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\headline={\hfill{\fivepoint Sept 26/93 }}
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\tabsatz
\centerline{\hbf Transitivity of Locality}
\sabsatz
\centerline{\hbf and Duality in Quantum Field Theory.}
\sabsatz
\centerline
{\hbf Some Modular Aspects}
\vskip 0.4cm
\centerline{\caps H.J. Borchers}
\mabsatz
\centerline{Institut f\"ur Theoretische Physik}
\centerline{Universit\"at G\"ottingen}
\centerline{Bunsenstrasse 9, D 3400 G\"ottingen}
\vskip 0.3cm
\centerline{and}
\vskip 0.3cm
\centerline{\caps Jakob Yngvason}
\mabsatz
\centerline{Science Institute}
\centerline{University of Iceland}
\centerline{Dunhaga 3, IS 107 Reykjavik, Iceland}
\babsatz
\centerline {\bf ABSTRACT}
\bigskip
Duality conditions for Wightman fields are formulated in terms of the
Tomita conjugations $S$
associated with algebras of unbounded operators. It is shown
that two fields that are relatively local to an irreducible
field fulfilling a condition of this type are relatively local to each
other.
Moreover, a local net of von Neumann algebras associated with such a
field satisfies (essential) duality. These results do not rely on
Lorentz covariance but follow from the observation that two
algebras of (un)bounded operators with the same Tomita conjugation have the
same (un)bounded weak commutant if one algebra is contained in the other.
\babsatz\vfill\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent{\bbf 1. Introduction}
\bigskip
The modular theory of Tomita and Takesaki [To, Ta]
is (in its simplest version) the
study of the antilinear map $S: A\Omega\mapsto A^*\Omega$ where $A$ runs
through the elements of a von Neumann algebra $\cm$ having $\Omega$ as a
cyclic and separating vector. In this paper we shall call $S$ a {\it
Tomita conjugation}. The basic result of the theory is Tomita's
theorem which states that the polar decomposition of $S$
gives rise to an antiunitary isomorphism between $\cm$ and its commutant
$\cm^\prime$, and a one-parameter group of automorphisms of $\cm$. Quantum
field theory in the algebraic setting of Haag, Kastler and Araki [H1] is a
natural client of Tomita-Takesaki theory because analytic vectors for the
energy (in particular the vacuum) are cyclic and and separating for the
local von Neumann algebras. Indeed, several recent developments in
algebraic quantum field theory are based on modular theory, see, e.g.,
[B2-3,BGL,BAL1-2,BSch,BSu,GF,Wie1-2].
In their pioneering work [BW1-2] Bisognano and Wichmann analyzed the
modular structure of algebras generated by Lorentz covariant Wightman field
operators localized in wedge shaped regions. They showed that the
modular conjugation is essentially given by the PCT operator and the modular group
is the one parameter group of Lorentz-boosts defined by the wedge. Using
this result they then proved essential duality [Ro] of local nets of von
Neumann algebras generated by Lorentz covariant Wightman fields. The
connection between modular structure and duality was further studied in
[Ri] within the Haag-Kastler-Araki framework. In two
space-time dimensions a geometrical interpretation of the modular operators
of wedge algebras has recently been derived from the spectrum condition for
the translation group alone [B2]. This result has been used to investigate
duality properties in conformal field theory by algebraic
methods [BGL, GF], without reference to Wightman fields.
Coming back to the work of Bisognano and Wichmann, one may
distinguish between three
separate issues in their papers.
The first is the identification of modular operators
associated with the wedge algebras of the {\it unbounded} field operators.
This part relies on the Lorentz covariance of the quantum field. The second
is the use of the modular structure to derive criteria for the existence of
local nets of {\it bounded} operators associated with the field. This
aspect has been further studied by Driessler, Summers and Wichmann [DSW],
Buchholz [Bu1] and Inoue [I3]. The third is the derivation of duality
properties
of local nets of von Neumann algebras associated with the field, also
discussed in [W], [DSW], [B] and [I3].
In a sequence of papers [BY1-4] we have studied the connection
between Wightman
fields and local nets of von Neumann algebras from a somewhat different
point of view, stressing the relation to noncommutative moment problems. In
[BY4] we formulated a criterion for the existence of a local net of von
Neumann algebras associated with a given Wightman field in terms of a
certain positivity property of the Wightman functions (see Theorem 4.4
below). This criterion does not rely on any modular structure, a fact that
was somewhat hidden in the original exposition of this result in [BY1]. In
particular, it applies to fields without Lorentz covariance, and can even
be extended in a straightforward way to fields with more general
localization properties than Wightman fields (e.g., fields localized in
space-like cones [BF] and to fields on curved space-time).
In view of this result it appears natural to regard the modular structure
of algebras of unbounded field operators as a separate ingredient and ask
which additional properties of the field and associated nets of von Neumann
algebras
can be derived from informations about this structure. In the
present paper we consider two closely related properties of this sort.
The first is the transitivity of relative locality, that
was established in [B1] for
Lorentz covariant Wightman fields. The second is the condition of
(essential) duality referred to above.
It turns out that only the elementary part of modular theory is needed for
the present purpose, namely the part which does not make use of the polar
decomposition of the Tomita conjugation $S$. Such an operator may be defined for
linear $*$-invariant families (not only algebras) of unbounded operators
with a cyclic vector, provided the vector is also cyclic for the unbounded
weak commutant of the family. In the next section we discuss this general
mathematical framework. The main observation, on which all subsequent
results are based, is the following one: If one operator family is
contained in another and both have the same Tomita conjugation, then the
(un)bounded weak commutants of the families are identical. In the case of
von Neumann algebras and bounded commutants, this statement follows
immediately from Tomita's theorem. However, it is in fact an elementary
result with a simple proof that applies also to unbounded
operators. Because of slight complications due to domain questions we
bring this result in two versions below (Propositions 2.7 and 2.9).
In Section 3 we consider nets of unbounded operator algebras generated by
Wightman fields. In order to avoid domain problems we require mild
energy bounds for the field operators.
We formulate general duality conditions for such nets in
terms of the Tomita conjugations, and show how they lead to the concept of
equivalence classes of relatively local fields (Theorem 3.5). We also point
out that duality itself is really not the essential point, but rather the
identity of the Tomita conjugations that follows from it (Theorem 3.6). Since
applications to more general objects than Wightman fields are
also of potential interest (e.g., to fields
with different localization properties, or fields on curved space time),
we have strived to isolate those aspects of the
Wightman framework that are really needed to obtain these results.
%We find such a reduction to the bare bones also helpful for a better
%understanding
%of the questions involved, although we
%realize that driving abstraction too far may have the opposite effect.
In Section 4 we discuss the connection between modular properties of
an unbounded field net and a net of von Neumann algebras associated with it,
extending results of Bisognano and Wichmann and of Driessler, Summers and
Wichmann to our general framework. In particular we show that duality of
the unbounded net leads to duality of the net of von Neumann algebras
(Theorem 4.5), and that if an irreducible field within an equivalence class
of relatively local fields can be associated with a local net of von
Neumann algebras, then the whole equivalence class can be associated with
such a net (Theorems 4.6 and 4.7). In the final Section 5 we generalize the
previous considerations to include Fermi fields as well as Bose fields.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\babsatz
\noindent {\bbf 2.\ Tomita conjugations and unbounded commutants}
\mabsatz
We begin by fixing the terminology and introducing some notation. Let $\CH$
be a Hilbert space and $\CD$ a linear subdomain of $\CH$. The family of all
closable linear operators $X$, $\ch\supset D(X)\to\ch$, such that
$\cd\subset D(X)\cap D(X^*)$ will as in [Sch] be denoted by
$\cc^\dagger(\cd,\ch)$. More generally, if $\cd_1\subset\ch$, we denote by
$\cc^\dagger(\cd,\cd_1)$ the subfamily of all operators in
$\cc^\dagger(\cd,\ch)$ with range in $\cd_1$. A {\it $*$-operator family}
on
$\CD$ is a family $\cf$ of closable, linear operators $\CD\to\CH$ such that
if
$X\in\CF$, then $X^*\mid\cd\in\cf$. A $*$-operator family $\cf$ is called
{\it linear}, if all linear combinations of elements in $\cf$ belong to
$\cf$. A linear $*$-operator family $\cp$ is a $*$-operator {\it algebra}
if the operators $X\in\cp$ leave $\cd$ invariant and $\cp$ is also closed
under operator multiplications. The largest $*$-operator algebra on $\cd$
consists of all operators $X$ with domain $\cd$ such that $\cd\subset
D(X^*)$ and $X$ and
$X^*$ leave $\cd$ invariant. This algebra is denoted by
$\cl^\dagger(\cd)$. If $\cd_0\subset\cd$ is a linear subdomain of $\cd$ we
use the notation $(\cf,\cd_0)$ for the family consisting of the
restrictions of the operators in $\cf$ to $\cd_0$. It is clear that
$(\cf,\cd_0)$ is a $*$-operator family on $\cd_0$.
The {\it graph topology\/} induced on a
domain $\cd$ by a family $\cf$ of operators on $\cd$ is defined by
the set of all seminorms of the form
$$\Vert\psi\Vert_{X_1,\dots,X_N}=\Vert
\psi\Vert+\sum_{i=1}^N\Vert X_i\psi\Vert\eqno(2.1)$$ with
$X_1,\dots,X_N\in\cf$. If $\cd_0\subset\cd$ is dense in $\cd$ in the
graph topology induced by $\cf$, then it follows in particular that
$\cd_0$ is a {\it core} for each operator $X\in\cf$, i.e., the closure
$\bar X=X^{**}$ of $X$ as an operator on $\cd$ is equal to the closure
of the restriction $X\mid\cd_0$. In fact, the domain of
$(X\mid\cd_0)^{**}$ is the closure of $\cd_0$ with respect to the graph
topology induced by the single operator $X$. If $\cf$ is an {\it algebra},
and $\cd_0$ is a core for each $X\in\cf$, then it follows conversely that
$\cd_0$ is dense in $\cd$ in the graph topology induced by $\cf$ [Sch].
For general families $\cf$, however, this need not be true, i.e.,
$\cd_0$ may be a
core for each single operator $X\in\cf$ without being dense in
the graph topology induced by the whole family.
\smallskip
Weak and strong commutants of $*$-operator families play an important role
in subsequent considerations. For weak commutants we use the following
terminology:
\medskip
\noindent {\bf 2.1 Definition.} Let $\cf$ be a $*$-operator family on $\cd$
and let $\cd_0$ be a linear subdomain of $\cd$. The {\it unbounded weak
commutant} of $\cf$ on $\cd_0$ consists of all operators
$Y\in\cc^\dagger(\cd_0,\ch)$, such that the equality $$\la
X^*\phi,Y\psi\ra=\la Y^*\phi,X\psi\ra\eqno(2.2)$$ holds for all $\phi$,
$\psi\in\cd_0$. The unbounded weak commutant is denoted by
$(\cf,\cd_0)^{uw}$.
The {\it bounded part\/} of $(\cf,\cd_0)^{uw}$ is called the {\it weak
commutant\/} and denoted by $(\cf,\cd_0)^{w}$.
\smallskip For later reference we state without proofs
some elementary properties of the
unbounded and bounded weak
commutants.
\medskip
\goodbreak
\noindent
{\bf 2.2 Lemma.}{\it
{\rm (i)} The restriction of the unbounded weak commutant to $\cd_0$, i.e.,
$((\cf,\cd_0)^{uw},\cd_0)$, is a linear $*$-operator family on $\cd_0$.
{\rm (ii)} $(\cf,\cd_0)^{w}$ is a weakly closed and $*$-invariant linear
subspace of the algebra $\CB(\ch)$ of bounded operators on $\ch$.
{\rm (iii)} If $\cp$ is a $*$-algebra on $\cd$ and $\cg$ is a $*$-family of
algebraic generators\footnote\dag{\rm By this we simply mean that $\cp$
consists of finite sums of products of elements in $\cg$. It
is, however, clearly sufficient to require that these linear combination
are dense in the topology on $\cp$ generated by the seminorms
$X\mapsto\Vert X\psi\Vert$ and $X\mapsto\Vert X^*\psi\Vert$ with $\psi\in\cd$.} for $\cp$, then $(\cg,\cd)^{uw}=(\cp,\cd)^{uw}$.
{\rm (iv)} If $\cf$ is a $*$-operator family on $\cd$ and $\cd_0\subset\cd$
is a
core for every operator in $\cf$, then $(\cf,\cd_0)^{w}=(\cf,\cd)^{w}$.
{\rm (v)} Let $\cf$ be a $*$-operator family on $\cd$. If $\cd_0\subset\cd$
and $Y\in(\cf,\cd_0)^{uw}$ are such that for each $X\in\cf$
the domain $\cd_0$
is dense
in $\cd$ in the graph topology induced by $\{Y, Y^*, X, X^*\}$, then
$Y\in(\cf,\cd)^{uw}$.}
\smallskip
Besides weak commutants we also consider strong commutants:
\noindent{\bf 2.3 Definition.} Let $\cf$ be a family of closable linear
operators on $\cd$. The {\it unbounded strong commutant} of
$\cf$, denoted $(\cf,\cd)^{us}$, consist of those
closable operators $Z$ that satisfy
(a) $\cd\subset D(Z)$, $Z\cd\subset\cd$ and $\cf\cd\subset D(Z)$,
(b) $[X,Z]\psi=0$ for all $X\in\cf$, $\psi\in \cd$.
\noindent The bounded part of $(\cf,\cd)^{us}$ is called the {\it strong
commutant} and denoted by $(\cf,\cd)^{s}$.
\smallskip
The following properties are easily verified, using Lemma 2.2 (iii) and
(v):
\medskip
\noindent
{\bf 2.4 Lemma.}{\it
{\rm(i)} Let $\cp$ be a $*$-algebra on a domain $\cd$. If
$Y\in(\cp,\cd)^{uw}$ and $Y$ leaves $\cd$ invariant, then
$Y\in(\cp,\cd)^{us}$. Conversely, if $Y\in(\cp,\cd)^{us}$ and $\cd\subset
D(Y^*)$, then $Y\in(\cp,\cd)^{uw}$.
{\rm (ii)} Let $\cp$ be a $*$-algebra on a domain $\cd$ and $\cg\subset\cp$
a $*$-family of algebraic generators for $\cp$. Let $Y\in\cl^\dagger(\cd)$.
If $Y\in(\cg,\cd_0)^{uw}$ for some $\cd_0\subset\cd$, and
for each $X\in\cg$
the domain $\cd_0$
is dense
in $\cd$ in the graph topology induced by $\{Y, Y^*, X, X^*\}$,
then $Y\in(\cp,\cd)^{us}$.}
\smallskip
Next we consider the definition of Tomita conjugations for $*$-families of
unbounded operators.
Let $\cf$ be a linear $*$-operator family on a dense domain $\cd\subset\ch$
and let $\Omega$ be a vector in $\cd$ that is cyclic for $\cf$ as well as
for $(\cf,\cf\Omega)^{uw}$. We consider only such families containing
the unit operator, so $\Omega\in\cf\Omega$. For every $X\in\cf$ and $Y\in
(\cf,\cf\Omega)^{uw}$ we have
$$\la Y\Omega,X^*\Omega\ra=\la X\Omega,
Y^*\Omega\ra.\eqno(2.3)$$
Because $\{Y\Omega\mid Y\in (\cf,\cf\Omega)^{uw}\}$ is dense
in $\ch$, we can define an antilinear operator $S^{(0)}:\ \cf\Omega\to\ch$
by $$S^{(0)}X\Omega=X^*\Omega.$$ It follows also from (2.3) that
${S^{(0)}}^*Y\Omega=Y^*\Omega$ if $Y\in(\cf,\cf\Omega)^{uw}$. In
particular, $S^{(0)}$ has a densely defined adjoint and is hence
closable. We denote its closure by $S$, or by $S_\cf$ when the dependence
on $\cf$ is important. The dependence of $S$ on the vector $\Omega$ can be
suppressed in the notation because $\Omega$ will be fixed in each case
considered in the sequel.
The closed, antilinear operator $S$ satisfies $S^2={\rm id}$. It has the
polar decomposition $$S=J\Delta^{1/2}=\Delta^{-1/2}J\quad\hbox{with
}\Delta=(S^*S)\geq 0\quad\hbox{and } J=J^*,\quad J^2=1.$$ We shall not make
use of this decomposition of $S$, however, except for the trivial remark
that
since $J$ is bounded, the domain $D(S)$ is the closure of $\cf\Omega$ with
respect to the graph topology induced by $\Delta^{1/2}$. The papers of
Inoue [I1-2] contain several results on the action of the modular group
generated by $\Delta$ in the case $\cf$ is an algebra satisfying some
additional hyptheses.
The domain of the adjoint $S^*=J\Delta^{-1/2}$ contains
$(\cf,\cf\Omega)^{uw}\Omega$, and
$${S}^*Y\Omega=Y^*\Omega\eqno(2.4)$$
if $Y\in(\cf,\cf\Omega)^{uw}$. If $\cf$ is an algebra of bounded operators,
then $S^*$ is equal to the Tomita conjugation of the von Neumann
algebra $\cf^\prime$, see, e.g., [KR]. For unbounded operator families the
domain of
$S^*$ is in general strictly larger than $(\cf,\cf\Omega)^{uw}\Omega$.
There are important cases in quantum field theory, however, where
equality holds as discussed in the next setion.
Let $\cf_1$ and $\cf_2$ be two linear $*$-operator families on a common
domain $\cd$ with $\cf_1\subset\cf_2$ and thus $(\cf_2,\cd)^{uw}\subset
(\cf_1,\cd)^{uw}$. If $\Omega$ is cyclic for both $\cf_1$ and
$(\cf_2,\cd)^{uw}$, then $S_{\cf_1}$ and $S_{\cf_2}$ are well defined and
$S_{\cf_1}\subset S_{\cf_2}$. Since $S_{\cf_i}$ is, by definition, the
closure of $S_{\cf_i}\vert \cf_i\Omega$, $i=1,2$, one immediately shows:
\noindent{\bf 2.5 Lemma.} {\it Let $\cf_1$ and $\cf_2$ be as above. The
following are equivalent:
{\rm (i)} $S_{\cf_1}=S_{\cf_2}$.
{\rm (ii)} $\cf_2\Omega\subset D(S_{\cf_1})=D(\Delta_{\cf_1}^{1/2})$.
{\rm (iii)} $\cf_1\Omega$ is a core for $\Delta_{\cf_2}^{1/2}$.}
\smallskip A useful sufficient criterion for (iii) is that
$\cf_1\Omega\subset D(\Delta_{\cf_2}^{1/2})$ and $\cf_1\Omega$ is
invariant under the unitary group $\Delta_{\cf_2}^{it}$ generated by
$\Delta_{\cf_2}$. For later use we also note the following simple result.
\noindent{\bf 2.6 Lemma.} {\it Let $\cf_1$ and $\cf_2$ be as above. Assume
there exists a $*$-operator family $\cq\subset (\cf_2,\cf_2\Omega)^{uw}$
such that $\Omega$ is cyclic for $\cq$ and $S_{\cf_1}^*=S_\cq$. Then
$S_{\cf_1}=S_{\cf_2}$.}
\sabsatz{\it Proof.\/} Since $\cq\subset (\cf_2,\cf_2\Omega)^{uw}$ we
have $S_\cq\subset S_{\cf_2}^*$ by (2.4) and thus
$S_{\cf_2}\subset S_\cq^*=S_{\cf_1}$.\hfill$\Box$
\smallskip
We now assume that $\cf_1$ and $\cf_2$ have the same Tomita conjugation and
want to draw some conclusions about the unbounded weak commutants of the
operator families $\cf_1$ and $\cf_2$. Our first result applies to the case
that the larger family is an {\it algebra}.
\medskip
\noindent {\bf 2.7 Proposition.} {\it Let $\CP$ be a $*$-operator algebra
on a dense domain $\CD$ and $\CF\subset
\CP$ a linear $*$-operator subfamily containing the unit operator. Suppose
$\Omega\in\CD$ is cyclic for $\CF$ and $(\CP,\cp\Omega)^{uw}$ (and hence
also for $\cp$ and $(\CF,\cp\Omega)^{uw}$). If the corresponding Tomita
conjugations for $\cf$ and $\cp$ are equal, $S_{\CF}= S_{\CP}$, then
$(\CF,\CP\Omega)^{uw}\subset (\CP,\CF\Omega)^{uw}.$ If in addition
$\cf\Omega$ is a core for each $X\in\cp$, then
$(\cf,\cp\Omega)^{uw}=(\cp,\cp\Omega)^{uw}$.}
\sabsatz {\it Proof.} Let $Y\in (\CF,\CP\Omega)^{uw}$, $X\in \CP$ and
$Z_1$, $Z_2\in
\CF$. Then $$\langle Y^*Z_1\Omega, XZ_2\Omega\rangle=\la Z_1\Omega,
YXZ_2\Omega\ra=
\la Y^*\Omega, Z_1^*XZ_2\Omega\ra,$$ where we have used that $\cp$ is an
algebra and $Y\in (\CF,\CP\Omega)^{uw}$. If we insert $1=S_\CP^2=S_\CF
S_\CP$ and use that $S_\CF^*Y^*\Omega=Y\Omega$, we obtain $$\la Y^*\Omega,
Z_1^*XZ_2\Omega\ra=\la Z_2^*X^*Z_1\Omega,Y\Omega\ra=
\la Y^*X^*Z_1\Omega, Z_2\Omega\ra=\la X^*Z_1\Omega,YZ_2\Omega\ra.$$ Hence
$Y$ commutes weakly with $X$ on $\CF\Omega$ and the first part of the
proposition is established.
Since $X\Omega\in\cp\Omega\subset D(Y)$, weak commutativity of $X$ and $Y$
on $\cf\Omega$ implies also that $$\langle X^*Z\Omega, Y\Omega\rangle=
\la Y^*Z\Omega,X\Omega\ra=
\la Z\Omega,YX\Omega\ra$$ for all $Z\in\cf$. This means that $Y\Omega\in
D((X^\dagger\vert\cf\Omega)^*)$, where we have denoted
$X^*\vert\cp\Omega=X^\dagger$. If $\cf\Omega$ is a core for $X^\dagger$ we
thus conclude, using that $\cf\Omega$ is dense,
$$[Y,(X^\dagger)^*]\Omega=0\qquad\hbox{for all $X\in\cp$,
$Y\in(\cf,\cp\Omega)^{uw}$}.\eqno(2.5)$$ Next, pick $X_1$, $X_2\in\cp$.
Since $\cp$ is an algebra, we obtain, using (2.5) twice, $$\eqalign{\la
X^*X_1\Omega,YX_2\Omega\ra&=
\la X^\dagger X_1\Omega,(X_2^\dagger)^*Y\Omega\ra =\la (XX_2)^\dagger
X_1\Omega, Y\Omega\ra\cr=\la X_1\Omega, ((XX_2)^\dagger)^*Y\Omega\ra&=\la
X_1\Omega, YXX_2\Omega\ra=\la Y^* X_1\Omega, XX_2\Omega\ra.\cr}$$ Hence $X$
commutes weakly with $Y$ on $\cp\Omega$ if $\cf\Omega$ is a core for each
operator in $\cp$.
\hfill$\Box$
\smallskip
\noindent{\bf 2.8 Remark.} In the case of two von Neumann
algebras $\CM_1\subset\CM_2$ with a common cyclic and separating vector,
Proposition 2.7 expresses the well known fact (see, e.g., [KR] Theorem
9.2.36) that $\CM_1=\CM_2$ if and
only if $\Delta_{\CM_1}=\Delta_{\CM_2}$. For two
Neumann algebras $\CM$ and $\CN$ with $\CN\subset\CM^\prime$
this means that $\CN=\CM^\prime$ if and only if
$\Delta_\CM={\Delta^{-1}_\CN}$, or equivalently, if and
only if $S_\CN^*=S_\CM$.
\smallskip
We now state and prove a variant of Proposition 2.7. In this version the
larger family is not necessarily an algebra, and the relevant domain is
generated by a subfamily of its strong commutant. We employ the following
notation: If $\cq$ is a $*$-operator family, then $D^*(\cq)$ denotes the
intersection of the domains of the adjoints, $D(Y^*)$, with $Y\in\cq$.
\medskip
\noindent {\bf 2.9 Proposition.} {\it Let $\CF_1\subset\cf_2$
be two linear
$*$-operator families (containing the unit operator) on a dense domain
$\CD$. Let $\cq$ be another $*$-operator family on
$\cd$ with $\cq\subset (\cf_2,\cd)^{us}$.
Suppose $\Omega\in\CD$ is cyclic for $\CF_1$ and
$\cq$ (and hence also for $\cf_2$ and $(\cf_1,\cd)^{us}$). If the
corresponding Tomita conjugations of $\cf_1$ and $\cf_2$ are equal,
$S_{\cf_1}=S_{\cf_2}$,
then $(\cf_1,\cq\Omega)^{uw}\cap \cc^\dagger(\cq\Omega, D^*(\cq))=
(\cf_2,\cq\Omega)^{uw}\cap
\cc^\dagger(\cq\Omega, D^*(\cq))$.
In particular,
$$(\cf_1,\cq\Omega)^{uw}\cap\CL^\dagger(\cd)\subset(\cf_2,\cq\Omega)^{uw}$$
and
$$(\cf_1,\cq\Omega)^{w}\cap(\cq,\cq\Omega)^w\subset(\cf_2,\cq\Omega)^{w}.$$}
{\it Proof.} Pick $Y\in (\cf_1,\cq\Omega)^{uw}\cap\cc^\dagger(\cq\Omega,
D^*(\cq))$ and $Z_1$, $Z_2\in\cq$. We first show that
$Z_2^*Y^*Z_1\Omega\subset D(S^*_{\cf_1})$, and $$S^*_{\cf_1}
Z_2^*Y^*Z_1\Omega =Z_1^*YZ_2\Omega.\eqno(2.6)$$ Indeed, if $X\in\cf_1$ we
have $$\eqalign{\la Z_2^*Y^*Z_1\Omega,S_{\cf_1} X\Omega\ra&= \la
Z_2^*Y^*Z_1\Omega,X^*\Omega\ra =\la Y^*Z_1\Omega,X^*Z_2\Omega\ra\cr=\la
XZ_1\Omega,YZ_2\Omega\ra&=
\la
Z_1X\Omega, YZ_2\Omega\ra=\la X\Omega,Z_1^*YZ_2\Omega\ra,\cr}$$
which proves the statement. Next, take $X\in\cf_2$.
Since $S_{\cf_1}=S_{\cf_2}$ by assumption, we have
$S_{\cf_1} X\Omega=X^*\Omega$.
Using this and (2.6) we now
obtain for $Z_1$, $Z_2\in\cq$ and $Y$ as above:
$$\eqalign {\langle Y^*Z_1\Omega, XZ_2\Omega\rangle&=
\la Y^*Z_1\Omega, Z_2X\Omega\ra=
\la Z_2^*Y^*Z_1\Omega, X\Omega\ra
=\la Z_2^*Y^*Z_1\Omega, S_{\cf_1}^2 X\Omega\ra\cr=
\la X^*\Omega, Z_1^*YZ_2\Omega\ra&=\la Z_1X^*\Omega,YZ_2\Omega\ra =\la
X^*Z_1\Omega,YZ_2\Omega\ra.\cr}$$ Hence $Y$ commutes weakly with $\cf_2$ on
$\cq\Omega$. The rest of the statement is a consequence of the fact that
$\CL^\dagger(\cd)$ and $(\cq,\cq\Omega)^w$ are both contained in
$\cc^\dagger(\cq\Omega, D^*(\cq))$.
\hfill$\Box$
\babsatz
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
{\bbf 3.\ Transitivity of locality and duality for quantum fields}
\mabsatz
Let $\{\Phi_\iota\}_{\iota\in I}$ be a family of translationally covariant
Bose-Wightman fields defined on a common invariant domain $cd$ in a Hilbert space $\ch$. Each field
${\Phi_\iota}$ is an operator valued tempered distribution on Minkowski space ${\bf
R}^d$; the index set $I$ may be finite or infinite. If $\co$ is an
open subset of ${\bf R}^d$ we denote by $\cp(\co)$ the $*$-algebra
generated algebraically by all smeared field operators $\Phi_\iota(f)$,
$\iota\in I$,
with support of the test functions $f$ in $\co$.
The algebras $\cp(\co)$ have certain properties that follow from the
Wightman axioms. Not all of these properties will be needed here,
and we list only those that are
important for the present purpose.
\smallskip\noindent
(A) The $\cp(\co)$ are $*$-operator algebras (containing the unit operator)
on a common dense domain $\cd\subset\ch$.
\smallskip\noindent
(I) If $\co_1\subset \co_2$, then $\cp(\co_1)\subset\cp(\co_2)$.
\smallskip\noindent
(L) If $\co_1$ and $\co_2$
are space-like separated, then $\cp(\co_1)$ and $\cp(\co_2)$ commute on
$\cd$.
\smallskip\noindent
{F}or convenience, we shall refer to a net of $*$-algebras $\cp(\co)$,
satisfying (A) and (I) as a {\it field net}, and if (L) is also fullfilled
we speak of a field net satisfying locality or a {\it local field net}.
%A field net $\cp(\cdot)$ is called translationally covariant if the
%following holds:
%\smallskip\noindent
%
%(T) There is a continuous, unitary representation $U$ of the translation
%group $\BR^d$ such that $\cd$ is invariant under $U(a)$ and
%$U(a)\cp(\co)U(a)^{-1}=\cp(\co+a)$ for all $a\in\BR^d$.
%\smallskip\noindent
%
%In this paper we shall in fact only make use of covariance with respect
%to time translations. The generator of the time translations is the
%Hamiltonian $H$ that will be assumed to fulfill the spectrum condition:
%\smallskip\noindent
%
%(S) The spectrum of $H$ is nonnegative.
%\smallskip\noindent
%
%Moreover, we assume that a vacuum vector exists:
%
%(V) There is a vector $\omega\in\cd$ such that $H\Omega=0$.
%\smallskip
Translational covariance and spectrum condition imply in Wightman
theory the Reeh-Schlieder property: If
$\Omega$ denotes the vacuum vector and $\co$ a nonempty open set, then
one has
\smallskip\noindent
(RS) For all $\co$, $\cp(\co)\Omega$
is dense in $\ch$ .
\smallskip
Under the same premises one can even derive a considerably stronger
property for any translationally covariant
subnet $\cp_0(\cdot)$ such that $\cp_0(\BR^d)$ is generated by
$\cup_{x\in\BR^d}\cp_0(\co+x)$ for all $\co$:
\smallskip\noindent
(BZ) $\cp_0(\co)\Omega$ is
dense in
$\cp_0(\BR^d)\Omega$ in the
graph topology induced by $\cp(\BR^d)$.
\smallskip
This result was essentially proved in [BZ], Theorem 1. Since the version
above is slightly stronger than the statement in [BZ], we prove it in
the Appendix.
Instead of requiring that the algebras $\cp(\co)$ are from the outset
defined for all open sets $\co$, it
is often natural to consider to begin with some particular class $\ck$ of
such subsets,
e.g.\ all (open) double cones. For the purpose of this paper a specific
geometrical interpretation of these sets is not needed, but only that
the class $\ck$ is sufficiently rich in the following sense:
\smallskip\noindent
(K) For each pair of space-like separated points $x$,
$y\in\BR^d$, there
are space-like separated $K_1$, $K_2\in\ck$ with $x\in K_1$ and
$y\in K_2$. Morover, if $K\in\ck$, then $K^\prime$ contains some
$K_1\in\ck$.\smallskip
When such a class $\ck$ has been singled out one may extend the net to
arbitrary regions $\co$ by defining $\cp(\co)$ as the algebra generated by
$\cp(K)$ with $K\subset \co$, $K\in\ck$:
$$\cp(\co)=\bigvee_{K\subset \co, K\in\ck}\cp(K)\eqno(3.1)$$ with
the understanding that $\cp(\emptyset)=\BC\cdot 1$. The operators
in $\cp(\co)$ are in general unbounded and we prefer not to close the
algebras in any particular topopolgy. Hence (3.1) is meant in the purely
algebraic sense. If $\ck$ is the class of open double cones and the net
$\cp(\cdot)$ is generated by Wightman fields smeared with test functions of
compact support, then the definition (3.1) agrees with the previous one,
i.e. $\cp(\co)$
is the algebra generated by field operators whose test functions have
support in $\co$. This follows by partition of unity. One can envisage
more general cases, however, where (3.1) has to be included as a separate
definition, e.g., when $\ck$ is the class of
space-like cones and $\cp(K)$ is generated by cone-localized operators,
or when $\ck$ contains only double cones of some minimal size.
In these cases (RS) and (BZ) should not be required for all nonempty, open
sets but only for sets containing a (nonempty) $K\in\ck$.
Besides $\ck$ we shall consider a further class of sets, $\cw$,
related to (K) in the following way:
\smallskip\noindent
(KW) Each $K\in \ck$ is contained in some $W\in \cw$, and if
$K_1$, $K_2\in\ck$ are space-like separated, then there are space-like
separated $W_1$, $W_2\in\cw$, such that $K_i\subset W_i$, $i=1,2$.
Moreover, each $W\in\cw$ contains some $K\in\ck$ and the same holds for
$W^\prime$.
\smallskip
For concreteness sake one
might think of $\cw$ as the class of all (open) space-like wedges, but
again such a
specific interpretation is not necessary. In fact, the
possibility $\cw=\ck$
is not excluded.
If conditions (K) and (KW) are fulfilled, then locality of the whole field
net $\cp(\cdot)$ is equivalent to locality of the net restricted to either
one of the classes $\ck$ or $\cw$. Denoting the space-like complement of
the closure of $\co$ by $\co^\prime$, locality can thus be stated as the
condition $$\cp(\co^\prime)\subset (\cp(\co),\cd)^{us}\eqno(3.2)$$ for all
$\co\in\ck$, or equivalently, all $\co\in \cw$.
Next we want to discuss the concept of {\it duality} for
local field nets.
If the algebras $\cp(\co)$ are von Neumann algebras on $\ch=\cd$
the usual duality
condition [H2] means that the inclusion in (3.2) is an equality.
Motivated by the
considerations in the preceeding section we generalize this in the
following way.
Let $\cp(\cdot)$ be a local field net with a vector $\Omega$ satisfying
(RS) for the class $\ck$. Then for each $\co$ belonging to $\ck$ or $\cw$
the vector $\Omega$ is
cyclic for $\cp(\co)$ and $\cp( \co^\prime)$. From (3.2) it follows that
the
Tomita conjugations $S_{\cp(\co)}$ and $S_{\cp(\co^\prime)}$ are well defined
with $S_{\cp(\co^\prime)}\subset S_{\cp(\co)}^*.$
\smallskip
\noindent{\bf 3.1 Definition.} The local field net $\cp(\cdot)$ satisfies
duality for the class $\ck$, or {\it $\ck$-duality}, if
$$S_{\cp(K^\prime)}= S_{\cp(K)}^*\eqno(3.3)$$ for all $K\in\ck$. In the
same way, we say that net satisfies duality for the class $\cw$, or {\it
$\cw$-duality}, if
$$S_{\cp(W^\prime)}= S_{\cp(W)}^*\eqno(3.4)$$
holds for all $W\in \cw$.
If the algebras $\cp(\co)$ consist of bounded operators this definition is
equivalent to the usual one, i.e., $\cp(\co)^{\prime\prime}=
\cp(\co^\prime)^\prime$ by Remark 2.8. It appears here as a natural
generalization, circumventing the problem of extending von Neumann's
commutator theorem to the case of algebras of unbounded operators. For
Wightman fields transforming covariantly with respect to a finite
dimensional representation of the Lorentz group the duality condition is
always fulfilled for
the class $\cw$ of space-like wedges in Minkowski space [BWI-II]. In this
case $S_{\cp(W)}$ can be explicitly computed for all $W\in \cw$ and by
inspection one sees that $S_{\cp(W)}^*=S_{\cp(W^\prime)}$.
Duality may be interpreted as a kind of maximality condition for local
field nets. In fact, as we shall now show, a local field net satisfying
$\cw$-duality leads to a unique equivalence class of fields that are
relatively local to each other.
To be precise, we must distinguish between two kinds of relative locality,
strong and weak. Let $\cp_1(\cdot)$ and $\cp_2(\cdot)$ be two field nets
with a common invariant dense domain $\cd$ in a Hilbert space
$\ch$. We say that the nets are {\it strongly relatively local} on $\cd$
if
$\cp_1(\co_1)$ and $\cp_2(\co_2)$ commute on $\cd$ for space like separated
$\co_1$ and $\co_2$. If $\cd_0\subset\cd$ and $\cp_1(\co_1)$,
$\cp_2(\co_2)$ commute weakly on $\cd_0$ in the sense of (2.1), we say that
the nets are {\it weakly relatively local} on $\cd_0$. When checking (weak)
relative locality, it is as above sufficient to take $\co_1=W\in\cw$ and
$\co_2=W^\prime$,
because of (K) and (KW).
A connection between the two notions of relative locality follows
immediately from
Lemma 2.4 (ii):
\noindent{\bf 3.2 Lemma.\ }{\it Suppose $\cp_1(\cdot)$ and $\cp_2(\cdot)$
are
weakly local to each other on $\cd_0\subset\cd$. If there exist
$*$-families of
generators, $\cg_1(W)$ and $\cg_2(W^\prime)$, for the algebras $\cp_1(W)$
and
$\cp_2(W^\prime)$, $W\in\cw$, such that $\cd_0$ is
dense in $\cd$ in the graph topology induced by $\cg_1(W)\cup
\cg_2(W^\prime)$,
then $\cp_1(\cdot)$ and $\cp_2(\cdot)$ are strongly relatively local to
each
other on $\cd$.}
A sufficient condition for the hypotheses of Lemma 3.2 to hold can be
stated in
terms
of mild energy bounds for the generators.
\smallskip
\noindent {\bf 3.3 Definition.} Let $X$ be a closable operator and $H$ a
self-adjoint operator on $\ch$. Let
$E_I$ denote the spectral projector of $H$ correspondinging to $I\subset
\BR$. We say that $X$ obeys
{\it compact $H$-bounds}, if $E_I\ch\subset D(\bar X)$ and
$\bar X E_I$ is a bounded operator
for all bounded $I$.
\smallskip
An equivalent condition is that $\bar XF(H)$
is a bounded operator for {\it some} measurable function $F$ which
vanishes nowhere on the spectrum of $H$.
\smallskip
\noindent {\bf 3.4 Proposition.} {\it Suppose $\cg$ is a family of
operators on a dense
domain $\cd\subset \ch$. Let $H$ be self adjoint and suppose that
\noindent
{\rm (i)} The operators in $\cg$ obey compact $H$-bounds
\noindent
{\rm (ii)} $\cd$ is invariant under $e^{itH}$, $t\in \BR$, and for each
$\psi\in
\cd$, $X\in \cg$, the function $t\mapsto Xe^{itH}\psi$ is continuous and
polynomially bounded.
\noindent Then every dense domain $\cd_0\subset\cd$ that is invariant under
the unitary group $e^{itH}$, $t\in \BR$, is dense in $\cd$ in
the graph topology induced by
$\cg$.}
\smallskip
This result is a generalization of Lemma 5.2(b) in [DSW]. We prove it
in the Appendix.
After this preparation we now come to the main result of this section.
We consider three field nets, $\cp_0(\cdot)$,
$\cp_1(\cdot)$ and $\cp_2(\cdot)$ on a common dense domain $\cd$
containing a vector $\Omega$ such that $\cd_0=\cp_0(\BR^d)\Omega$ is
dense and (BZ) holds for $\co\in\ck$ if
$\cp(\cdot)$ is defined as the net generated by
$\cp_0(\cdot)$, $\cp_1(\cdot)$ and $\cp_2(\cdot)$. We
denote by $\cp_{0i}(\co)=\cp_0(\co)\vee\cp_i(\co)$
the $*$-algebra generated
by $\cp_0(\co)$ and $\cp_i(\co)$, and define
$\cd_{0i}=\cp_{0i}(\BR^d)\Omega$, $i=1,2$.
It is assumed that $\cd$, $\cd_0$
and $\cd_{0i}$ are invariant under time translations $e^{itH}$. This holds
automatically if the field nets transform covariantly under time
translations and $\Omega$ is invariant.
\smallskip
\noindent {\bf 3.5 Theorem.\ }{\it Assume
\noindent{\rm (a)} $\cp_0(\cdot)$ is local and satisfies $\cw$-duality.
\noindent {\rm (b)} $\cp_i(\cdot)$ is weakly relatively
local to
$\cp_0(\cdot)$ on $\cd_{0i}$, $i=1,2$.
\noindent Then $\cp_1(\cdot)$ and
$\cp_2(\cdot)$ are weakly relatively local to each other on
$\cd_0$.
\noindent If the algebras $\cp_i(W)$, $i=0,1,2$,
$W\in\cw$, possess families of generators, $\cg_i(W)$, satisfying the
conditions of Proposition 3.4, then the field nets are strongly relatively
local to each other on the common invariant domain $\cd$.}
{\it Proof.} Pick $W\in\cw$. We use Lemma 2.6 and Proposition 2.7 with
$\cf_1\equiv\cf=\cp_0(W)$, $\cf_2\equiv\cp=\cp_{02}(W)$ and
$\cq=\cp_0(W^\prime)$. Since $\cp_0(\cdot)$ is local and weakly relatively
local to $\cp_2(\cdot)$ we know that $\cf_2$ and $\cq$ commute weakly on
$\cd_{02}$. Duality of $\cp_0(\cdot)$ thus implies $S_{\cf_1}^*=S_\cq$, so
$S_{\cf_1}=S_{\cf_2}$ by Lemma 2.6. Since $\cp_1(W^\prime)\subset
(\cp_0(W),
\cp_{01}(W^\prime)\Omega)^{uw}$, it follows by Proposition 2.7 that
$\cp_1(W^\prime)\subset (\cp_2(W), \cp_0(W^\prime)\Omega)^{uw}$. From (BZ)
we know that $\cp_0(W^\prime)\Omega$ is dense in $\cd_0$ in the
graph topology induced by $\cp_1(W^\prime)\cup\cp_2(W)$. Hence, by Lemma
2.2 (v), $\cp_1(W^\prime)$ and $\cp_2(W)$ are weakly relatively local to
each other on $\cd_0$. The last statement of the theorem
follows from Lemma
3.2.\hfill$\Box$
\medskip
It is clear from the proof that the only role of the duality assumption is
to ensure that $\cp_0(W)$ and $\cp_{0i}(W)$ have the same Tomita
conjugation. If this is known by other means the conclusions of the theorem
thus also hold. If one applies Proposition 2.9 instead of Proposition 2.7
one obtains the following variant of Theorem 3.5.
\medskip\noindent {\bf 3.6 Theorem.\ }{\it
Assume
\noindent {\rm(a)} $\cp_0(\cdot)$ is local.
\noindent {\rm (b)} $\cp_i(\cdot)$ is strongly relatively
local to
$\cp_0(\cdot)$ on $\cd$, $i=1,2$.
\noindent {\rm (c)} The algebras $\cp_i(W)$ , $i=1,2$, $W\in \cw$,
have families of generators $\cg_i(W)$ such that $\cg_i(W)\Omega\subset
D(S_{\cp_0(W)})$.
\noindent Then $\cp_1(\cdot)$ and
$\cp_2(\cdot)$ are weakly relatively local to each other on
$\cd_0$.
\noindent If the generators satisfy the conditions of Proposition 3.4,
then the field nets are strongly relatively local to
each other on the common invariant domain $\cd$.}
{\it Proof.\/} As in the proof of Theorem 3.5 we put
$\cf_1=\cp_0(W)$
and $\cq=\cp_0(W^\prime)$, but this time we define
$\cf_2=\cp_0(W)+\cg_2(W)$. Hypothesis (c)
and Lemma 2.5 imply that the Tomita conjugations of $\cf_1$ and $\cf_2$ are
equal. Since $\cp_1(W^\prime)\subset
(\cf_1,\cq\Omega)^{uw}\cap\CL^\dagger(\cd)$ we conclude from Proposition
2.9 that $\cp_1(W^\prime)$ and $\cp_2(W)$ commute weakly on
$\cp_0( W^\prime)\Omega$. Property (BZ) then ensures commutativity on
$\cd_0$. The energy bounds then imply commutativity on $\cd$ as in
Theorem 3.5\hfill$\Box$
\babsatz\hfill\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
{\bbf 4.\ Duality for local von Neumann algebras generated by quantum
fields}
\mabsatz
In this section we start by reviewing some facts concerning duality and
essential duality for local nets of von Neumann algebras. We then discuss
the question when a Wightman field can be associated with such a net, and
link duality properties of the Wightman field with those of the von Neumann
net.
We consider a net of von Neumann algebras $\cm(K)$,
defined for some specified class $\ck$ of bounded open
subsets of $\R^d$ satisfying condition (K) of the last section, and assume
that
conditions (I), (A) and (RS) are fulfilled (with $\cm(\cdot)$ in
place of $\cp(\cdot)$).
For more general regions ${\co}$ (in particular for $\co=W$ or
$W^\prime$ with $W\in\cw$, or $\co=K^\prime$ with $K\in\ck$) we {\it define}
$$\cm({\co}):=\big(\bigcup_{K\subset{\co},\ K\in\ck}
\cm(K)\big)^{\prime\prime}.\eqno(4.1)$$(Since the algebras $\cm(K)$
consist of bounded operators it is convenient to include the weak closure
in the definition, which was not done in (3.1).) One says that the net
satisfies {\it duality} (more precisely $\ck$-{\it duality}) if
$$\cm(K)=\cm(K^\prime)^\prime\eqno(4.2)$$ for all $K\in\ck$. As remarked in
the last section, (4.2) is equivalent to the condition
$$S_{\cm(K)}=S_{\cm(K^\prime)}^*\eqno(4.3)$$ for the Tomita conjugations, if
the algebras are von Neumann algebras.
If duality does not hold for a net $\cm(\cdot)$, it is natural to ask
whether $\cm(\cdot)$ can at least be embedded into a net fulfilling this
condition. For a local net the answer is affirmative in the particular case
that the net satisfies the {\it essential duality} condition introduced
by Roberts
[Ro]. This means that besides $\cm$ the net
$$K\mapsto\cm^d(K):=\cm(K^\prime)^\prime\eqno(4.4)$$ is also local. In
fact,
locality of $\cm$ means that $\cm(K)\subset\cm^d(K),$ which implies
$\cm(K^\prime)\subset\cm^d(K^\prime),$ and thus
$\cm^d(K^\prime)^\prime\subset\cm(K^\prime)^\prime=\cm^d(K)$. If
$\cm^d$ is local the opposite inclusion holds and $\cm^d$ satisfies
duality. It is also easy to see that $\cm^d$ is the only net containing
$\cm$ with this property: If $\CN$ is a local net containing $\cm$, then
$\cm(K^\prime)\subset\CN(K^\prime)$, and thus
$\CN(K)\subset\CN(K^\prime)^\prime
\subset\cm(K^\prime)^\prime=\cm^d(K)$.
This implies also $\CN(K^\prime)\subset\cm^d(K^\prime)$, so if $\CN$
satisfies duality we have
$\cm^d(K)=\cm^d(K^\prime)^\prime\subset\CN(K^\prime)^\prime=\CN(K)$.
Hence any local extension of $\cm$ lies between $\cm$ and $\cm^d$, and if
essential duality holds for the net $\cm$ then it has a unique extension to
a net satisfying duality, namely $\cm^d$.
In [BW1,BW2] Bisognano and Wichmann established a link between essential
duality for the algebras localized in double cones and duality for algebras
localized in wedge-shaped regions. For convenience of the reader and later
reference we formulate this as a Lemma below. Our version involves only the
general conditions (K) and (KW) of Section 3.
Let $\ck$ and $\cw$ be two families of open subsets of
Minkowski space satisfying conditions (K) and (KW).
Let $\{\cm(K)\}_{K\in\ck}$ be a net of von Neumann algebras
and define $\cm(W)$ and $\cm(W^\prime)$ as in (3.1) for $W\in\cw$.
Moreover,
for $K\in \ck$ define
$$\widehat\cm(K):=\bigcap_{W\supset K,\, W\in\cw} \cm(W).\eqno(4.5)$$
By (KW) this is
always a local net if $\cm(\cdot)$ is local.
It is in general unrelated to the other extension
$\cm^d(\cdot)$ defined in (4.3), but if $\cm(\cdot)$ satisfies
$\cw$-duality
both extensions coincide:
\smallskip
\noindent
{\bf 4.1 Lemma.}
{\it If the duality condition $$\cm(W)^\prime=\cm(W^\prime)\eqno(4.6)$$
holds for all $W\in\cw$, then the net $\{\cm(K)\}_{K\in\ck}$ satisfies
essential duality and $$\cm^d(K)=\widehat\cm(K)\eqno(4.7)$$ for all
$K\in\ck$.}
{\it Proof.} The net $\widehat \cm$ can be extended to general regions
${\cal O}$ in Minkowski space by defining $\widehat\cm({\cal O})$ in the
same way as in (4.1) with $\cm$ replaced by $\widehat\cm$. We first show
that $$\widehat\cm(W)=\cm(W)\quad\hbox{and}\quad\widehat\cm(W^\prime)=
\cm(W^\prime)\eqno(4.8)$$ for all $W\in\cw$. It is clear that
$\cm(K)\subset\widehat\cm(K)$ for all $K\in\ck$ and thus $\cm({\cal
O})\subset\widehat\cm({\cal O})$ for all regions ${\cal O}$. Hence, by
(4.3), $\cm(W)=\cm(W^\prime)^\prime\supset\widehat\cm(W^\prime)^\prime.$ On
the other hand, by locality of $\widehat\cm$, $\widehat\cm(
W^\prime)^\prime\supset\widehat\cm(W)\supset\cm(W),$ so $\cm(W)=\cm(
W^\prime)^\prime=\widehat\cm(W^\prime)^\prime=\widehat\cm(W)$.
The next assertion is that
$$\cm(K^\prime)=\widehat\cm(K^\prime)\eqno(4.9)$$ for all $K\in\ck$. The
inclusion $\cm(K^\prime)\subset\widehat\cm(K^\prime)$ is clear, and if
$K_1\subset K^\prime$, there exist by assumption space like separated $W_1,
W\in\cw$ such that $K_1\subset W_1$ and $K\subset W$. This implies
$\widehat\cm(K_1)\subset\widehat\cm(W^\prime)=\cm(W^\prime)\subset
\cm(K^\prime)$ for all $K_1\subset K^\prime$, and thus
$\widehat\cm(K^\prime)\subset\cm(K^\prime)$.
Finally, we show that the net $\{\widehat\cm(K)\}_{K\in\ck}$ satisfies
duality. In fact, by (4.6) and (4.8) we have
$$\eqalign{\widehat\cm(K)=\bigcap_{W\supset K} \cm(W)=
\bigcap_{W\supset K} \widehat\cm(W^\prime)^\prime&=\cr
\big(\bigcup_{W\supset K} \widehat\cm(W^\prime)\big)^{\prime}&\supset
\big(\bigcup_{K^\prime\supset W^\prime} \widehat\cm(W^\prime)\big)^{\prime}
\supset\widehat\cm(K^\prime)^\prime,\cr}$$
and the opposite inclusion follows
from locality of $\widehat\cm$.
Altogether we have shown that
$\cm(K^\prime)^\prime=\widehat\cm(K^\prime)^\prime=\widehat\cm(K)$,
so $\cm^d$ is equal to $\widehat\cm$ and satisfies duality.\hfill$\Box$
\bigskip
We now turn to the connection between local nets of von Neumann
algebras and local nets of algebras of unbounded operators.
As in [DSW] and [BY4] we employ the following
notion.
%%%%VID-BO'T
\mabsatz
{\bf 4.2 Definition.\/}
A field net $\{\cp(K)\}_{K\in\ck}$ is associated with a net
of von Neumann algebras, $\{\cm(K)\}_{K\in\ck}$, if every $X\in\cp(K)$ has
an
extension to a
closed operator $\widetilde X$ with $\widetilde {X^\dagger} \subset \widetilde
X^{*} $, such that $\widetilde X$ is affiliated with
the von Neumann algebra
$\CM(K)$.
\sabsatz
Affiliation of $\widetilde X$ with $\CM(K)$ means precisely that the von
Neumann algebra $m(\widetilde X)$ generated by the polar decomposition of
the extended operator is a subalgebra of $\CM(K)$. It is in fact sufficient
to require this for the operators in some $*$-invariant family $\cg(K)$ of
algebraic generators for $\cp(K)$; hence the field net is associated with
some {\it local\/} net of von Neumann algebras if and only if these
generators have
closed extensions that commute strongly for space-like separated regions.
By (3.1) and (4.1) it then follows that the operators
in $\cp(\co)$ have extensions affiliated with
$\cm(\co)$ for arbitray regions $\co$.
As pointed out in [BY4] the extensions $\tilde X$ might in general have to
be
defined in a larger Hilbert space than the original operators $X$. In this
paper, however, we shall only consider fields with the property that the
weak commutants $\cp(K)^w$ are algebras for $K\in\ck$, and for such fields
an
enlargement of the Hilbert space is not necessary. In fact, $\cp(\cdot)$ is
then always associated with the net $$\CM_{\rm
min}(K):=\CP(K)^{w\prime},\eqno(4.10)$$ which is a subnet of any other net
of
von Neumann algebras with this property. Note also that $\CM_{\rm
min}(\co)=\CP(\co)^{w\prime}$ for arbitrary regions $\co$, by (3.1) and
(4.1).
If the weak commutants $\cp(K)^w$ are algebras
the question whether $\cp(\cdot)$ can be associated with a {\it local} net
of
von Neumann algebras is the same as asking whether $\CM_{\rm
min}(\cdot)$ is local.
Our criterion for locality of $\CM_{\rm min}(\cdot)$ involves the following
notion of positivity [P]:
\mabsatz {\bf 4.3 Definition.\/} Let $\GA$ be a $*$-algebra with unit over
${\bf C}$ and suppose $X_0\in\GA$ commutes with all elements in $\GA$. A
linear functional $\omega$ on $\GA$ is said to be {\sl centrally
positive\/} with respect to $X_0$ if $\omega$ is positive on all elements
of the form $\sum_{n=1}^N X_0^nY_n$ with $Y_n\in\GA$ such that
$\sum_{n=1}^N \lambda^nY_n\in\GA^+$ for all $\lambda\in{\bf R}$.
Here
$\GA^+$ denotes the cone of elements of the form $\sum_i Z_i^*Z_i$,
$Z_i\in\GA$.
\mabsatz
The following result on the connection between bounded and unbounded nets
was proved in [BY4]:
\sabsatz
\goodbreak
{\bf 4.4 Theorem.}
{\it Let $\cp(K)_{K\in\ck}$ be a local field net satisfying (BZ)
and assume $\CP(K)^w$
is a von Neumann algebra for every $K\in\ck$. Then the following
conditions are equivalent.
\item{\rm (1)} The field net is associated in the sense of Definition 4.2
with some
local net of von Neumann
algebras acting on the same Hilbert space.
\item{\rm (2)} The minimal net $\CM_{\rm min}(K)=\CP(K)^{w\prime}$ is
local.
\item{\rm (3)} For every open set $\co$
and every hermitian operator $X\in\CP(\co)$
the state defined by the vacuum vector $\Omega$ on the
algebra generated by $X$ and $\CP(\co^\prime)$ is
centrally positive with respect to $X$.
\item{\rm (4)} If $K, K_1\in\ck $ with $K_1\subset K^\prime$
and $X$ belongs to a set of hermitian generators $\cg(K)$ for
$\CP(K)$, then
the state defined by the vacuum vector $\Omega$ on the
algebra generated by $X$ and $\CP(K_1)$ is
centrally positive with respect to $X$.}
\mabsatz
In [DSW] it is shown that $\CP(\co)^w$ is a von Neumann algebra for all
open regions $\co$ if the net
is generated by Wightman fieds satisfying {\sl generalized H-bounds} of
order $<1$, i.e. if $\Phi(f)^{**}e^{-(1+H^2)^{\alpha/2}}$ is a bounded
operator for all field operators $\Phi(f)$ and some $\alpha<1$, where $H$
denotes the Hamiltonian. In fact, such generalized $H$-bounds imply that
every operator in $\cp(\co)^w$ commutes {\it strongly} with the closures of
the operators in $\cp(\co)$. Hence the extension $\tilde X$
in Def.\ 4.2 may in this case simply be taken to be the closure $\bar X$.
Let us now consider a field net $\cp(\cdot)$ satisfying the premises of
Theorem 4.4 and assume the positivity condition (4) is fulfilled so
$\cm_{\rm min}(\cdot)$ is local. It is natural to ask whether essential
duality holds for $\CM_{\rm
min}(\cdot)$. If this is the case a partial answer can be given to the
question of uniqueness: All local nets of von Neumann algebras, with which
$\cp(\cdot)$ is associated, lie between $\cm_{\rm min}(\cdot)$ and the net
$$\cm_{\rm max}(K):=\CM_{\rm min}^d(K)=\cp(K^\prime)^w.\eqno(4.11)$$
In particular, $\cp(\cdot)$ is associated with a unique net of von
Neumann algebras (on the same Hilbert space) if and only if
$\cp(K)^{w\prime}=\cp(K^\prime)^w$ for all $K$. We now relate these
questions to properties of the Tomita conjugations of $\cp(\cdot)$.
We first note that from the definition (3.1) it follows that
for arbitrary regions $\co$
$$\cp(\co)^w=\bigcap_{K\subset\co,
K\in\ck}\cp(K)^w.$$
Hence $\cp(\co)^w$ is an algebra if
$\cp(K)^w$ is an algebra for all $K\in\ck$. Let $\co$ belong to $\ck$ or
$\cw$, so the Tomita conjugations $S_{\cp(\co)}$
and $S_{\cm_{\rm min}(\co)}$ are defined. Since
$S_{\cp(\co)^{w\prime}}=S^*_{\cp(\co)^w}$ and $S_{\cp(\co)}\subset
S^*_{\cp(\co)^w}$, we have
$$S_{\cp(\co)}\subset S_{\CM_{\rm
min}(\co)}.\eqno(4.12)$$
Replacing $\co$ by $\co^\prime$ and taking adjoints
we obtain
$$S_{\CM_{\rm min}(\co^\prime)^\prime}\subset
S_{\cp(\co^\prime)}^*.\eqno(4.13)$$
Locality of $\cp(\cdot)$ implies that
$S_{\cp(\co)}\subset S_{\cp(\co^\prime)}^*$. If the stronger condition
$S_{\cp(\co)}= S_{\cp(\co^\prime)}^*$ holds we deduce from (4.12) and
(4.13)
that $S_{\CM_{\rm min}(\co^\prime)^\prime}\subset S_{\CM_{\rm min}(\co)}$.
Locality of $\cm_{\rm min}(\cdot)$ and Proposition 2.7 then imply that
$\CM_{\rm min}(\co^\prime)^\prime=\CM_{\rm min}(\co)$. Combining this with
Lemma 4.1 we have thus obtained the following sharpening of Theorem 4.4:
\sabsatz {\bf 4.5 Theorem.\ }{\it Let $\cp(\cdot)$ satisfy the premises
and the positivity condition (4) of Theorem 4.4. If $\cw$-duality holds for
$\cp(\cdot)$, i.e. $S_{\cp(W)}= S_{\cp(W^\prime)}^*$ for all $W\in\cw$,
then
essential duality holds for the minimal net $\CM_{\rm
min}(K)=\CP(K)^{w\prime}$.
$\cp(\cdot)$ is then associated with a unique
net of von Neumann algebras
satisfying $\ck$-duality, namely $$\cm_{\rm max}(K)=\cm_{\rm
min}^d(K)=\widehat\cm_{\rm min}(K)=\cp(K^\prime)^w.$$ If $\ck$-duality
holds for $\cp(\cdot)$, then $\cm_{\rm min}$ also satisfies $\ck$-duality,
i.e.
$\CM_{\rm max}=\CM_{\rm min}$.}
\bigskip
As a last subject of this section we consider variants of Theorems 4.4 and
4.5, where the positivity and duality conditions are only made for a
subnet $\cp_0(\cdot)$
of $\cp(\cdot)$. These results should be compared with earlier results of a
similar kind by Driessler, Summers and Wichmann, see in particular Theorems
4.6 and 6.1 in [DSW]. The main point is that $\cp_0(\cdot)$
already suffices to generate a net of von Neumann algebras for
$\cp(\cdot)$ if the Tomita conjugations of $\cp_0(W)$ and $\cp(W)$ are
identical, $W\in\cw$.
{\bf 4.6 Theorem.\/} {\it Let $\{\cp(K)\}_{K\in \ck}$ be a local net
satisfying
(BZ).
Assume $\cp(K)^w$ is a von Neumann algebra for every $K\in\ck$, and that
the algebras $\cp(K)$ are generated by operators fulfilling the
conditions of Proposition 3.4. Let $\cp_0(\cdot)$ be a subnet satisfying
the same conditions as $\cp(\cdot)$ for the weak commutants and assume
$\Omega$ is cyclic
for $\cp_0(W)$ and $\cp_0(W^\prime)$, $W\in\cw$. If moreover
$S_{\cp_0(W)}=S_{\cp(W)}$
for all $W\in\cw$, then the following conditions are equivalent:
\item{\rm(1)} The field net $\cp(\cdot)$ is associated with some local net
of von Neumann algebras on the Hilbert space of the field.
\item{\rm (2)} The field net $\cp_0(\cdot)$ is associated with some local
net of von Neumann algebras on the Hilbert space of the field.
\item{\rm (3)} If $K, K_1\in\ck $ with $K_1\subset K^\prime$,
and $X$ belongs to a set of hermitian generators $\cg_0(K)$ for
$\CP_0(K)$, then
the state defined by the vacuum vector $\Omega$ on the
algebra generated by $X$ and $\CP_0(K_1)$ is
centrally positive with respect to $X$.
\noindent If this holds, then $\cp(\cdot)$ is associated with the net
$\widehat \cm_{\rm 0, min}(\cdot)$ with $\cm_{\rm 0, min}(K)=
\cp_0(K)^{w\prime}$.}
{\it Proof.} Equality of the $S$-operators implies that
$\cp(W)^w=\cp_0(W)^w$ for all $W\in \cw$ by Propositions 2.7 and 3.4.
Hence $\widehat \cm_{\rm 0, min}(\cdot)=\widehat \cm_{\rm min}(\cdot)$.
But (2) is equivalent to locality of $\cm_{\rm 0, min}(\cdot)$, which is
equivalent to locality of $\widehat \cm_{\rm 0, min}(\cdot)$ by condition
(KW). Hence (1) and (2) are equivalent.
The equivalence of (2) and (3) follows from Theorem 4.4.\hfill$\Box$
\medskip
If the subnet $\cp_0(\cdot)$ satisfies $\cw$-duality,
$S_{\cp_0(W)}^*=S_{\cp_0(W^\prime)}$, then
$S_{\cp_0(W)}=S_{\cp(W)}$
for all $W\in\cw$ by Lemma 2.6.
(Take $\cf_1=\cp_0(W)$,
$\cq=\cp_0(W^\prime)$ and $\cf_2=\cp_0(W)$.) In particular, $\cp(\cdot)$
is also $\cw$-dual. Combining Theorems 4.5 and 4.6 we thus also have
\sabsatz {\bf 4.7 Theorem.\ }{\it Let $\cp(\cdot)$ satisfy the premises of
Theorem 4.6. Let $\cp_0(\cdot)$ be a subnet satisfying the same condition
for the weak commutants and in addition the positvity condition $(3)$. If
$\cw$-duality holds for
$\cp_0(\cdot)$, then the minimal net $\CM_{\rm min}(K)=\CP(K)^{w\prime}$
satisfies essential duality, and $\CM_{\rm max}(\cdot)=
\widehat \cm_{\rm 0, min}(\cdot)$.}
\babsatz
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent {\bbf 5.\ Fermi fields and twisted duality}
\mabsatz
In this last section we discuss briefly how
the preceeding results can be generalized to include Fermi fields.
The basic concept in this case is a field net
$\cp(\cdot)$ (in the sense of Section 3), where
the locality condition (L) is replaced by the $\BZ^2$-graded local
structure described below. Using the same device as in [BW2] to change
anticommutators into commutators all previous results carry over
with minor modifications to the new situation. We shall therefore only
describe the necessary changes without formulating the statements anew.
\smallskip
The graded structure is defined by a unitary operator $U_0$ (\lq\lq
rotation by
2$\pi$"), such that $U_0^2=I$, $\cd$ is invariant under $U_0$ and
$U_0\cp(\co)U_0^{-1}=\cp(\co)$ for all $\co$. Morever, for
translationally covariant fields we assume that $U_0$ commutes with the
translations and that $U_0$ leaves the vacuum $\Omega$ invariant.
Every $X\in\cp(\BR^d)$ has a
unique decomposition into a \lq\lq Bose" and a \lq\lq Fermi" part with
respect to $U_0$: $$X=X_b+X_f\quad\hbox{\rm with}\quad
U_0X_bU_0^{-1}=X_b,\quad U_0X_fU_0^{-1}=-X_f,\eqno(5.1)$$ where
$$X_{b}:=\hbox{${1\over 2}$}(X+U_0XU_0^{-1})\quad\hbox{and}\quad
X_{f}:=\hbox{${1\over 2}$}(X-U_0XU_0^{-1}).\eqno(5.2)$$ In this way each
$\cp(\co)$ becomes a $\BZ^2$-graded algebra with unit:
$$\cp(\co)=\cp(\co)_b\oplus\cp(\co)_f\eqno(5.3)$$ with
$$\cp(\co)_f\cdot\cp(\co)_f\subset\cp(\co)_b =\cp(\co)_b\cdot
\cp(\co)_b\eqno(5.4)$$ and $$\cp(\co)_b\cdot
\cp(\co)_f=\cp(\co)_f\cdot\cp(\co)_b=\cp(\co)_f.\eqno(5.5)$$ As a
generalization of local commutativity we assume {\it graded local
commutativity}, i.e. $$\left[\cp(\co)_b,\cp(\co^\prime)_b\right]=
\left[\cp(\co)_b,\cp(\co^\prime)_f\right]=
\left[\cp(\co)_f,\cp(\co^\prime)_f\right]_
+
=\{0\}\eqno(5.6)$$
for all $\co$.
In the same way as in [BW2] one may avoid the use of anticommutators by
introducing the operator
$$Z:=(1+i)^{-1}(U_0+iI)$$
with
the property
$$ZX_bZ^{-1}=X_b \quad \hbox{\rm and}\quad ZX_fZ^{-1}=iU_0X_f\eqno(5.7)$$
for all $X\in\cp(\BR^d)$.
If $\cq$ is a set of operators on $\cd$ we employ the notation
$$\cq^z:=\{ZXZ^{-1}\mid X\in \cq\}.\eqno(5.8)$$
We remark that because $Z$ leaves $\cd$ invariant, the operation of
changing $\cq$ into $\cq^z$ commutes with the
formation of weak and strong commutants, i.e.
$$(\cq^{uw})^z=(\cq^{z})^{uw},\quad (\cq^{w})^z=(\cq^{z})^{w,}\quad
(\cq^{us})^z=(\cq^{z})^{us},\quad (\cq^{s})^z=(\cq^{z})^{s}.\eqno(5.9)$$
Note also that in spite of the fact that $Z^2\neq 1$, one has
$(\cq^z)^z=\cq$,
if $\cq$ is invariant under ad $U_0$,
i.e. if $X_b$ and $X_f$ belong to $\cq$ for $X\in\cq$.
Because of (5.7) and (5.1), we can write
(5.6) in analogy to (3.2) as
$$\cp(\co^\prime)^z\subset (\cp(\co),\cd)^{us}.\eqno(5.10)$$
If (5.10) (or equivalently (5.6)) holds
we call the net $\co\mapsto\cp(\co)$ a {\it $\BZ_2$-graded
local field net}.
As in Section 3 we single out two classes of subsets $\ck$ and $\cw$
satisfying (K) and (KW). If $\cp(K)$ is to begin with only
defined for $K\in\ck$, and $\cp(\co)$ is defined by (3.1) for more
general regions, then a $\BZ_2$-graded
local structure for $\{\cp(K)\}_{K\in\ck}$
extends naturally to the algebras $\cp(\co)$ for general $\co$.
If a field net $\cp(\cdot)$ satisfies (RS), or the stronger condition (BZ),
for a $U_0$-invariant
vector $\Omega$, then the same holds for
the net $\cp(\cdot)^z$. The Tomita conjugations $S_{\cp(\co)}$
and $S_{\cp(\co^\prime)^z}$ are then well defined for $\co\in\ck$ or
$\co\in\cw$,
and (5.10) implies that
$$S_{\cp(\co)}^*\supset S_{\cp(\co^\prime)^z}.\eqno(5.11)$$
The duality conditions of Section 3 for such a
field net generalize to {\it twisted $\ck$-duality} and
{\it twisted $\cw$-duality},
which by definition mean that
$$S_{\cp(\co)}^*= S_{\cp(\co^\prime)^z}\eqno(5.12)$$
for all $\co\in\ck$ or all $\co\in\cw$, respectively.
With these definitions Theorems 3.5 and 3.6 now carry over to the
case of graded nets with the only change that \lq\lq local\rq\rq\ has to be
repalced by
\lq\lq $\BZ_2$-graded local\rq\rq\ , and \lq\lq dual\rq\rq\ by \lq\lq
twisted dual\rq\rq\ . Notice that because of (5.9) and the fact that
$(\cp(\co)^{z})^z=\cp(\co)$ there is a symmetry between the algebras
$\cp(\co)$ and their $z$-transforms in all results involving these
algebras and their strong or weak commutants. Thus one can interchange
$\cp(\co)$ and $\cp(\co^\prime)^z$ in equations like (5.10)-(5.12),
and also in the generalizations of Theorems 3.5 and 3.6.
\medskip
The concept of a $\BZ_2$-graded local net of von Neumann
algebras, $\co\mapsto\cm(\co)$, is defined in the same way as a
$\BZ_2$-graded local net
with the additional requirement that $\cm(\co)$ is a von Neumann
algebra for all $\co$. For the association of a $\BZ_2$-graded
net of unbounded operator algebras with such a net we have to require
that the association is compatible with the graded structure:
\medskip
{\bf 5.1 Definition.} A $\BZ_2$-graded
net $\cp(\cdot)$ of (unbounded) operator algebras is associated with a
$\BZ_2$-graded net $\cm(\cdot)$ of von Neumann algebras
(with the same operator
$U_0$), if the following
holds:
Every $X\in\cp(\co)$
has an extension to a closed operator $\tilde{X}$ on $\ch$ with
$\tilde {X}^*\supset \widetilde{X^\dagger}$
and $U_0\tilde {X}U_0^{-1}=(U_0XU_0^{-1})\widetilde{\phantom {x}}$,
such that $\tilde {X}$ is affiliated with the von Neumann
algebra $\cm(\co)$.
\medskip
Since all extensions considered are supposed to satisfy the
condition
$U_0\widetilde XU_0^{-1}=(U_0XU_0^{-1})\widetilde{\phantom{x}}$ it follows
that
the isometric
operator $V$ and the absolute value $\vert\widetilde X\vert$ appearing in
the
polar decomposition $\widetilde X=V\vert\widetilde X\vert$ have the correct
Bose or Fermi character, i.e.\ $V$ is Bose/Fermi
if $X$ is Bose/Fermi and $\vert \widetilde X\vert$ is in both cases a Bose
operator. This
assertion is a simple consequence of the uniqueness of the polar
decomposition of a closed operator.
\smallskip
The criterion for the the association of a graded local net of unbounded
operator algebras with a
graded local net of von Neumann algebras is an obvious modification of
Theorem 4.4.
As in the Section 4 we assume
that the weak commutants
$\cp(\co)^w$
are algebras. The unbounded net
$\cp(\cdot)$ is then associated with a unique
minimal $\BZ_2$-graded
net of von Neumann algebras on the the same Hilbert space, namely
$$\cm_{{\rm min}}(\co):=\cp(\co)^{w\prime}.\eqno(5.13)$$
In fact, since $U_0$ leaves $\cd$ invariant and $U_0^2=1$, one has
$U_0\cm_{{\rm min}}(\co)U_0^{-1}=\cm_{{\rm min}}(\co)$, so $U_0$ defines
a grading of this minimal net, and one checks that the extensions defined
as in the proof of Lemma 4.3 in [BY4] satisfy the requirements of
Definition 5.1. Moreover, by (5.9) and since $\cp(\co^\prime)^w$ is an algebra,
one has
$$\cm_{{\rm min}}(\co^\prime)^{z\prime}=
\cp(\co^\prime)^{zw}.\eqno(5.14)$$
Hence graded local
commutativity for the net $\cm_{{\rm min}}$ is equivalent to the
requirement
that
$$\cp(\co)^{w\prime}\subset(\cp(\co^\prime)^{z})^w
\eqno(5.15)$$
for all $\co$. We can now proceed in an analogous way as
in [ BY4], with (5.15) replacing Eq. (4.1) in that paper.
This leads to a generalization of Theorem 4.4 (Theorem 4.6 in [BY4]):
\smallskip
The net $\cp(\cdot)$ is associated with a graded local net of von Neumann
algebras if and only if the following generalization of the positivity
condition (4) holds:
\smallskip
\item{\rm ($4^\prime$)} If $K, K_1\in\ck $ with $K_1\subset K^\prime$
and $X$ belongs to a set of hermitian generators $\cg(K)$ for
$\CP(K)$, then
the state defined by the vacuum vector $\Omega$ on the
algebra generated by $X$ and $\CP(K_1)^z$ is
centrally positive with respect to $X$.
\medskip
Analogs of Theorems 4.5, 4.6 and 4.7 also hold for $\BZ_2$-graded nets
if $\cm^d$ is defined as
$\cm^d(K)=\cm(K^\prime)^{z\prime}$.
The modifications are as follows: Repalce \lq\lq local\rq\rq\ by
\lq\lq $\BZ_2$-graded local\rq\rq\ ,\lq\lq dual\rq\rq\ by \lq\lq
twisted dual\rq\rq\ , and $\cp_0(K_1)$ in Theorem 4.6 by
$\cp_0(K_1)^z$.
%\end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip \bigskip
\noindent{\bbf Appendix A}
\bigskip
We derive here property (BZ) for a net $\cp(\cdot)$ of unbounded operator
algebras generated by Wightman fields. The crucial fact is the
following consequence
of translational covariance, spectrum condition and temperedness of the
Wightman fields: Suppose $A, B_1,\dots, B_n\in\cp(\BR^d)$ and put
$B_i(x_i)=U(x_i)B_iU(-x_i)$, with $x_i\in\BR^d$, $i=1,\dots,n$, where
$U(x)$ denotes the representation of the translation group. Then the
function $\Phi(x_1,\dots,x_n)=AB_1(x_1)\cdots B_n(x_n)\Omega$ has an
analytic continuation $\Phi(z_1,\dots,z_n)$ for
$z_1,z_{2}-z_1,\dots,z_{n}-z_{n-1}$ in the forward tube, as a function with
values in the Hilbert space $\ch$ of the operators (Jost vector). Note that
this property is not a consequence of the
translational covariance and spectrum condition alone, for although
$x\mapsto U(x)\psi$ has an analytic continuation into the forward tube as
an $\ch$-valued function $z\mapsto U(z)\psi$ for all $\psi\in\ch$,
it is not a priori clear that $U(z)\psi$ belongs to the domain of the
closure of each of
the unbounded operators in $\cp(\BR^d)$ if $\psi\in\cd$. For
tempered Wightman fields, however, $\Phi(x_1,\dots,x_n)$ is a polynomially
bounded, continuous function, whose Fourier transform in
distributional sense,
$\tilde\Phi(p_1,\dots,p_n)$, has support for $p_i+\cdots + p_n$ in the forward
light cone, $i=1,\dots,n$, and $\Phi(z_1,\dots,z_n)$ can be defined by
smearing
$\tilde\Phi(p_1,\dots,p_n)$ with $\exp(i\sum_jz_j\cdot p_j)$.
It is
not difficult to extend this argument to any translationally covariant
field net
with spectrum condition, provided $\bar AU(x)\psi$ is polynomially bounded
for all $A\in\cp(\BR^d)$, $\psi\in\bar\cd$, where $\bar\cd$ denotes
the closure of the domain $\cd$ in the graph topology induced by
$\cp(\BR^d)$. In fact, the analytic
continuation $\Phi(\zeta)$ of $\Phi(x)=\bar AU(x)\psi$ for $\zeta$ in
the forward tube is obtained in the same way as for
Wightman fields by
integrating the Fourier transform $\tilde \Phi$ with
$e^{i\zeta\cdot p}$. In order to
be able to iterate this procedure and define
$\bar AU(\zeta_1)\bar B_1\cdots U(\zeta_n)B_n\Omega$,
it suffices to show that $U(\zeta)\psi$ belongs to $\bar\cd$ for all
$\psi\in \bar\cd$ and $\zeta$ in the forward tube,
because the closures of the operators in $\cp(\BR^d)$
leave $\bar \cd$ invariant. To prove $U(\zeta)\psi\in\bar\cd$
we consider for $A\in \cp(\BR^d)$ the vector valued function $
\Psi(\zeta)=\Phi(\zeta)/P(\zeta)$, where $\Phi(\zeta)$ is the
analytic continuation of $\Phi(x)=\bar AU(x)\psi$ as before, and
$P(\zeta)$ is a polynomial without
zeros in the closure of the forward tube and of sufficiently high
degree so that $\Phi
(x)/P(x)$ is integrable over
$\BR^d$. The function $\Psi(\zeta)$
is in the same way as $\Phi(\zeta)$ the
Fourier-Laplace transform of a distribution with support in the
forward light cone $\bar V^+$. From this we obtain the integral
representation
$$\Phi(\zeta)=P(\zeta)\int K(\zeta-x){\bar AU(x)\psi\over P(x)}dx$$
where $K(\zeta)=(2\pi)^{-d}\int_{\bar V^+} \exp(i\zeta\cdot p) dp$.
An approximation of the integral by a Riemannian sum provides a sequence
$\psi_n$ of vectors in $\bar \cd$ such that $\psi_n\to U(\zeta)\psi$
and $\bar A\psi_n\to\Phi(\zeta)$. Hence $U(\zeta)\psi\in\bar\cd$.
Given the analytic continuation of $\Phi(x_1,\dots,x_n)$
the derivation of (BZ)
follows the same pattern as in the Reeh-Schlieder Theorem: Pick
$A_1,\cdots A_N\in\cp(\BR^d)$ and assume
$\varphi_1,\dots,\varphi_n\in\ch$ are such that
$$\sum_{j=1}^N\langle\varphi_j,A_jB_1(x_1)\cdots B_n(x_n)\Omega\rangle=
0\eqno(A1)$$
for all $B_i\in\cp_0(\co)$ and $x_i$ in some neigbourhood $\CN$ of zero.
Then (A1) holds by analytic continuation for all $x_i\in\BR^d$. This
implies that $\cp_0(\co+\CN)\Omega$ and
$(\vee_{x\in\BR^d}\cp_0(\co+x)\Omega$
have the same closure in
the graph topology induced by $\cp(\BR^d)$. Now we may replace $\co$
by a smaller
set $\co_0$ such that $\co_0+\CN\subset\co$, and by assumption
$\vee_{x\in\BR^d}\cp_0(\co_0+x)=\cp(\BR)^d$, so (BZ) holds for all
(nonempty, open) $\co$.
\bigskip
\bigskip
\noindent{\bbf Appendix B}
\bigskip
{\it Proof of Proposition 3.4:\/}
Let $\chi_n$ be the characteristic function of the interval
$[-n,n]$, and define $h_n=\chi_n*\gamma$ where $\gamma$ is some fixed
$C^\infty$-function of compact support with $\int \gamma(t)dt=1$. If $\tilde h_n(t)=\hbox{${1\over
2\pi}$}\int
\exp(-ist)h_n(s)ds$ is the Fourier transform of $h_n$, then $\int \tilde
h_n(t)g(t)\to g(0)$ for all continuous, polynomially bounded functions $g$.
If $X\in\cg$ and $\psi\in \cd$, we thus obtain, using (i) and (ii):
$$X\psi=\lim_{n\to\infty}\int \tilde
h_n(t) X\exp(itH)\psi\, dt=\lim_{n\to\infty}\bar Xh_n(H)\psi.$$
Now because $\cd_0$ is dense and $\bar Xh_n(H)$ bounded, we can for each
$n$
approximate $\bar Xh_n(H)\psi$ arbitrarily closely in norm with vectors of the form
$\bar Xh_n(H)\varphi$ with $\varphi\in\cd_0$. Writing
$\bar Xh_n(H)\varphi=\int \tilde
h_n(t) X\exp(itH)\varphi\, dt$ and approximating the integral by a
Riemannian
sum we see that
$\bar Xh_n(H)\psi$, and hence $X\psi$, can be approximated arbitrarily
well by a finite linear combination of translates of $\varphi$,
i.e.\ by vectors of the form
$\sum_j c_j\exp(it_jH)\varphi$ with $c_j\in\BC$, $t_j\in\BR$.
Because $\cd_0$ is invariant by assumption, we can thus for any
$\varepsilon >0$ obtain an
$\varepsilon$-approximation of $X\psi$ by $X\xi$ for some $\xi\in\cd_0$. It
is clear
that the same argument applies to any finite set of operators satisfying
conditions (i) and (ii), so $\cd_0$ is dense in $\cd$ in the graph
topology induced by $\cg$.
\bigskip
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%
\vskip 0.4cm\noindent
{\bf Acknowledgements\ } This work was partially supported by the
Heraeus-Stiftung and the
Icelandic Science Foundation.
\vskip 0.6cm\noindent
{\bbf References}
%\babsatz
%\vskip 0.2cm
\def\ref{\par\vskip 10pt \noindent \hangafter=1
\hangindent 22.76pt}
\parskip 5pt
{\baselineskip=3ex\eightpoint\smallskip
\font\eightit=cmti8
\font\eightbf=cmbx8
\def\it{\eightit}
\def\bf{\eightbf}
%
\ref {[BW1]} J. Bisognano and E.H. Wichmann,
{\it On the duality condition for a Hermitean scalar field},
J. Math. Phys. {\bf 16}, 985-1007 (1975).
%
\ref {[BW2]} J. Bisognano and E.H. Wichmann,
{\it On the duality condition for quantum fields},
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%
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%}
\bye