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%%%%%%%%%%%%%%%%%%%%%%%%% A NOTE ON ESSENTIAL DUALITY %%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BY JAKOB YNGVASON %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\tabsatz
\centerline{\hbf A Note on Essential Duality}
\vskip 0.4cm
\centerline{\caps Jakob Yngvason}
\mabsatz
\centerline{Science Institute}
\centerline{University of Iceland}
\centerline{Dunhaga 3, IS 107 Reykjavik, Iceland}
\babsatz
\babsatz \centerline {\bf ABSTRACT} \bigskip
By considering some simple models it is shown that the essential duality
condition for local nets of von Neumann algebras associated with Wightman
fields need not be fulfilled if Lorentz covariance is dropped. These models
illustrate a point made by Borchers in the proof of his two dimensional CPT
theorem for local nets: The Lorentz covariant net constructed from the
wedge algebras of a given two dimensional net may not be unique. It is also
shown that in higher dimensions the Lorentz boosts constructed by means of
the modular groups of wedge algebras may act nonlocally in the directions
parallel to the edge of the wedge.
\babsatz\vfill\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent{\bbf 1. Introduction}
\bigskip
One of the most remarkable results in algebraic quantum field theory in
recent years is Borchers' derivation of the CPT theorem for local nets in
two dimensional space-time [B]. The main ingredient in this proof is a
certain converse of the Bisognano-Wichmann analysis of modular structures
in quantum field theory [BW1,2], (see also [BY]): Whereas Bisognano and
Wichmann assume covariance with respect to Lorentz transformations and show
that the Lorentz boosts coincide with the modular group of the algebra
corresponding to a wedge-domain, Borchers proves that in two dimensions the
modular group of a wedge algebra can be used to define a representation af
the Poincar\'e group, if the net is covariant with respect to a
representation of the translation group satisfying the spectrum condition.
Moreover, the initial net can be embedded into a net that is covariant with
respect to this representation af the Poincar\'e group, has CPT symmetry
and satisfies Haag-duality.
As pointed out in [B], the Poincar\'e covariant net obtained by this method
may not be unique unless the initial net satisfies wedge duality [BW1,2].
This condition of wedge duality is also important in another context: It
implies essential duality [Ro], which is one of the chief assummptions on
which the analysis of superselection rules is based [H]. By the results of
Bisognano and Wichmann wedge duality always holds for local nets generated
by Wightman fields transforming with respect to finite dimensional
representations of the Lorentz group, but examples of local nets violating
this condition have apparently not been known so far.
In the present note we
discuss a class of very simple fields in two dimensions, whose duality
properties can be completely analyzed. Among these fields are both such
that violate essential duality (and hence, {\it a fortiori} wedge
duality), as well as fields that
satisfy wedge duality without being Lorentz covariant. The former provide
examples where the net has infinitely many different extensions
satisfying duality. Higher dimensional
fields of a similar type
demonstrate clearly
that some additional assumptions are needed to
extend Borchers' result to space-time dimensions larger
than 2, because the action of the
modular group of a wedge algebra in directions parallel to the
edge of the wedge is not local in these examples.
%It seems advisable to keep these
%examples in mind in
%the search for a general CPT theorem within the framework of local
%observables.
\vfill\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent{\bbf 2. Generalized free fields on a light ray}
\bigskip
Let $(x^0,x^1)$ denote the usual time and space coordinates in two
dimensional Minkowski space and $x_\pm=x^0\pm x^1$ the corresponding light
cone coordinates. Suppose $\Phi$ is a hermitian Wightman field that
transforms covariantly under space-time translations, but not necessarily
under Lorentz transformations, and depends only on one light cone
coordinate, say $x_+$. Locality implies that the commutator
$[\Phi(x_+),\Phi(y_+)]$ has support only for $x_+=y_+$. Moreover, from the
spectrum condition it follows that the generator for translations of
$\Phi$ in the $x_+$-direction, $P^0-P^1$, is positive semidefinite.
This implies readily that the Fourier transform of the two point function,
${\cal W}_2$, defined by $\la\Omega,
\Phi(x_+)\Phi(y_+)\Omega\ra=(1/2\pi)\int \exp[ip(x_+-y_+)]\tilde {\cal
W}_2(p) dp$, with $\Omega$ the vacuum vector, has the form $$\tilde {\cal
W}_2(p)=\theta(p) p Q(p^2)+c\delta(p),\eqno(2.1)$$ where $Q(p^2)$ is a
positive, even polynomial in $p\in\BR$, $\theta(s)=1$ for $s\geq 0$ and
zero else, and $c=\la\Omega,\Phi(x_+)\Omega\ra^2\geq 0$ is a constant.
Subtracting $c^{1/2}$ from $\Phi$ if necessary, we may drop the
$\delta(p)$-term. For simplicity of notation we also from now on write
$x,y$ instead of $x_+,y_+$.
The models we consider are generalized free fields with the two point
function (2.1) (without the $\delta$-term). They are characterized by the
commutation relations $$[\Phi(x),\Phi(y)]=DQ(D^2)\delta(x-y){\bf
1},\eqno(2.2)$$ where we have for convenience denoted $id/dx$ by $D$. The
simplest case is the chiral $U(1)$-current $\Phi_0$, given by
$$[\Phi_0(x),\Phi_0(y)]=D\delta(x-y){\bf 1}.\eqno(2.3)$$ The other models
can be obtained from $\Phi_0$ by differentiation. To see
this, one notes that since $Q(p^2)$ is positive and an even function of
$p$, one can
write
$$Q(p^2)=L(p)L(-p),\eqno(2.4)$$
where $L(p)$ is a polynomial satisfying
$$L(-p)=L(p)^*.\eqno(2.5)$$
The polynomial $L(p)$ is uniquely fixed (up to a sign) for a given $Q(p^2)$
by the requirement
that its zeros lie in the closed upper half plane. The most
general form of $L(p)$ is then
$$L(p)=R(p)P(p),\eqno(2.6)$$
where $R(p)$ has zeros only on the real axis and can be written as
$$R(p)=(ip)^n \prod_{j=1}^{k}(p^2-r_j^2),\eqno(2.7)$$
with $r_j>0$, while $P(p)$ is a polynomial whose zeros have a strictly
positive imaginary part. Note that $P(p)$ satisfies the
reality condition (2.5) and $R(-p)=(-1)^nR(p)$. From (2.4) and (2.5) it is
clear that the field
$\Phi$ defined by
$$\Phi(x)=L(D)\Phi_0(x)\eqno(2.8)$$
satisfies the commutation relation (2.2). It also follows that the smeared
field operators $\Phi(f)$ are well defined and essentially self adjoint
on their natural domain for every tempered real distribution $f$
such that $L(p)\tilde{f}(p)$ is square integrable with respect to the
measure $\vert p\vert dp$. Such $f$ are necessarily measurable functions,
but they may have singularities and grow at infinity if $L(p)$ has zeros
on the real axis.
Since on a light ray Lorentz boosts are the same as dilatations, the field
$\Phi$ is Lorentz covariant if and only if $Q(p^2)=p^{2n}$ for some natural
number $n$, i.e. $$\Phi(x)=(d/dx)^n\Phi_0(x).\eqno(2.9)$$ In fact, if
$U_0(\lambda)$ is the unitary operator implementing dilatation by
$\lambda>0$ for $\Phi_0$, i.e.,\
$U_0(\lambda)\Phi_0(x)U_0(\lambda)^{-1}=\lambda\Phi_0(\lambda x)$, then
$$U_0(\lambda)\Phi(x)U_0(\lambda)^{-1}=\lambda^{n+1}\Phi(\lambda
x)\eqno(2.10)$$ for $\Phi$ as in (2.9).
The two point function for the field (2.9) is $${\cal W}_2(x-y)=(2\pi)^{-1}
(x-y+i0)^{-2(n+1)}.\eqno( 2.11)$$ A M\"obius transformation, $x\mapsto
x^\prime={ax+b\over cx+d}$ with $a,b,c,d\in\BR$ and $ad-bc>0$, changes
${\cal W}_2$ according to the formula $${\cal
W}_2(x^\prime-y^\prime)dx^\prime dy^\prime= \left[{(cx+d)(cy+d)\over
ad-bc}\right]^{2n} {\cal W}_2(x-y)dx dy.\eqno(2.12)$$ It follows that there
is a unitary representation of the M\"obius group implementing the
transformations $\Phi(f)\mapsto\Phi(f^\prime)$ with
$$f^\prime(x^\prime)={(ad-bc)^n\over
(cx+d)^{2n}}f(x)={(cx^\prime-a)^{2n}\over
(ad-bc)^n}f(-(dx^\prime-b)/(cx^\prime-a)).\eqno(2.13)$$ As a consequence
the field (2.9) can be extended to a conformally covariant field on the
compactified light ray [BSch].
Returning to the general case (2.8) we define the unitary Weyl operators as
usual by
$$W(f)=e^{i\Phi(f)},\eqno(2.14)$$
where $f$ is a real function such that $L(p)\tilde{f}(p)$ is square
integrable with respect to the
measure $\vert p\vert dp$. The Weyl relations are
$$W(f)W(g)=e^{-K(f,g)/2} W(f+g)\eqno(2.15)$$
with
$$K(f,g)=\la\Omega,[\Phi(f),\Phi(g)]\Omega\ra=
\int_{-\infty}^{\infty}p\, Q(p^2)\tilde{f}(-p)
\tilde{g}(p) dp.\eqno(2.16)$$
It follows that $W(f)$ commutes with $W(g)$ if and only if $K(f,g)=0$, in
particular if $f$ and $g$ have disjoint
supports.
Using the concrete realization of $\Phi$ as a derivative of $\Phi_0$ we
can write
$$W(f)=W_0(L(D)f)\eqno(2.17)$$
with $W_0(g)=\exp(i\Phi_0(g))$, and
$$K(f,g)=K_0(L(D)f,L(D)g)=K_0(Q(D^2)f,g)=K_0(f,Q(D^2)g)\eqno(2.18)$$
where $K_0$ is the vaccum expectation value of the commutator for $\Phi_0$.
We note also that the Weyl operators are weakly continuous in
$f$ with respect to the norm
defined by the two point function, {\it viz},
$$\Vert \tilde{f}\Vert^2=\int_0^\infty p \vert L(p)\tilde{f}(p)\vert^2
dp.\eqno(2.19)$$
The Weyl operators generate on Fock space a local net of von Neumann
algebras ${\cal M}(I)=\{W(f)\mid {\rm supp\ }f\subset I\}^{\prime\prime}$
associated with bounded intervals $I$ of the light ray $\BR$.
Alternatively, one may think of these algebras as a net over
two-dimensional Minkowski space. If $ {K}$ is an open double cone on
Minkowski space we define ${\cal A}(K)={\cal M}(I)$, where $I$ is the
projection of $K$ onto the light ray $x_-=0$. Locality of the net ${\cal
M}(\cdot)$ for disjoint intervals is equivalent to the locality of ${\cal
A}({\cdot})$ for space like separated double cones.
Duality conditions for the net ${\cal A}({\cdot})$ also translate
immediately into corresponding conditions for ${\cal M}(\cdot)$. {\it
Duality} for double cones, i.e., the condition ${\cal A}(K)^\prime={\cal
A}(K^\prime)$ for all doble cones $K$, is equivalent to ${\cal
M}(I)^\prime={\cal M}(I^\prime)$ for all intervals $I$. Here $K^\prime$
denotes as usual the space like complement of the closure of $K$, whereas
$I^\prime$ denotes the set theoretical complement of the closure of $I$ in
$\BR$. The algebras corresponding to unbounded subsets of $\BR^2$ or $\BR$
are by definition generated by the subalgebras corresponding to double
cones or intervals, repectively, contained in the unbounded sets. {\it
Essential duality} for ${\cal A}(\cdot)$ means that the net $K\mapsto{\cal
A}(K^\prime)^\prime$ satisfies locality; this is equivalent to
$I\mapsto{\cal M}(I^\prime)^\prime$ being local. Finally, {\it wedge
duality}, which implies essential duality [BW1,2] (see also [BY]), is the
condition ${\cal A}(W)^\prime={\cal A}(W^\prime)$ for all space like wedges
$W$. This is equivalent to ${\cal M}(\BR^+)^\prime={\cal M}(\BR^-)$, where
$\BR^{\pm}$ denotes the positive or negative half axis.
%
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
{\bbf 3. The Algebras ${\cal M}(I)$, ${\cal M}(I^\prime)$ and ${\cal
M}(\BR^\pm)$} \bigskip
The net ${\cal M}(\cdot)$ generated
by $\Phi=L(D)\Phi_0$ may be regarded as a subnet of the net
${\cal M}_0(\cdot)$ generated by the
chiral $U(1)$-current $\Phi_0$. One may ask to what extent the algebras
${\cal M}(I)$, ${\cal M}(I^\prime)$ and ${\cal
M}(\BR^\pm)$
depend on the polynomial $L(p)$, in particular when they are equal to the
corresponding algebras generated by $\Phi_0$.
The answer given below can be summarized as follows: If $I$ is a bounded
interval, then polynomials differing by more than a constant factor
lead to different ${\cal
M}(I)$, while
${\cal M}(I^\prime)$ depends only on the roots of $L(p)$ in the open
upper half plane,
and the same holds for ${\cal
M}(\BR^\pm)$.
To discuss this in more detail let ${\cal M}_1(\cdot)$ and ${\cal M}_2
(\cdot)$ denote two such nets corresponding to different polynomials
$L_1(p)$ and $L_2(p)$. It is clear that if $L_1(p)$ is a factor of
$L_2(p)$, then ${\cal M}_2(\cdot)$ is a subnet of ${\cal M}_1(\cdot)$. We
now show that conversely, if $L_1(p)$ is not a factor of $L_2(p)$, then
${\cal M}_2(I)$ is not a subalgebra of ${\cal M}_1(I)$ for a bounded
interval $I$. In particular, ${\cal M}_1(I)\neq{\cal M}_2(I)$, unless $L_1$
and $L_2$ are proportional to each other.
To see this, note that if $L_1$ is not a factor of $L_2$, then one
can find a real test function $g$ such that $DL_1(-D)L_2(D)g(x)=0$ on $I$,
while $DL_2(-D)L_2(D)g$ is nonzero on some open
subinterval of $I$. (Recall that $L_1$ and $L_2$ satisfy the reality
condition (2.5).) By the first property it is clear that
$W_2(g)=W_0(L_2(D)g)$ commutes with all $W_1(f)=W_0(L_1(D)f)$ with ${\rm
supp }f\subset I$, so $W_2(g)\in {\cal M}_1(I)^\prime$. On the other hand, if
$DL_2(-D)L_2(D)g$ is nonzero on
a subinterval of $I$ then one has for some $f$ with support in $I$ that
$$K_0(L_2(D)f,L_2(D)g)=\int f(x)DL_2(-D)L_2(D)g(x)dx\neq 0,\eqno(3.1)$$
and thus $W_2(g)\notin {\cal M}_2(I)^\prime$. Hence ${\cal
M}_1(I)^\prime\not\subset{\cal M}_2(I)^\prime$.
By an analogous argument, with $I$ replaced by $I^\prime$ or $\BR^\pm$, one
sees that ${\cal M}_2(I^\prime)\not\subset{\cal M}_1(I^\prime)$
and ${\cal M}_2(\BR^\pm)\not\subset{\cal M}_1(\BR^\pm)$ if the
polynomial $P_1(p)$, obtained by cancelling the real zeros of $L_1(p)$,
is not a
factor of the corresponding polynomial $P_2(p)$ for
$L_2(p)$. For instance, suppose $P_1(\alpha)=0$ while $P_2(\alpha)\neq
0$, for some $\alpha$ in the open upper half plane. The test function
$g(x)=h(x)\exp(i\alpha x)$, where $h$ is a $C^\infty$-function with
$h(x)=1$ on
$\BR^+$, satisfies the equation $DL_1(-D)L_2(D)g=0$ on $\BR^+$, but
$DL_2(-D)L_2(D)g$ is nonzero on some interval in $\BR^+$. As above, we
conclude from this that $W_2(g)\in\CM_1(\BR^+)^\prime$ and
$W_2(g)\not\in\CM_2(\BR^+)^\prime$.
If $L_1$ and
$L_2$ have the same zeros in the open upper half plane, we assert that
${\cal M}_1(\BR^\pm)={\cal M}_2(\BR^\pm)$, and hence also
${\cal M}_1(I^\prime)={\cal M}_2(I^\prime)$ for bounded intervals
$I$. This may be seen as follows.
Suppose $L_i(p)=R_i(p)P(p)$, where $R_i$ has only real zeros, $i=1,2$,
while the zeros of $P(p)$ have
strictly positive imaginary parts. If $f$ is a test function with support
in $\BR^\pm$ and
$\varepsilon>0$ we define $g_\varepsilon$ by
$\tilde{g}_\varepsilon(p)=R_1(p)\tilde{f}(p)/R_2(p\pm i\varepsilon)$. Since
$\tilde{g}_\varepsilon$ has no poles in the upper (lower) half plane,
$g_\varepsilon$ has again support in $\BR^\pm$. Since $L_2(p)\tilde
g_\varepsilon(p)$ obviously converges to $L_1(p)f(p)$ in the one particle
space of $\Phi_0$, it follows that
$W_2(g_\varepsilon)=W_0(L_2(D)g_\varepsilon)$ converges weakly to
$W_1(f)=W_0(L_1(D)f)$ as $\varepsilon\to 0$. Hence
${\cal M}_2(I^\prime)\subset{\cal M}_1(I^\prime)$. Interchanging the role
of 1 and 2 one then obtains
${\cal M}_1(\BR^\pm)={\cal M}_2(\BR^\pm)$.
\bigskip
\vfill\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent{\bbf 4. Violation of Duality and Essential Duality}
\bigskip
It is well known that the chiral $U(1)$-current $\Phi_0$ satisfies
duality, i.e.,
$${\cal M}_0(I)^\prime={\cal M}_0(I^\prime)\eqno(4.1)$$
for all bounded intervals $I$.
This follows from the analysis in [HL], see also [BSch]. If the
polynomial $L$ is nonconstant with only real zeros, then
${\cal M}(I)\neq {\cal M}_0(I)$
and ${\cal M}(I^\prime)={\cal M}_0(I^\prime)$, by the results
above. Thus duality
is violated for
such fields. They still satisfy wedge duality, however,
because ${\cal M}_0$ does, and ${\cal M}(\BR^\pm)={\cal M}_0(\BR^\pm)$.
Note also that the conformally covariant fields (2.9) can be extended
to the compactified
light ray, and duality holds for this extension by the general analysis
in [BSch] (see also [BGL] and [GF]).
To complete the picture we shall now show directly (i.e., without using the
duality properties of ${\cal M}_0(\cdot)$) that duality is always violated
for the fields (2.8) with a nonconstant $L(p)$, and that essential duality
is violated if $L(p)$ has zeros with a nonvanishing imaginary part.
Let us consider duality first. Let $I\subset\BR$ be a bounded interval. If
$L(p)$ and hence $Q(p)$ is not a constant, we can obviously find a test
function $g$ such that $DQ(D^2)g(x)=0$ for $x\in I$, while $DL(D)g(x)\neq
0$ for all $x$ in some open subinterval of $I$. The first property
implies that $W(g)\in {\cal
M}(I)^\prime$ because of (2.2).
On the other hand, by the latter property of $g$
one can find a test function $h$ with support in $I$ such that
$$\int \tilde{h}(-p)p L(p)\tilde{g}(p)dp= \int h(x)DL(D)g(x) dx\neq
0.\eqno(4.2)$$
Let $f$ be the tempered distribution whose Fourier transform
is $\tilde{f}(p)=\lim_{\varepsilon\to
0^+}\tilde{h}(p)/L(p+i\varepsilon)$. Since $\int \vert p\vert
\vert L(p)\tilde{f}(p)\vert^2 dp<\infty$, $W(f)$ and $K(f,g)$ are well
defined. By (2.16), (2.4) and (4.2) we have
$$K(f,g)=\int p L(-p)\tilde{f}(-p)L(p)\tilde{g}(p) dp=
\int \tilde{h}(-p)p L(p)\tilde{g}(p)dp\neq 0.\eqno(4.3)$$
Hence $W(f)$ and $W(g)$ do not commute. Since $W(g)\in {\cal
M}(I)^\prime$, it is enough to check that $W(f)\in {\cal
M}(I^\prime)^\prime$ in order verify our assertion that duality is violated.
But if $u(x)$ is a test function with support in $I^\prime$, then
$$K(f,u)=\int pL(p)L(-p)\tilde{f}(-p)\tilde u(p)dp=\int
pL(p)\tilde{h}(-p)\tilde u(p)dp=K(-DL(-D)h,u)=0\eqno(4.4)$$
because $DL(-D)h$ has support in $I$. Hence $W(f)\in {\cal
M}(I^\prime)^\prime$, so duality does not hold.
To check for essential duality let $h$ be a nonzero test function with
support in a bounded interval $I$ and define $f$ in the same way as
above, i.e., $\tilde{f}(p)=\lim_{\varepsilon\to
0^+}\tilde{h}(p)/L(p+i\varepsilon)$. As before, $W(f)\in {\cal
M}(I^\prime)^\prime$, and hence also $W(f_a)\in {\cal
M}(I_a^\prime)^\prime$, where $f_a$ is defined by
$f_a(x)=f(x-a)$ and $I_a=I+a$. Define $g$ by $\tilde{g}(p)=
\lim_{\varepsilon\to 0^+}\tilde{h}(p)/L(-p-i\varepsilon)$. Since
$DL(D)h$ has support in $I$, a computation analogous to (4.4) shows
that $W(g)\in {\cal
M}(I^\prime)^\prime$.
Since $I_a\cap I=\emptyset$ for large
enough $\vert a\vert$, essential duality implies that the function
$F(a)=K(f_a,g)$ vanishes for large enough $\vert a\vert$.
Hence the Fourier
transform of $F$ must in that case be entire
analytic. From (2.16), (2.6) and (2.7) it follows that this
Fourier transform is given by
$$\tilde{F}(p)=\frac{pL(p)L(-p)\tilde{h}(p)\tilde{h}(-p)}{L(p)^2}=
(-1)^n\frac{P(-p)}{P(p)}\tilde{h}(p)\tilde{h}(-p).\eqno(4.5)$$
If $P(p)$ is not constant, then $P(p)$ has zeros in the upper half plane
while the zeros of $P(-p)$ lie in the lower half plane.
The function $\tilde{F}$ is then not entire analytic and essential duality
is violated. \bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent{\bbf 5. Modular Structures of the Wedge Algebras}
\bigskip
We shall now compute the modular groups defined by the vacuum state for the
wedge algebras ${\cal M}(\BR^+)$ and ${\cal M}(\BR^-)$. In order to compare
the modular groups for different $\Phi$'s it is convenient to realize all
fields on the same Hilbert space as above, namely the Hilbert space ${\cal
H}_0$ of $\Phi_0$. This is the Fock space over the one particle space
${\cal H}_{01}=L^2(\BR, \theta(p)pdp)$. As before we write $L(p)=R(p)P(p)$,
where $R(p)$ has only real zeros, $P(p)$ has only zeros with a strictly
positive imaginary part, $P(-p)=P(p)^*$, and
$R(-p)=(-1)^n R(p)$ for
real $p$.
For each $\lambda>0$ we define
two unitary operators, $U_+(\lambda)$ and $U_-(\lambda)$ by
$$\eqalignno{U_+(\lambda)\psi(p)&=\lambda\frac{P(
p)}{P(-p)}\frac{P(-\lambda
p)}{P(\lambda p)}\psi(\lambda p)&(5.1)\cr
U_-(\lambda)\psi(p)&=\lambda\frac{P(-
p)}{P( p)}\frac{P(\lambda
p)}{P(-\lambda p)}\psi(\lambda p)&(5.2)\cr}$$
for $\psi\in{\cal H}_{01}$ and canonical extension to the Fock space
${\cal H}_0$. That $U_\pm(\lambda)$ is unitary follows from the reality
condition $P(-p)=P(p)^*$ for $p\in\BR$, which means that ${P(\lambda
p)}/{P(-\lambda p)}$ is just a phase factors for real $p$. It is clear that
$U_\pm(\cdot)$ is a representation of the multiplicative group $\BR^+$.
Moroever, $U_+=U_-$ if and only if $P(p)$ is a constant, in which case
$U_\pm(\lambda^{-1})$
is the representation $U_0(\lambda)$ of the Lorentz boosts for
$\Phi_0$, cf.\ (2.10).
If $\psi$ is of the form $\psi(p)=L(p)\varphi(p)$, then
$U_+(\lambda)\psi(p)=L(p)V_+(\lambda)\varphi(p)$ with
$$V_+(\lambda)\varphi(p)=\lambda{L(-\lambda p)\over L(-p)}\varphi(\lambda
p).\eqno(5.3)$$
Here we have used that $R(p)/R(-p)=\pm 1$ for real $p$. In the same way
$U_-(\lambda)\psi(p)=L(p)V_-(\lambda)\varphi(p)$ with
$$V_-(\lambda)\varphi(p)=\lambda{L(\lambda p)\over L(p)}\varphi(\lambda
p).\eqno(5.4)$$
$V_\pm$ is of course just the unitarily equivalent realization of
$U_\pm$ in the Fock space constructed over the one particle space of
$\Phi$, {\it viz.\/} ${\cal H}_1=L^2(\theta(p)pQ(p^2)dp)$.
Since ${L(\lambda
p)}/{L(-\lambda p)}$ is analytic in $p$ in the upper half plane for
$\lambda>0$ it is clear that
if $\tilde f$ is the Fourier transform of a test function with support in
the right wedge $\BR^+$, then the same holds for $V_+(\lambda)\tilde f$.
The analogous statement for the left wedge $\BR^-$ and $V_-(\lambda)$ is
also clear.
It follows that one can define one paprameter groups
$\sigma^\pm_t$ of automorphisms of ${\cal M}(\BR^\pm)$ (realized as
algebras on ${\cal H}_0$) by
$$\sigma^+_t(W(f))=U_+(e^{-2\pi t})W(f)U_+(e^{2\pi t})\eqno(5.5)$$
for ${\rm supp\ } f\subset\BR^+$, and
$$\sigma^-_t(W(f))=U_-(e^{2\pi t})W(f)U_-(e^{-2\pi t})\eqno(5.6)$$
for ${\rm supp\ } f\subset\BR^-$.
In order to show that the groups defined by (5.5) and (5.6) are indeed the
modular groups associated with the vacuum state on ${\cal M}(\BR^+)$ and
${\cal M}(\BR^-)$ respectively, it is sufficient to verify the KMS
condition. We discuss this for the algebra ${\cal M}(\BR^+)$, the other is
treated in the same way. One must show that for test functions $f$, $g$
with support
in $\BR^+$ the function $F(t):=\la \Omega, \sigma^+_t(W(f))W(g)\Omega\ra$
has an analytic
continuation from the real axis into the half
strip $\{t+is\mid 0~~From the Weyl relation (2.15) and the equation $\la\Omega,
W(h)\Omega\ra=\exp(-\Vert\tilde h\Vert^2/2)$, with $\Vert \cdot\Vert$ as in
(2.19), it follows that (5.7) is equivalent to a corresponding relation
for the two point function, where we find it convenient to use the variable
$\lambda=e^{2\pi t}$ instead of $t$:
The function $G(\lambda):=\la\Omega,\Phi(f)U_+(\lambda)\Phi(g)\Omega\ra$
has an
analytic continuation in $\lambda$ from the positive real axis up
into the cut plane $\BC\backslash
\BR^+$, and
$$\lim_{\theta\uparrow 2\pi}G(e^{i\theta})=
\la\Omega,\Phi(g)\Phi(f)\Omega\ra.\eqno(5.8)$$
We shall now verify (5.8). Since $\Phi(f)=\Phi_0(L(D)f)$ we
have by (5.3) for
$\lambda>0$:
$$\eqalignno{G(\lambda)&=\la\Omega,\Phi(f)U_+(\lambda)\Phi(g)\Omega\ra
=\int_{0}^{\infty}\tilde
f(-p)\lambda{L(-\lambda p)\over L(- p)}\tilde g(\lambda
p)pQ(p^2)dp\cr&=\lambda\int_{0}^{\infty}\tilde
f(-p)\tilde g(\lambda
p)pL(p)L(-\lambda p)dp.&(5.9)\cr}$$
Since $g$ is a test function with support in $\BR^+$,
$\tilde g(\lambda p)$ is analytic in
$\lambda$ in the upper
half plane and rapidly decreasing in $p\in\BR^+$, together with
$d\tilde g(\lambda p)/d\lambda$, for fixed $\lambda$. Hence we may
continue $G$ analytically, obtaining
$$G(\lambda e^{i\pi})=-\lambda\int_{0}^{\infty}\tilde
f(-p)\tilde g(-\lambda
p)pL(p)L(\lambda p)dp.\eqno(5.10)$$
Now the integrand is analytic in the lower half plane in $p$ with rapid
decrease at infinity, so we may
rotate the integration contour clockwise by $\pi$, obtaining
$$G(\lambda e^{i\pi})=-\lambda\int_{0}^{\infty}\tilde
f(p)\tilde g(\lambda
p)pL(-p)L(-\lambda p)dp.\eqno(5.11)$$
Finally, we again continue analytically in $\lambda$ and obtain
$$\lim_{\theta\uparrow 2\pi}G(\lambda e^{i2\theta})=
\lambda\int_{0}^{\infty}\tilde
f(p)\tilde g(-\lambda p)pL(-p)L(\lambda p)dp.\eqno(5.12)$$
For $\lambda=1$ the right side is just
$$\int_{0}^{\infty}\tilde
g(-p)\tilde f(p)pQ(p^2)dp=
\la\Omega,\Phi(g)\Phi(f)\Omega\ra.\eqno(5.13)$$
Having identified $\sigma_t^\pm$ as the modular groups of ${\cal
M}(\BR^\pm)$ we can now write the polar decomposition of the Tomita
operators $S_\pm$ that map $A\Omega$ into $A^*\Omega$ for $A\in{\cal
M}(\BR^\pm)$: We have $S_\pm=J_\pm\Delta_\pm^{1/2}$ with
$\Delta_\pm^{1/2}=U_\pm(e^{i\pi})$, in particular
$$\Delta_+^{1/2}\psi(p)=-{P(p)^2\over P(-p)^2}
\psi(-p)\eqno(5.14)$$
for $\psi\in{\cal H}_{01}$ of the form $\psi(p)=L(p)\tilde f(p)$, where
$f$ has support in $\BR^+$, so $\tilde f$ is analytic in the upper half
plane, and
$$J_+\psi(p)=-{P(p)^2\over P(-p)^2}{\psi(p)}^*\eqno(5.15)$$
for all $\psi\in{\cal H}_{01}$. This defines the operators on the one
particle space; the extension to the whole Fock space is by sums of tensor
products. The corresponding formulae for ${\cal
M}(\BR^-)$ are
$$\Delta_-^{1/2}\psi(p)=-{P(-p)^2\over P(p)^2}
\psi(-p)\eqno(5.16)$$
and
$$J_-\psi(p)=-{P(-p)^2\over P(p)^2} {\psi(p)}^*\eqno(5.17)$$
We see that $J_+=J_-$ if and only if $P(p)^2=P(-p)^2$,
i.e, if and only if $P(p)$ is constant, because $P(p)$ and $P(-p)$ have
their zeros in different half planes. This shows anew that ${\cal
M}(\BR^+)^\prime={\cal M}(\BR^-)$ if and only if $L(p)$ has only real zeros.
Finally we remark that by (5.15) the algebras
${\cal M}_1(\BR^+)$ and ${\cal M}_2(\BR^+)$
generated by two different fields $\Phi_1$ and $\Phi_2$
have the same $J$-operators if and only if the corresponding
polynomials $P_1$, $P_2$ differ at most by a constant factor.
Since $\Omega$ is separating for
${\cal M}_1(\BR^+)
\vee{\cal M}_2(\BR^+)$ (the commutant contains ${\cal M}_0(\BR^-$)),
Tomita's Theorem implies that the algebras ${\cal M}_1(\BR^+)$ and
${\cal M}_2(\BR^+)$
are equal if and only if their $J$-operators are equal. Hence
${\cal M}_1(\BR^+)={\cal M}_2(\BR^+)$ if and only if
$P_1$, $P_2$ are proportional, as already noted in Section 3.
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent{\bbf 6. Connection with the Two Dimensional CPT Theorem}
\bigskip
In [B] it is proved that any two dimensional local net with
translational covariance and spectrum condition can be embedded into a
Poincar\'e covariant net that is CPT invariant and satisfies
duality. As pointed out in [B], it is not to be expected in
general that the extended net is unique, in fact the construction in [B]
gives rise to infinite families of such nets. If the
original net satisfies wedge duality, all these nets coincide, but
in general they will all be different. We shall now show that this is
indeed the case for the field (2.8) if $L(p)$ has a root
away from the real axis.
The construction of [B] applied to a net ${\cal M}(\cdot)$
on a light ray defines to begin with two local
nets ${\cal N}_+(\cdot)$ and ${\cal N}_-(\cdot)$ as follows: If $I_{a,b}$
denotes the interval $]a,b[$ and $\BR^+_a=]a,\infty[$, $\BR^-_a=]-\infty,
a[$, then one defines
$$\eqalignno{{\cal N}_+(I_{a,b})&={\cal M}(\BR^+_a)\cap{\cal
M}(\BR^+_b)^\prime&(6.1)\cr
{\cal N}_-(I_{a,b})&={\cal M}(\BR^-_b)\cap{\cal
M}(\BR^-_a)^\prime&(6.2)}$$
Both nets contain the original net ${\cal M}(\cdot)$ as a subnet
and they are
Poincar\'e covariant with CPT symmetry. The Lorentz boosts for
${\cal N}_\pm(\cdot)$ are given by the modular group of ${\cal
M}_\pm(\BR^\pm)={\cal N}_\pm(\BR^\pm)$, and the CPT operator is the
$J$-operator of ${\cal M}_\pm(\BR^\pm)$.
The nets ${\cal N}_\pm(\cdot)$ generated by the field (2.8) can be
decribed explicitly in terms of the net ${\cal M}_0(\cdot)$ of
the $U(1)$-current $\Phi_0$. For this purpose we define two unitary
operators $Y_\pm$ on the Fock space ${\cal H}_0$ by
$$\eqalignno{Y_+\psi(p)&={P(p)\over P(-p)}\psi(p)&(6.2)\cr
Y_-\psi(p)&={P(-p)\over P(p)}\psi(p)&(6.3)\cr}$$
for $\psi\in{\cal H}_{01}$ and canonical extension to ${\cal H}_0$. We
assert that
$${\cal N}_+(I)=Y_+{\cal M}_0(I)Y_+^{-1}\qquad\hbox{\rm and}\qquad
{\cal N}_-(I)=Y_-{\cal M}_0(I)Y_-^{-1}\eqno(6.4)$$
for all $I$. To show this for ${\cal N}_+(\cdot)$ (the other case is
analogous) we note first that
since ${\cal M}_0(\cdot)$ satisfies duality we have
${\cal M}_0(I_{a,b})={\cal M}_0(\BR^+_a)\cap{\cal
M}_0(\BR^+_b)^\prime$. This implies the corresponding relation for
the net $Y_+{\cal M}_0(\cdot)Y_+^{-1}$. Moreover,
we have ${\cal
N}_+(\BR^+)=\CM(\BR^+)$ and hence also ${\cal
N}_+(\BR_a)=\CM(\BR_a^+)$ for all $a$ by translational covariance.
It therefore suffices to check that $Y_+{\cal M}_0(\BR^+)Y_+^{-1}={\cal
M}_+(\BR^+)$. But this is an easy consequence of the fact that $Y_+$
establishes a one to one correspondence between test functions $\tilde
f(p)$ that are
analytic in the upper half plane and functions of the form $L(p)\tilde
g(p)$ with $\tilde g(p)$ analytic in the upper half plane: Since
$L(p)=R(p)P(p)$ with $R(-p)=\pm R(p)$, we have $Y_+\tilde f(p)=L(p)\tilde
g(p)$ with $\tilde g(p)=\pm \tilde f(p)/L(-p)$.
The fact that ${\cal N}_+(I)=Y_+{\cal M}_0(I)Y_+^{-1}$ is reflected
in the corresponding relations for the Lorentz boosts and the CPT
operators. The Lorentz boosts for $\CM_0(\cdot)$ are given by (2.10) and
hence for ${\cal N}_+(\cdot)$ by
$$Y_+U_0(\lambda)Y_+^{-1}=U_+(\lambda)\eqno(6.5)$$
where $U_+(\lambda)$ is the modular group for ${\cal
N}_+(\BR^+)=\CM(\BR^+)$, cf.\ (5.1). The CPT operator for $\CM_0(\cdot)$
is the $J$ operator for $\CM_0(\BR^\pm)$, given by
$$J_0\psi(p)=-\psi(p)^*\eqno(6.5)$$
for $\psi\in{\cal H}_{01}$, and the CPT operator for ${\cal N}_+(\cdot)$ is thus
$$Y_+J_0Y_+^{-1}=Y_+^2J_0=J_+,\eqno(6.6)$$
cf.\ (5.15). The corresponding operators for $\CN_-(\cdot)$
are given by (5.2)
and (5.17). It is clear that the nets $\CN_+(\cdot)$ and $\CN_-(\cdot)$
are equal if and only if $\CM(\cdot)$ satisfies wedge duality, or
equivalently, if and only if $J_+=J_-$. As noted above, this holds if
and only if $P(p)$ is a constant, i.e. $L(p)$ has only real roots.
As discussed in [B] one may for each of the nets $\CN_\pm(\cdot)$ produce
an infinite family of local nets
$$\CN_\pm^k(I)=(J_-J_+)^k\CN_\pm(I)(J_-J_+)^{-k},\quad k\in\BZ.\eqno(6.7)$$
All these nets are again Poincar\'e covariant, satisfy CPT invariance and
duality and extend the original net
$\CM(\cdot)$. In the present case we have
$$J_-J_+=Y_+^4=Y_-^{-4}\eqno(6.8)$$
and thus
$$\CN_\pm^k(I)=Y_+^{4k\pm 1}\CM_0(I)Y_+^{-(4k\pm 1)}\eqno(6.9)$$
because $\CN_\pm(I)=Y_\pm\CM(I)_0Y_\pm^{-1}$ and $Y_-=Y_+^{-1}$. The CPT
operator for $\CN_\pm^k(I)$ is
$$Y_+^{4k\pm 1}J_0Y_+^{-(4k\pm 1)}=Y_+^{2(4k\pm1)}J_0.\eqno(6.10)$$
It is clear that unless $P(p)$ is a constant these operators are all
different, and the same is therefore true for
the nets $\CN_\pm^k(\cdot)$.
The models considered above have the special
feature that the nets $\CN_+(\cdot)$ and $\CN_-(\cdot)$ are unitarily
equivalent to each other, although they are represented differently on the
Hilbert space of the net $\CM(\cdot)$. In fact, since $\CN_\pm(I)=Y_\pm
\CM_0(I)Y_\pm^{-1}$ and $Y_+=Y_-^{-1}$, we have $\CN_+(I)=
Y_+^{2}\CN_-(I)Y_+^{-2}$. More generally, the unitary equivalence of
$\CN_\pm(\cdot)$ holds for all nets such that $\CM(\BR^-)$ is unitarily
equivalent to $\CM(\BR^+)^\prime$, provided the equivalence is implemented
by an operator commuting with translations. It is not to be expected that
this holds in general.
A last point worth mentioning is that the nets $\CN_\pm^k(\cdot)$ do not
exhaust the list of extensions of $\CM(\cdot)$ satisfying duality in
the examples considered. In fact, $\CM(\cdot)$ is a subnet of
$\CM_0(\cdot)$, which is not equal to any of $\CN_\pm^k(\cdot)$
if $P(p)$ is not constant, although these nets are all unitarily equivalent to
$\CM_0(\cdot)$.
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent{\bbf 7. Higher Dimensional Examples}
\bigskip
In this last section we consider fields in $n$-dimensional
Minkowski space, $n>2$. The most general two-point function consistent
with positivity, translational covariance, spectrum condition and locality
has in Fourier space the form
$${\cal W}_2(p)=\sum_{i=1}^N M_i(p)d\mu_i(p)\eqno(7.1)$$
where $d\mu_i$ is a positive Lorentz-invariant measure with support in
the forward light cone and $M_i$ is a polynomial that is positive on the
support of $d\mu_i$, $i=1,\dots,N$. Guided by the low-dimensional
examples considered above we
shall compute the modular groups
of the wedge algebras for generalized free fields on $\BR^n$
in the special case that
the sum in (7.1) contains only one term, i.e.,
$${\cal W}_2(p)=M(p)d\mu(p),\eqno(7.2)$$
and the polynomial $M$ allows a factorization,
$$M(p)=F(p)F(-p)\eqno(7.3)$$
where $F(p)$ is a function (in general not a polynomial)
with certain analyticity properties to be
specified below.
To describe the properties of $F$ we use the light cone coordinates
$x_\pm=x^0\pm x^1$ for $x=(x^0,\dots,x^{(n-1)})\in\BR^n$ and denote
$(x^2,\dots,x^n)$
by $\hat x$. The Minkowski scalar product is
$$\la x,y\ra=\mfr 1/2 (x_+y_-+x_-y_+)-\hat x\cdot\hat y.\eqno(7.4)$$
The right wedge, $W_R$, is characterized by $x_+>0$, $x_-<0$;
hence the Fourier transform,
$\tilde f(p)=\int \exp (-i\la p,x\ra) f(x) d^n x$ of a test
function $f$ with support in $W_R$ has for fixed $\hat p\in\BR^{n-2}$
an analytic continuation in $p_+$ and $p_-$ into the half planes
${\rm Im\, } p_+>0$,
${\rm Im\, } p_-<0$. The property required for $F$ is that $F(\pm p)$
is analytic and $F(-p)$ {\it without zeros} in this domain, with
$F(-p)=F(p)^*$ for $p\in\BR^n$. There is no lack of polynomials
$M$ allowing such a
factorization; one example (suggested by H.J. Borchers) is
$$M(p)=(p^1)^2+\cdots+(p^n)^2+m^2\eqno(7.5)$$
with
$$F(p)=\sqrt{\hat p\cdot \hat p+m^2}+ip^1=\sqrt{\hat p\cdot \hat
p+m^2}+\mfr
i/2(p_+-p_-).\eqno(7.6)$$
If $d\mu(p)=\theta(p^0)\delta(\la p,p\ra-m^2)$ we can
replace the polynomial (7.5) by $(p^0)^2$, hence the corresponding
generalized free field is nothing but the time derivative
$(d/dx^0)\Phi_m(x)$, where $\Phi_m$ is the free field of mass $m$.
In analogy with (5.3) we now define for $\lambda>0$ the unitary operators
$V_R(\lambda)$
on the Fock space ${\cal H}$ over the one-particle space
${\cal H}_1=L^2(\BR^n, M(p)d\mu(p))$ by
$$V_R(\lambda)\varphi(p)={F(-\lambda p_+,-\lambda^{-1}p_-,-\hat
p)\over F(- p_+,- p_-,-\hat
p)}\varphi(\lambda p_+,\lambda^{-1}p_-,\hat p)\eqno(7.7)$$
for $\varphi\in{\cal H}_1$ and canonical extension to ${\cal H}$.
By means of $V_R(\lambda)$ we then define a one parameter group of
automorphisms of the von Neumann algebra $\CM(W_R)$ on
${\cal H}$ generated by the Weyl
operators $W(f)$ with ${\rm supp\, }f\subset W_R$:
$$\sigma^R_t(W(f))=V_R(e^{-2\pi t})W(f)V_R(e^{2\pi t}).\eqno(7.8)$$
Note that we are working on the Fock space constructed over
$L^2(\BR^n, M(p)d\mu(p))$
and not $L^2(\BR^n,d\mu(p))$, hence we do not make use of the
analogues of the operators $U_+$ defined in (5.1).
By essentially the same computation that verified (5.8) one
shows that (7.8) satisfies the KMS condition and is therefore the modular
group defined by the vacuum state on $\CM(W_R)$. The corresponding
modular operator is given by
$$\Delta_R^{1/2}\varphi(p)={F(p_+,p_-,-\hat
p)\over F( -p_+, -p_-,-\hat
p)}\varphi(-p_+,-p_-,\hat p)\eqno(7.9)$$
where $\varphi$ is analytic in $p_+$ in the upper half plane and in
$p_-$ in the lower half plane. The modular conjugation is
$$J_R\varphi(p)={F(p_+,p_-,-\hat
p)\over F( -p_+, -p_-,-\hat
p)}\varphi(p_+,p_-,-\hat p)^*.\eqno(7.10)$$
Note that (7.9) and (7.10) are written for $\varphi$ in
the space $L_2(M(p)d\mu(p),\BR^n)$
that depends on the field,
while $\psi$ in (5.14) and (5.15) belongs to
one particle space $L_2(\theta(p)pdp,\BR)$
of the chiral $U(1)$-current. This is the reason why there is a square
in the phase
factor in (5.14)-(5.15), while (7.9)-(7.10) has only the
first power of $F$.
For the left wedge
$W_L=\{x\mid x_+<0, x_->0\}$ the corresponding operators are
$$V_L(\lambda)\varphi(p)={F(\lambda p_+,\lambda^{-1}p_-,\hat
p)\over F( p_+, p_-,\hat
p)}\varphi(\lambda p_+,\lambda^{-1}p_-,\hat p),\eqno(7.11)$$
$$\Delta_L^{1/2}\varphi(p)={F(-p_+,-p_-,\hat
p)\over F( p_+, p_-,\hat
p)}\varphi(-p_+,-p_-,\hat p)\eqno(7.12)$$
and
$$J_L\varphi(p)={F(-p_+,-p_-,\hat
p)\over F( p_+, p_-,\hat
p)}\varphi(p_+,p_-,-\hat p)^*.\eqno(7.13)$$
By comparing
(7.10) and (7.13) we see that the field satisfies the wedge
duality condition
$\CM(W_R)^\prime=\CM(W_L)$ if and only if $F(p)=F(-p)$ on the support of
$d\mu$. This condition is, e.g., violated in the example (7.6), and the
arguments of Sect.\ 4 are easily generalized to show that neither does
essential duality hold in this case.
The example (7.6) demonstrates also that the
modular group of $\CM(W_R)$ may
act nonlocally in the $\hat x$-directions. In fact,
let $f$ be a test function with compact support in $W_R$. Under the
transformation (7.7) the Fourier transform $\tilde f$ is mapped into
$$\tilde f_\lambda(p)={\sqrt{\hat p\cdot \hat p+m^2}-\frac
i2(\lambda p_+-\lambda^{-1} p_-)\over\sqrt{\hat p\cdot \hat p+m^2}-\frac
i2(p_+-p_-)}\tilde f(\lambda p_+,\lambda^{-1} p_-, \hat p).\eqno(7.10)$$
This is no longer the Fourier transform of a function of compact support
in the $\hat x$-directions,
because it is not analytic in $\hat p$. From this lack of analyticity it
is not difficult to deduce that
$W(f_\lambda)$ does not belong to any wedge algebra generated by the
field unless the wedge is a translate of $W_R$ or $W_L$,
but we refrain
from presenting a formal proof of this. The
operator $W(f_\lambda)$
is still localized in the $x^0,x^1$-directions in the sense that it is
contained in $\CM(W_R+a)\cap\CM(W_R+b)^\prime$ for some $a,b\in W_R$,
in accordance with the theorem of Borchers [B].
\bigskip\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%
\vskip 0.4cm\noindent
{\bf Acknowledgements\ } I am grateful to professors H.J. Borchers and D.
Buchholz for discussions and to the
Heraeus-Stiftung for financial support during my stay in
G\"ottingen in the
summer 1993.
\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}
%
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%
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\end
~~