p^{-iz / \beta} \pi_\beta ( b )
\Delta_p^{\alpha + i z / \beta}   E_\Lambda 
\Delta_p^{\alpha} E_\Lambda 
%
\over
%
\Tr \, \bigl( \Delta_p^{\alpha} E_\Lambda \bigr)^2 }  }
%
for $0 < \Im z  < \alpha \beta$. The function $F_{a,b}^{(1)} (z)$ is analytic in its domain and 
continuous at the boundary. We recall that 
%
$\Delta_p^{-it / \beta } \pi_\beta (b) \Delta_p^{it / \beta} = \pi_\beta \bigl( \tau_{t} (b) \bigr)
\in \pi_\beta \bigl( \A(\O_\circ) \bigr)$ for $|t | < \delta_{\circ \circ} $. 
%
Using once again 
the cyclicity of the trace and $E_\Lambda \in \pi_\beta \bigl( \A(\O_\circ) \bigr)'$, 
we conclude that 
%
\& { \lim_{\Im z \searrow 0} F_{a,b}^{(1)} (z) & = { \Tr \, \pi_\beta (a) E_\Lambda    
\pi_\beta \bigl( \tau_{\Re z} (b) \bigr)
\Delta_p^{\alpha}   E_\Lambda 
\Delta_p^{\alpha} E_\Lambda 
%
\over
%
\Tr \, \bigl( \Delta_p^{\alpha} E_\Lambda \bigr)^2 }   
\cr
& = { \Tr \,     
\pi_\beta \bigl( a \tau_{\Re z} (b) \bigr) \bigl( E_\Lambda
\Delta_p^{\alpha}   E_\Lambda \bigr)^2
%
\over
%
\Tr \, \bigl( \Delta_p^{\alpha} E_\Lambda \bigr)^2 }   
\cr
&= \omega_\Lambda \bigl( a \tau_{\Re z} (b) \bigr)  \qquad \qquad 
\forall |{\Re z}| < \delta_{\circ \circ}.}
%
On the other hand, for $|{\Re z}| < \delta_{\circ \circ}$,
%
\& { \lim _{\Im z \nearrow \alpha \beta} F_{a,b}^{(1)} (z ) & = { \Tr \, \pi_\beta (a) E_\Lambda    
\Delta_p^{\alpha} \pi_\beta \bigl( \tau_{\Re z} (b) \bigr)
 E_\Lambda 
\Delta_p^{\alpha} E_\Lambda 
%
\over
%
\Tr \, \bigl( \Delta_p^{\alpha} E_\Lambda \bigr)^2 } 
\cr
& = { \Tr \, \pi_\beta (a) E_\Lambda    
\Delta_p^{\alpha}  E_\Lambda \pi_\beta \bigl( \tau_{\Re z} (b) \bigr) 
\Delta_p^{\alpha} E_\Lambda 
%
\over
%
\Tr \, \bigl( \Delta_p^{\alpha} E_\Lambda \bigr)^2 } .}
%
For $ \alpha \beta < \Im z <  2 \alpha \beta$ we set 
% 
\# {F_{a,b}^{(2)} (z ) := { \Tr \, \pi_\beta (a) E_\Lambda    
\Delta_p^{\alpha} E_\Lambda  
\Delta_p^{- \alpha -iz / \beta} \pi_\beta (b)  
\Delta_p^{2\alpha + i z / \beta}  E_\Lambda  
%
\over
%
\Tr \, \bigl( \Delta_p^{\alpha}  E_\Lambda \bigr)^2 } .}
%
The function $F_{a,b}^{(2)} (z )$ is analytic in its domain and 
continuous at the boundary. 
By definition,
% 
\& { \lim_{\Im z \searrow \alpha \beta} F_{a,b}^{(2)} (z ) 
& = 
{ \Tr \, \pi_\beta (a) E_\Lambda    
\Delta_p^{\alpha} E_\Lambda  
\pi_\beta \bigl( \tau_{\Re z} (b) \bigr)
\Delta_p^{\alpha}   E_\Lambda  
%
\over
%
\Tr \, \bigl( \Delta_p^{\alpha} E_\Lambda \bigr)^2 } 
\cr
& = \lim_{\Im z \nearrow \alpha \beta} F_{a,b}^{(1)} (z) 
\qquad \forall |{\Re z}| < \delta_{\circ \circ}.}
%
Furthermore,  $F_{a,b}^{(2)}$ satisfies  
%
\& { \lim_{ \Im z \nearrow 2 \alpha \beta} F_{a,b}^{(2)} (z) 
&= { \Tr \, \pi_\beta (a) E_\Lambda    
\Delta_p^{\alpha} E_\Lambda
\Delta_p^{\alpha}  \pi_\beta \bigl( \tau_{\Re z} (b) \bigr)  
E_\Lambda  
%
\over
%
\Tr \, \bigl( \Delta_p^{\alpha} E_\Lambda \bigr)^2 }
\cr
&= { \Tr \, \pi_\beta (a) \bigl( E_\Lambda    
\Delta_p^{\alpha} E_\Lambda \bigr)^2 \pi_\beta \bigl( \tau_{\Re z} (b) \bigr)  
%
\over
%
\Tr \, \bigl( \Delta_p^{\alpha} E_\Lambda \bigr)^2 }
\cr
&= \omega_\Lambda  \bigl( \tau_{\Re z} (b) a  \bigr) 
\qquad \forall |{\Re z}| < \delta_{\circ \circ}.}
%
Using the Edge-of-the-Wedge theorem [SW] we conclude
that $F_{a,b}^{(1)}$ and $F_{a,b}^{(2)}$ are  
the restrictions to the upper (resp.\ lower)
half of the double cut  strip 
%
\# { {\cal G}_{2,\alpha} = \{ z \in \C : 0 < \Im z < 2 \alpha \beta \} \setminus 
\{ z \in \C : | \Re z | \ge \delta_{\circ \circ} , \Im z = \alpha \beta \} }
% 
of a function 
%
\# {
F_{a,b} (z) 
:= 
\left\{
\eqalign{
&  { F_{a,b}^{(2)} (z)  }
\cr
&
{ F_{a,b}^{(1)} (z) } 
}
\right\} 
\hbox{\rm for}
\left\{
\eqalign{
& { \alpha \beta < \Im z < 2 \alpha \beta ,}  
\cr
& {0 < \Im z < \alpha \beta,}  }  
\right\} 
}
%
defined and continuous on the closure of ${\cal G}_{2, \alpha}$
and analytic for $z \in {\cal G}_{2, \alpha}$.
From (98) and (102) we infer
%
\# { F_{a,b} (t) = \omega_\Lambda  \bigl( a \tau_{t} (b) \bigr)  
\quad \hbox{and} \quad F_{a,b} (t + i 2 \alpha \beta ) 
= \omega_\Lambda  \bigl( \tau_{t} (b) a  \bigr) 
\quad \hbox{for} \quad |t| < \delta_{\circ \circ}.}
%
Analogous results for arbitrary $n \in \N$ can be established by the same line of arguments
but with considerable more effort.}

\vskip  1cm
\Hl{Bounds on the Quasi-Partition Function}

\noindent
Let $\Lambda_i = \bigl( \O_\circ^{(i)} , \O^{(i)} , \hat{\O}^{(i)} \bigr)$ be a sequence of 
triples of double cones with diameters~$({\breve r}_i , r_i , \hat{r}_i)$. 
In order to ensure that 
(for $0 < \alpha < 1 /2$ and $n \in \N$ fixed) the
`quasi-partition function' 
%
\# { Z_{\Lambda_i}(\alpha, n) := \Tr \, \bigl( E_{\Lambda_i} 
\Delta_{p, i}^\alpha E_{\Lambda_i} \bigr)^n , \qquad \Lambda_i = 
\bigl( \O_\circ^{(i)}, \O^{(i)} , \hat{\O}^{(i)} \bigr), } 
% 
is bounded from below as $i \to \infty$ it is necessary that $\O^{(i)}$ grows rapidely 
with $\O_\circ^{(i)}$. Otherwise the energy contained 
in the boundary, which is necessary to decouple the 
local region from the outside, lessens the eigenvalues
of $E_{\Lambda_i} \Delta_{p, i}^\alpha E_{\Lambda_i}$ so drastically that
it might outrun the increase in the number of states 
contributing to the trace by enlarging~$\O_\circ^{(i)}$. 




\Th{Assume that the net of local observables $\O \to {\cal R}_\beta (\O)$ 
is regular from the outside.
Furthermore assume there exist positve constants
$m, c, d$, such that the nuclear norm of
$ \Theta_{\alpha, {\cal O}_\circ^{(i)}}^\sharp $ --- this map was introduced in (24) ---
is bounded by
%
\# { \|  \Theta_{\alpha, {\cal O}_\circ^{(i)}}^\sharp  \|_1 
\le  c {\breve r}_i^d  \cdot \alpha^{-m}  , \qquad m , 
c , d > 0 .}
%
It follows that there exists a sequence of triples of space-time regions
$\bigl\{ \Lambda_i \bigr\}_{i \in \n}$  with 
%
\# {r_i = {\breve r}_i^\gamma, \qquad \gamma >  \max \{ 1, d(m+1) / m^2 \}, }
%
and $\hat {r}_i \searrow r_i $
sufficiently fast, such that
%
\# { Z_{\Lambda_i} (\alpha, n) := \Tr \, \bigl( E_{\Lambda_i} 
\Delta_{p, i}^\alpha E_{\Lambda_i} \bigr)^n > C > 0 \qquad  \forall i > i_\circ, \quad 
i_\circ \in \N .}
%
}


\vskip .5cm
The proof proceeds in several steps: we show that

\vskip .3cm

\noindent
i.) If $s-\lim_{i \to \infty}
E_{\Lambda_i} = \1 $, then 
%
\# { \lim \inf_i \Tr \, \bigl( E_{\Lambda_i} 
\Delta_{p, i}^\alpha E_{\Lambda_i} \bigr)^n > 0 .}
%
\vskip .1cm
\noindent
ii.) If $\|  \eta_i - \Omega_\beta  \|  \to 0$ as $i \to \infty$, then
%
\# { s-\lim_{i \to \infty} 
E_{\Lambda_i} = \1 .}
%

\vskip .1cm

\noindent
iii.) Let $\chi_i \in {\cal P}^\natural
\bigl({\cal M}_\beta ( \O_\circ^{(i)}) \vee {\cal M}_\beta ( \O^{(i)})' , 
\Omega_\beta \bigr)$ denote the product vector which satisfies
%
%
%
\# { ( \chi_i \, , \, M N  \, \chi_i) = (\Omega_\beta \, , \, M \Omega_\beta)
(\Omega_\beta \, , \, N \Omega_\beta) \qquad
%
\forall M_i \in {\cal M}_\beta ( \O_\circ^{(i)}), 
\quad N \in {\cal M}_\beta ( \O^{(i)})' .}
% 
Combining a result of Araki [A 74]
concerning the distance of vectors in the positive
natural cone ${\cal P}^\natural
\bigl({\cal M}_\beta ( \O_\circ^{(i)}) \vee {\cal M}_\beta ( \O^{(i)})' , 
\Omega_\beta \bigr)$ with a generalized cluster theorem [J\"a c], we conclude that
%
\# { \| \chi_i - \Omega_\beta \|^2 \le
c {\breve r}_i^d  \cdot \bigl( r_i - {\breve r}_i \bigr)^{-{m^2 \over m+1} } ,}
%
where $c>0$ is a constant which does not depend on $r_i$ or ${\breve r}_i$.  
If we put $r_i = {\breve r}_i^\gamma$ with $\gamma > \max \{ 1, d(m+1) / m^2 \}$, then
%
\# { \| \chi_i - \Omega_\beta \| \to 0 \qquad \hbox{as} \quad {\breve r}_i  \to \infty.}
%  
\vskip .1cm

\noindent
iv.) Let $\eta_i \in {\cal P}^\natural
\bigl({\cal M}_\beta ( \O_\circ^{(i)}) \vee {\cal M}_\beta ( \O^{(i)})' , 
\Omega_\beta \bigr)$ denote the product vector which satisfies
%
%
%
\# { ( \eta_i \, , \, M N  \, \eta_i) = (\Omega_\beta \, , \, M \Omega_\beta)
(\Omega_p^{(i)} \, , \, N \Omega_p^{(i)})  
%
\qquad \forall M_i \in {\cal M}_\beta ( \O_\circ^{(i)}), \quad 
N \in {\cal M}_\beta ( \O^{(i)})' ,} 
%
where $\Omega_p^{(i)}$
is the product vector which satisfies
%
\# { ( \Omega_p^{(i)} \, , \, A_i B_i  \, \Omega_p^{(i)}) = (\Omega_\beta \, , \, A_i \Omega_\beta)
(\Omega_\beta \, , \, B_i \Omega_\beta) \qquad
\forall A_i \in {\cal R}_\beta ( \O^{(i)}), \quad 
B \in {\cal R}_\beta ( \hat{\O}^{(i)})' .}
%
Taking advantage of the assumption that the net $\O \to {\cal R}_\beta (\O)$ is regular 
from the outside we can adjust $\hat{\O}^{(i)}$
such that
%
\# { \| \chi_i - \eta_i \| \le  \| \chi_i - \Omega_\beta \| \qquad \hbox{as} \quad i \to \infty.}
%
\vskip .1cm
\noindent
v.) Combining  iii.) and  iv.) we conclude that
%
\# { \| \eta_i - \Omega_\beta \| \to 0 \qquad \hbox{as} \quad  {\breve r}_i  \to \infty.}
%  
if $r_i = {\breve r}_i^\gamma$ with $\gamma >  \max \{ 1, d(m+1) / m^2 \}$ 
and $\hat {r}_i \searrow r_i $ sufficiently fast.

%


\vskip .5cm
The details are as follows

\Lm{Assume that  $s-\lim_{i \to \infty}
E_{\Lambda_i} = \1 $. It follows that 
%
\# { \lim \inf_i \Tr \, \bigl( E_{\Lambda_i} 
\Delta_{p, i}^\alpha E_{\Lambda_i} \bigr)^n 
> 0 \qquad \forall n \in \N.}
%
}

\Pr{$\Delta_{p,i}^{1/2}$ is by definition a positive operator. 
$\Omega_p^{(i)}$ is the unique eigenvector of $K_\beta$ for the simple eigenvalue
$\{ 0 \}$. Thus any other vector $\Psi \in \H_\beta$ satisfies $( \Psi \, , \,
\Delta_{p,i}^{2\alpha} \Psi) > 0$ for all $\alpha >0$.
Let $\Psi_j$, $j = 1,2$, be any pair of orthogonal and normalized
vectors. We consider the case $n=2$. Since  $s-\lim_{i \to \infty}
E_{\Lambda_i} = \1 $, it follows that 
%
\& { \lim \inf_i \Tr \, \bigl( E_{\Lambda_i} 
\Delta_{p, i}^\alpha E_{\Lambda_i} \bigr)^2 
&\ge \lim \inf_i \sum_{j=1}^2 \Bigl( E_{\Lambda_i}  \Psi_j \, , \,
\Delta_{p, i}^\alpha  E_{\Lambda_i} 
\Delta_{p, i}^\alpha  E_{\Lambda_i}  \Psi_j \Bigr)
\cr
&= \lim \inf_i \sum_{j=1}^2 \Bigl( \Psi_j \, , \,
\Delta_{p,i}^{2\alpha} \Psi_j \Bigr)
> 0 .}
%
}


\Lm{(Buchholz and Junglas). Let $\{ \Lambda_i \}_{i \in \n}$ be an 
increasing sequence of triples of space-time regions such that 
%
\# {  \|  \eta_i - \Omega_\beta  \|  \to 0  \qquad \hbox{as} \quad i \to \infty.}
%
It follows that
$\H_{\Lambda_i}$ tends to the whole Hilbert space $\H_\beta$, i.e., $s-\lim_{i \to \infty} 
E_{\Lambda_i} = \1 $. }

\Pr{By assumption $\eta_i = W^*_i  (\Omega_\beta \otimes \Omega_p^{(i)}) $ 
converges to $\Omega_\beta$. Therefore
the unitary operators~$W_i$ --- see (61) --- fulfill
%
\# { W^*_i  (\Phi \otimes \Omega_p^{(i)}) \to \Phi 
\qquad \forall \Phi \in \H_\beta }
%
as $i \to \infty$. Recall that $E_{\Lambda_i} = W_i^* (\1 \otimes P_{\Omega_p^{(i)}})
W_i$, where $P_{\Omega_p^{(i)}}$ denotes the projection onto~$\C \cdot \Omega_p^{(i)}$.
Hence
%
\& { E_{\Lambda_i}  \Phi &=
W^*_i  (\1 \otimes P_{\Omega_p^{(i)}}) W_i
\bigl( \Phi - W^*_i  (\Phi \otimes \Omega_p^{(i)}) \bigr)
\cr
& \qquad  + W^*_i  (\Phi \otimes \Omega_p^{(i)}) \to \Phi 
\qquad
\forall \Phi \in \H_\beta ,}
%
as $i \to \infty$. I.e.,  $s-\lim_{i \to \infty } E_{\Lambda_i}  = \1 $.}


We now apply the following generalized cluster
theorem [J\"a  c]:

\Th{Assume there exist positve constants
$m, c, d$, such that the nuclear norm of
$ \Theta_{\alpha, {\cal O}_\circ^{(i)}}^\sharp $ --- this map was introduced in (24)
--- is bounded by
%
\# { \|  \Theta_{\alpha, {\cal O}_\circ^{(i)}}^\sharp  \|_1 
\le  c \breve{r}_i^d  \cdot \alpha^{-m}  , \qquad m , 
c , d > 0 .}
%
Let 
$M_i \in {\cal M}_\beta ( \O_\circ^{(i)})$ and 
$N \in {\cal M}_\beta ( \O^{(i)})'$. Then --- for $(r_i - {\breve r}_i)$ large compared to $1/2$ ---
%
\# {\Bigl| \sum_{j=1}^{j_\circ}( \chi_i \, , \,  M_j N_j\chi_i )
- ( \Omega_\beta \, , \,  M_j N_j\Omega_\beta )
\Bigr|  
\le
c {\breve r}_i^d  \cdot \bigl( r_i - {\breve r}_i \bigr)^{-{m^2 \over m+1} }
\cdot \Bigl\| \sum_{j=1}^{j_\circ} M_j N_j \Bigr\|. }
%
where $c> 0$ is a constant which does not depend on ${\breve r}_i$ or $r_i$. } 

\Cor{Assume there exist positve constants
$m, c, d$, such that the nuclear norm of
$\Theta_{\alpha, {\cal O}_\circ^{(i)}}^\sharp$ is bounded by
%
\# { \|  \Theta_{\alpha, {\cal O}_\circ^{(i)}}^\sharp  \|_1 
\le  c \breve{r}_i^d  \cdot \lambda^{-m}  , \qquad m , 
c , d > 0 .}
%
Let $(\O_\circ^{(i)}, \O^{(i)}) $ denote a sequence of pairs of double cones
with 
%
\# {r_i = {\breve r}_i^\gamma, \qquad \gamma >  \max \{1 , d(m+1) / m^2 \}.  }
%
It follows that
%
\# {  \| \chi_i - \Omega_\beta  \|  \to 0 
\qquad \hbox{as} \quad i \to \infty.}
}

\Pr{Since $\chi_{i}  \in {\cal P}^\natural
\bigl({\cal M}_\beta ( \O_\circ^{(i)}) \vee {\cal M}_\beta ( \O^{(i)})', \Omega_\beta \bigr)$,
we can apply a result of Araki [A 74] concerning the distance of two vectors
which belong to ${\cal P}^\natural
\bigl({\cal M}_\beta ( \O_\circ^{(i)}) \vee {\cal M}_\beta ( \O^{(i)})', \Omega_\beta \bigr)$: 
%
\# { \| \chi_{i} - \Omega_\beta  \|^2 
\le \sup_{  
\| D_i \| = 1 } \bigl| (\chi_i , D_i \chi_i )  
- (\Omega_\beta , D_i 
\Omega_\beta) \bigr|    ,}
%
where the supremum has to be evaluated over all 
%
$D_i \in {\cal M}_\beta ( \O_\circ^{(i)}) \vee {\cal M}_\beta ( \O^{(i)})' $. 
%
Thus   
%
\# { \| \chi_i - \Omega_\beta \|^2 \to 0 \qquad \hbox{as} \quad {\breve r}_i  \to \infty,}
%  
follows from the previous theorem, if $r_i = {\breve r}_i^\gamma$ with $ \gamma > 
\max \{1 , d(m+1) / m^2 \}$. }
 
Sofar there was no restriction on the relative size of the regions
$\O^{(i)}$ and $\hat {\O}^{(i)}$. We can exploit this freedom 
if the net of local observables $\O \to {\cal R}_\beta (\O)$ is regular from the outside.
Note that regularity from the outside implies  
%
\# { {\cal M}_\beta (\O) ' \cap {\cal M}_\beta (\hat{\O}) \to \C \cdot \1   
\qquad \hbox{as} \quad \hat{\O} \searrow \O. }
%


\Lm{Let
$\bigl( \O_\circ^{(i)} , \O^{(i)} \bigr)$ be a sequence of 
pairs of double cones with diameters~$({\breve r}_i , r_i)$. 
Let $\chi_i \in {\cal P}^\natural
\bigl({\cal M}_\beta ( \O_\circ^{(i)}) \vee {\cal M}_\beta ( \O^{(i)})' , 
\Omega_\beta \bigr)$ denote the product vector which satisfies
%
%
%
\# { ( \chi_i \, , \, M N  \, \chi_i) = (\Omega_\beta \, , \, M \Omega_\beta)
(\Omega_\beta \, , \, N \Omega_\beta) }
%
for all $M_i \in {\cal M}_\beta ( \O_\circ^{(i)})$ and 
$N\in {\cal M}_\beta ( \O^{(i)})'$ and let $\eta_i \in {\cal P}^\natural
\bigl({\cal M}_\beta ( \O_\circ^{(i)}) \vee {\cal M}_\beta ( \O^{(i)})' , 
\Omega_\beta \bigr)$ denote the product vector which satisfies
%
%
%
\# { ( \eta_i \, , \, M N  \, \eta_i) = (\Omega_\beta \, , \, M \Omega_\beta)
(\Omega_p^{(i)} \, , \, N \Omega_p^{(i)}) }
%
for all $M_i \in {\cal M}_\beta ( \O_\circ^{(i)})$ and 
$N \in {\cal M}_\beta ( \O^{(i)})'$. 
%
Then we can adjust $\hat{\O}^{(i)}$, i.e., the size of $\hat{r}_i$, such that
%
\# { \| \chi_i - \eta_i \| \le \| \chi_i - \Omega_\beta \|  \qquad \hbox{as} \quad i \to \infty.}
%
}


\Pr{Consider a sequences of pairs of double cones   
$\{ ( \O_\circ^{(i)} , \O^{(i)} ) \}_{ i \in \n} $  
eventually exhausting all of $\R^4$. For each $i \in \N$ fixed we consider a sequence 
of double cones $\{ \hat{\O}^{(i,k)}\}_{ k \in \n} $ such that
%
\# { \hat{\O}^{(i,k)} \searrow \O^{(i)} , \qquad k \to \infty .}
%
In order to ease the notation we set
%
\# { {\cal A}_i = {\cal M}_\beta \bigl( \O_\circ^{(i)} \bigr),
\qquad
{\cal B}_i = {\cal M}_\beta \bigl( {\cal O}^{(i)} \bigr),
\qquad 
{\cal C}_{i,k} = {\cal M}_\beta \bigl( \hat{\cal O}^{(i,k)} \bigr) }
%
and
%
\# { {\cal D}_{i} = {\cal A}_i \vee {\cal B}_i',
\qquad 
{\cal E}_{i,k} = {\cal A}_i \vee {\cal C}_{i,k}'  .}
%
Note that for each $i \in \N$ fixed, the sequence $\{ {\cal E}_{i,k}  \}_{k \in \n}$ of algebras 
satisfies 
${\cal E}_{i,k+1}  \subset {\cal E}_{i,k} $ and  
%
\# { \cap_k {\cal E}_{i,k} =  {\cal D}_i . } 
%
Let $\Omega_p^{(i,k)}$ denote the unique product vector in the natural positive 
cone ${\cal P}^\natural \bigl({\cal R}_\beta (\O^{(i)}) \vee {\cal R}_\beta (\hat {\O}^{(i,k)})', 
\Omega_\beta \bigr) $
satisfying
%
\# { (\Omega_p^{(i,k)} \, , \, AB   \Omega_p^{(i,k)}) =  (\Omega_\beta \, , \, A \Omega_\beta) 
(\Omega_\beta \, , \, B \Omega_\beta)}
%
for all $A \in {\cal R}_\beta (\O^{(i)})$ and $B \in {\cal R}_\beta (\hat {\O}^{(i,k)})'$.
Note that
for $C_{i,k} \in {\cal C}_{i,k}'$
%
\# { (\Omega_p^{(i,k)} \, , \, C_{i,k} \Omega_p^{(i,k)}) 
= (\Omega_\beta \, , \, C_{i,k} \Omega_\beta) .}
%
If we choose product vectors $\eta_{i,k}$  and $\chi_i$ 
in the natural cone ${\cal P}^\natural ({\cal D}_i, \Omega_\beta)$
such that
%
\# {  ( \eta_{i,k} \, , \, MN \eta_{i,k}) = (\Omega_\beta \, , \, M \Omega_\beta)
(\Omega_p^{(i,k)} \, , \, N \Omega_p^{(i,k)}) }
%
and
%
\# {( \chi_i \, , \, MN \chi_i) = (\Omega_\beta \, , \, M \Omega_\beta)
(\Omega_\beta \, , \, N \Omega_\beta)}
%
for all $M \in {\cal A}_i$, $N \in {\cal B}_i'$, 
then by the result of Araki [A 74] 
%
\# { \| \eta_{i,k} - \chi_i  \|^2 
\le \sup_{ D_i  \in {\cal D}_i , \| D_i \| = 1 } \bigl| (\eta_{i,k} , D_i \eta_{i,k} )  
- (\chi_i , D_i 
\chi_i) \bigr|    .}
%
Now assume that for each $i \in \N$ fixed there exist a sequence $\{  E_{i,k} \in {\cal E}_{i,k} : 
\| E_{i,k} \| = 1 \}_{k \in \n}$ such 
that
%
\# { \lim_{k \to \infty} \bigl| (\eta_{i,k} \, , \, E_{i,k} \eta_{i,k}  )  
- (\chi_i \, , \, E_{i,k} \chi_i) \bigr| \ge \| \chi_i - \Omega_\beta \| . }
%
The linear functional $(\eta_{i,k} \, , \, \,. \, \eta_{i,k}  )  
- (\chi_i \, , \, \,. \, \chi_i)$ is ultra-weakly continuous on the von Neumann 
algebra ${\cal D}_i$. 
Therefore the sequence $\{   E_{i,k} \in {\cal E}_{i,k} : 
\| E_{i,k}  \| = 1 \}_{k \in \n}$ has a weak limit point:
%
\# { w-\lim_{k \to \infty} E_{i,k}  =: D_i, \qquad D_i \in {\cal D}_{i} =
\cap_k {\cal E}_{i,k} ,} 
%
such that 
%
\# {  \bigl| (\eta_{i,k} , D_i \eta_{i,k} ) - (\chi_i , D_i \chi_i)  \bigr| 
> {1 \over 2 }
\| \chi_i - \Omega_\beta \|  
\qquad \forall k > k_i,} 
% 
and some $k_i \in \N$, in contradiction to 
%
\# {  \bigl| (\eta_{i,k} , E_{i,k} \eta_{i,k} ) - (\chi_i , E_{i,k} \chi_i)  \bigr| = 0 
\qquad \forall E_{i,k} \in {\cal E}_{i,k}, \quad \forall k \in \N .} 
% 
%
Therefore, the assumption (144)
can not hold true.
It follows that there exists some $k_i \in \N$ such that 
--- note that $\| \chi_i - \Omega_\beta \| \to 0$ as $i \to \infty$ ---
%
\# { \sup_{D \in {\cal D}_{i} , \| D  \| = 1}
 \bigl| (\eta_{i, k} \, , \, D  \eta_{i,k}  )  
- (\chi_i \, , \, D  \chi_i)  \bigr|  <  \| \chi_i - \Omega_\beta \|  
\qquad \forall k \ge k_i. }
% 
We conclude that if we set $\hat{\O}^{(i)} : = \hat{\O}^{(i, k_i)}$, then
%
\# {  \| \eta_{i} - \Omega_\beta \| \le \| \eta_i - \chi_i  \|  + \| \chi_i - \Omega_\beta  \| 
\le 2 \| \chi_i - \Omega_\beta  \|  
\qquad \hbox{as} \quad i \to \infty.} 
% 
}



\vskip 1cm

\Hl{The Thermodynamic Limit}


\noindent
The unit ball in $\A^*$ is weak$^*$-compact, thus for every net of states
%
\# { \Lambda(\O_\circ, \O, \hat {\O}) \to \omega_\Lambda  ,
\qquad n , \alpha  \quad \hbox {fixed}, }
% 
there exists  a subsequence  $\{ \omega_{\Lambda_i} \}_{i \in \n}$ converging to 
some state $\omega$. Whether or not this state is a $(\tau, n \alpha \beta)$-KMS
state depends on the energy contained in the boundary, i.e.,
the choice of the relative size of~${\cal O}_\circ$,~$\O$, and~$ \hat {\O}$.
We show that the necessary quantitative
information restricting the surface energy can be drawn from the bounds on the 
nuclear norms of the maps $\Theta_{\alpha, {\cal O}}$ 
and $\Theta^\sharp_{\alpha, {\cal O}}$ introduced in (22) and (24).

\vskip .5cm
Let
$\Lambda_i = \bigl( \O_\circ^{(i)} , \O^{(i)} , \hat{\O}^{(i)} \bigr)$ be a sequence of 
triples of double cones with diameters~$({\breve r}_i , r_i , \hat{r}_i)$. 
We will now exploit the fact that the elements of $\A_p$, $p \in \N$, introduced at 
the end of Section 2,
have good localization properties: we show that 
there exists some $p \in \N$ such that
%
\# {  \Bigl| \omega_{\Lambda_i} \bigl( a \tau_{i n\alpha \beta} (b) \bigr) -
\omega_{\Lambda_i} (b a ) \Bigr| < \epsilon_i \qquad \forall a,b \in \A_p,}
%
where $\epsilon_i \searrow 0$ as $i \to \infty$. 
The surface energy is controlled 
by adjusting the relative size of~${\breve r}_i$, $r_i$  and $\hat{r}_i$.
Inspecting the definition (93) of $\omega_{\Lambda_i}$ 
we recognize that in order to prove (151) it is sufficient to control
%
\# { \Tr \, \,  \rho_{\Lambda_i} \,\,  \pi_\beta (a) 
\bigl[ \pi_\beta \bigl( \tau_{i k \alpha \beta} (b) \bigr) , E_{\Lambda_i} \bigr]  , \qquad
k = 1, \ldots, n.}
%
Let us consider the case $n=2$.
Let $a, b \in \A_p$, $p \in \N$ fixed.
It follows that
%
\# { \tau_{ik\alpha \beta} (b) \in \A_p  \qquad \hbox{for} \quad k = 1, 2.}
%
Since $a$ and $b$ as well as $ c:= \tau_{i\alpha \beta} (b)$ and $ d := \tau_{2i\alpha \beta} (b)$ 
are almost localized in $\O_\circ^{(i)}$ for $i$ sufficiently large,
they almost commute with $E_{\Lambda_i}$. 
Thus, for example,
%
\& {  \Tr \, \rho_{\Lambda_i} \, \pi_\beta (a) [ \pi_\beta \bigl( \tau_{i 2\alpha \beta} (b) \bigr) , 
E_{\Lambda_i} ]  =
& {  \Bigl| \Tr \, [ \pi_\beta \bigl( \tau_{i 2\alpha \beta} (b) \bigr) , 
E_{\Lambda_i} ]   \cdot 
\bigl( \Delta_{p, i}^\alpha   
E_{\Lambda_i} \bigr)^2 \pi_\beta (a)  \Bigr|
%
\over 
%
\Tr \, \bigl( \Delta_{p, i}^\alpha E_{\Lambda_i} \bigr)^2 } 
\cr
& 
\le  { \| \,  [ \pi_\beta \bigl( \tau_{i 2\alpha \beta} (b) - d_i \bigr),  E_{\Lambda_i} ] \,  \|
\cdot
\Tr \, | \bigl( \Delta_{p, i}^\alpha 
E_{\Lambda_i} \bigr)^2| 
\cdot
\| a  \|  \over \Tr \, 
\bigl( \Delta_{p, i}^\alpha E_{\Lambda_i} \bigr)^2} 
\cr
& 
\le  { 2  \| a  \|  \over \Tr \, \bigl( \Delta_{p, i}^\alpha E_{\Lambda_i} \bigr)^2} 
\,
\|  \tau_{i 2\alpha \beta} (b) - d_i \| 
\cdot
\bigl( \Tr \, | \Delta_{p, i}^\alpha E_{\Lambda_i} | \bigr)^2  \cr
& 
\le {c_1 \over \Tr \, \bigl( \Delta_{p, i}^\alpha E_{\Lambda_i} \bigr)^2}
\cdot {\rm e}^{- c_2 {\breve r}_i^{2p}} \cdot {\rm e}^{ c_3 r_i^d},  }
%
for certain positive constants $c_1$, $c_2$ and $c_3$.
Here $d_i \in \A (\O_\circ^{(i)})$ denotes the local approximation of 
$d:=\tau_{i 2\alpha \beta} (b) \in \A_p$ 
such that $[E_{\Lambda_i},d_i] = 0$. 
In the last inequality we made use  of Proposition 4.3 and  
the second part of Lemma 2.1  We conclude that the numerator in (154)
vanishes in the thermodynamic limit, if
we set $p > d \gamma /2$, $\gamma > \max \{1 , d(m+1) / m^2 \}$  and
%
\# {  r_i = {\breve r}_i^\gamma,    \qquad  i \in \N. }
%
As has been show in the previous section,
the denominator does not vanish as $i \to \infty$, but is bounded from below by some 
positive constant.

\vskip 1cm

We will now  
establish the KMS-property for all weak limit points of~$\{ \omega_{\Lambda_i} \}_{i \in \n}$,
provided the regions $\Lambda_i = ( \O_\circ^{(i)}, \O^{(i)}, \hat {\O}^{(i)})$ 
tend to the whole space-time in 
agreement with the restrictions imposed on the relative size of 
${\breve r}_i$,  $r_i$ and $\hat{r}_i$ in Theorem 6.1.

\Th {Let $n \in \N$ and $0 < \alpha < 1/2$ be fixed and let
$\Lambda_i = \bigl( \O_\circ^{(i)}, \O^{(i)}, \hat {\O}^{(i)} \bigr)$ be a sequence of 
triples of double cones 
with diameters ${\breve r}_i$ $r_i$ and $\hat{r}_i$ such that
%
\# {r_i = {\breve r}_i^\gamma, \qquad \gamma > \max \{ 1, d(m+1) / m^2 \}, }
%
and 
%
\# { \hat{r}_i \searrow r_i  \qquad \hbox{as} \quad i \to \infty, }
%
sufficiently fast such that
%
\# { \lim_{i \to \infty} \| \eta_{i} - \Omega_\beta \|  = 0 .}  
%
Then every weak limit point of the sequence $\{ \omega_{\Lambda_i} \}_{i \in \n}$ is a 
$\tau$-KMS-state at inverse temperature $n \alpha \beta > 0$.}


\Pr {Let $a,b \in \A_p$, $p > d \gamma /2$.
We consider the case $n= 2$.  
\vskip .2cm
\noindent
i.)  Let
$\omega_{2 \alpha \beta}$ denote the limit state 
of a convergent subsequence $\{ \omega_{\Lambda_i} \}_{i \in \n}$.  
For every~$\epsilon > 0$ we can find an index $i \in \N$ such that
%
\# { \bigl| \omega_{2 \alpha \beta} \bigl( a \tau_{i 2 \alpha \beta} (b) -ba \bigr) \bigr|  
 \le \bigl|   
\omega_{\Lambda_i}    \bigl( a \tau_{i 2 \alpha \beta} (b) -ba \bigr)  
\bigr| 
+ \epsilon .}
%
\vskip .2cm
\noindent
ii.) We approximate $\tau_{i 2 \alpha \beta} (b) , \tau_{i \alpha \beta} (b)$ and
$b$ by local elements in $\A \bigl( \O_\circ^{(i)} \bigr)$ and
apply the  commutator estimate (154) several times: for~$ \epsilon > 0$ and 
suitable (large) $i \in \N$ we find
%
%
\& {  \bigl|  \omega_{2 \alpha \beta}  \bigl( a \tau_{i 2 \alpha \beta} (b) -ba \bigr)  \bigr| &
\le 
\Bigl| {  
\Tr \, \pi_\beta \bigl( a \tau_{i 2 \alpha \beta} (b)\bigr)    
\bigl( E_{\Lambda_i} \Delta_{p, i}^\alpha E_{\Lambda_i} \bigr)^2
\over \Tr \, \bigl(  \Delta_{p, i}^\alpha E_{\Lambda_i} \bigr)^2} - \omega_{\Lambda_i}  (ba) \Bigr| 
+ \epsilon \cr
& \le \Bigl| {  
\Tr \, \pi_\beta (a) E_{\Lambda_i} \pi_\beta \bigl(\tau_{i 2 \alpha \beta} (b)\bigr)    
\bigl( \Delta_{p, i}^\alpha E_{\Lambda_i} \bigl)^2 \over \Tr \, \bigl(  
\Delta_{p, i}^\alpha E_{\Lambda_i} \bigr)^2}
- \omega_{\Lambda_i}  (ba) \Bigr| 
+ 2 \epsilon \cr
& =  \Bigl| {  
\Tr \, \pi_\beta (a) E_{\Lambda_i} \Delta_{p, i}^\alpha 
\pi_\beta \bigl(\tau_{i \alpha \beta} (b)\bigr)    
 E_{\Lambda_i}\Delta_{p, i}^\alpha
E_{\Lambda_i} \over \Tr \, \bigl(   
\Delta_{p, i}^\alpha E_{\Lambda_i} \bigr)^2}
- \omega_{\Lambda_i}  (ba) \Bigr| 
+ 2 \epsilon \cr
& \le  \Bigl|  { 
\Tr \, \pi_\beta (a) \bigl( E_{\Lambda_i} \Delta_{p, i}^\alpha \bigr)^2
\pi_\beta (b) E_{\Lambda_i} \over \Tr \, \bigl(  
\Delta_{p, i}^\alpha E_{\Lambda_i} \bigr)^2}
- \omega_{\Lambda_i}  (ba) \Bigr| 
+ 3 \epsilon \cr
& \le  \Bigl| {  
\Tr \, \pi_\beta (a) \bigl( E_{\Lambda_i} \Delta_{p, i}^\alpha
E_{\Lambda_i} \bigr)^2 \pi_\beta (b)
\over \Tr \, \bigl(   
\Delta_{p, i}^\alpha E_{\Lambda_i} \bigr)^2}
- \omega_{\Lambda_i}  (ba) \Bigr| 
+ 4 \epsilon \cr
& = 4 \epsilon .}
%
Thus we have found a norm dense,
$\tau$-invariant $\ast$-subalgebra of the set $\A_\tau$ of analytic elements for $\tau$,
namely $\A_p$, $p > d \gamma /2$, such that the state $\omega_{2 \alpha \beta}$ satisfies
% 
\# { \omega_{2 \alpha \beta} \bigl (a \tau_{2 i \alpha \beta}(b) \bigr) 
= \omega_{2 \alpha \beta} (b a)  
\qquad \forall a, b \in \A_p . }
%
Consequently $\omega_{2 \alpha \beta} $ is a $(\tau, 2 \alpha \beta)$-KMS-state.
Similiar results for arbitrary $n \in \N$ can be established by the same line of arguments
but with considerable more effort.}

 
Once we have constructed a $(\tau, \beta')$ KMS-state $\omega_{\beta'}$,
the GNS-representation $\pi_{\beta'}$ gives us a new  
thermal field theory
%
\# { \O \to {\cal R}_{\beta'} (\O) := \pi_{\beta'} \bigl( \A(\O) \bigr)'' , 
\qquad \O \in \R^4,}
%
acting on a new Hilbert space
$\H_{\beta'}$ with GNS-vector $\Omega_{\beta'}$. If $\beta \ne \beta'$, then
the new thermal field theory will not
be unitarily equivalent to the old one [T]. In fact,
there might be several extremal $(\tau, n \alpha \beta)$-KMS-states,
which induce unitarily inequivalent representations, i.e., ``disjoint
thermal field theories'', at the same inverse temperature $\beta' = n \alpha \beta$.

\vskip .5cm

\noindent
{\it  Acknowledgements.\/} The present work started in collaboration with D.\ Buchholz. 
The final formulation is strongly influenced by his constructive criticism and
by several substantial hints. For critical reading of the manuscript many thanks are due 
to M.~M\"uger. Kind hospitality of the 
II.\ Institute for theoretical physics, University of Hamburg,
the Institute for theoretical physics, University of Vienna, the 
Erwin Schr\"odinger Institute (ESI), Vienna, and the Dipartimento di 
Matematica, Universita di Roma ``Tor Vergata" is gratefully acknowleged. 
This work was financed by 
the Fond zur F\"orderung der Wissenschaft\-lichen 
Forschung in Austria, Proj.\ Nr.\ P10629 PHY and 
a fellowship of the Operator Algebras Network, EC TMR-Programme.




\vskip .8cm

\noindent
{\fourteenrm References}

\vskip .4cm

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\cr}



\bye




