\documentstyle[twoside,11pt]{article}
\pagestyle{myheadings}
\markboth{ }{ }
\def\greaterthansquiggle{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}}
\def\lessthansquiggle{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\beqa}{\begin{eqnarray}}
\newcommand{\eeqa}{\end{eqnarray}}
\newcommand{\beqan}{\begin{eqnarray*}}
\newcommand{\eeqan}{\end{eqnarray*}}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}
\newcommand{\no}{\nonumber}
\newcommand{\bob}{\hspace{0.2em}\rule{0.5em}{0.06em}\rule{0.06em}{0.5em}\hspace{0.2em}}
\newcommand{\grts}{\greaterthansquiggle}
\newcommand{\lets}{\lessthansquiggle}
\def\dddot{\raisebox{1.2ex}{$\textstyle .\hspace{-.12ex}.\hspace{-.12ex}.$}\hspace{-1.5ex}}
\def\Dddot{\raisebox{1.8ex}{$\textstyle .\hspace{-.12ex}.\hspace{-.12ex}.$}\hspace{-1.8ex}}
\newcommand{\Un}{\underline}
\newcommand{\ol}{\overline}
\newcommand{\ra}{\rightarrow}
\newcommand{\Ra}{\Rightarrow}
\newcommand{\ve}{\varepsilon}
\newcommand{\vp}{\varphi}
\newcommand{\vt}{\vartheta}
\newcommand{\dg}{\dagger}
\newcommand{\wt}{\widetilde}
\newcommand{\wh}{\widehat}
\newcommand{\br}{\breve}
\newcommand{\A}{{\cal A}}
\newcommand{\B}{{\cal B}}
\newcommand{\C}{{\cal C}}
\newcommand{\D}{{\cal D}}
\newcommand{\E}{{\cal E}}
\newcommand{\F}{{\cal F}}
\newcommand{\G}{{\cal G}}
\newcommand{\Ha}{{\cal H}}
\newcommand{\K}{{\cal K}}
\newcommand{\cL}{{\cal L}}
\newcommand{\M}{{\cal M}}
\newcommand{\N}{{\cal N}}
\newcommand{\R}{{\cal R}}
\newcommand{\cS}{{\cal S}}
\newcommand{\W}{{\cal W}}
\newcommand{\dfrac}{\displaystyle \frac}
\newcommand{\hy}{${\cal H}\! \! \! \! \circ $}
\newcommand{\h}[2]{#1\dotfill\ #2\\}
\newcommand{\tab}[3]{\parbox{2cm}{#1} #2 \dotfill\ #3\\}
\def\lint{\int\limits}
\voffset=-24pt
\textheight=21cm
\textwidth=15.9cm
\oddsidemargin 0.0in
\evensidemargin 0.0in
\normalsize
\sloppy
\frenchspacing
\raggedbottom
\renewcommand{\baselinestretch}{1.5}
\begin{document}
\bibliographystyle{plain}
\begin{titlepage}
\begin{flushright}
UWThPh-1992-29\\
\end{flushright}
\vspace{4cm}
\begin{center}
{\Large \bf
Stability of Pure Thermodynamic Phases \\[5pt]
in Quantum Statistics}\\[50pt]
Heide Narnhofer \\
Institut f\"ur Theoretische Physik \\
Universit\"at Wien
\vfill
{\bf Abstract} \\
\end{center}
We discuss how measurements can distinguish between pure and mixed
thermodynamical phases and look for arguments in favour of pure
thermodynamical phases.
\vfill
\end{titlepage}
\section{Introduction}
It is a well accepted fact that in quantum mechanics the equilibrium states are
described by the KMS condition [1,2]. States satisfying the KMS condition form a
simplex so there exists a unique decomposition into extremal KMS states
that coincide with the factorial decomposition. Furthermore, extremal
KMS states cluster for every asymptotic abelian automorphism, so
especially for space translations and therefore in extremal KMS states
global densities carry no dispersion.
In classical theory we are accustomed with this fact, but already before we
have taken the thermodynamic limit. A superposition of pure thermodynamic
phases corresponds to our incomplete knowledge of the system. But also
the pure thermodynamic phases are defined by probability arguments. The
pure state corresponds to a point in configuration space, and refinement
of the measurements can improve our knowledge which point it is.
Quantum mechanics produces dispersions as a consequence of noncommutativity
and we have first to discuss how superpositions of states for large
systems influences the theory of measurement in quantum mechanics.
\section{Theory of Measurement for KMS States}
Let $A$ be an observable in quantum mechanics of a single particle with
nondegenerate pure point spectrum corresponding to eigenvalues $a_i$ and
eigenvectors $\vp_i$. If the system is in a state corresponding to the
vector $\psi$ then a measurement of $A$ has an outcome $a_i$ with
probability $|\langle \psi|\vp_i\rangle|^2$ so that in the average
$E(A) = \langle \psi |A|\psi\rangle$. If we repeat the measurement
immediately then with security we will measure $a_i$, which means
that by the measurement of $A$ $\psi$ has turned into the state $\vp_i$
[3,4].
Now we assume $\omega$ to be a superposition of states. For instance,
$A = \sigma_z$ and $\omega$ corresponding to the density matrix
$\left( \begin{array}{cc} 1/2 & \\ & 1/2 \end{array}\right)$.
Either we can consider the state as superposition of states corresponding
to $\left| \begin{array}{c} 1 \\ 0 \end{array} \right\rangle$ and
$\left| \begin{array}{c} 0 \\ 1 \end{array} \right\rangle$ with
probability 1/2, but also as superposition of
$\left| \begin{array}{c} 1 \\ 0 \end{array} \right\rangle$ and
$\left| \begin{array}{c} 1/2 \\ \sqrt{3}/2 \end{array}\right\rangle$
and
$\left| \begin{array}{c} 1/2 \\ - \sqrt{3}/2 \end{array}\right\rangle$
with probability 1/3. In all cases the outcome of the measurement of
$\sigma_z$ will be 1 with probability
$$
\sum \alpha_i \left|\langle \psi_i
\left| \begin{array}{c} 1 \\ 0 \end{array} \right\rangle \right|^2 =
\mbox{Tr }
\left| \begin{array}{c} 1 \\ 0 \end{array} \right\rangle
\left\langle \begin{array}{c} 1 \\ 0 \end{array} \right| \rho
\frac{1}{2}, \eqno(2.1)
$$
therefore independent of the chosen decomposition.
Now take $A$ to have degenerate pure point spectrum, e.g.,
$$
A = \frac{1}{2} |\vp_1 + \vp_2 \rangle \, \langle \vp_1 + \vp_2| +
\frac{1}{2} |\vp_1 - \vp_2\rangle \, \langle \vp_1 - \vp_2| =
|\vp_1\rangle \, \langle \vp_1| + |\vp_2\rangle \, \langle \vp_2|.
$$
If a measurement of $A$ gives the value {\bf 1} the corresponding projection
is $P = \sum_i |\vp_i\rangle \langle \vp_i|$ and the original state
corresponding to a vector $\psi$ changes to a state corresponding to the
vector $A\psi$, given by the density matrix
$$
\sum_{i,j} |\vp_i\rangle \; \langle \vp_j| \langle \psi|\vp_j\rangle \;
\langle \vp_i|\psi\rangle = |P\psi\rangle \langle P\psi| \eqno(2.2)
$$
so independent of the basis taken in the space of degeneracy.
In the thermodynamic limit observables are local and therefore have a highly
degenerate spectrum. On the other hand, the decomposition of a state
into pure ones is not only nonunique but also less meaningful, since in
contradistinction to the classical situation a pure state of the quasilocal
algebra
$$
\A = \ol{\bigcup_\Lambda \A_\Lambda}
$$
is in general not pure for a local subalgebra which contains the real
observables. Avoiding to refer to pure states the outcome of a
measurement can be described in the following way:
Let $\omega$ be the state of an algebra. Take
$A = \sum_{i=1}^n a_i P_i$ as observable, so with discrete finite
spectrum. Define
$$
\omega_i(B) = \frac{\omega(P_i B P_i)}{\omega(P_i)}. \eqno(2.3)
$$
After the measurement of $A$ the system will be in the state $\omega_i$ with
probability $\omega(P_i) = \lambda_i$ and will give the result
$\omega_i(A) = a_i$, such that the average expectation of
$A = \sum \lambda_i \omega_i(A) = \omega(A)$. This is in agreement with the
previous example.
Among the states that are linear superpositions of states with
$\xi_i(A^2) = \xi_i(A)^2$ the state $\sum \lambda_i \omega_i$ minimizes
the free energy $S(\sum \mu_i \xi_i|\omega)$ and is in this sense
optimal.
In fact, fix $\{\xi_i\}$ corresponding to density matrices $\sigma_i =
P_i \sigma_i P_i$ and optimize $\{\mu_i\}$. This gives
$$
\frac{\partial}{\partial \mu_j} (\mbox{Tr }(\rho \log \rho - \rho \log
\sum \mu_i \sigma_i) - \sum \alpha \mu_i) =
- \frac{\partial}{\partial \mu_j} \sum_i (\mbox{Tr } P_i \rho P_i
\log \mu_i \sigma_i + \alpha \mu_i) = 0,
$$
$$
\mbox{Tr } P_i \rho P_i = \alpha \mu_i, \qquad
\sum \alpha \mu_i = 1 \qquad \mbox{or} \qquad
\mu_i = \omega(P_i).
$$
Next we optimize $\sigma_i$ to
$$
\inf S(\omega|\sigma_i) = \inf (\mbox{Tr } \rho \log \rho -
\mbox{Tr } P_i \rho P_i \log P_i \sigma_i P_i) =
\mbox{Tr } \rho \log \rho - \mbox{Tr } P_i \rho P_i \log P_i \rho P_i.
$$
Repeating the measurement immediately gives the same result. If, however,
we let time pass and if we assume that in the thermodynamic limit the
time evolution is asymptotic abelian, if, in addition, we assume that
$\omega$ is extremal time invariant, so a pure thermodynamic phase,
then $\omega_i$, which belongs to the same folium as $\omega$, satisfies
$\lim_{t \ra \infty} \omega_i \circ \alpha_t = \omega$. Thus repeating the
same measurement in long enough intervals we can find by experiment the
probability distribution $\{ \lambda_i\}$ corresponding to $\omega$
(and therefore $\omega(A) = \sum_i \lambda_i a_i$). Here we do not need an ensemble to determine
$\omega$, where insecurity in preparing the probe can give a
probability character to $\omega$, but we keep observing the same system.
Now consider $\omega$ to be a superposition of, let us say, two
extremal invariant states, $\omega = \alpha_1 \omega_1 + \alpha_2
\omega_2$. If we now measure $A$, and find $a_i$, then according to our rule the state
$\omega$ is changed into the state
$$
\vp_i(B) = \frac{\alpha_1 \omega_1(P_i B P_i) + \alpha_2 \omega_2(P_i B P_i)}
{\alpha_1 \omega_1 (P_i) + \alpha_2 \omega_2 (P_i)} \eqno(2.4)
$$
with probability $\alpha_1\omega_1(P_i) + \alpha_2\omega_2(P_i)$.
In the course of time this state evolves to
$$
\omega_{i1} = \frac{\alpha_1 \omega_1(P_i)\omega_1 + \alpha_2 \omega_2
(P_i)\omega_2}{\alpha_1 \omega_1(P_i) + \alpha_2 \omega_2(P_i)}
$$
which by a new measurement of $A$ which gives, say $a_j$,
and after waiting long enough turns into
$$
\omega_{ij} = \N (\alpha_1 \omega_1(P_i) \omega_1(P_j)\omega_1 +
\alpha_2 \omega_2 (P_i) \omega_2 (P_j)\omega_2). \eqno(2.5)
$$
Therefore we can observe a different probability distribution for the
outcome of measurements.
Repeating the measurement for $n$ times,
$$
\prod \omega_1 (P_j)^{\ell_j} \qquad \mbox{or} \qquad
\prod \omega_2 (P_j)^{\ell_j}
$$
with $\sum \ell_j = n$ will for a given $\{\ell_j/n\}$ dominate so that
we press by infinitely many measurements the system into the pure
thermodynamic phase $\omega_k$ with probability $\alpha_k$. Therefore
apart from fluctuations at the beginning we cannot observe by
measurement of the same system whether we have started with a mixed
phase. To determine the probabilities $\alpha_k$ we have to make
copies of the sample and the mixed state can be interpreted as the
ensemble of pure phases with the probability distribution $\alpha_k$.
We can conclude that in quantum mechanics it seems meaningful to assign
to a pure phase more reality than a probabilistic one.
In the following we will concentrate
which arguments might support pure thermodynamic phases.
\section{Decomposition into Pure Thermo\-dynamic Phases and the
Algebra at Infinity}
We consider as observable algebra the quasilocal algebra
$$
\A = \ol{\bigcup \A_\Lambda}, \qquad \A_\Lambda \subset \A_{\bar \Lambda}
\mbox{ if } \Lambda \subset \bar \Lambda. \eqno(3.1)
$$
Every state $\omega(A)$ corresponds to a GNS representation
$$
\omega(A) = \langle \Omega |\pi (A) |\Omega\rangle. \eqno(3.2)
$$
In this representation we can define the algebra at infinity [5],
$$
\A_\infty = \bigcap_\Lambda \left( \bigcup_{\bar\Lambda \subset \Lambda^c}
\pi(\A_{\bar \Lambda})\right)''. \eqno(3.3)
$$
If $\A$ is simple -- what we can assume -- then $\A_\infty$ corresponds
to the center ${\cal Z} = \pi(\A)'' \cap \pi(\A)'$.
Now we assume that the time evolution is given by an automorphism group
$\alpha_\tau$. The equilibrium states are those that satisfy the KMS
condition
$$
\omega (AB) = \omega (B \tau_{i\beta} A). \eqno(3.4)
$$
If $\bar \omega < \omega$ also satisfies the KMS condition, then it can
be written as
$$
\bar \omega(A) = \langle \Omega|z \pi(A)|\Omega\rangle \quad
\mbox{ with } \quad z \in {\cal Z}
$$
and equivalently every such $z$ defines a KMS state $\omega_z$.
Therefore the decomposition of a KMS state into extremal KMS states
is unique, coincides with the central decomposition and physically
can be interpreted as the roughest decomposition into states with
trivial algebra at infinity.
\section{Pure KMS States are Stable under Perturbation}
By a local operator $P$ time evolution can be perturbed to
$$
\tau_t^P(A) = \tau_t(A) + \sum_{n \geq 1} i^n \int_{0 \leq s_1 \ldots
\leq s_n} ds_1 \ldots ds_n [\tau_{s_1}(P), \ldots, [\tau_{s_n}(P),
\tau_t A]] \eqno(4.1)
$$
such that in a time covariant representation with
$$
\pi(\tau_t A) = e^{iHt}\; \pi(A)\; e^{-iHt} \eqno(4.2)
$$
we have
$$
\pi(\tau_t^P A) = e^{i(H + \pi(P))t}\; A \;e^{-i(H + \pi(P))t}.
\eqno(4.3)
$$
Then analyticity in the KMS states allows to define a perturbed state
$\omega^P$ KMS with respect to $\tau_t^P$ by
$$
\omega^P(A) = \frac{\langle \Omega|\pi(A) \; e^{-(H+\pi(P))}|\Omega
\rangle}{\langle \Omega |e^{-H-\pi(P)}|\Omega\rangle}. \eqno(4.4)
$$
This idea was used in Ref. 6 to favour pure KMS states:
Consider a sequence $\{ P_\alpha\}$ such that
$$
\lim_{\alpha \ra \infty} \| \tau_t^{P_\alpha} A - \tau_t A\| = 0.
\eqno(4.5)
$$
Arrange $P_\alpha$ such that it converges to $z \in {\cal Z} = \A_\infty$.
Then
$$
\lim_{\alpha \ra \infty} \omega^{P_\alpha}(A) = \frac{\langle \Omega|\pi(A) \; e^{-z}|
\Omega\rangle}{\langle \Omega| e^{-z}|\Omega\rangle} \neq
\omega(A), \eqno(4.6)
$$
if $z \neq \lambda {\bf 1}$, which is possible in mixed KMS states, but not
in pure ones. Therefore pure KMS states seem to be more stable under
small changes of the dynamics. But we will argue in the next section that
the perturbation theory offered here is not the really relevant one.
\section{Dynamical Stability as Criterium for Equilibrium}
In Ref. 7 it is proposed to characterize equilibrium by stability under
perturbations of the dynamics and it is shown that this stability under
appropriate additional assumptions demands that the equilibrium state
satisfies the KMS condition.
Again we consider the perturbed dynamics $\tau_t^P$. But we do not start
with a KMS state but just assume that we have an invariant state
$\omega = \omega \circ \tau_t$ and can do some kind of perturbation
theory so that there exist states $\omega^P = \omega^P \circ \tau_t^P$
with
$$
\mbox{n-}\lim_{P \ra 0} \omega^P = \omega. \eqno(5.1)
$$
In addition we assume that $\tau_t$ is strongly asymptotically abelian
and that $\omega$ is a factor state. Then we expand
$\omega^{\lambda P} =\omega^{\lambda P} \circ \tau_t^{\lambda P}$ in $\lambda$.
Asymptotically abelianness together with norm convergence implies
$$
\lim_{t \ra \infty} \left.\frac{d \omega^\lambda(\tau_t A)}{d\lambda}
\right|_{\lambda = 0} = 0 . \eqno(5.2)
$$
The other term in the expansion to order $\lambda$ gives
$$
\lim_{T \ra \infty} \int_{-\infty}^{+\infty} \omega([P, \tau_t A])dt = 0.
\eqno(5.3)
$$
Using once more that $\omega$ is a factor state and therefore clustering
in $t$ it is shown in Ref. 7 that $\omega$ has to satisfy the KMS
condition for some temperature $\beta^{-1}$.
If we examine the assumptions we notice that $\omega$ to be a factor
is only of technical importance. Otherwise we could have a superposition
of two KMS states with two different temperatures without violating
dynamic stability. Conversely, we notice that following the
construction in Sect. 4 every pure KMS state is dynamically stable
and so is every superposition of KMS states.
It remains to check if an enlargement of the permitted perturbations to
nonlocal ones can favour pure thermodynamic phases, for sure such
perturbations will lead to states that are not normal with respect to
the unperturbed one, and we have to clarify whether norm convergence of
the states can be relaxed.
\section{Adiabatic Stability}
Dynamic stability can be interpreted in the following way: We note that we
do not know the time evolution to all accuracy, and we can only rely on those
observations that are only mildly affected by our ignorance.
Adiabatic stability now accepts that time evolution itself undergoes small
fluctuations due to the surroundings and equilibrium states are those
where these fluctuations do not add to a global effect. So especially
if we switch on and off a local perturbation slowly, the we should
return to the initial state. In this way the probabilistic argument
for dynamic stability can be turned to a dynamic one.
In Ref. 8 again it was assumed that the time evolution is asymptotic
abelian in the sense that a scattering transformation between locally
perturbed and unperturbed automorphisms exists, i. e.,
$$
\mbox{n-}\lim_{t \ra \pm \infty} \tau_t^P \tau_{-t} a =
\gamma^P_\pm a. \eqno(6.1)
$$
Now adiabatic switching on and off allows the system to react completely
on the perturbed dynamics so that the effect of switching and the
invariance of the state $\omega$ under this operation can be expressed as
$$
\omega \circ (\gamma_-^P)^\dg \gamma_+^P (a) = \omega(a) \eqno(6.2)
$$
or
$$
\omega \circ \gamma_+^P = \omega \circ \gamma_-^P \eqno(6.3)
$$
which again implies to first order in $P$
$$
\lim \int_{-T}^{+T} \omega([P,\tau_t a])dt = 0 \eqno(6.4)
$$
so that we are back to dynamic stability. The difference is that there we
assumed a perturbation of the state -- $\omega \circ \gamma^P$
demanding normality. Here we obtain the perturbed state as time limit and
require independence of the time direction.
In this context we also notice that (4.4) is not really the relevant
perturbation. Of course, it is mathematically perfectly well defined,
but the perturbation of a mixed KMS state that reflects the dynamic
reaction of the system on a perturbation is given by
$$
\omega(a) = \int d\mu(z) \omega_z(a) = \int d\mu(z)
\langle \Omega_z|\pi_z(a)|\Omega_z\rangle \longrightarrow
$$
$$
\longrightarrow \bar \omega^P(a) = \int d\mu(z)
\frac{\langle \Omega_z|\pi_z(a) e^{-H_z-\pi_z(P)}|\Omega_z\rangle}
{\langle \Omega_z|e^{-H_z - \pi_z(P)}|\Omega_z\rangle} \eqno(6.5)
$$
and here $\lim_\alpha \bar \omega^{P_\alpha} = \omega$, iff
$\lim \tau_t^{P_\alpha} = \tau_t$
and there is no condition on the purity of $\omega$.
\section{Global Perturbation}
Local perturbation of the dynamics can be expressed on the level of the
$C^*$ algebra, so linear superposition of states cannot find its
reflection in less stability. On the other hand, at least on the lattice
we know how to treat global perturbations and know their effect on the
KMS states. In Ref. 2 we find the following characterization of KMS
states for lattice systems:
We introduce the Banach space of translationally invariant interactions,
which has a norm dense intersection with the strictly local operators
of the quasilocal $C^*$ algebra. On this Banach space we define the free
energy density functional which is convex and continuous. Its tangent
functionals that at every point form a simplex define states on $\A$,
that satisfy the KMS condition, and even more, all translationally
invariant KMS states (which satisfy the Gibbs condition and maximize
the $S$-$H_\phi$ functional on $\omega$, whose maximum coincide with the
free energy density) can be obtained in this way. Furthermore, theory
of convex functionals tells us that every extremal tangent functional
can be obtained as limit of tangent functionals that are unique. For a
convex function $R \ra R$ also the converse is true: only the extremal tangent
functionals can be obtained in this way (right and left derivatives). In
higher dimensions mixed phases can be limits of pure phases only if
they correspond to phase transition points.
If we translate these observations into response to time dependent
perturbation, we obtain the following:
Let $P$ be a local operator, let $P_n = \sum_{x \in \Lambda_n} \sigma_x P$
and $\Lambda_n \nearrow Z^d$.
Let $\omega$ be a $\tau_t$-KMS state, $\tau_t$ asymptotic abelian. Define
$$
\omega^0(A) = \lim_{\lambda \ra 0} \lim_{n \ra \infty} \lim_{t \ra \infty}
\omega(\tau_t^{\lambda P_n}A). \eqno(7.1)
$$
Then $\omega^0$ is almost surely an extremal $\tau_t$-KMS state.
\paragraph{Proof:} $\lim_{t \ra \infty} \omega(\tau_t^{\lambda P_n}A)$ is a
$\tau_t^{\lambda P_n}$-KMS state in the same folium as discussed in
Sects. 5 and 6.
Further, $\lim_{n \ra \infty} \| \tau_t^{\lambda P_n} A - \tau_t^{\lambda
\bar P} A\| = 0$, where $\tau_t^{\lambda \bar P}$ is a globally
disturbed time evolution.
This guarantees that the weak limit points $\omega^{\lambda \bar P}$
of $\omega^{\lambda P_n}$ are
$\tau_t^{\lambda \bar P}$ KMS. Now almost surely
$\omega^{\lambda \bar P}$ is unique.
Again for local $A$
$$
\lim_{\lambda \ra 0} \| \tau_t^{\lambda \bar P} A - \tau_t A\| = 0
$$
so $\lim_{\lambda \ra 0} \omega^{\lambda \bar P}$ is an extremal KMS state
almost surely by the properties of a convex functional.
Of course, one would prefer to exchange $\lim_{n \ra \infty} \lim_{t \ra
\infty}$ or even more treat adiabatic switching of global perturbations.
To do this we would need additional assumption on space-time correlations.
The M\o ller automorphisms $\gamma_\pm$ do for sure not exist and would
only hinder to favour pure states.
We have concentrated here on quantum statistics. Since the algebra at
infinity has its counterpart also in classical statistical mechanics,
equivalent results can be obtained also in this context as is discussed in
Ref. 9 in chapter 7, Corollary 7.30, though not expressed in terms of
time evolution.
\section{Surface Perturbation}
The shortcoming in the above argument lies in the fact that our result
is restricted to translation invariant KMS states. Therefore we miss
crystals -- to restrict the previous arguments on a sublattice seems
artificial, because we would have to know the lattice size a priori --
and also local coexistence of e.g. water and vapor in separated
regions.
Global perturbation can be expressed on the local level in the following
way:
We start with $\omega^{\lambda \bar P}$ and restrict it to $\A_\Lambda$.
There it is a KMS state -- $\Omega$ is cyclic and separating -- with
respect to some Hamiltonian
$$
H_\Lambda + \lambda \sum_{x \in \Lambda} \sigma_x P + W_\Lambda.
\eqno(8.1)
$$
We know that
$$
\lim_{|\Lambda | \ra \infty}
\frac{1}{|\Lambda|} \omega^{\lambda P}(W_\Lambda(\lambda)) = 0 \eqno(8.2)
$$
and in this sense is a surface term. Whether it is really concentrated
on the surface is harder to estimate due to noncommutativity effects.
$$
H_\Lambda + \lim_{\lambda \ra 0} W_\Lambda(\lambda) = H_\Lambda +
W_\Lambda \eqno(8.3)
$$
determines $\omega^0$ restricted to $\A_\Lambda$. In this sense our
last result means that almost all $W_\Lambda$ which lead to
translationally invariant states lead to pure phases. This should be
generalized to arbitrary $W_\Lambda$ where of course one has to
specify what is meant with almost all. So far $W_\Lambda$ has to be
small with respect to the volume, but we would prefer finite with respect
to the surface. Also we should find some insight how weak decay of local
correlations -- as it happens, if a continuous symmetry is broken --
influences the stability of pure thermodynamic phases. So far results
mainly exist for the Ising model where we have enough mathematical
tools to ask more detailed questions. Compare e.g. R. Dobrushin [10] in
this volume where a global change in the boundary conditions of a
pure phase of length $n$ can only influence an area of depth
$\sqrt{n}$ and thus cannot produce a mixed phase.
\section*{Acknowledgements}
I want to thank the auditorium for the encouraging discussion,
especially Joel Lebowitz for insisting to clarify the theory of
measurement for thermal states. Also I thank Walter Thirring for
critical remarks on the manuscript.
\vfill
\begin{thebibliography}{99}
\item R. Haag, N.M. Hugenholtz, M. Winnink, {\em Commun. Math. Phys.}
{\bf 5} (1967) 215.
\item O. Bratteli and D.W. Robinson, {\em Operator Algebras and Quantum
Statistical Mechanics I and II} (Springer, New York, 1981).
\item J. von Neumann, {\em Mathematische Grundlagen der Quantenmechanik}
(Berlin, 1932).
\item J.S. Bell, {\em Speakable and Unspeakable in Quantum Mechanics}
(Cambridge University Press, 1987).
\item D. Ruelle, in {\em Carg\`ese Lectures in Physics} (Gordon and
Breach, New York, 1970).
\item H. Narnhofer and D.W. Robinson, {\em Commun. Math. Phys.} {\bf 41}
(1975) 89.
\item R. Haag, D. Kastler, E.B. Trych-Pohlmeyer, {\em Commun. Math. Phys.}
{\bf 38} (1974) 173.
\item H. Narnhofer and W. Thirring, {\em Phys. Rev.} {\bf A26} (1982) 3646.
\item H.O. Georgii, {\em Gibbs Measures and Phase Transitions} (Walter de
Gruyter, Berlin, 1988).
\item R. Dobrushin, this volume.
\end{thebibliography}
\end{document}