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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Now read in reference file LMDR.TEX
% so that all citations are defined. This file would be identical
% with the reference list following at the end of the paper.
% In order not to demand special action from the users of MPARC,
% we include a shortened version here.
\let\REF\labref
\REF AMR \AbMa \par
\REF AM \AM \par
\REF Ara \Ara \par
\REF Bo1 \Bona \par
\REF Bo2 \Bonb \par
\REF BR \Rob \par
\REF Dav \Dav \par
\REF Du1 \Dua \par
\REF Du2 \Dub \par
\REF DRW \IMD \par
\REF DW1 \DWa \par
\REF DW2 \MFH \par
\REF HL \HL \par
\REF HS \HS \par
\REF Kos \Kos \par
\REF Lin \Lind \par
\REF Lla \Lla \par
\REF MS \MorStro \par
\REF Pet \Pet \par
\REF PW \PW \par
\REF RW1 \RWa \par
\REF RW2 \RWb \par
\REF Rue \Ruelle \par
\REF Tak \Tak \par
\REF Ume \Umega \par
\REF Un1 \Unner \par
\REF Un2 \Unnb \par
\REF Un3 \Unna \par
\REF Wei \Weinberg \par
\REF Wer \DELHI \par
\let\MFQ\Dua
\let\MFD\DWa
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Beginning of paper proper. No macro definitions from here on
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$\eqno{\sl DIAS-STP-91-35}$$
\vskip 2.0cm
%
\font\BF=cmbx10 scaled \magstep 3
{\BF \baselineskip= 25pt
\centerline{\BF Local dynamics of mean-field}
\centerline{\BF Quantum Systems} }
\vskip 1.0cm
\centerline{{\bf N.G. Duffield }
\footnote {$\11$}
{{\sl School of Mathematical Sciences,
Dublin City University, Dublin 9,
Ireland.
}}
{\bf and R.F. Werner.}
\footnote{$\12$}
{{\sl FB Physik, Universit\"at Osnabr\"uck,
Postfach 4469, D-4500 Osnabr\"uck, Germany.
}}}
\vskip 1.0cm
{\baselineskip=12pt
\midinsert\narrower\narrower\noindent
{\bf Abstract.}
In this paper we extend the theory of \mf-dynamical semigroups given
in \tref{\MFD,\Dua} to treat the irreversible \mf\ dynamics of
quasi-local \mf\ observables. These are observables which are site
averaged except within a region of tagged sites. In the thermodynamic
limit the tagged sites absorb the whole lattice, but also become
negligible in proportion to the bulk. We develop the theory in detail
for a class of interactions which contains the \mf\ versions of
quantum lattice interactions with infinite range. For this class we
obtain an explicit form of the dynamics in the thermodynamic limit. We
show that the evolution of the bulk is governed by a flow on the
one-particle state space, whereas the evolution of local perturbations
in the tagged region factorizes over sites, and is governed by a
cocycle of completely positive maps. We obtain an $H$-theorem which
suggests that local disturbances typically become completely
delocalized for large times, and we show this to be true for a dense
set of interactions. We characterize all limiting evolutions for
certain subclasses of interactions, and also exhibit some
possibilities beyond the class we study in detail: for example, the
limiting evolution of the bulk may exist, while the local cocycle
does not. In another case the bulk evolution is given by a diffusion
rather than a flow, and the local evolution no longer factorizes
over sites.
\endinsert
}
\vfill\break
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\beginsection 1. Introduction.
The characteristic feature of \mf\ systems can be expressed by
saying that each particle or elementary subsystem interacts in an
equal way with every other such subsystem, and responds to the
average of these interactions. In this paper we will be concerned
with the limiting dynamics of such systems as their size becomes
infinite. Therefore we will consider a sequence of models comprising
an increasing collection of copies of the basic subsystem. When we
speak of an interaction between the subsystems, we mean that for
each model in the sequence a (generally) irreversible dynamics is
specified. The \mf\ nature of the models entails first of all that
the interaction is invariant with respect to permutation of the
subsystems; the idea that each subsystem responds to an average is
made precise by the property that the generator of the dynamics of a
large system can be approximated by taking a generator involving
only a few (often just two) subsystems, averaging it over all
permutations of the subsystems, and multiplying it by the number of
subsystems.
This is in close analogy to lattice systems with translation
invariant interaction: there one obtains the Hamiltonian for a
finite region approximately by averaging terms involving only a few
sites over all translations which map these sites into the given
region, and by multiplying with the volume of the region. In this
analogy \mf\ systems are just lattice systems, whose underlying
lattice has permutation symmetry rather than translation symmetry.
This analogy suggests a canonical way of obtaining a ``\mf\
approximation'' of an arbitrary lattice model with translation
invariance: one merely has to take the Hamiltonians of the lattice
model for some sequence of regions going to infinity in the sense of
van Hove \tref{\Ruelle}, and symmetrize each with respect to all
permutations of the lattice sites.
{\chg}We do not attempt to justify this procedure as an approximation
to the original lattice system. Our aim is rather to obtain as
complete an analysis of the \mf\ theory as possible.
The description of \mf\ systems in terms of their permutation symmetry
becomes more transparent if one looks at the intensive rather than the
extensive observables. As described above the Hamiltonian of a \mf\
system divided by the number of subsystems, \ie the intensive
variable ``Hamiltonian density'', has the property that for a large
system it is approximately equal to the Hamiltonian density of a
smaller version of the system, symmetrized over all permutations.
Sequences of observables (indexed by the system size) with this
property were called ``approximately symmetric'' in \tref\RWa, and
have become the central notion of a research programme on \mf\
systems. The basic result in \tref\RWa\ concerns the thermodynamics of
Hamiltonian \mf\ systems, and is a formula for the free energy density
in the thermodynamic limit in terms of a Gibbs variational principle
in one-particle quantities. This result was later extended to
``inhomogeneous \mf\ systems'' in which the permutation symmetry is
restricted to sites with approximately equal external or random
parameters \tref\RWb. If one starts from a lattice model with
translation invariant interaction, the thermodynamics of its \mf\
version can be written down directly by evaluating the mean energy and
the mean entropy for homogeneous product states.
This prescription is often taken as the definition of the \mf\
approximation. However, it would be impossible to define the dynamics
``in the \mf\ approximation'' if this is only understood as a class of
variational states. In contrast, in our programme \mf\ models are
treated as quantum systems in their own right. The dynamics of \mf\
models was treated in \tref\MFD\ from the point of view that the
dynamics should map the set of \mf\ intensive variables, \ie it
should map the approximately symmetric sequences into itself. A
corresponding study of the inhomogeneous case was undertaken in
\tref\IMD, and the special properties of Hamiltonian dynamics, as
opposed to general irreversible dynamics, were described in \tref\MFH:
in this case one obtains in the limit a flow on the state space of the
one-particle algebra, which is Hamiltonian in the full sense of
classical mechanics with respect to a canonical Poisson bracket
structure on the state space. In earlier approaches \tref\Bona\
beginning with \tref\HL\ this had been noted only in the case when the
Hamiltonian is written in terms of the generators of a Lie group
representation so that a symplectic structure can be imported from the
coadjoint orbits.
The works described so far focussed entirely on the properties of
the intensive observables, which in the \mf\ limit become
completely delocalized. This leaves open the question how the
evolutes of a localized observable behave under a \mf\ dynamics.
Intuitively, the picture is that under a completely
delocalized evolution such as a \mf\ dynamics the observable would
instantaneously develop a completely delocalized tail, while
initially still exhibiting a strong dependence on the original
localization region. For very large times one might expect that this
dependence on the original localization becomes weaker, especially
when the dynamics is dissipative. It is therefore natural to use a
concept analogous to the approximately sequences in which the
symmetrization operations leave out all the sites of the original
localization region. Put differently these sites are given a ``tag''
and one aims to study the motion of the tagged subsystems under the
averaged influence of the remaining ones. This programme has been
carried out in \tref\MFQ\ for any fixed set $I$ of tagged sites. In
this paper we further extend this approach allowing more and more
tagged sites in thermodynamic limit, as long as the proportion of
tagged sites goes to zero. The above intuitive picture is confirmed
by our analysis.
A closely related programme for the study of \mf\ systems has been
based on the work of Morchio and Strocchi \tref\MorStro. Their aim
was to show how the dynamics of a system with long range
interactions can be defined in the thermodynamic limit even though
the \ql\ local algebra in the usual sense cannot be invariant under
such an evolution due to appearance of delocalized tails. Their
proposal is to enlarge the \ql\ algebra by suitable weak limits of
observables capable of describing delocalized intensive quantities.
It is clear that these limits exist only with respect to a suitably
chosen set of states, and consequently much of the theory centers on
this choice. For the case of \mf\ theories their programme was
carried out by B\'ona \tref\Bona\ and Unnerstall \tref{\Unner,\Unnb}. In a
sense their approach is dual to ours, in focussing on the states
rather than on the observables. In particular, the permutation
symmetry, which is as central to their approach as to ours, is built
in by choosing the folium of permutation symmetric states on the
\ql\ algebra, whereas in our approach it determines the connection
between observables of systems of different sizes. The thermodynamic
limit of observables in our approach is always taken in norm, whereas
in the picture of Morchio and Strocchi it is typically taken in the
$s$-topology associated with the chosen folium of states.
Consequently, our limiting object is a C*-algebra, whereas they
arrive more naturally at a W*-algebra or a von Neumann algebra.
The paper is organized as follows. In section 2 we define \ql\ \mf\
observables. These are what we call the \qs\ sequences of
observables: those which are delocalized (\ie site-averaged) except
over local regions of tagged sites which become proportionately
negligible in the thermodynamic limit. Such sequences of observables
have well defined ``thermodynamic limits'' in a space which we
construct explicitly.
In section 3 we formulate the notion of a \mf\ dynamical semigroup
as a sequence of dynamical semigroups which preserves the set of
\qs\ observables, and which furthermore gives rise to contraction
semigroup on the inductive limit space. We demonstrate that a wide
class of evolutions has this property, this class being considerably
wider than in \tref{\HL,\Bona,\Unna}. In particular, we include the
\mf\ versions of arbitrary translation invariant, possibly
dissipative lattice interactions. The existence of the limiting
dynamics is subject to a growth condition which is far weaker than
that required for the original translation invariant interactions
\tref\Rob\ . For this class of models the limiting dynamics is shown
to have the following special form: on initially localized
observables it factorizes over the individual sites of the region of
localization, while the global evolution of the delocalized tail is
implemented by a flow on the one-site state space of the system. The
non-linear differential equation for this flow is just the Hartree
equation. Such a form was obtained in \tref\Bona, but only for
Hamiltonian interactions between finite numbers of sites.
{\chg}More recently this type of dynamical evolution has been
considered by B\'ona \tref\Bonb\ as a generalization of quantum
mechanics itself, and was linked to a modification of quantum
mechanics recently proposed by Weinberg \tref\Weinberg.
As a special case, our theory can be applied to classical Markov
processes: the factorization of the local evolutions has been
used to investigate the Poissonian approximation in queueing
networks \tref\Dub.
{\chg}In section 4 we consider some properties of the limiting
evolution in some general cases. Firstly, we show that if the finite
volume dynamics is Hamiltonian, then the limiting dynamics is
completely determined by the energy density function appearing in the
Gibbs variational principle for the equilibrium states: as a
Hamiltonian function in the sense of classical mechanics it generates
the flow which describes the global evolution via a Poisson structure
on the one-particle state space. Its gradient is the Hamiltonian
operator (depending on the global state) generating the local unitary
cocycle. This description is complete in the sense that any
Hamiltonian function can be approximated by one arising from our class
of models.
The next level of complexity is given by the sequences of generators
which can be written in Lindblad form in terms of approximately
symmetric observables. Here the local dynamics is still given by a
state dependent Hamiltonian. However, it can no longer be expressed
as the gradient of single function. We show that up to approximation
any state dependent Hamiltonian arises from a model of this type.
The global flow is no longer Hamiltonian, and is essentially arbitrary
in the class considered. The flow, and indeed the whole limiting
evolution in this subclass is reversible (exists for negative times),
while all evolutions for finite size systems are strictly
dissipative.
Finally, in the full class studied in section 3 we obtain an (up to
approximations arbitrary) state-dependent Lindblad generator.
However, we observe that such evolutions
do not exhaust the set of \mf\ dynamical semigroups. This is
illustrated by describing a sequence of dynamical semigroups whose
\mf\ limiting dynamics exists in our sense, but lacks some of the
fundamental features established for the lattice class: the global
limiting dynamics is given by a diffusion on the one-particle state
space rather than a flow, and the evolution of local observables
does not reduce to a product of one-site evolutions.
In one of the classes mentioned above the local dynamics is still
Hamiltonian, while the global evolution is not. The converse can also
happen in the sense that any generator (e.g.\ a Hamiltonian one) may
be perturbed in such a way that the global evolution is unchanged, but
the local evolution becomes dissipative. We construct such
perturbations explicitly in terms of permutation operators.
In section 5 we study the relation between the local and the global
dynamics. In fact we are able to construct an example of a sequence
of semigroups which is a \mf\ dynamical semigroup in the global, but
not local, sense. A limiting dynamics exists for the fully site
averaged observables {\it only}. Finally, we investigate the
delocalization of initially localized observables for lattice class
evolutions. We prove an $H$-Theorem which suggests that in the
dissipative case all local information should be lost as the local
states are drawn towards the flow of the global state. We show that
under the addition of an arbitrarily small perturbation any lattice
class generator has such an evolution.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\beginsection 2. \Capital\qs\ Observables
In this section we describe the notion of \qs\
observables, which generalizes on the one hand the usual quasi-local
observables known from lattice models, and on the other hand the
\mf\ intensive variables introduced in \tref\RWa.
In order to define the thermodynamic limit of a physical quantity it
is always necessary to define the observable in question for all
system sizes occurring on the way to the thermodynamic limit. For
example, for the usual interactions of lattice systems it is the
translation invariance of the potential which determines the
connection between the energy observables at different system sizes.
\Capital\qsy\ as defined here is a property not of an observable of
a single system of finite size but of a \net\ of observables
indexed by the size. Associated with this notion is a definition of
the thermodynamic limit of a \qs\ observable, and much of the work
in this section will go into the identification of the space in
which these limits lie.
Before taking up the formal development let us clarify the aim of
this section by relating it to a standard construction in functional
analysis, the inductive limit of Banach spaces. There one has a
sequence $(\A_N)$ of spaces with a system of isometric
``inclusion maps'' $\j NM:\A_M\to\A_N$ (defined for $N\geq M$)
satisfying the chain relation $\j NR=\jj NMR$. The term ``inclusion
map'' indicates that the elements $X_R\in\A_R$ and $\j NRX_R\in\A_N$
will eventually be identified. In other words, we are interested
only in the sequence $N\mapsto X_N$, which is defined for
sufficiently large $N$ (e.g. $N\geq R$) and satisfies
$X_N=\j NMX_M$ for all $N,M$ for which $\j NM$ and $X_M$ are
defined. The space of such sequences is then called the ``union'' of
the $\A_N$ with respect to the inclusions $\j NM$. It is clear that
this set of sequences forms a vector space under $N$-wise
operations. If we work in the category of Banach spaces the limit
space $\Ay$ of the system $(\A_N,\j NM)$ is taken as the completion
of this union. The elements of the completion can also be
represented by sequences, namely by those for which
$\norm{X_N-\j NMX_M}$ becomes arbitrarily small as both $N$ and $M$
become sufficiently large. Note that in the trivial case where all
$\A_N$ are equal and $\j NM$ is always the identity these sequences
are precisely the Cauchy sequences. So we might call sequences with
this property ``\jC''. Sequences $X,X'$ for which
$\norm{X_N-X'_N}\to0$ represent the same element of the completion.
Thus $\Ay$ is equal to the quotient of the space of \jC\ sequences
up to equality under the seminorm $\norm{X}=\lim_N\norm{X_N}$.
The \ql\ algebra of a lattice system is an example of this
construction. Here the $\A_N$ are the observable algebras of an
increasing family of regions, and the embedding $\j NM$ is by
tensoring with the identity element on all sites of $\sdiff NM$.
Since the $\j NM$ in this case are homomorphisms of C*-algebras,
the union becomes a *-algebra, and the limit space $\Ay$ is also a
C*-algebra, called the C*-inductive limit of the $\A_N$.
$\Ay$ is usually called the \ql\ algebra of the lattice system, and
we will denote it by $\Aloc$, reserving the symbol ``$\Ay$'' for
other limit spaces to be discussed below.
A very similar construction was used in \tref{\RWa} to define
the algebra of intensive observables of \mf\ systems. Here one uses
the same spaces $\A_N$, but the inclusions $\j NM$ are modified by
averaging over all permutation automorphisms of the larger region.
It is easy to check that the resulting maps $\j NM$ again satisfy
the chain relation, but they are no longer isometric, nor even
injective. Nevertheless, the notions of \jC\ sequences (called
``approximately symmetric'' in \tref\RWa) and the limit space $\Ay$
make sense even in this case. It turns out that the $N$-wise
product of \jC\ sequences is again \jC\ so that the limit space
becomes an (abelian) C*-algebra even though the $\j NM$ are no
longer homomorphisms. In this paper we generalize the construction
still further: we will allow the chain relation to be not strictly
satisfied but only asymptotically for large indices. In fact it
suffices for a sensible definition of \jC\ sequences and the limit
space to have that $\limls MN\norm{(\j NR-\jj NMR)X_R}=0$ for every
fixed $R$ and $X_R\in\A_R$. We will not, however, develop an
abstract theory of ``fuzzy inductive limits'' along these lines, but
instead will focus on the case at hand, the physical motivation for
the choice of the $\j NM$, and the concrete representation of the
limit space $\Ay$.
We will consider systems composed of many ``particles'', each of which
has observables described by the same C*-algebra with unit $\A$. For
most of the general theory we do not need any further assumptions on
this algebra but in many models of interest $\A$ is just a finite
dimensional matrix algebra describing a ``spin''. {\chg}In section 3,
in the discussion of \mf\ dynamics in the full \lc\ of generators we
will make this assumption for simplicity. By $K(\A)$ or simply by
$\KA$ we denote the state space of this algebra.
We equip $\KA$ with the weak* topology. The evaluation of a
continuous linear functional $\sigma$ on any C*-algebra $\B$ on
$X\in\B$ will be written as $\bra\sigma,X>$. To each particle we
associate a ``site'' of a lattice $\Lat$, e.g. $\Lat=\Ir\1d$ for
systems on a $d$-dimensional cubic lattice. Denoting by $\A_{\set
x}$ the isomorphic copy of $\A$ ``at site $x$'', we write
$\A_I=\bigotimes_{x\in I}\A_{\set x}$ for the observable algebra of
the subsystem localized in the finite subset $I\subset\Lat$. Here and
below we always use the the minimal C*-tensor product, although in
applications the algebras concerned are usually finite dimensional
matrix algebras, for which all C*-tensor products coincide. Mappings
between finite regions induce homomorphisms between the associated
obervable algebras. Explicitly, if $\eta:I\to J$ is an injective map
we define $\heta:\A_I\to\A_J$ by
$$ \heta\bigl(A_1\otimes A_2\cdots\otimes A_{\card{I}}\bigr)
= A_{\eta\1{-1}(1)}\otimes A_{\eta\1{-1}(2)}
\cdots A_{\eta\1{-1}(\card J)}
\eqno(2.1)$$
with the understanding that on the right hand side
$A_{\eta\1{-1}(x)}=\idty$, whenever $x$ is not in the range of $\eta$.
Note that if $\eta$ is the inclusion map of $I$ into $J\supset I$,
$\heta$ is just the usual embedding between the subalgebras $\A_I$ and
$\A_J$ used in the construction of the \ql\ algebra of the lattice
system as a C*-inductive limit. Since we will be interested in yet
another kind of inductive limit it will be convenient to suppress the
inclusion maps $\heta:\A_I\to\A_J$, and similarly the inclusion of
each $\A_I$ into the \ql\ algebra $\Aloc$. Thus for $I\subset J$ we
shall simply write $\A_I\subset\A_J\subset\Aloc$.
There are $\card N!/(\card N-\card M)!$ injective maps from a set
of $\card M$ elements into a set of $\card N\geq\card M$ elements. In
\tref{\RWa,\DWa} the identification between the intensive \mf\
observables at different system sizes was made by the average of all
$\heta$, where $\eta$ runs over all injective maps. In contrast,
only a single map (namely the natural injection
$\eta:M\hookrightarrow N$) is used in the construction of the \ql\
algebra. Here we will use an average over a subset of injective
maps, which generalizes both of these possibilities: for $I\subset
M\subset N$ we define $\Ima NM\1I$ as the set of all injective maps
$\eta:M\to N$ such that $\eta(i)=i$ for all $i\in I$, which is a set
of ${\card{\sdiff NM}!/\card{\sdiff MI}!}$ elements.
The corresponding average is
$$ \j NM\1I={\card{\sdiff MI}!\over\card{\sdiff NM}!}
\sum_{\eta\in\Ima NM\1I}\heta
\quad:\A_M\to\A_N
\quad.\eqno(2.2)$$
Thus for $I=\emp$ we recover the map used in the ``global'' theory of
\mf\ systems \tref{\RWa,\DWa}, and for $I=M$ we get the injection
used for the \ql\ algebra. The family $\j NM\1I$ for fixed $I$ was used
in \tref\Dua\ to set up a theory of \mf\ systems with a fixed set $I$
of ``tagged particles''. In this paper we go one step further, by
allowing the set of tagged particles to become infinite in the
thermodynamic limit.
Thus we will take the limit not only over an increasing family of
regions, we will also consider in each region a subset of tagged
sites, such that in the limit every site of the lattice eventually
becomes tagged. We formalize this by using the notion of \ts s: a
{\bf\ts\/} is a finite subset $N\subset\Lat$ of the lattice under
consideration, together with a subset $\tag N\subset N$ of ``tagged
sites''. Rather than denoting a \ts\ by the pair $(N,\tag N)$ we
will just use the symbol $N$, in much the same way as a vector space
is usually denoted by the same letter as its underlying set, without
explicit reference to the operations defined on it. For \ts s we
define an inclusion relation $M\tsubs N$ as
``$M\subset N$ and $\tag M\subset \tag N$''. For \ts s $M\tsubs N$
we now define
$$ \j NM=\j NM\1{\tag M}:\A_M\to\A_N
\quad.\eqno(2.3)$$
This is the basic family of inclusions on which our inductive limit
construction is built. In applications one usually does not take the
observables to be defined for all regions $N$, but only along some
subsequence of regions (e.g. cubes). Therefore we will assume some
\net\ $(N_\alpha)_{\alpha\in\aleph}$ of \ts s to be
given, and we will only consider limits along this \net.
Allowing only sequences at this point would not introduce a
simplification in anything we do in this paper. On the other hand it
is convenient to be able to state the theory for a general net of
regions in $\Lat$ going to the lattice in the sense of van Hove,
without being forced to specify a particular enumeration. Therefore we
allow the index set $\aleph$ to be an arbitrary directed set. Readers
who feel more at home with sequences are invited to take $\aleph=\Nl$,
and to substitute ``sequence'' for ``net'' throughout. This will be
sufficient (though perhaps not convenient) for all applications. Our
only assumptions on the \net\ $(N_\alpha)_{\alpha\in\aleph}$ are that
it is increasing with respect to the relation $\tsubs$, that the
tagged subsets absorb the lattice, \ie $\bigcup_\alpha\tag
N_\alpha=\Lat$, and that in the limit the tagged sites are relatively
few, i.e.
$$ \lim_\alpha {\card{\tag N_\alpha}\over\card{N_\alpha}}=0
\quad.\eqno(2.4)$$
Since the \net\ of regions will be fixed once and for all there is no
ambiguity in writing $N\to\infty$ for $\alpha\to\infty$, and
$\lim_Nf(N)$ for $\lim_\alpha f(N_\alpha)$ for the limit of any
$N$-dependent quantity. We will adopt this convention from now on, so
in the sequel we will never refer to the labels $\alpha$ or the set
$\aleph$.
We now single out the \jC\ \net s in the sense mentioned in the
introduction to this section. These \net s $N\mapsto X_N$ with
$X_N\in A_N$ are the basic observables we consider. $X_N$ will be
symmetrized over most sites in $\notag N$, \ie over all sites with
the exception of the relatively small subset $\tag N$.
Intuitively, $X_N$ is a local observable with a
symmetrized (or completely delocalized) tail. One should think of
$X_N$ as a \net\ of observables ``converging to a \ql\ \mf\
limit''. Our formal definition is given below, together with the
corresponding notion \tref\Dua\ for a fixed set of tagged sites.
\iproclaim 2.1 Definition.
Let $X_N\in\A_N$ for every $N$ in the given fixed \net\ of tagged
sets. Then
\item{(1)}
the \net\ $N\mapsto X_N$ is called a {\bf \qs}, or a \qs\
observable, if
$$ \limls MN \norm{X_N-\j NM X_M}=0 \quad.$$
The set of such \net s will be denoted by $\Yl$.
\item{(2)}
the \net\ $N\mapsto X_N$ is called {\bf \Isymm}, if
$$ \limls MN \norm{X_N-\j NM\1I X_M}=0 \quad.$$
The set of such \net s will be denoted by $\Yl\1I$.
\eproclaim
As noted before the crucial property of the maps $\j{}{}$
for making \qsy\ a notion of ``convergent \net'' is the
approximate chain relation $\j NR\approx \jj NMR$. This
relation will now be proven together with some other basic
combinatorial facts.
\iproclaim 2.2 Proposition.
Let $I\subset J\subset R\subset M\subset N\subset\Lat$.
Then
\item{(1)} $\j NR\1I=\j NM\1I\circ\j MR\1J$.
\item{(2)} $\displaystyle
\norm{\j NR\1I-\j NM\1J\circ\j MR\1I}
\leq 2\card R\, \card J
{\card N+\card M\over\card N\,\card M}
\leq 4\card R {\card J\over\card M}
\quad.$
\item{(3)} $\limls MN\norm{\j NR-\jj NMR}=0$\quad.
\eproclaim
\proof:
\def\IM{\Ima{}{}\,} \def\IMt{\tilde\IM}
\def\card#1{\vert#1\vert}
All maps appearing in (1) and (2) act like the identity on $\A_I$, and
like their counterparts with $I=\emp$ on the remining sites. Therefore
it suffices to show (1) for $I=\emp$. Suppose (2) had been proven for
this special case. Then we would obtain for the general case a bound
of the same form, but with $\card I$ subtracted from the numbers
appearing in it. The bound as stated then follows from the
monotonicity of the function $x\mapsto(a+x)(b+x)(c+x)\1{-1}$ when
$(a+x)$ and $(b+x)$ are positive, and $c\geq\max\set{a,b}$. It
therefore suffices to show both (1) and (2) only in the case
$I=\emp$.
(1) $\j NM\1\emp(A)$ can be computed by taking $\heta(A)$ for any
injective map $\eta:M\to N$ and then symmetrizing over all
permutations of $N$. It follows that $\j NM\1\emp\circ\heta=\j NR\1\emp$
for {\it any} injective $\eta:R\to M$. Equation (1) thus follows by
taking the appropriate average over $\eta$.
(2) Consider the map $\jt NR$ (resp. $\jt MR$) defined as the
equal-weight averages over all $\heta$ with $\eta:R\to N$ (resp.
$\eta:R\to M$) such that in addition $\eta(R)\cap J=\emp$. Let
$\pr N\1J=\j NN\1J$ denote the average over all permutation
automorphisms of $\A_N$ of permutations leaving $J$ pointwise fixed.
Then $\jt NR=\pr N\1J\circ\heta$ and $\j NM\1J=\pr N\1J\circ\heta_1$,
where $\eta$ and $\eta_1$ are any of the maps over which $\jt NR$ and
$\j NM\1J$ are averages. Hence
$\jt NR=\pr N\1J\circ\heta_1\circ\heta_2=\j NM\1J\circ\heta_2$, where
$\eta_2:R\to M$ is injective with $\eta_2(R)\cap J=\emp$. By
averaging over all $\eta_2$ we find
$$ \jt NR=\j NM\1J\circ\jt MR\quad.$$
The rest of the proof consists in establishing the estimate
$$ \norm{\jt NR-\j NR\1\emp}
\leq 2{\card R\, \card J \over\card N}
\quad.$$
Applying the same estimate to $\jt MR$, and inserting into the above
equation then yields the result. The second form of the estimate
follows because $\card M\leq\card N$.
Let $\IM\equiv\Ima NR\1\emp$ denote the set of all injective
$\eta:R\to M$, and $\IMt$ the subset with $\eta(R)\cap J=\emp$. Note
that for large $N$ the ``probability'' $\eta(R)$ meeting $J$ goes to
zero. More precisely, by Lemma IV.1 of \tref\RWa we have that
$$ \epsilon\equiv{\card{\sdiff\IM\IMt}\over\card\IM}
\leq{\card R\,\card J\over\card N}
\quad.$$
Now both $\j NR\1\emp$ and $\jt NR$ are averages of $\heta$ with
different weights. Since $\norm{\heta}=1$ for all $\eta$ we can
estimate their norm difference by the sum of the absolute differences
of these weights. For $\eta\in\IMt$ the weight in $\j NR\1\emp$ is
$\card\IM\1{-1}$, and in $\jt NR$ it is $\card\IMt\1{-1}$. The
difference is $\epsilon\card\IMt\1{-1}$. Thus multiplied with the
number $\card\IMt$ of terms we get the contribution $\epsilon$ to the
error. For the remaining $\card{\sdiff\IM\IMt}=\epsilon\card\IM$
terms the weight in $\j NR\1\emp$ is still $\card\IM\1{-1}$, but is zero
in $\jt NR$. Hence these terms also contribute $\epsilon$ to the error
estimate, and putting the contributions of these two types of terms
together, we obtain the required estimate for
$\norm{\jt NR-\j NR\1\emp}$.
(3) Taking $J=\tag M$ and $I=\tag N$ in (2) we get
$\limsup_N\norm{\j NR-\jj NMR}
\leq 2\card R{\card{\tag M}\over\card M}$,
which goes to zero as $M\to\infty$ by our standing assumption (2.2) on
the net of \ts s.
\QED
In the following Lemma we establish a standard way of showing
that a given \net\ $X_\dt$ is \qs, namely by showing that $X_N$ can
be uniformly approximated for large $N$ by a \net\ of the special
form $N\mapsto\j NRY$ for $Y\in\A_R$. We will call such \net s
{\bf basic \net s}, and denote the set of such \net s by
$\Yb$. In an ordinary inductive limit $\Yb$ corresponds to the
union $\bigcup_N\A_N$, which is dense in the limit Banach space $\Ay$
by definition. This density statement carries over to general ``fuzzy
inductive limits'', that is, whenever the chain relation holds
approximately. Here we establish it first on the level of \net s.
Since by Proposition 2.2(1) the chain relation holds for $\j NM\1I$
with fixed $I$ we can hence apply the same reasoning to the inductive
system $(\A_N,\j NM\1I)$.
\iproclaim 2.3 Lemma.
Let $X_N\in\A_N$ for all $N$ in the given \net\ of \ts s. Then
$X_\dt$ is \qs\ iff for all $\epsilon>0$ there are a \ts\ $R$ and
$Y\in\A_R$ such that
$$ \limsup_N\norm{X_N-\j NRY}\leq\epsilon
\quad.$$
$X_\dt$ is \Isymm\ iff in addition one can choose
$\tag R=I$.
\eproclaim
\proof:
(1) Let $X_\dt$ be \qs. Then by definition there is for any
$\epsilon>0$ some \ts\ $M$ such that
$\limsup_N\norm{X_N-\j NMX_M}\leq\epsilon$. Hence we can set $R=M$
and $\hat Y=X_M$.
Conversely, suppose that $\norm{X_N-\j NRY}\leq\epsilon$ for
$N\tsups N_\epsilon$. Then
$\norm{X_N-\j NMX_M}\leq 2\epsilon
+\norm{\j NRY-\jj NMRY}$
for $N\tsups M\tsups N_\epsilon$.
Taking in this estimate the limit $\limsup_M\limsup_N$ and using the
approximate chain relation Lemma 2.4(2) we find that this limit is
less than $2\epsilon$ for any $\epsilon$.
Exactly the same arguments work for \Isymm\ \net s, with
all $\j NM$ replaced by $\j NM\1I$.
\QED
With the help of this Lemma we can clarify the relations between
\qsy\ and $I$-symmetry for different values of $I$. Intuitively,
$\Yl$ is the limit of $\Yl\1I$ of $I\nearrow\Lat$, \ie the limit of
allowing more and more tags. It will be useful also to have a
systematic way of removing tags, \ie to include sites previously
exempted from all symmetrizations back into the bulk. The operator
of ``removing all tags except those in $I$'' is given by
$$ \pr N\1I:=\j NN\1I:\A_N\to\A_N
\quad.\eqno(2.5)$$
By Proposition 2.2(1) $\pr N\1I$ clearly is a projection.
$\pr N\1\emp$ is the operation of removing all tags.
\iproclaim 2.4 Proposition.
\item{(1)} For $I\subset J$, $\Yl\1I\subset\Yl\1J\subset\Yl$.
\item{(2)}
The map $\pr{}\1I:X_\dt\mapsto(\pr\dt\1IX_\dt)$ projects $\Yl$ onto
$\Yl\1I$.
\eproclaim
\proof:
(1) The inclusion $\Yl\1I\subset\Yl$ for any $I$ is obvious from Lemma
2.3. What remains to be shown is that any basic \net\ of the form
$N\mapsto \j NR\1IY$ can be approximated by one of the form
$\j NM\1J\tilde Y$. By Proposition 2.2(2) we can set
$\tilde Y=\j NM\1IY$ for some $M\subset\Lat$, and get
$\sup_N\norm{\j NR\1IY-\j NM\1J\tilde Y}
\leq 2(\card R\,\card J)/\card M$,
which can be made arbitrarily small by taking $M$ large enough.
(2) It is evident that the operation $\pr\dt\1I$ on \net s is a
projection. By Proposition 2.2(1) with $N=M$ we have
$\pr N\1I\circ\j NM\1J=\j NM\1I$ for $I\subset J$. Hence on basic \net s
$\j NR\1JY$ with $J\supset I$ the projection operation produces again
basic \net s. Since we can approximate any \qs\ \net\ by basic
\net s $\j NR$ with $\tag R=J$ sufficiently large, Lemma 2.3 says
that $\pr\dt\1I$ maps $\Yl$ into $\Yl\1I$. Taking $I=J$ it is clear that
basic \Isymm\ \net s are invariant under the projection,
hence $\pr\dt\1I(\Yl)=\Yl\1I$.
\QED
We can now proceed to identify the inductive limit space of the system
$(\A_N,\j NM)$. We will use the following notation: for any \ts\ $N$,
and any $\rho\in\KA$ we introduce the conditional expectation $\Ern
N:\A_{\notag N}\to\A_{\tag N}$ with respect to the product state
$\rho\1{\untag N}$ on the untagged sites. Thus
$$ \bra\sigma,\Ern N(A)>
=\bra\sigma\otimes\rho\1{\untag N},A>
\quad,\eqno(2.5)$$
where $\sigma$ is an arbitrary state of $\A_{\tag N}$, and
$A\in\A_N$. Since we identify $\A_{\tag N}$ with a subalgebra of
$\A_{\notag N}$ we can consider $\Ern N$ as a projection of norm one
on $\A_N$, \ie a conditional expectation in the sense of Umegaki
\tref\Umega.
If we identify $\A_N$ in turn with a subalgebra of $\Aloc$ we can
also consider $\Ern N$ as a map $\Ern N:\A_N\to\Aloc$. This is the
point of view taken in the following Theorem. We recall at this point
that $\KA$, being the state space of a unital C*-algebra, is
weak*-compact. For any C*-algebra $\B$, $\C(\KA,\B)$ will denote the
space of weak*-continuous functions on $\KA$, taking values in $\B$,
and topologized with the supremum norm
$\norm{f}=\sup_{\rho\in\KA}\norm{f(\rho)}_{\B}$.
\iproclaim 2.5 Theorem.
\item{(1)}
Let $X$ be a \qs\ \net. Then for all $\rho\in\KA$ the norm limit
$$ X\y(\rho)\equiv \lim_N\Ern N(X_N) \quad\in\Aloc $$
exists uniformly for $\rho\in\KA$ and
$\rho\mapsto X\y(\rho)$ is weak*-to norm continuous.
\item{(2)}
The map $X\in\Yl\mapsto X\y\in\CKAy$ is onto, and isometric in the
sense that
$$ \norm{X\y}=\lim_N\norm{X_N}\quad.$$
It is also a homomorphism taking the $N$-wise product of \net s
into the product of $\CKAy$.
\item{(3)} A \qs\ \net\ $X$ is \Isymm\ if and only if
$X\y(\rho)\in\A_I\subset\Aloc$ for all $\rho$.
\eproclaim
\proof:
\def\CKAI{\C(\KA,\A_I)}
\def\SUM#1{\sum_{{\scriptstyle\nu \atop\scriptstyle#1}}}
The core of this result has been proven in section IV of \tref\RWa.
There ``approximately symmetric \net s'' ( in our terminology
``$\emp$-symmetric'' \net s) were allowed to take values in a
\net\ of algebras of the form $\B\otimes\A_N$ for a fixed
``initial algebra'' $\B$, and $\A_N$ as above. Symmetrizations were
only to be applied to the tensor factors of $\A_N$, and not to $\B$.
But taking $\B=\A_I$, this is precisely a description of
\Isymm\ \net s. Therefore we can immediately apply the
results of \tref\RWa (Compare also Theorem 2.1 in \tref\Dua).
Thus for \Isymm\ \net s the limit in (1) exists, and is a
weak*-continuous
function $X\y:\KA\to\B\equiv\A_I\hookrightarrow\Aloc$.
Moreover, every $f\in\CKAI$ is of the form $f=X\y$ for some
\Isymm\ $X$. The isometry and homomorphism properties are also
shown in \tref\RWa.
Since every \qs\ \net\ is uniformly approximated by \Isymm\
ones with finite $I$, existence and continuity of the limit,
isometry property and homomorphism property immediately carry over
from the \Isymm\ case. It remains to prove (3) and that
$X\mapsto X\y$ is onto. We have already seen that on \Isymm\
\net s this map is onto $\CKAI$. Hence suppose that $X$ is
\qs\ and $X\y\in\CKAI$. Hence there is an \Isymm\
\net\ $Y$ such that $X\y=Y\y$. By (2) this means that
$\norm{X\y-Y\y}=\lim_N\norm{X_N-Y_N}=0$. Hence $X$ is approximated
uniformly for large $N$ by an \Isymm\ \net, and must be
\Isymm\ by Lemma 2.3.
To see that $X\mapsto X\y$ is onto, let $f\in\CKAy$.
Since $\bigcup_I\CKAI$ is dense in $\CKAy$ we can find for any
summable sequence $\epsilon_\nu$ a sequence of \ts s $R_\nu$ and
$X\1\nu\in\A_{R_\nu}$ such that
$$ f=\sum_\nu \jy{R_\nu}X\1\nu
\qquad\hbox{with }
\quad\norm{\jy{R_\nu}X\1\nu}\leq\epsilon_\nu
\quad,$$
where $\jy RX_R$ denotes the limit $Y\y$ for the basic \net\
$Y_\dt=\j \dt RX_R$.
The idea of the proof is to pick a sequence $S_\nu$ of \ts s which
increases sufficiently fast, and to set
$$ X_N=\sum_{{\nu\atop S_\nu\tsubs N}}\j N{R_\nu}X\1\nu
\quad.$$
Note that every $\nu$ is eventually included in this sum because the
tagged subsets $\tag N$ absorb $\Lat$ as $N\to\infty$.
Since $\norm{\jy{R_\nu}X\1\nu}=\lim_N\norm{\j N{R_\nu}X\1\nu}$ we can
pick $S_\nu$ such that for $N\tsups S_\nu$ we have
$\norm{\j N{R_\nu}X\1\nu}\leq2\epsilon_\nu$. The sum defining $X_N$
is then convergent for every $N$. For later use we note that
the numbers
$$ \delta_N=\SUM{S_\nu\tsubs N}\norm{\j N{R_\nu}X\1\nu}
$$
converge to a finite limit.
We now have to show that for sufficiently rapidly growing $S_\nu$
the \net\ $X$ becomes \qs. With the estimate Proposition 2.2(2)
we get
$$\eqalign{ \norm{X_N-\j NMX_M}
&\leq \SUM{S_\nu\tsubs M}
\norm{\bigl(\j N{R_\nu}-\jj NM{R_\nu}\bigr)X\1\nu}
+ \SUM{ S_\nu\tsubs N ;\, S_\nu\notsubs M}
\norm{\j N{R_\nu}X\1\nu} \cr
&\leq \SUM{S_\nu\tsubs M}\norm{X\1\nu}\cdot
4\card{R_\nu}{\card{\tag M}\over\card M}
+ (\delta_N-\delta_M)
\quad.}$$
If $S_\nu$ is chosen large enough the $\nu\th$ term in the sum is
only present if $M$ is large in the sense of the basic \net\
along which we take all limits. Since $\card{\tag M}/\card M\to0$ as
$M\to\infty$ in that \net, we can pick $S_\nu$ such that the
$\nu\th$ term is bounded by $\epsilon_\nu$ for all $N,M$. Hence the
sum converges absolutely, and vanishes in the limit $\limls MN$. The
second term vanishes because the $\delta_N$ converge. It is evident
from the construction that
$X\y=\lim_N\jy NX_N=\sum_\nu \jy{R_\nu}X\1\nu=f$.
Hence $X\mapsto X\y$ is surjective.
\QED
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\beginsection 3. The dynamics of \qs\ observables.
In the previous section we have identified the \qs\ \net s as the
appropriate \mf\ \net s of observables. Suppose a dynamics for the
\mf\ system is given. By this we mean that for each $N$ in our fixed
\net\ of subregions of $\Lat$ there is specified a semigroup
$\Ttn:t\ge0$ of completely positive unit preserving linear maps on
$\A_N$. We can say that the dynamics has good mean-field properties if
at least it maps the set of \qs\ \net s into itself. In the first
part of this section we shall formalize the notion of a mean-field
dynamical semigroup as a dynamics which in addition gives rise to a
well-defined limiting semigroup in the inductive limit space $\A\y$.
The dynamical semigroups considered in \tref\Dua\ had the prima
facie weaker property that they preserved only $I$-symmetry for each
finite $I\subset\Lat$. We will show that this is in fact an
equivalent property to the preservation of \qsy\ under the
additional hypothesis that each $\Ttn$ is permutation symmetric.
In physical models it is a set of generators $G_N$ of the dynamics
$\Ttn=e\1{tG_N}$ which will usually be provided; this by way of a
\net\ of Hamiltonians or a \net\ of dissipative maps. Thus one will
want to determine whether a given \net\ of generators exponentiates
to form a \mf\ dynamical semigroup, and in that case to compute the
limiting semigroup on $\Ay$.
Our aim in this section is to demonstrate that a wide class of
dissipative interactions in quantum lattice systems do indeed
generate \mf\ dynamical semigroups. These can be thought of as the
\mf\ version of interactions with infinite range, but subject to a
relatively weak decay condition. Indeed, we are able to show that
the decay conditions required for the existence of a limiting
dynamics are strictly weaker those required for the corresponding
translation invariant interaction. Of course, this class includes
interactions involving no more than a fixed finite number of sites
as a special case. Apart from proving the existence of the limiting
dynamics for the class of lattice models, we obtain a form for the
limiting dynamics which shows that observables living on different
tagged sites evolve independently according to the (time-dependent)
average state of the system. This conforms with the intuitive
physical picture of \mf\ dynamics. We stress, however, that \mf\
dynamical limits need not in general have this property. Indeed, in
section 4.5 of this paper we construct examples of \mf\ dynamical
limits which do not.
We will start the section by generalizing the \mf\ dynamics of
\Isymm\ sequences as described in \tref\Dua\ to that of \qs\ \net s.
We then describe the dynamics of \qs\ \net s under the influence of
generators of a fixed polynomial degree, and demonstrate the
factorization property of the dynamics in the thermodynamic limit.
Finally, we show that the dynamics of the lattice class of models
can be approximated by those with polynomial generators (\ie those
in which only a finite number of sites interact) and show that the
factorization of the dynamics is preserved by this approximation.
We will call a \net\ of operators $T_\dt$ {\bf \qsy\ preserving}
if it maps the set of \qs\ \net s onto itself, that is if
$X_\dt\in\Yl\ \Rightarrow\ T_\dt X_\dt\in\Yl$.
The proof of the following Lemma is a straightforward modification
of Lemma 2.2 of \tref\DWa.
\iproclaim Lemma 3.1.
Let $T_\dt$ be a uniformly bounded \net\ operators which is
\qsy\ preserving.
Then there exists a unique operator $T\y$ on $\A\y$
such that for all \qs\ \net s $X_\dt$
$(T_\dt X_\dt)\y=T\y X\y\ $.
\eproclaim
\iproclaim Definition 3.2.
A \net\ $\Tt\dt:\ t\ge0$ of completely positive unital
(\ie identity preserving) contractions is called a {\bf \mf\
dynamical semigroup} if
\item{(1)} for each $t\ge 0$, $\Tt\dt$ is \qsy\ preserving,
\item{(2)} $\nullinf\ni t\mapsto \Tt\infty$ is a
stongly continuous contraction semigroup on $\A\y$.
\eproclaim
The requirement of strong continuity for the limit semigroup
$\Tt\infty$ can be seen as a statement about uniformity of the
continuity of the $\Ttn$ with $N$. Indeed, it can be shown (c.f.\
Theorem 2.3 of \tref\DWa\ ) that 3.2(2) is implied by 3.2(1) under
the additional requirement that
$$\lim_{{t\to0\atop N\to\infty}}\norm{\Ttn X_N-X_N}=0$$ for all
$X_\dt\in\Yl$.
For any \Isymm\ \net\ $X_\dt$ (for example, a \net\ which is
$J$-symmetric for some $J\subset I$), we will find it useful to
refer explicitly to its \mf\ limit as an element of $\C(\KA,\A_I)$,
rather than the injection into $\C(\KA,\Aloc)$. We will use the
symbol $X\y\1I$ for this purpose.
Corresponding to Lemma 3.1 we have for each finite $I\subset\Lat$ a
notion of $I$-symmetry preservation for \net s of maps. Moreover, as
is detailed in \tref\Dua, a uniformly bounded $I$-symmetry
preserving \net\ of maps $T_\dt$ has a unique limit $T\y\1I$ on
$\C(\KA,\A_I)$ such that for all \Isymm\ \net s
$X_\dt$, $(T_\dt X_\dt)\y\1I=T\y\1I X\y\1I$. For $I\subset R$,
$\jy R\1I X_R$ will
denote the limit function $X\y\1I$ corresponding to the basic
$I$-symmetric \net\ $\j\dt R\1I X_R$.
Suppose that a \net\ of maps $T_\dt$ is $I$-symmetry preserving for
all finite $I\subset\Lat$. Since we view $\A_I$ as a subalgebra of
$\Aloc$, we canonically regard $T\y\1I$ as a map on the subalgebra
$\C(\KA,\A_I)\subset\C(\KA,\Aloc)\equiv\Ay$. Now the union over $I$ of
the subalgebras $\C(\KA,\A_I)$ is dense in $\A\y$. Thus we might
expect to construct from the maps $T\1I\y$ a map $T\y$ as a limit of
\qsy\ preserving maps on $\Yl$.
It will be the case in all examples which we treat that $T_\dt$ is
permutation symmetric in the sense that for all \ts s $N$, $T_N$
commutes with any automorphism $\hat\pi$ of $\A_N$ induced by a
permutation $\pi$ of $\notag N$. Note that this means that $\Ttn$ is
independent of the tagging $\tag N$. With permutation invariance
the notions of ``\qsy\ preservation'' and ``$I$-symmetry preservation
for all finite $I\subset\Lat$'' become equivalent.
\iproclaim Theorem 3.3.
Let $T_\dt$ be a \net\ of unital permutation-symmetric
contractions. Then the following are equivalent:
\item{(1)} $T_\dt$ is $I$-symmetry preserving for each finite
$I\subset\Lat$
\item{(2)} $T_\dt$ is \qsy\ preserving.
\eproclaim
\proof:
(1)$\Longrightarrow$(2)
Since $T_\dt$ is $I$-symmetry preserving for all finite
$I\subset\Lat$, it is \qsy\ preserving on the dense subset of
\basic\ \net s in $\Yl$. Approximating any \qs\ \net\ as closely as
desired by a \basic\ \net\ we see that $T_\dt$ is \qsy\ preserving
on the whole of $\Yl$.
(2)$\Longrightarrow$(1)
Let $X_\dt$ be \Isymm. Then $X_\dt$ and hence $T_\dt X_\dt$
are \qs. But by permutation symmetry of $T_\dt$ we have
$T_\dt X_\dt=T_\dt\pr\dt\1I X_\dt=\pr\dt\1I T_\dt X_\dt$, which by
Proposition 2.4(2) is \Isymm.
\QED
It is worth remarking at this point by analogous reasoning to that
used in the proof of the above Theorem, one can compare the \Isymm\
limits and $J$-symmetric limits of $T_\dt X_\dt$ for an \Isymm\
\net\ when $I\subset J$. Since $T_\dt X_\dt$ is \Isymm, it is also
$J$-symmetric with limit $(T\y\1I X\y\1I)\otimes \idty_{J\setminus
I}$. But from Proposition 2.4(1) $X$ is $J$-symmetric and
$X\y\1J=X\y\1I\otimes\idty_{J\setminus I}$. Thus the family of
operators $T\y\1I$ obeys the {\bf consistency relation}
$$ T\y\1J(X\y\1I\otimes\idty_{J\setminus I})
=T\y\1I X\y\1I\otimes\idty_{J\setminus I}
\quad.$$
\iproclaim Corollary 3.4.
Replace definition 3.2 by the
weaker statement that for all finite $I\subset\Lat$, $\Tt\dt$ is
$I$-symmetry preserving and has a strongly continuous limit
$\Tt\infty\1I$ on $\C(\KA,\A_I)$. If each $\Ttn$ is permutation
symmetric, then $\Tt\dt$ is a \mf\ dynamical semigroup.
\eproclaim
\proof: By Theorem 3.3, for each $t\ge0$, $\Tt\dt$ is \qs\
preserving. Since for each finite $I$,\ $t\mapsto\Tt\infty\1I$ is
strongly continuous,
$\Tt\infty$ is strongly continuous on the dense
set $\cup_I \C(\KA,\A_I)$; and
since $\norm{\Tt\infty\1I}\le 1$, $\Tt\infty$ extends to a strongly
continuous contraction semigroup on the whole of $\A\y$.
\QED
We now turn to the question of finding \net s of operators which
generate \mf\ dynamical semigroups. We deal first with perhaps the
simplest class of generators: those which are constructed for each $N$
by resymmetrization of an interaction of a fixed finite number of
sites, and rescaled by the system size $\absn{N}$. For any
C*-algebra $\V$ let $\B(\V)$
denote the set of bounded linear operators on $\V$.
Define the symmetrization operator
$\Sym_N:\bigcup_{M\subset N}\B(\A_M)\to\B(\A_N)$ by setting
setting $\Sym_N G_M$ to be the average over all bijective maps
$\eta:N\to N$ of
$\heta\1{-1} \bigl(G_M\otimes\id_{\notag N\setminus\notag M}\bigr)\heta$.
Thus $\Sym_N G_M$ is the average over the copies $G_{\eta(M)}$ of $G_M$
acting on all possible subsets $\A_{\eta(M)}$ of $\A_N$.
\iproclaim Definiton 3.5.
A \net\ of operators $G_\dt$
will be called a {\bf bounded polynomial generator}
of {\bf degree} $R$ if for some $R\subset\Lat$ and
all $N\supset R$,
$$\eqalignno{
G_N&={\absn{N}\over\absn{R}}\Sym_N G_R
\quad,&\cr}$$
where $G_R$ is the generator of a semigroup of completely positive
unital maps on $\A_R$, and $\norm{G_R}\equiv\gamma<\infty$.
\eproclaim
One sees by use of the Trotter product formula that each
$\Ttn=e\1{tG_N}$ is completely positive.
The scaling $(\absn{N}/\absn{R})$ in Definition 3.5
means that for each $N$, each site responds to a mean of its
interaction with all other sites. For example if $\nabsn{R}=2$ then
for all $A\in\A$,
$$G_N(A\otimes\idty_{\eleminus{N}{1}})
={1\over 2(\nabsn N-1)}\sum_{x\in N}
\idty_{\eleminus{N}{{1,x}}}\otimes
(G_{\set{1,x}}+G_{\set{x,1}})(A\otimes\idty)
\quad.$$
The \Isymm\ properties of semigroups with bounded polynomial
generators have been investigated in \tref\Dua. We can extend these
as follows.
\iproclaim Theorem 3.6.
Let $G_\dt$ be a bounded polynomial generator of degree $R$, and set
$\Tt\dt=e\1{tG_\dt}:\ t\ge 0$. Then
\item{(1)} $T$ is a \mf\ dynamical semigroup.
\item{(2)} $T$ has the {\bf disjoint homomorphism property}, namely,
for all finite $I\subset\Lat$
$$\Tt\infty\1{I}=\bigotimes_{i\in I}\Tt\infty\1{\set i}\quad,$$
where the tensor product is to be understoood in the range space
$\A_I$ of $\C(\KA,\A_I)$, and each $\Tt\infty\1{\set i}$ is an
isomorphic copy of
the same map.
\item{(3)} The restriction of $\Tt\infty$ to the intensive (i.e.
$\emptyset$-symmetric) observables
is implemented by a weak*-continuous flow $\Ft:\ t\ge0$ on $\KA$,
\ie for $X\y$ intensive and $t\ge0$,
$$\Tt\infty X\y=X\y\circ \Ft\quad.$$
where
$\KA\times\nullinf\ni(\rho,t)\mapsto \Ft\rho\in\KA$
is jointly continuous and $\Ft\flow_s=\flow_{t+s}$.
\eproclaim
\proof: (1) By section 5 of \tref\Dua, for each finite
$I\subset\Lat$, $t\mapsto\Tt\dt$ is $I$-symmetry preserving with a
strongly continuous limit $t\mapsto\Tt\infty\1I$on $\C(\KA,\A_I)$.
Thus by Corollary 3.4 $\Tt\dt$ is a \mf\ dynamical semigroup.
(2) is proved in section 5 of \tref\Dua\ and (3) in Proposition
3.4(4) of \tref\DWa.
\QED
We now come on to discuss the exact form of $\Tt\infty$ when
$\Tt\dt$ has a bounded polynomial generator $G_\dt$ of degree $R$.
For each $\rho\in\KA$ and $R\ni y$ define
the bounded linear operator $\Lrb {y}$ on $\A_{\set y}$ by
$$\Lrb y A
=\Er{\eleminus Ry}
G_R(A\otimes\idty_{\eleminus Ry})
\qquad\hbox{and set}\qquad\Lr=\Lrb 1\quad.\eqno(3.1)$$
Thus for a fixed $\rho$ the $\Lrb y$ are isomorphic
copies of the operator $\Lr$ on $\A$.
In Proposition 3.4 of \tref\DWa\ it was seen that
$\Lr$ is the generator of the
implementing flow $\flow$, i.e.
$$\ddt\Ft\rho=\Ft\rho\circ \locgen\1{\Ft\rho}
\quad.$$
This is the sense in which it is said in \tref\AM\ that $\Lr$ is the
generator of a non-linear dynamical semigroup for \mf\ models. But
we now observe that $\Lr$ plays a more general role: it generates
{\it local} dynamics in \mf\ models. For let $X_N=\j NR X_R$, making
$X_\dt$ \Isymm\ for any $I\subset\tag R$. Then according to
Proposition 5.2 of \tref\Dua,
$$(G\y\1{I}X\y\1{I})(\rho)=(\jy R\1{I}\sum_{x\in\notag R}
\Lrb x X_R)(\rho)=\Er{\notag R\setminus I}\sum_{x\in\notag R}\Lrb
x X_R\quad.$$
We shall prove below that $t\mapsto L\1{\rho_t}$
is the generator of what we term the {\bf local cocycle}
$t\mapsto\lrt$ in $\B(\A)$ which (i) implements the flow
$\Ft\rho=\rho\circ\lrt$; and (ii) has products which implement
the local evolutions:
$(\Tt\infty\1I X\y\1I)(\rho)=(\lrt)\1I(X\y\1I)(\Ft\rho)$.
We start be considering the cocycle. In Lemma 3.7 we establish the
existence of solutions to the differential equation
$\dot\lrt=\lrt\circ\locgen\1{\Ft\rho}$. The topological Lemma 3.8
is required to determine continuity of the solution in Proposition
3.9.
\iproclaim Lemma 3.7.
\item{(1)}
The equation
$$\ddt \lrt=\lrt\circ \locgen\1{\Ft\rho}\quad,$$
with initial condition $\Lambda\1\rho_0=\id$
has a unique solution
$\nullinf\times\KA\ni(t,\rho)\mapsto\lrt\in\B(\A)$.
\item{(2)}
The local cocycle $\Lambda$ of (1) has the composition law
$$\Lambda\1\rho_s\circ\Lambda\1{\flow_s\rho}_t=\Lambda\1\rho_{s+t}
\quad.$$
\eproclaim
\proof: (1) $\norm{ \Lr}\le\gamma$.
Thus, existence and uniqueness of a norm-continuous
solution of the integral equation
$$\lrt=\id+\int_0\1t ds\Lambda\1\rho_s L\1{\flow_s\rho}\eqno(3.2)$$
follows by standard methods (see e.g. \tref\HS\ ).
We clearly have the norm estimates
$$\norm{\lrt}\le e\1{\gamma t}\qquad\hbox{and}\qquad
\lim_{t\to0}\sup_{\rho\in\KA}\norm{\lrt-\id}=0\quad.\eqno(3.3)$$
(2) For all $\rho\in\KA$ and $t\ge s\ge 0$ define
$\Gamma\1\rho(s,t)=\Lambda\1\rho_s\Lambda\1{\flow_s\rho}_{t-s}$
Then
$${d\over dt}\Gamma\1\rho(s,t)
=\Gamma\1\rho(s,t)L\1{\Ft\rho}
\qquad\hbox{and}\quad
\Gamma\1\rho(s,s)
=\Lambda\1\rho_s
\quad.$$
So for fixed $s$ and $\rho$ we have that for $t\ge s$ the map
$t\mapsto\Gamma\1\rho(s,t)$ obeys the same differential equation as
$\lrt$, and has the same boundary value at the point $t=s$. Thus by
uniqueness in part (1) above, $\Gamma\1\rho(s,t)=\Lambda\1\rho_t$ for
all $t\ge s$.
\QED
\iproclaim Lemma 3.8.
Let $\Omega_0$ be a compact set in $\A$. Then there exists a compact
set $\Omega\supset\Omega_0$ such that for any $\gamma'>\gamma$,
$$\rho\in\KA,\ A\in\Omega\Longrightarrow \Lr\in\gamma'\Omega\quad.$$
\eproclaim
\proof: Since for any $X\in\A_R$, $\rho\mapsto\Er{\eleminus R1}X$
is weak*-to-norm continuous and bounded,
$\KA\times\A\ni(\rho,A)\mapsto \Lr A$ is jointly continuous.
Thus the set
$\Omega_1=\set{\Lr A\stt\rho\in\KA,\ A\in\Omega_0}$,
being the continuous image of the compact
set $\KA\times\Omega_0$, is compact.
Furthermore, $\sup_{A\in\Omega_1}\norm{ A}\le
\gamma\sup_{A\in\Omega_0}\norm{ A}$.
Proceed by iteration and construct in a like manner the sequence of
compact sets $\Omega_2,\Omega_3$ and so on.
For any $\gamma'>\gamma$, construct the set
$$\tilde\Omega
=\set{A\in\A\stt A=\sum_{1=i}\1\infty(\gamma')\1{-i}t_iA_i:\
A_i\in\Omega_i,\ t_i\in\bracks{0,1} }
\quad.$$
Then $\tilde\Omega$ is bounded and
$\set{\Lr\tilde\Omega\mid \rho\in\KA}\subset\tilde\Omega$.
Furthermore, by construction, $\tilde\Omega$ can be approximated to
within $\eps$ by finite sums from the compact sets
$(\Omega_n)_{n\in\Nl}$ and is hence pre-compact. Taking the closure
$\Omega$ of $\tilde\Omega$ we obtain the required set.
\QED
\iproclaim Proposition 3.9.
For each $A\in\A$ the map
$(\rho,t)\mapsto \lrt A$ is jointly continuous.
\eproclaim
\proof: Since by eq (3.3) $t\mapsto\lrt$ is norm-continuous,
uniformly in $\rho$, it is enough to prove that for each $t$,
$\rho\to\lrt A$ is weak*-to-norm continuous. Now
$(t,\rho)\mapsto \Ft\rho$ and $(\rho,A)\mapsto \Lr A$ are both
jointly continuous.
Thus by composition $(t,\rho,A)\mapsto L\1{\Ft\rho}A$ is jointly continuous.
For a fixed $A\in\A$, let $\Omega$ be the compact set $\Omega$
corresponding to $\Omega_0=\set A$ in
Lemma 3.8. Then since
$$\eqalignno{
(\lrt-\Lambda\1\sigma_t)A
&=\int_0\1t ds (\Lambda\1\rho_s-\Lambda\1\sigma_s)L\1{\flow_s\rho}A
+ \Lambda\1\sigma_s(L\1{\flow_s\rho}-L\1{\flow_s\sigma})A&\cr
\noalign{\hbox{we have that}}
\sup_{B\in\Omega}\norm{(\lrt-\Lambda\1\sigma_t)B}
&\le \gamma\int_0\1t ds \sup_{B\in\Omega}
\norm{(\Lambda\1\rho_s-\Lambda\1\sigma_s)B}
+\gamma\1{-1}(e\1{\gamma t}-1)\eps_t(\rho,\sigma)
&\cr}$$
where
$\eps_t(\rho,\sigma)=\sup_{0\le s\le t}\sup_{B\in\Omega}
\norm{ (L\1{\flow_s\rho}-L\1{\flow_s\sigma})B}$.
Thus, by Gronwall's Lemma (see e.g. \tref\HS)
$$\sup_{B\in\Omega}\norm{(\lrt-\Lambda\1\sigma_t)B}
\le \gamma\1{-1}(e\1{\gamma t}-1)\eps_t(\rho,\sigma)
\quad.$$
Since $\Omega,\KA$ and
$\lbrack 0,t\rbrack$ are compact, then by the joint continuity of
$(\rho,t,A)\mapsto L\1{\Ft\rho}A$,\
$\eps_t(\rho,\sigma)\to0$ as $\sigma\to\rho$ weak*.
Thus $(\lrt-\Lambda\1\sigma_t)A\to 0$ as $\sigma\to\rho$ weak*.
\QED
Now according to Theorem 3.6(2) $\Tt\infty\1I$ is constructed as
a tensor product (in $\Aloc$) of $\Tt\infty\1{\set i}:\ i\in I$. Thus
to know $\Tt\infty$ it suffices to calculate one of the
$\Tt\infty\1{\set i}$. The purpose of Proposition 3.9 is that it
enables us to verify that a possible candidate for $\Tt\infty\1{\set i}$
is indeed a strongly continuous contraction semigroup on
$\C(\KA,\A)$. With no loss of generality we take $i=1$. We define
for each finite $I\subset\Lat$ the algebra
$$\PP\1I=\bigcup_{R\subset\Lat}\set{\jy R\1I X\stt X\in \A_R}\qquad.$$
Thus $\PP\1I$ can be thought of as a dense polynomial subalgebra of
$\C(\KA,\A_I)$ comprising the \mf\ limits of \basic\ \Isymm\
\net s.
\iproclaim Theorem 3.10.
Let $X_\dt$ be $\set1$-symmetric. Then
$$(\Tt\infty\upone X\y\upone)(\rho)=
\lrt X\y\upone(\Ft\rho)$$
\eproclaim
\proof: Define
$$(\TTt\infty\upone X\y\upone)(\rho)=\lrt
X\y\upone(\Ft\rho)$$
We show that $\TTt\infty\upone$ is a strongly continuous
contraction semigroup on $\C(\KA,\A)$.
By the joint continuity of
$(\rho,t)\mapsto\lrt$ into the strong-operator topology on $\B(\A)$,
and the joint continuity of $(\rho,t)\mapsto \Ft\rho$,
then for each $X\y\upone\in\C(\KA,\A)$ we
have that $(\rho,t)\mapsto\lrt X\y\upone(\Ft\rho)$ is jointly
continuous, uniformly for $\rho\in\KA$ compact and $t$ in compacta.
Hence we conclude that $\thone t X\y\upone\in\C(\KA,\A)$ and
that $t\mapsto\thone t$ is strongly continuous.
Furthermore we have the composition law
$$(\thone t \thone s X\upone_\infty)(\rho)
=\Lambda_t\1\rho\Lambda_s\1{\Ft\rho}X\upone_\infty(\Ft\rho)
=\Lambda_{t+s}\1\rho X\upone_\infty(\flow_{t+s}\rho)
=(\thone {t+s}X\upone_\infty)(\rho)
\quad,$$
where the second equality uses the composition law of
$\Lambda$. Since
$\norm{\thone t}\le\norm{\lrt}\le e\1{\gamma t}$ we conclude from
Proposition 1.17 of \tref\Dav\ that $\thone t:\ t\ge0$ is a strongly
continuous semigroup on $\C(\KA,\A)$.
We calculate the action of the generator of $t\mapsto\thone{t}$ on
a $\set1$-symmetric \basic\ function of degree $R\ni 1$. By Lemma
3.7, $t\mapsto\lrt$ is differentiable uniformly in $\rho$, and by
Proposition 3.4 of \tref\DWa, so is $t\mapsto \Ft\rho$ (in the weak*
sense). So we can differentiate:
$$\eqalignno{
\ddt (\thone t X\y\upone)(\rho)\atzero
&=\ddt \lrt\E\1{\Ft\rho}_{\eleminus R1}X_R{}\atzero &\cr
&=\Er{\eleminus R1}\sum_{x\in\notag R}\Lrb x X_R &\cr
&=(G\y\upone X\y\upone)(\rho)
\quad.&\cr}$$
Thus the generator, $\ghone$, of $t\mapsto \thone{t}$ agrees with
$G\upone_\infty$ on $\PP\upone$.
Since $\norm{\thone t}\le e\1{\gamma t}$, any
$\kappa>\gamma$ lies in the resolvent set of $\ghone$. For such
$\kappa$, \
$(\kappa-\ghone)\PP\upone=(\kappa-G\upone_\infty)\PP\upone$.
But it is
proved in Proposition 5.3(3) of \tref\Dua\ that $\PP\upone$ is a
core for $G\upone_\infty$, and consequently
$(\kappa-\ghone)\PP\upone$ must be dense in $\C(\KA,\A)$.
By Proposition 2.1 of \tref\Dav, $\PP\upone$ is also as core for
$\ghi$. Since the generators $\ghone$ and $G\upone_\infty$ agree on
a core, they are equal, and so $\thone t=\Tt\infty\upone$ for all
$t\ge0$.
\QED
Using our formalism the positivity and flow-implementing properties
of $\lrt$ follow straightforwardly.
\iproclaim Proposition 3.11.
\item{(1)} Each $\lrt$ is completely positive and unital.
\item{(2)} $\Ft\rho=\rho\circ\lrt$.
\eproclaim
\proof: (1) For any $R$ with $1\in\notag R$
$$\lrt X=(\Tt\infty\upone\jy{\set1} RX\otimes
\idty_{\eleminus R1})(\rho)
=\lim_{N\to\infty}
\Er{\eleminus N1}\Tt N (X\otimes\idty_{\eleminus N1})
\quad.$$
Since $X\mapsto X\otimes\idty_{\eleminus N1}$, $\Tt N$
and $\Er{\eleminus N1}$ are all completely
positive unital maps, $\lrt$, as a limit of such maps, is also
completely positive and unital.
(2) For $A\in\A$,
$$\eqalignno{
\la\rho\circ\lrt,A\ra&
=\la\rho,\lrt (\jy{\set 1}\upone A)(\Ft\rho)\ra
=\la\rho,(\Tt\infty\upone\jy{\set1} A)(\rho)\ra &\cr
&=\lim_{N\to\infty}\la\rho\1{\notag N},\Tt N\ \j N{\set 1} A\ra
=(\jy{\set1} A)(\Ft\rho)
=\la \Ft\rho,A\ra
&\cr}$$
\QED
Before extending Theorem 3.10 to treat the evolution of \qs\
observables, note that since each $\lrt$ is completely positive and
unital, then by Theorem 4.23 of \tref\Tak\ the product map
$\lrt\otimes\ldots\otimes\lrt$ (with $\nabs{I}$ factors) on the
$\abs{I}$-fold
algebraic tensor product $\A\1{\odot I}$ extends to a completely
positive unital map on $\A_I$. We denote this latter map by
$(\lrt)\1I$. Being positive and unital $\norm{ (\lrt)\1I}=1$. We can
extend each $(\lrt)\1I$ to $\B(\Aloc)$ by tensoring with the identity
map, and construct the infinite tensor product
$\lrti=\lim_{I\nearrow\Lat}(\lrt)\1I$, the limit being in the strong
operator topology of $\B(\Aloc)$. Our final theorem for bounded
polynomial generators is as follows.
\iproclaim Theorem 3.12.
Let $\Tt\dt=e\1{tG_\dt}$ with $G_\dt$ a bounded polynomial generator.
Then $\Lambda_t$ {\bf locally implements} $\Tt\infty$ in the sense
that for all $X\in\Yl$,
$$\bigl(\Tt\infty X\y\bigr)(\rho)
= \lrti X\y(\Ft\rho)
\quad.\eqno(3.4)$$
\eproclaim
\proof: Combining Theorem 3.10 with Theorem 3.6(3) we see that
equation (3.4) holds for \Isymm\ \net s $X_\dt$ with limits of the
form $X\y\1I=A\y\1{i_1}\otimes\ldots Z\y\1{i_{\abs{I}}}$. Since
$(\lrt)\1I$ is bounded, one obtains the stated result for any
function in $\C(\KA,\A_I)$ by approximation with limits of sums of
such terms. The final form is obtained by approximating \net s in
$\Yl$ by \basic\ \net s.
\QED
Recalling that $(\Tt\infty X_\infty)(\rho)=\lim_{N\to\infty}
\Er{\untag N}\Ttn X_N\in\Aloc$, the form of $\Tt\infty$ given above
shows that $\lrt$ implements the one-site evolution of tagged sites
when the bulk (of untagged sites) is in the product state formed
from $\rho$. In the remainder of this section we extend Theorem 3.12
beyond the bounded polynomial generators. Consider the following
\net s of generators.
\iproclaim Definition 3.13.
A \net\ of operators $G_\dt$ will be called {\bf \lc}
if for each finite $M\subset\Lat$ there exists \net\
$N\mapsto\Gamma\1{N}_M\in\B(\A_M)$ such that
following condtions hold.
\item{(1)} $\Gamma\1{N}_M=0$ for all $N\subset M$.
\item{(2)} $\Gamma_M=\lim_{N\to\infty}\Gamma\1{N}_M$ exists in the
strong operator topology.
\item{(3)} The bounds $\gamma_M\equiv\sup_{N\supset M}\norm{
\Gamma\1N_M}$ are summable so that
$\sum_M\absn{M}\gamma_M\equiv\gamma<\infty$.
\item{(4)}
For each $N$
$$G_N\equiv\sum_{M\subset N}{\nabsn{N}\over\nabsn{M}}
\Sym_N(\Gamma\1{N}_M)$$
is the generator of a norm-continuous semigroup of completely
positive unital contractions on $\A_N$.
\falsepar
\eproclaim
This definition makes sense not only for \net s of generators,
but also of general bounded operators on $\A_N$. For $G_N$ to
generate it is sufficient, but by no means necessary, that each
$\Gamma\1N_M$ generates on $\A_M$.
The polynomial generators (resp. operators) are
the special case, where $\Gamma\1N_M$ is non-zero only for some $M$,
and independent of $N$. The next level of complexity is to allow the
$N$-dependence, but to retain only one fixed $M$. A generator
constructed in this way is
asymptotically equal to the polynomial generator constructed from
$\Gamma_M=\lim_M\Gamma\1N_M$. In this case the ``\lc\ bound'' $\gamma$
is $\gamma=\nabsn M\sup_N \norm{\Gamma_M\1N}$. If for each $i$ in some
index set $G\1i$ is a \lc\ \net\ of operators on $\B(\A)$ with
\lc\ bounds $\gamma\1i$ such that $\sum_i\gamma\1i<\infty$ the
sum $G_N\equiv\sum_i G_N\1i$ exists for all $N$, and defines again a
\lc\ \net\ with bound $\gamma\leq\sum_i\gamma\1i$.
It is useful to note that the sets $M$ in this definition enter only
via their cardinality: due to the symmetrization over $M$ implicit in
$\Sym_N$ the labelling of the set $M$ becomes completely irrelevant.
By adding up all terms coming from $M$'s of the same cardinality we
can reduce the sum over $M$ to a sum over only one standard set $M$,
say $\oneto{\card M}$.
The \lc\ generators can be seen to arise in the following way.
Let $\Lat=\Ir\1d$, and let the fixed \net\ of regions
be such that $N\to\infty$ in the sense of Van Hove \tref\Ruelle.
$\S$ will denote the set of finite subsets of $\Ir\1d$.
Suppose that a translation invariant family of generators
$M\mapsto\Gamma_M\in\B(\A_M)$ is specified.
Construct the generator \net\
$$\hat G_N=\sum_{M\ni0}{1\over\nabsn{M}}
\sum_{{x\in\Ir\1d \atop M+x\subset N}}\Gamma_{M+x}
\quad.$$
$\hat G_\dt$ is, of course, translation invariant rather than
permutation invariant. When
$G_M(\cdot)=i\com{\Phi_M,\cdot}$ for some family $(\Phi_M)$ of
self adjoint potentials, it can be shown \tref\Rob\ that a
limiting dynamics exists provided that
$\sum_{M\in\S}e\1{\nabsn{M}}\norm{ \Gamma_M}$ is finite. But it is
shown in \tref\DWa\ that the symmetrized version of this interaction
$N\mapsto G_N=\Sym_N \hat G_N$ is \lc. For \lc\ interactions it is
then proved in \tref\DWa\ that a limiting
dynamics for intensive (\ie $\emptyset$-symmetric) observables
exists. We see from Definition 3.13(3) of the \lc\ that this means
that this dynamics exists under the condition that
$\gamma=\sum_{M\in\S}\nabsn{M}\norm{ \Gamma_M}$ is finite, a
considerably weaker condition than that of \tref\Rob.
In the remainder of this section we will show that for \lc\
generators, the limiting dynamics exists for {\it all} \qs\ \net s,
and furthermore that this dynamics is locally implemented as in
Theorem 3.12.
With the $\Gamma_M$ as in Definition 3.13,
define the bounded polynomial generator \net\
$G\1M_\dt$ by
$$ G\1M_N=\sum_{\hat M\subset M}{\nabsn{N}
\over\nabsn{{\hat M}}} \Sym_N \Gamma_{\hat M}
\quad.$$
We aim to show that $G_\dt$ generates a \mf\ dynamical semigroup by
showing that it can be approximated by those generated by the
$G\1M_\dt$. When we assume that $\A$ is finite dimensional this
turns out to be quite
easy to prove. In view of the calculation of the
$\emp$-symmetric mean-field dynamics for \lc\ generators in section
4 of \tref\DWa,
we expect that the proof for $\A$ infinite dimensional is possible,
albeit lengthy.
Denote by $\Lambda\1{M,\rho}_t$ the cocycle which locally implements
the \mf\ dynamical semigroup generated by $G\1M_\dt$, and denote by
$\locgen\1{M,\rho}$ its generator. $\Ft\1M$ will be the corresponding flow
on $\KA$.
\iproclaim Lemma 3.14.
Let $G_\dt$ be \lc, and let $\A$ be finite dimensional. Then the
norm limits
$$ \Lr A=\lim_{M\to\infty} \locgen\1{M,\rho}A
= \sum_M \Er{\eleminus M1}\Gamma_M(A\otimes\idty_{\eleminus M1})
$$ and
$\lrt\equiv\lim_{M\to\infty}\Lambda\1{M,\rho}_t$ exist, are
continuous functions of $\rho$, and satisfy
equation (3.2). $\lrt$ is completely positive and unital.
\eproclaim
\proof: Summing the terms in $G\1M_\dt$ we see by comparison with
equation (3.1) that
$$ \locgen\1{M,\rho}A
=\sum_{\hat M\subset M}\Er{\eleminus{\hat M}1}
G_{\hat M}(A\otimes\idty_{\eleminus{\hat M}1})
\quad.$$
$\norm{\locgen\1{M,\rho}A}
\le\sum_{M'\subset M}\norm{\Gamma_{M'}} \norm{ A}$.
By 3.13(3) this is bounded uniformly in $M$ and $\rho$ by $\gamma$,
and the tail $\sum_{M'\supset M}\norm{\Gamma_{M'}}\to0$
as $M\to\infty$. Hence $\locgen\1{M,\rho}A$ is convergent
as $M\to\infty$ to the form of $\locgen\1{\rho}A$ given.
Since convergence is uniform in
$\rho$ and for each $M$\ $\rho\mapsto\locgen\1{M,\rho}$ is
continuous, then $\rho\to\locgen\1\rho A$ is continuous.
According to Theorem 4.11 and Proposition 4.6(2) of \tref\DWa, the
flows $\Ft\1M\rho$ converge weak* as $M\to\infty$, uniformly for $t$
in compacta, to some $\Ft\rho\in\KA$, where $t\mapsto\Ft$ is a
weak*-continuous flow on $\KA$. Since $\A$ is finite dimensional
this holds in the norm topology of $\KA$ as well. It is now a
straightforward matter to show that $\Lambda\1{M,\rho}_t$ converges
uniformly to the unique norm-continuous solution $\lrt$ of the
equation
$\lrt=\id+\int_0\1t ds \Lambda\1{\rho}_s \locgen\1{\flow_s\rho}$. Since
convergence is uniform, $\rho\mapsto\lrt$ is continuous.
As a limit of completely positive unital maps, $\lrt$ is completely
positive and unital.
\QED
\iproclaim Theorem 3.15. Let $G_\dt$ be of \lc, with $\A$ finite
dimensional. Then $G_\dt$ is the generator of \mf\ dynamical
semigroup which is locally implemented by the $\lrt$ of Lemma 3.14,
and which hence has the disjoint homomorphism property.
\eproclaim
\proof:
Since we work in the norm topology of $\KA$ it is a simple matter to
show that for all finite $I\subset\Lat$,
$(\TTt\infty\1I f)=(\lrt)\1I f(\Ft\rho)$
defines a strongly continuous contraction
semigroup on $\C(\KA,\A_I)$. One differentiates to find the action
of its generator $\ghi$ on \basic\ \Isymm\ \net s $X_\dt=\j
\dt R\1I X_R$ with $I\subset R$ as
$$(\hat G\y\1{I}X\y\1{I})(\rho)=
\Er{\notag R\setminus I}\sum_{x\in\notag R}\Lrb
x X_R\quad.$$
But this is equal to $(G\y\1I X\y\1I)(\rho)$. For
$G_\dt \j\dt RX_R=\sum_M Y\1{(M)}_\dt$ where
for for each $M$, $Y\1{(M)}_\dt$ is the \qs\ \net\
$N\mapsto Y\1{(M)}_N=(\absn{N}/\absn{M})(\Sym_N \Gamma\1N_M)\j
NR X_R:\ N\supset M$. By 3.13(3), $M\mapsto\nnorm{Y\1{(M)}}$ is summable,
so that for each $\eps>0$ there exists $M_\eps$ such that
$\nnorm{G_\dt \j\dt RX_R-\sum_{M\subset M_\eps} Y\1{(M)}_\dt}<\eps$.
Hence
$G_\dt \j\dt RX_R$ is \qs\ and
$$\eqalignno{
(G_\dt \j\dt RX_R)\y\1I(\rho)
&=\lim_{M\to\infty}\sum_{M'\subset M} Y\y\1{(M'),I}(\rho)&\cr
&=\lim_{M\to\infty}\Er{\notag R\setminus I}
\sum_{x\in\notag R}\locgen\1{\rho,M} X_R&\cr
&=\lim_{M\to\infty}\Er{\notag R\setminus I}
\sum_{x\in\notag R}\locgen\1{\rho} X_R&\cr
&=(\hat G_\dt \j\dt RX_R)\y\1I(\rho)
\quad.&\cr}$$
In Proposition 3.16 below we show that $\PP\1I$ is a core for
$\ghi$. Then the above argument shows that for $s\in\Cx:\ \Re(s)>0$,
$((s-G_\dt)\Yb)\y\1I=(s-G\y\1I)\PP\1I=(s-\hat G\y\1I)\PP\1I$ is dense in
$\Yl\1I$. So by the implication (4)$\Rightarrow$(5) of Theorem 2.3 of
\tref\DWa, and Theorem 3.2 of \tref\Dua, $G\y\1I$ is well-defined and
$G_\dt$ has an $I$-symmetry preserving \mf\ limit which is generated
by $G\y\1I$. This is true for all $I$, thus $G_\dt$ generates a \mf\
dynamical semigroup $\Tt\infty$, and
$(\Tt\infty X_\infty)(\rho)=(\lrt)\1\infty X(\Ft\rho)$.
\QED
It remains to show that $\PP\1I$ is a core for $\ghi$. Our strategy
is to express $\ghi$ in terms of a derivative on $\C(\KA,\A_I)$, and
then use standard methods to show firstly that the set of
differentiable functions is preserved by $\thone{t}$ and is hence a
core for $\ghi$, and secondly that each differentiable function can
be approximated, along with its derivatives, by an element of
$\PP\1I$.
For a unital C*-algebra $\V$ and $f\in\C(\KA,\V)$ we define the gradient
$\d f(\rho)$ of $f$ at $\rho\in\KA$ by
$$ \bra \sigma-\rho, \d f(\rho)> \equiv
\ddt f(t\sigma-(1-t)\rho)\atzero\quad,\eqno(3.5)$$
and say that $f$ is differentiable whenever this exists as a
continuous function on $\KA$.
Equation (3.5) must be understood as being $\V$-valued in the
sense that the duality $\bra\cdot,\cdot>$ is between $\A$ and $\KA$,
leaving $\bra\sigma-\rho,\d f(\rho)>\in\V$.
Equation (3.5) fixes the gradient only up to a multiple of the
identity. We remove this ambiguity and fix $\d f$ as an element of
$\C(\KA,\V\otimes\A)$ by imposing
the convention that $\bra\rho,\d f(\rho)>=0$.
$\C\11(\KA,\V)$ will denote the set of
differentiable functions in $\C(\KA,\V)$.
Clearly $\PP\1I\subset\C\11(\KA,\A_I)$.
This notion of a derivative also lifts to $\B(\V)$. Let
$H\in\C(\KA,\B(\V))$. Then we define $\d H$ to be the element of
$\C(\KA,\B(\V))$ such that $(\d H)X=\d (HX)$ for each $X\in\V$.
For example, take $\V=\A$, and
let $\hat\locgen$ be the local generator corresponding to a
bounded polynomial generator $G_\dt$ of degree $M$.
Let $A\in\A$, $\rho,\sigma\in\KA$, and for
$h\in\lbrack0,1\rbrack$ set
$\rho_h=\rho+h(\sigma-\rho)$. Then
$$\eqalignno{
\bra\sigma,\d {\hat\locgen}\1\rho A>
&=\lim_{h\to0}(\E\1{\rho_h}_{M\setminus\set1}-
\E\1{\rho}_{M\setminus\set1})
G_M(A\otimes\idty_{M\setminus\set1}) &\cr
&=(\nabs M-1)\E\1{\sigma-\rho}_{\set2}
\E\1\rho_{M\setminus\set{1,2}}
G_M(A\otimes\idty_{M\setminus\set1})
\quad.&\cr}$$
According to Theorem 4.11 and Proposition 4.3 of \tref\DWa, the
limit flow $\Ft\rho=\rho\circ\lrt=\rho\circ\lim_M\Lambda\1{M,\rho}_t$
is differentiable and hence preserves the set of differentiable
complex-valued functions. In particular
$$\d (f\circ\Ft)(\rho)=J\1\rho_t (\d f)(\Ft\rho)$$
for a suitable Jacobian $J\1\rho_t\in\B(\A)$, and furthermore there
exists a bound $\norm{ J\1\rho_t}\le e\1{\gamma t}$. We require now
to prove a similar result for $\lrt$. Since we work with $\A$ finite
dimensional, the proof is quite simple. Item (5) of the following
proposition also provides the last remaining step in the proof of
Theorem 3.15.
\iproclaim Proposition 3.16.
Let $\A$ be finite dimensional. Then
\item{(1)} $L$ is differentiable
\item{(2)} $\Lambda_t$ is differentiable for all $t\ge0$.
\item{(3)} For all finite $I\subset\Lat$,
$\C\11(\KA,\A_I)$ is invariant under
$\thi t$ for all $t\ge0$.
\item{(4)}
For all finite $I\subset\Lat$, $\C\11(\KA,\A_I)$ is a core for $\ghi$.
\item{(5)}
For all finite $I\subset\Lat$, $\PP\1I$ is a core for $\ghi$.
\eproclaim
\proof:
(1)
$$\eqalignno{
h\1{-1}\norm{(L\1{\rho_h}-L\1\rho)-(L\1{M,\rho_h}-L\1{M,\rho})}
&\le h\1{-1}\sum_{M'\supset M}
(\nabs{M'}-1)\norm{(\E_{M'\setminus\set1}\1{\rho_h}
-\Er{M'\setminus\set1})\Gamma_{M'}}&\cr
&\le\nnorm{\sigma-\rho}\sum_{M'\supset M}\nabs{M'}
\nnorm{\Gamma_{M'}}
\quad.&(3.6)\cr }$$
By Definition 3.13(3) this bound is the tail of a covergent sum. Thus
the limit of the LHS of inequality (3.6) as $M\to\infty$ is zero,
uniformly in $h$. We showed above that each $L\1{M}$ is
differentiable, and so
$\d L\1\rho$ exists and is equal to
$\lim_{M\to\infty}\d L\1{M,\rho}$.
(2) Since $\A$ is finite dimensional,
we consider $t\mapsto(\Ft\rho,\lrt)$ as an
integral curve of the vector field
$(\dot\rho,\dot\Lambda)=(\rho\circ\Lr,\Lambda\circ\Lr)$ on the
Banach space $\KA\times\B(\A)$ with norm
$\norm{(\rho,\Lambda)}=\norm{\rho}+\norm{\Lambda}$.
Since $\locgen$ is bounded and
$\rho\mapsto\locgen\1\rho$ is
differentiable, one sees (from section 4.1 of \tref\AbMa )
that $\rho\mapsto\lrt$ is differentiable,
at least locally in time. In fact, since
$\norm{(\dot\rho,\dot\Lambda)}\le\gamma\norm{(\rho,\Lambda)}$,
then in fact these integral curves exists for all time and are
differentiable.
(3) Let $f\in\C\11(\KA,\A_I)$. Then clearly
$$(\d \thi t f)(\rho)
=(\d(\lrt)\1I)f(\Ft\rho)+(\lrt)\1I J\1\rho_t\d f(\Ft\rho)
\quad.$$
(4) Let $f\in\C\11(\KA,\A_I)$. Then
$$\eqalignno{
\ghi f(\rho)
&=\ddt(\lrt)\1I f(\Ft\rho)\atzero &\cr
&=\Er{\notag R\setminus I}\sum_{x\in\notag I}
\Lrb x f \bra\rho\circ L\1\rho,\d f(\rho)>
\quad.&(3.7)\cr}$$
Thus $\C\11(\KA,\A_I)$ is a subset of $\dom(\ghi)$, which by (2) and
(3) is $\thi t$ invariant. Furthermore, $\C\11(\KA,\A_I)$ is dense in
$\C(\KA,\A_I)$ (it contains the dense subset of polynomials $\PP\1I$)
and so it is a core for $\ghi$.
(5) We complete the proof by showing any $f\in\C\11(\KA,\A_I)$ there
is a sequence of polynomials $(f_n)_{n\in\Nl}\subset\PP\1I$ such that
$\lim_{n\to\infty}f_n=f$ and $\lim_{n\to\infty}\d f_n=\d f$.
For then from equation (3.7) one sees that
$\lim_{n\to\infty}\ghi f_n=\ghi f$ and so $\PP\1I$ is a core for
$\ghi$.
Consider the set $\linear$ of linear functions
$\set{\rho\mapsto\Er{\set 1}A\stt A\in\A_{\nabs I+1}}$ in $\PP\1I$.
Clearly the algebra generated by $\linear$ is dense in $\PP\1I$ and
hence dense in $\C(\KA,\A_I)$.
Furthermore for $\rho\ne\sigma\in\KA$ we can choose and $g$ and $h$ in
$\linear$ such that
$g(\rho)\ne0$ and $\bra\sigma-\rho,\d h(\rho)>\ne0$.
So, by Nachbin's Theorem stated in Theorem 1.2.1 of \tref\Lla, the
algebra generated by $\linear$ is dense in $\C\11(\KA,\A_I)$ in the
norm $\norm{ f }_1=\norm{ f}+\norm{ \d f}$, as required.
\QED
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\beginsection 4. Properties of the the limiting evolution
\beginsubsection 4.1. Hamiltonian systems
In many examples the semigroups $\Tt N$ are reversible in the sense
that the generator is of the form
$$ G_N(X)=\absn{N}i\com{H_N,X}
\eqno(4.1)$$
with a Hamiltonian density $H_N=H_N\1*\in\A_N$. For the
thermodynamics of \mf\ systems it is sufficient for $H$ to be
$\emp$-symmetric \tref\RWa. For the dynamics one needs to
assume more, e.g. that the generator be of \lc\ as in Definition
3.13. This is readily written in terms of $H$: we want that
$$\eqalignno{
H_N=&\sum_{M\subset N}\j NM\1\emp H_M\1N
\qquad\hbox{with} \quad H_M\1N\in\A_M &(4.2)\cr
&\hbox{such that } \quad
\sum_M\absn{M}\12\,\sup_N\norm{H_M\1N} <\infty \cr
&\hbox{and}\qquad
H_M\1\infty =\norm{\cdot}-\lim_N H_M\1N
\qquad \hbox{exists.}\cr}$$
Then Definition 3.13 is satisfied with
$\Gamma\1N_M(\cdot)=\absn{M}i\com{H_M\1N,\cdot\, }$.
For Hamiltonian dynamics each $\Tt N$ is an automorphism. Since the
$N$-wise products of \qs\ \net s are again \qs\ we conclude
immediately that $\Tty(X\y Y\y)= \bigl(\Tt\dt(X_\dt Y_\dt)\bigr)\y
=\bigl(\Tt\dt(X_\dt)\Tt\dt(Y_\dt)\bigr)\y=\Tty(X\y)\Tty(Y\y)$. Thus
$\Tty$ is a homomorphism. Within the \lc\ of
generators we can say more: the local evolutions are themselves
Hamiltonian, with a $\rho$-dependent Hamiltonian:
$$ \Lr(A)=i\com{\Hr,A}
\qquad\hbox{with}\quad
\Hr=\sum_M\absn{M}\, \Er{\eleminus M1}(H_M\1\infty)
\eqno(4.3)$$
The growth condition on $\sup_N\norm{H_M\1N}$ ensures that
$\norm{\Hr}$ is bounded on $\KA$, and $\Hr$ has continuous first
derivatives with respect to $\rho$.
This form of $\Lr$ has the consequence that each $\lrt$ is unitarily
implemented: we have
$$ \lrt(A)=U\1\rho_tAU\1{\rho*}_t
\qquad\hbox{with}\quad
\ddt U\1\rho_t = iU\1\rho_t H\1{\Ft\rho}
\quad\hbox{and}\quad U\1\rho_0=0
\quad.\eqno(4.4)$$
The Hamiltonian $\Hr$ is closely related to the energy density
function $H\y:\KA\to\Rl$, which enters the Gibbs variational
principle for the limiting free energy of the \mf\ system \tref\RWa.
In the Euler-Lagrange equations for this variational principle one
needs the gradient of this function, \ie the derivatives along
directions in the state space. The gradient $\d H\y(\rho)$ in the
sense of equation (3.5) is an element of $\A$, also called the
``effective Hamiltonian''. The thermal equilibrium states are then
infinite product states with a one-particle state $\rho$ which is an
equilibrium state for $\Hr$. This amounts to an implicit non-linear
equation for $\rho$ known as the ``gap equation''\tref{\RWa,\DELHI}.
Assuming $H_N$ to be of the form (4.2) we obtain
$$\eqalignno{
\bra \sigma-\rho, \d H\y(\rho)>
&= \ddt H\y(t\sigma-(1-t)\rho)\atzero &(4.5)\cr
&=\ddt \sum_M\bra(t\sigma-(1-t)\rho)\1{\notag M},
H_M\1\infty>\atzero \cr
&=\sum_M \absn{M}
\bra(\sigma-\rho)\otimes\rho\1{\absn{M}-1},
H_M\1\infty>\cr
&= \sum_M \absn{M}
\bra\sigma-\rho,\Er{\eleminus N1}H_M\1\infty>\cr
&= \bra\sigma-\rho,\Hr>
\quad.&(4.6)\cr}$$
Here the first equality in (4.5) is the definition of the gradient
as an element $\d H\y(\rho)\in\A$, and the last line shows that
$\Hr$ satisfies this definition. It is clear, however, that equation
(4.5) fixes the gradient only up to a multiple of the identity. As in
section 3, we can get rid of this ambiguity by imposing the convention
$\bra\rho,\d H\y(\rho)>=0$. Then the above equation becomes
$\d H\y(\rho)=\Hr-\bra\rho,\Hr>\idty$.
The identification of $\Hr$ with the gradient of $H\y$ is also
important for establishing an important property of the flow $\Ft$
in the Hamiltonian case: it is itself Hamiltonian in the sense of
classical mechanics \tref\MFH. In order to make sense of this
statement we have to introduce a symplectic structure on the state
space $\KA$. The state space itself has no natural symplectic
structure (it may be odd dimensional). However, each of the leaves
of the foliation of the state space into unitary equivalence classes
of states allows a non-degenerate symplectic stucure \tref\MFH.
Since $\lrt$ is unitarily implemented we already know that the flow
$\Ft\rho=\rho\circ\lrt$ respects this foliation. The easiest way to
define the symplectic strucure on all leaves simultaneously is to
define the Poisson bracket of two differentiable functions
$f,g:\KA\to\Rl$. Using the definition (4.5) of the gradient we set
$$ \Poisson fg(\rho)=\bra\rho,i\com{\d f(\rho),\d g(\rho)}>
\quad.\eqno(4.7)$$
Note that the convention for the gradient is irrelevant here, since
multiples of the identity drop out of the commutator anyway.
One now checks easily \tref\MFH\ that the flow
satisfies Liouville's equation in the form
$$\ddt f(\Ft\rho)\atzero
= \Poisson{H\y}f(\rho)
\quad.\eqno(4.8)$$
The possibility of writing the limiting evolution as a classical
Hamiltonian flow was noticed a long time ago in \tref\HL. However,
in order to state this, Hepp and Lieb used the natural symplectic
structure on the coadjoint orbits of a Lie group. Therefore the
Hamiltonian had to be written as a function of the generators of a
group representation. This approach was also adopted by \tref\Bona.
It has the disadvantage of introducing an additional auxiliary
object (the group representation) which becomes unnecessary as soon
as the symplectic structure is established on the state space
itself. For the dissipative evolutions discussed below the
disadvantage becomes even more pronounced.
To summarize: if each $\Tt N$ is generated by a Hamiltonian, then
the global dynamics is given by a {\it Hamiltonian} flow, and the
local dynamics is also generated by a {\it Hamiltonian}.
\beginsubsection 4.2. Lindblad generators from symmetric \net s
It is well known \tref\Lind\ that the generator of a dynamical
semigroup can be written as a sum of a commutator and terms of the
form $G(X)=\Lindblad VX$.
If we want to turn this into a \net\ of generators a natural
possibility is to insert for $V$ a $\emp$-symmetric \net\
like the Hamiltonians in the previous subsection, and to multiply
the result by the system size. It is this class that we would like
to study here. We mention that the only type of dissipative
inter-particle interaction included in some previous work
\tref\Unna\ was a single term of this type.
More precisely, we demand that the generators are of the form
$$\eqalign{
G_N(X)&= \card N \sum_{\alpha} \Lindblad{\Va N}X \cr
\hbox{where}\qquad
\Va N &= \sum_{M\subset N} \j NM\1\emp \Va M\1N \cr
\hbox{where}\qquad
\gamma_{\alpha,M}&= \sup_N\norm{\Va M\1N} < \infty \quad, \cr
\Va M\1\infty &= \lim_N \Va M\1N \hbox{exists in norm}\quad,\cr
\hbox{and}\qquad
\sum_\alpha&\left(\sum_M \card M\12\gamma_{\alpha,M}\right)\12
\left(\sum_M \gamma_{\alpha,M}\right) <\infty \quad.
}\eqno(4.9)
$$
It is clear that under these circumstances the \net s $\Va\dt$ are
$\emp$-symmetric, and
$$ \Va\infty(\rho)= \sum_M \bra\rho\1M,\Va M\1\infty>
\quad.\eqno(4.10)$$
Moreover, the functions $\Va\infty:\KA\to\Cx$ are differentiable, and
$\d \Va\infty(\rho)= \sum_M\Er{M\setminus1}(\Va M\1\infty)\in\A$.
We can then compute the local dynamics as follows:
\iproclaim 4.1 Proposition.
Generators of the form (4.9) are \lc\ in the sense of definition
3.13, and hence define a \mf\ dynamical semigroup.
The generator of the local dynamics is
$$\eqalign{
\Lr(A)&=i\com{\Hr,X} \quad,\cr
\hbox{where}\qquad
\Hr &= \sum_\alpha {1\over i}\left(\Va\infty \d \Va\infty\1*
-\Va\infty\1* \d \Va\infty\right)
\quad.}$$
\eproclaim
\proof:
\def\Dt{\,\dt\,}
\def\MMp{{(M,M')}}
\def\MaM{{M\&M'}}
By the remarks after Definition 3.13 it suffices to consider a single
term $\alpha$. Hence we will simply omit $\alpha$ from the above
formulas. Moreover, we may assume that $V_M\1N$ is non-zero only for
some standard set $\oneto{\card M}$ for each cardinality of $M$. Now
each of the two terms in $G_N=\nabsn N\Lindblad{V_N}X$ involves a
double sum over $M,M'$ of terms of the type
$$ G\1{\MMp}_NX=\nabsn N \,
(\j NM\1\emp V_M)\1*\com{X,(\j N{M'}\1\emp V_{M'})}
\quad.$$
We claim that $G\1{M,M'}$ is a \lc\ \net\ of operators with a \lc\
bound
$(\card M+\card M')\12\gamma_M\gamma_{M'}$. By the remarks after 3.13
this will be enough to complete the proof, since
$\sum_{m,m'}(m+m')\12\gamma_m\gamma_{m'}\leq
4(\sum_mm\12\gamma_m)(\sum_m\gamma_m)$.
The expression for $G\1{\MMp}_N$ is the average over all pairs
$(\pi,\pi')$ of permutations of $\oneto{\card N}$ of
$\nabsn N\ \hat\pi(V_M)\1*\com{X,\hat\pi'(V_{M'})}$, where we have
identified $V_M,V_{M'}$ with elements of $\A_N$ living at the sites
indicated.
Substituting $\pi'=\pi\1{-1}\pi''$ we can thus write
$$ G\1{\MMp}_N=\nabsn N\Sym_N\left({1\over N!}\sum_\pi
V_M\1*\com{\Dt,\hat\pi(V_{M'})}\right)
\quad.$$
It is easy to see that under the outer symmetrization all terms
coincide, for which the ``overlap'' $M\cap\pi(M')$ has the same
number of elements.
Let $N!c_k(N)$ denote the number of permutations of $\oneto{\card N}$
with $\card{M\cap\pi(M')}=k$. By definition we have $\sum_kc_k(N)=1$.
Then we can write
$G\1{\MMp}_N=\card N/(\card M+\card{M'})\Sym_N\Gamma\1N_M$ with
$\Gamma\1N_M$ an operator on
$\A_{\MaM}$, where $\MaM$ is a set of cardinality
$\card M+\card{M'}$, say $\oneto{\card M+\card{M'}}$, and
$$ \Gamma\1N_{\MaM}=\card{\MaM}\sum_k c_k(N)
(V_M\otimes\idty\1{\card {M'}})\1*\com{\Dt,
\idty\1{\card M-k}\otimes V_{M'}\otimes\idty\1k}
\quad.$$
This expression makes sense only for
$\card N\geq\card{\MaM}=(\card M+\card M')$,
but we can choose any definition of $\Gamma\1N_M$ for the finitely many
exceptional $N$ without changing the validity of our claim. Now by
Lemma IV.1 of \tref\RWa\ we have $c_0=1-\Order(N\1{-1})$, and hence
$\Gamma_\MMp=\lim_N\Gamma\1N_\MMp
=(V_M\otimes\idty\1{M'})\1*\com{\Dt,\idty\1{M}\otimes V_{M'}}$.
It remains to compute the the limiting generator $\Lr$. We could do
this by adding up the contributions $\Lr_{\MMp}$ from all pairs
$\MMp$.
A quicker way to see the result is to use the
results of the previous section: since $\Va N$ satisfies the
conditions (4.2) (apart from hermiticity) we know (by splitting $\Va
N$ into hermitian and skew-hermitian part) that
$N\mapsto X_N\equiv \absn N\com{\Va N,A\otimes\idty\1{\card N-1}}$
is a $\set1$-symmetric \net\ with
$X\y(\rho)=\com{\d\Va\infty(\rho),A}$.
Multiplying this $\set1$-symmetric \net\ with the $\emp$-symmetric
\net\ $\Va N\1*$ we get a $\set1$-symmetric \net\ with limit
$\Bar{\Va\infty(\rho)}\com{\d\Va\infty(\rho),A}$. Adding the
contribution from the conjugate term in the Lindblad form, and
summing over $\alpha$ we find that $G_N(A\otimes\idty\1{\absn N-1})$
is $\set1$-symmetric with limit
$\Lr(A)$ as stated in the Proposition.
\QED
Since the local dynamics is generated by a Hamiltonian it might be
suspected that this forces the global evolution to be Hamiltonian as
well, but this is not so. We demonstrate this with the following
elementary example:
\noindent{\bf Example:}\
Let $\A$ be the algebra of $2\times2$-matrices, and set
$V_N=\j N1\sigma\1+$, where
$\sigma\1+=\scriptstyle\pmatrix{0&1\cr0&0}$.
Then from the Proposition one readily verifies that
$$ \Hr={1\over i}\pmatrix{0 &-\rho_{12}\cr
\rho_{21} &0}
\quad.\eqno(4.11)$$
The flow is determined from the differential equation
$\dot\rho=i\com{\Hr,\rho}$. This equation can be written in terms of
the variables $x=\rho_{11}-\rho_{22}$, $y=\abs{\rho_{12}}\12$, and
the argument of $\rho_{12}$. The latter is constant, and we can
furthermore eliminate $y$ from the fact that $\Ft\rho$ is unitarily
equivalent to $\rho$, and consequently
$2\tr(\rho\12)-1=x\12+4y\equiv\lambda\12$ is a constant of the motion.
The resulting equation $\dot x=x\12-\lambda\12$ is readily solved, and
gives $x(t)=-\lambda\tanh(\lambda(t-t_0))$, where $t_0$ is
determined from the initial condition. For $t\to\infty$ we get
$x(t)\to-\lambda$, and consequently $\abs{\rho_{12}}\12=y\to0$. Thus
in the state space, which is identified with a ball in 3 dimensions,
the flow moves along the meridians on concentric spheres to the
southern half of the axis. It is thus certainly not Hamiltonian.
In this example, although the flow $\Ft$ is no longer Hamiltonian, it
is reversible in the sense that it also exists for negative times.
This is no coincidence. In fact, if we replace $\Va N$ by
$\tilde\Va N=\Va N\1*$ we obtain another generator $\tilde G_\dt$ of
the form (4.9), and from Proposition 4.1 we immediately get the
local Hamiltonian as $\tilde\Hr=-\Hr$. Thus in spite of the fact that
for finite $N$ no $\Tt N$ needs to have a positive inverse, $\Tty$
does.
We have seen that for the generators studied in this subsection the
local dynamics is generated by a state-dependent Hamiltonian $\Hr$. It
is natural to ask whether any more can be said about the generators of
the form (4.9), or whether {\it any} function $\rho\mapsto\Hr$ can
occur. Since we have not attempted to find exhaustive conditions under
which the \mf\ limit of a \net\ of generators exists, we cannot be
expected to show the latter result. However, we will show the only
slightly weaker statement that any function $\rho\mapsto\Hr$ may be
approximated by local Hamiltonians arising from generators satisfying
(4.9). In particular, any ordinary differential equation respecting
unitary equivalence classes is approximately the equation determining
the flow $\Ft$ of some \mf\ dynamical semigroup. This makes it
unnecessary for us to provide examples of various types of possible
behaviour of the flow: any structurally stable phase portrait of
dynamical systems, stable and unstable points and limit cycles, as
well as chaotic behaviour can occur.
The proof that approximately all $\Hr$ occur is simple. It is useful
for this purpose to think of $\rho\mapsto\Hr$ as a 1-form on $\KA$.
This is permissible since gradients, 1-forms and local Hamiltonians
are all defined only up to multiples of the identity.
By Proposition 4.1 $\Hr$ is a sum of terms of the form
${1\over i}(\Va\infty \d \Va\infty\1* -\Va\infty\1* \d \Va\infty)$.
It is useful to write $\Va\infty=f+ig$. Then the contribution to the
Hamiltonian is $2(g\d f -f\d g)$.
In this expression $f$ and $g$ can now be chosen as arbitrary real
valued polynomials on $\KA$, or even sums of polynomials converging
in $\C\12$-norm. (We do not need the latter fact, it suffices to use
the polynomials for the approximation argument). In particular,
setting $f=gh$, any 1-form $g\12\d h$ with polynomial $g,h$ can be
realized. Since on a compact set any differentiable function (of
finitely many variables) can be approximated uniformly together with
its derivatives by polynomials \tref\Lla, we can drop the constraint
that $g$ and $h$ should be polynomials. Since we can write any bounded
function as a difference of two squares (take the first square as a
constant larger than the upper bound), we conclude that by taking sums
we can uniformly approximate any 1-form.
To summarize, in the class of \mf\ dynamical semigroups studied in
this subsection the local dynamics is still {\it Hamiltonian}. The
flow $\Ft$ thus respects unitary equivalence classes and is
reversible, but {\it not Hamiltonian}. On any one equivalence class
essentially any flow is possible.
\beginsubsection 4.3. General lattice class
In the previous section we demonstrated that essentially any function
$\rho\mapsto\Hr$ can occur as the local Hamiltonian of the local
dynamics in a suitable \mf\ model in the class described. Here we
address the same question for the lattice class: we will show
that the functions $\rho\mapsto\Lr$, which can arise from \mf\
dynamical semigroups with \lc\ generators is dense in the set of
continuous functions associating with each state $\rho$ a generator
$\Lr$ of some dynamical senmigroup on $\A$. The purpose of this
question is to verify that we have not missed some structure
theorem for the local dynamics which would put a constraint on this
function. For simplicity we will always assume that $\A$ is finite
dimensional.
\iproclaim 4.2 Proposition.
Let $\A$ be finite dimensional, and let $\CKBA$ denote the space of
continuous functions on $\KA$ with values in the operators on $\A$.
Consider the cone $\Gen$ of functions $\locgen\in\CKBA$ such that for
all $\rho$, $\Lr$ generates a dynamical semigroup, and the subcone
$\pGen\subset\Gen$ of local generators $\rho\mapsto\Lr$ arising from
polynomial generators. Then $\pGen$ is norm dense in $\Gen$.
\eproclaim
\proof:
We consider first
polynomial generators $G_N=(\absn N/\absn R)\Sym_NG_R$ with $G_R$
extremal in the cone of permutation symmetric Lindblad generators on
$\A_R$, \ie we consider the form $G_R(\cdot)=\absn R\Sym_R\Lindblad
V{\,\cdot\,}$ with $V\in\A_R$. Note that we do not require $V$ itself to be
permutation symmetric. As a convenient expression for $\Lr$ in terms
of $V$ we use
$$ \Lr_V(A)=\sum_{x\in\notag R}\Er{\eleminus Rx}
\Set\Big{\Lindblad V{\heta_x(A)}}
\quad,\eqno(4.12)$$
where $\heta_x$ embeds an $\A$ as the copy of $\A$ at site $x$.
>From this expression it is clear that $\Lr_{V\otimes\idty}=\Lr_V$,
and more generally
$$ \Lr_{V\otimes W}= \bra\rho\1R,V\1*V>\Lr_W+\bra\rho\1S,W\1*W>\Lr_V
\quad,\eqno(4.13)$$
where $V\in\A_R$ and $W\in\A_S$. Note that
the coefficient of
$\Lr_W$ depends on $V$ and conversely. We want to get rid of this
dependence by finding suitable $W$ for which the first term becomes
negligible, while $\bra\rho\1S,W\1*W>$ approximates any desired
function. A subclass of the generators discussed in the previous
subsection precisely meets this description:
we set $W_S=\j S{S'}\1\emp F$ with $F=F\1*\in\A_{S'}$. Then by
Proposition 4.1 we have $\lim_S\Lr_{W_S}(A)=i\com{\Hr,A}$ with
$i\Hr=W\y\1*\d W\y-W\y \d W\y\1*$. But since $F$ is hermitian, $W\y$
is a real function, and hence $\Hr=0$. On the other hand,
$\lim_S \bra\rho\1S,W\1*W> =\abs{\jy{{{S'}}}F}\12$, which is the square
of an arbitrary real polynomial on $\KA$. By this we can approximate
an arbitrary positive continuous function, and consequently the
closure of $\pGen$ contains all functions of the form $\rho\mapsto
f(\rho)\Lr_V$ with $f\in\CKA$, $f\geq0$, and $\Lr_V\in\pGen$. Any
constant function $\Lr\equiv\locgen$ is in $\pGen$, since we can
take the corresponding one-site generator
$G_N=\absn N\Sym_N\locgen$.
Given now an arbitrary function $\locgen\in\Gen$ we can choose a
sufficiently fine continuous partition of the identity, i.e.
$f_\alpha\in\CKA$, $f_\alpha\geq0$, $\sum_\alpha f_\alpha\equiv1$,
such that $f_\alpha$ has its support only near some $\rho_\alpha$,
such that $\Lr$ is uniformly close to
$\sum_\alpha f_\alpha\locgen\1{\rho_\alpha}$. We have just shown that
the latter expression is in the closure of $\pGen$. Hence $\pGen$
is dense in $\Gen$.
\QED
\beginsubsection 4.4 Lindblad generators from permutation operators
For finite dimensional $\A$ any \net\ of generators is of the form
$G_N(X)=\absn Ni\com{H_N,X}
+ \absn N\sum_\alpha \bigl(\Lindblad{\Va N}X\bigr)$.
In this subsection we suppose that $\A$ is the algebra of $d\times
d$-matrices, and that $H_N$ and each $\Va N$ is a linear combination
of permutation operators. Then $G_N$ vanishes on any operator $X$
commuting with permutations, and dually $\rho\1N\circ G_N=0$ for any
state $\rho\in\KA$. Thus every homogeneous product state $\rho\1N$ is
invariant under the semigroups $\Tt N$. Since the generator of the
flow is expressed by evaluating $G_N$ in such states, it is clear that
if the $G_N$ define a quantum dynamical semigroup, every state $\rho$
will be invariant under the associated flow. Hence the flow $\Ft$ is
trivial. This does {\it not} mean, however, that the local dynamics
is also trivial. Indeed, we know from the previous section that
approximately we can realize any local generator $\rho\mapsto\Lr$, and
in particular any $\Lr$ such that $\rho\circ\Lr=0$. However, for the
\mf\ dynamical semigroups discussed in this section we do not have to
invoke this approximate result: the flow is exactly constant.
As a first example, consider the Hamiltonian case. For simplicity we
choose a polynomial generator of degree $R$, \ie we set
$H_N=\j NR\1\emp\hat H
=\j NR\1\emp\sum_{\pi\in\Perm_R}h(\pi)U_\pi$,
where $\Perm_R$ denotes the group of permutations of the sites $\notag
R$, $U_\pi$ the unitary operator implementing the permutation
$\pi$, and $h$ is any function on $\Perm_R$. The operator $\j NR\1\emp$
implies an averaging over all permutations hence we may suppose
without loss of generality that $\hat H$ is itself permutation
invariant. Equivalently, $h$ can be taken as an invariant function
($h(\pi\pi')=h(\pi'\pi)$), \ie it is in the center of the group
algebra. The complete information about the dynamics is contained in
the energy density function
$$ H\y=\bra\rho\1R,\hat H>
=\sum_\pi h(\pi)\bra\rho\1R,U_\pi>
\quad.\eqno(4.14)$$
Since every unitary $U\otimes U\cdots U=U\1{\otimes R}$ commutes with
$U_\pi$, $\pi\in\Perm_R$ it is clear that
$H\y(\rho)=H\y(\rho\circ\ad_U)$. Thus $H\y$ is constant on each
unitary equivalence class. The flow on each of the symplectic
submanifolds of $\KA$ is thus generated by a constant Hamiltonian,
\ie the flow is constant in accordance with the general remarks made
above. In order to evaluate (4.14) more explicitly we use the
formula
$\tr(A_1\cdots A_N)=\tr\bigl((A_1\otimes\cdots A_N)U_\pi)$, for $\pi$
the cyclic permutation of $\set{1,\ldots n}$, which is readily shown
by expanding both sides with respect to the same basis.
We get
$$ \bra\rho\1R,U_\pi>
=\prod_k \left(\tr(\rho\1k)\right)\1{n_k(\pi)}
\quad,\eqno(4.15)$$
where $n_k(\pi)$ is the number of cycles of length $k$ appearing in
the cycle decomposition of $\pi\in\Perm_R$, and where we have
used the symbol $\rho$ for both the state and its density matrix.
Thus $H\y$ is a polynomial in the $\absn R$ variables $\tr(\rho\1k)$.
Put differently, $H\y$ is a symmetric polynomial in the eigenvalues of
$\rho$. It is easy to check that all such polynomials can occur.
The Hamiltonian for the local dynamics is $\Hr=\d H\y(\rho)$. This is
non-zero, so the local dynamics is not trivial. From the form of $H\y$
it is clear that $\Hr$ is a polynomial in $\rho$ and the numbers
$\tr(\rho\1k)$. In particular, $\com{\Hr,\rho}=0$, confirming once
again that the flow is constant.
The simplest, though physically quite interesting example of this kind
of Hamiltonian is the \mf\ version of the Heisenberg model. There we
have $d=2$, $R=\set{1,2}$, and the Hamiltonian is
$H_R=\sum_{\nu=1}\13\sigma\1\nu\otimes\sigma\1\nu=2F-\idty$, where
$\sigma\1\nu$ denotes the Pauli matrices, and $F\equiv U_{(12)}$
denotes the flip operator. Then $H\y(\rho)=2\tr(\rho\12)$, and
$\Hr=4\rho$.
In the context of the class studied in subsection 4.2 the
assumptions made at the beginning of the present subsection amount to
postulating that each $\Va N\1M$ is a linear combination of permutation
operators. Thus $\Va\infty$ can be chosen as an arbitrary polynomial
in the variables $\tr(\rho\1k)$. Repeating the arguments in 4.2 we find
that $\Hr$ is now an arbitrary polynomial in $\rho$ whose coefficients
are symmetric polynomials in the eigenvalues of $\rho$. Taking the
flip $F$ and $V_N=\j N2\1\emp F$ gives a trivial dynamics because
$F=F\1*$, as noted in the previous subsection. So one has to go to
higher order permutations.
The next possibility is to use directly formula (4.12) for general
polynomial generators. With $V=F$ it is easily evaluated using the
formula $\tr(A\otimes B F)=\tr(AB)$. This gives
$\tr\bigl(\sigma\otimes\rho \Lr_F(A)\bigr)
=\tr\sigma\otimes\rho \bigl(2F\12(\idty\otimes A)
-F\12(A\otimes\idty)-(A\otimes\idty)F\12 \bigr)
=2\bigl(\tr(\sigma)\tr(\rho A)-\tr(\sigma A)\tr(\rho)\bigr)$.
Hence
$$\eqalignno{
\Lr(A) &= 2(\rho(A)-A ) &(4.16a) \cr
\lrt(A)&= e\1{-2t}A +(1-e\1{-2t})\rho(A)\idty
\quad,&(4.16b)}$$
\ie the local evolution contracts exponentially fast to multiples of
the identity.
\beginsubsection 4.5 Failure of the disjoint homomorphism property
We have shown in section 3 that for a \net\ of generators to
generate a \mf\ dynamical semigroup in the sense of Definition 3.2 it
is sufficient that they be of \lc. Here we give some simple examples
to show that this condition is by no means necessary. These examples
also show that some of the characteristic features of the limiting
semigroups derived above are not valid for arbitrary \mf\ dynamical
semigroups, but are consequences of the special \lc\ form.
There is a standard way of obtaining a dynamical semigroup from a
Hamiltonian evolution: for any Hamiltonian $H=H\1*$ we may consider the
generator
$$ G\1H(A)=\Lindblad HA=i\com{H,i\com{H,A}}
\eqno(4.17)$$
Thus $G\1H$ is nothing but the square of the generator
$i\com{H,\cdot\,}$
of the Hamiltonian evolution. It is well known (see Theorem 2.31 of
\tref\Dav) that squaring the generator of a group of isometries on a
Banach space produces the generator of a contraction semigroup,
which is just the integral of the group of isometries with respect
to the convolution semigroup of the heat equation. Explicitly, we
have
$$\eqalign{
e\1{tG\1H}(A)&=\int_{-\infty}\1{+\infty} ds\
\mu_t(s) e\1{isH}Ae\1{-isH}\cr
\hbox{with }\qquad
\mu_t(s)&= (4\pi t)\1{-1/2}e\1{-\textstyle{s\12\over 4t}}
\quad.\cr}\eqno(4.18)$$
It is important to note that in this integral both positive and
negative $s$ enter. Thus squaring the generator of a non-reversible
quantum dynamical semigroup will not in general produce the
generator of another.
We now apply this construction to a \mf\ dynamical semigroup,
generated by a \net\ $H_N$ of Hamiltonian densities satisfying
(4.2). Let us denote the resulting \mf\ dynamical group by
$S_{t,N}(A)=\exp(it\absn NH_N)A\exp(-it\absn NH_N)$. We now square the
generator for each $N$, getting
$$\eqalign{
G_N(A)&= {\absn N}\12\Bigl(\Lindblad{H_N}A\Bigr) \cr
\Tt N(A)&= \int ds\ \mu_t(ds)\, S_{s,N}(A)
\cr}\eqno(4.19)$$
Now let $X\in\Yl$ be \qs. Then so is $S_{s,\dt}(X_{\dt})$. Using the
strong continuity of $S_{\dt,\dt}$ we then find that $\Tt\dt(X_\dt)$
is again \qs. Hence $\Tt\dt$ preserves \qsy. We can take the limit
$N\to\infty$ under the integral and obtain
$$ \Tty=\int ds\ \mu_t(ds)\,S_{s,\infty}
\quad.\eqno(4.20)$$
Hence $\Tt\dt$ is a \mf\ dynamical semigroup. The generator $G_\dt$
is clearly not of lattice class, since $\norm{G_N}$ grows like
${\absn N}\12$ rather than like $\absn N$. We know that the evolution
described by $S\y$ on the intensive variables $\CKA$ is given by a
Hamiltonian flow. The generator of this flow is a first order
differential operator. Its square, which generates the restriction
of $\Tty$ to $\CKA$ is hence a second order differential operator.
We may put this in probabilistic terms saying that the evolution of
intensive variables under $\Tty$ is given by a diffusion on $\KA$
rather than a flow. More precisely, we get a diffusion along the
orbits of the flow generated by $H\y$. We could also add several
generators like $G_\dt$ and obtain diffusions along higher
dimensional submanifolds in $\KA$ \tref\DWa. We note that the
generator (4.17) is very similar to the form considered in section
4.2: There we would have taken
${\absn N}\bigl(\Lindblad{H_N}A\bigr)={\absn N}\1{-1}G_N$.
Since $G_N$ has a well defined limit it is clear that
${\absn N}\1{-1}G_N$ goes to zero. We have noted this consequence of
the hermitian nature of $H_N$ before and used it in the proof of
Proposition 4.2.
The integral formula (4.20) not only gives the evolution of the
intensive observables but also the local evolution. It can no longer
be given by a local cocycle $\lrt$, because the equation determining
$\lrt$ (Lemma 3.7) presupposes the existence of the flow. The root of
this difficulty is the failure of the disjoint homomorphism property
(Theorem 3.6(2))for $\Tt N$, which is easily verified from the form of
the squared generator of $S\y$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\beginsection{5. Local and Global Evolutions.}
\beginsubsection{5.1 Global \mf\ dynamical semigroups need not be
local.}
The notion of \mf\ dynamical semigroup which we have used in this
paper, namely a limiting evolution of \qs\ \net s, is {\it a
priori} stronger than the original formulation of \tref\DWa\ as
a limiting evolution for the subset of intensive (i.e.
$\emp$-symmetric) observables only. We constrast these by saying
that the latter comprises an evolution of global or fully
site-avearged quantities only, which the former gives the evolution
in local regions as well.
So far we have given examples of operator \net s which generate
in the stronger local sense. In fact we can adapt section 4.4 to
demonstrate an operator \net\ which for which there is a
limiting global evolution, but {\it not} a limiting local evolution.
Thus the present notion of a \mf\ dynamical semigroup is indeed
stronger than the former notion.
Assume for the fixed \net\ $(N_\alpha)_{\alpha\in\aleph}$ that
$\abs{N}$ takes odd and even values infinitely often. We shall call
$N$ itself odd or even accordingly.
>From the operator
$H_{\set{1,2}}=2F-\idty$ of section 4.4, form the
bounded polynomial generator
$\hat G_N(\cdot)={\nabsn N}\Sym_N
\com{H_{\set{1,2}}\ ,\ \cdot\ },$
and set $G_N=(-1)\1{\nabsn N}\hat G_N$.
Thus, $G_\dt$ is like a bounded polynomial generator, except that
the $N\th$ element is multiplied by the alternating quantity
$(-1)\1{\nabsn N}$.
Clearly the two \net s
$$\Tt\dt\1\odd=\{\Ttn\mid N\ \hbox{odd}\ \}
\qquad\hbox{and}\qquad
\Tt\dt\1\even=\{\Ttn\mid N\ \hbox{even}\ \}$$
are mean-field
dynamical semigroups in the local sense, although on different
\net s of regions. But the local generators for the odd and even
\net\ are
$\locgen\1{\rho,\odd}=-4i\ad\rho$ and
$\locgen\1{\rho,\even}=4i\ad\rho$ respectively. Hence the
full \net\ $\Tt\dt$ can have no local \mf\ limit. On the other
hand, examining the global evolution one sees that the limitng
flow is in both cases trivial since
$\rho\circ\locgen\1{\rho,\odd}=
\rho\circ\locgen\1{\rho,\even}=0$; so
$\Tt\infty\1\odd X_\infty=\Tt\infty\1\even X_\infty=
X_\infty$ for any $\emp$-symmetric \net\ $X_\dt$ and
$t\in\Rl$.
Since for $\emp$-symmetric $X_\dt$ the sub\net s $\Ttn X_N$
for $N$ odd and $N$ even are
$\emp$-symmetric, we need only compare odd and even terms in the
full \net\ in order
to demonstrate $\emp$-symmetry for the full \net.
But
$$\lim_{N\ \odd\ \to\infty}
\ \ \lim_{M\ \even\ \to\infty}
\norm{\Tt N X_N-\j NM\1\emp \Tt M X_M}
=\norm{\Tt\infty\1\even X_\infty-\Tt\infty\1\odd X_\infty}=0$$
as required.
\beginsubsection{5.2 Dynamical stability of local evolutions.}
As we have stressed earlier, for \mf\ dynamical semigroups with the
disjoint homomorphism property the implementing map $\Lambda$
plays a
dual role. It implements the evolution of local states
$\sigma\mapsto\sigma\circ\lrt$ on the state spaces of tagged algebras,
and also the flow
$\flow$ via the equation $\Ft\rho=\rho\circ\lrt$.
Now we have seen that initially localized observables (i.e.
\net s of the form $\j \dt R\1I X_R$) develop in time
a symmetrized tail
in the algebra over the untagged sites. Suppose that in the limit as
$t\to\infty$, this tail in fact becomes dominant, so that the time
developed observable loses all information about its initial
localization. Working in the dual picture with an intial state
$\rho$ on each of the untagged algebras, this would mean that any
initial local state $\sigma$ on a tagged algebra $\A$ would evolve
through $\sigma\mapsto\sigma\circ\lrt$ towards the \mf\ state
$\Ft\rho$. This motivates the following definition.
\iproclaim Definition 5.1. We shall say that a local cocycle
is {\bf \ag\ } in a topology $\tau$ of $\KA$ if for each
$\rho,\sigma\in\KA$,
$$\tau-\lim_{t\to\infty}\sigma\circ\lrt-\Ft\rho=0\quad.$$
\eproclaim
Of course, when the local generator is Hamiltonian one would not
expect this type of asymptotic result. However, it is relatively
easy to find an $H$-Theorem for the joint evolution of local and
global states. (In \tref\DWa\ we
were able to prove an $H$-Theorem for the flow alone, but only under the
assumption that for some $\rho\in\KA$ and all $N$, $\rho\1N$ is an
invariant state for $\Ttn$).
We shall show that the
relative entropy (recalled below) of an arbitrary local state
$\sigma\circ\lrt$ with respect to
the global state $\Ft\rho$ is non-increasing in time.
In the following we let $S(\omega_1 ,\omega_2)$ denote the
entropy of $\omega_2\in\KA$ relative to $\omega_1\in\KA$ as defined for
normal states on a von Neumann algebra in \tref\Ara, and extended
to states on C*-algebras in \tref{\PW,\Kos}\ and also in
\tref\Pet.
The crucial property we shall need here is that if $\gamma:\A\1n\to
A\1n$ is a completely positive unital map, then
$S(\omega_1,\omega_2)\ge S(\omega_1\circ\gamma,\omega_2\circ\gamma)$.
In the particular case where both states are given by
non-singular densities $D_{\omega_1}$ and $D_{\omega_2}$ with respect
to a trace $\trace$,
$$ S(\omega_1,\omega_2)
=\trace\bigl(D_{\omega_2}(\log D_{\omega_2}
-\log D_{\omega_1})\bigr)
\quad.$$
\iproclaim Proposition 5.2.
Let $\Tt\dt$ be a \mf\ dynamical semigroup whose limit has the disjoint
homomorphism property with local cocyle
$\Lambda$ and implementing flow $\flow$.
Then for each $(\rho,\sigma)\in\KA\times\KA$, the function
$\nullinf\ni t\mapsto S(\Ft\rho,\lrt\sigma)$ is
non-increasing.
\eproclaim
\proof: Since $\Ft\rho=\rho\circ\lrt$, and since by Lemma 3.14
$\lrt$ is completely positive and unital, we have that
$$S(\Ft\rho,\sigma\circ\lrt)=S(\rho\circ\lrt,\sigma\circ\lrt)
\le S(\rho,\sigma)\quad.$$
\QED
Since $t\mapsto S(\Ft\rho,\sigma\circ\lrt)$ is only shown to be
non-increasing, rather than strictly decreasing, we are unable to
infer that $\Lambda$ is \ag. {\chg}In fact, in the purely
Hamiltonian case discussed in section 4.1 $S(\Ft\rho,\sigma\circ\lrt)$
is even a constant of the motion.
Hence we have to make do with the intuitive
picture that the trajectories of the local state at least remain in a
neighbourhood of the global state.
Furthermore, nothing is said about
the stability, asymptotic or otherwise, of the global state itself.
Thus even {\it with} an \ag\ cocycle, it can happen that
trajectories of the flow take wild paths. In order to obtain an
example, we can take a generator with chaotic flow, which is possible
by the completeness result at the end of section 4.2. The proof of the
following then Theorem shows that we may find an arbitrarily small
perturbation which leaves the flow unchanged, but modifies the cocycle
to an \ag\ one.
\iproclaim Theorem 5.3.
The set of generators whose local cocycles are norm-\ag\
is dense in $\Gen$.
\eproclaim
\proof:
\def\drt{\Delta\1\rho_t} \def\prt{P\1{\rho_t}} \def\xrt{X\1\rho_t}
Let $L\in\Gen$ generate a local cocycle $\Lambda$. For any
$\eps>0$ let $\Delta$ be the local cocycle generated by $L+\eps W$,
where $W$ is (proportional to) the generator of equation (4.16a):
$W\1\rho A=\bra\rho,A>\idty-A$ for any $A\in\A$. Since $\rho\circ
W\1\rho=0$, the flows generated by $L$ and $L+\eps W$ are identical.
We denote this flow by $\flow$. Our claim is that
$L+\eps W$ is norm-\ag\ for all $\eps>0$.
It useful to introduce for any $\rho\in\KA$ the projection
$P\1\rho:\A\to\A$ with $P\1\rho(A)=\bra\rho,A>\idty$. Thus
$W\1\rho=-(\id-P\1\rho)$. Since $\lrt\idty=\drt\idty=\idty$, and
dually $\rho\circ\lrt=\rho\circ\drt=\Ft\rho\equiv\rho_t$ we have the
relations
$$ \prt=P\1\rho\lrt=P\1\rho\drt=\lrt \prt=\drt \prt
\quad.\eqno(5.1)$$
We can therefore restrict $\drt$ to the range of the projection
$\id-\prt$. More formally, we introduce the operators
$$ \xrt=\drt-\prt=\drt(\id-\prt)=(\id-P\1\rho)\drt
\quad.$$
>From equation (5.1) we find that $\ddt\prt=\prt L\1{\rho_t}$. Hence
$\xrt$ satisfies the differential equation
$$ \ddt\xrt= (\xrt+\prt)(L\1{\rho_t}-\eps(\id-\prt))-\prt L\1{\rho_t}
=\xrt(L\1{\rho_t}-\eps\id) $$
with the initial condition $X\1\rho_0=(\id-P\1\rho)$. Clearly, this is
the same equation satisfied by $e\1{-\eps t}\lrt(\id-\prt)$, and by
uniqueness we conclude that
$$ \drt=(1-e\1{-\eps t})\prt+e\1{-\eps t}\lrt\quad.$$
As $t\to\infty$ the second term goes to zero, so that $\drt$
is norm-\ag.
\QED
\let\REF\doref
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% This is the reference file, ordinarily to be read before the
% beginning of the paper as well
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\ACKNOW
This work was started while N.G.D.\ was a Research Scholar at the
Dublin Institute for Advanced Studies. N.G.D.\ thanks Yu.M. Suhov
for a useful discussion.
R.F.W\ would like to thank the Dublin Institute for Advanced Studies
for the hospitality during a stay in the summer of 1991. R.F.W.\
is supported by a Heisenberg fellowship of the DFG in Bonn.
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