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PCT-Theorem, Algebraic Quantum Field Theory
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\begin{document}
\title{\bf{On the PCT--Theorem\\ in the Theory of
Local Observables}}
\author{\vspace{5pt} H.J. Borchers$^{1}$ and J. Yngvason$^{2}$\\
\vspace{-4pt}\small{$1.$ The Erwin Schr\"odinger International
Institute for Mathematical Physics} \\ \small{Boltzmanngasse 9, A--1090 Wien}\\
\small{and}\\ \small{Institut
f\"ur Theoretische Physik,
Universit\"at G\"ottingen,}\\ \small{Bunsenstrasse 9, D--37073
G\"ottingen\thanks{permanent address}}\\
\vspace{-4pt}\small{$2.$ Institut f\"ur Theoretische Physik,
Universit\"at Wien}\\ \small{Boltzmanngasse 5, A 1090 Vienna,
Austria}\\ \\ \small{ \it Dedicated to Sergio Doplicher and John Roberts
on the occasion of their 60th birthday}}
\date{}
\maketitle
%%%%%%%%%%%%%
\begin{abstract}
We review the PCT-theorem and problems connected with its demonstration.
We add a new proof of the PCT-theorem in the theory of local observables
which is similar to that one of Jost in Wightman quantum field theory.
We also look at consequences in case the PCT-symmetry is given on the
algebraic level. At the end we present some examples which
answer general questions and throw some light on open problems.
\end{abstract}
\maketitle
\section{Introduction}
Before starting to discuss the different aspects of the PCT-theorem
we collect some terminology and notations that we shall be using.
This is necessary because
the notions in the literature are not unique.
\subsection{\bf On quantum field theory}
The term Lagrangian or Wightman field theory means quantum field
theory with \lq\lq point like\rq\rq\
localized fields. More precisely, the fields
are operator valued distributions and the field operators are supposed
to transform covariantly under a continuous unitary representation of
the connected part $\CP^\uparrow_+$ of the Poincar\'e group, or at
least of the subgroup of space-time translations. The representation
of the translations is required to fulfill the spectrum condition and
the representation space shall contain a vacuum vector (mostly assumed
to be unique). It is also assumed that this vacuum vector is cyclic
for the algebra generated by the field operators. For space-like
separation of the arguments the field operators shall either commute
or anti-commute. The Bose-fields have to commute with themselves and
all other fields. Fermi-fields anti-commute with Fermi-fields. The
Bose-fields represent particles with integer spin and the Fermi-fields
such with half-integer spin. Therefore, the phrase ``spin and
statistics'' means in this context the same as ``spin and commutation
relations''. Para-fields, introduced by O.W. Greenberg \cite{Gr61},
will not be considered. Para-statistics are usually only discussed in
the frame of quantum field theory of local observables, where a
reduction to ordinary Bose- or Fermi commutation relations has
been achieved by Doplicher and Roberts in \cite{DR90}.
Quantum field theory of local observables (QFTLO) in the sense of
Araki, Haag and Kastler \cite{Ha92} is concerned with $C^*$-algebras
$\mathcal{A}(O)$, associated with bounded open regions $O\subset
\mathbb R^d$. These algebras shall fulfill isotony, i.e., $O_1\subset
O_2$ implies $\CA(O_1)\subset\CA(O_2)$, and locality, i.e., if $O_1$
and $O_2$ are space-like separated, then the algebras $\CA(O_1)$ and
$\CA(O_2)$ commute element-wise. Space-like separation is here
defined in terms of the Minkowski metric on $\mathbb R^d$. For
unbounded domains $G$ the algebra $\CA(G)$ is defined as the
$C^*$-inductive limit of $\{\CA(O):\ O\subset G\}$.
%
Usually it is
assumed that the net $\{\CA(O)\}$ is covariant under a representation
of the proper, orthochronous Poincar\'e group $\CP^\uparrow_+$, or at
least some subgroup $\CG\subset\CP^\uparrow_+$, by
automorphism of
$\CA(\BR^d)$, i.e., to every $g\in\CG$ there is assigned an
automorphism $\alpha_g$ such that
$\alpha_{g_{1}}\alpha_{g_{2}}=\alpha_{g_{1}g_{2}}$, and for all $O$
\begin{equation}\label{1.0}\alpha_g\CA(O)=\CA(gO).\end{equation}
A QFTLO may often be defined in terms of Wightman fields by taking as
local algebras $\CA(O)$ the von Neumann algebras generated by the polar
decompositions of the smeared field operators with test functions
supported in $O$. The local commutativity of these algebras is
not a consequence of the locality of the field alone, but several
conditions which guarantee this are known. See \cite{BY92} for a
review.
\subsection{\bf Representations.}
A representation $\pi$ of $\CA(\BR^d)$ on a Hilbert space $\CH_{\pi}$
will always mean a non-degenerate representation. We shall denote the
von Neumann algebra $\pi(\CA(O))^{\prime\prime}$ by $\CM_{\pi}(O)$,
or simply by $\CM(O)$ if $\pi$ is a vacuum representation (see below).
We say that {\it weak additivity} holds in the representations
considered, i.e., the von Neumann algebra generated by
$\cup\{\CM_{\pi}(O+x):
x\in\BR^d\}$ is equal $\CM_{\pi}(\BR^d)$ for each bounded, open $O$,
independent of its size. Usually it is required only for vacuum
representations.
%Bemerkung:
When the QFTLO is
covariant under $\CP^\uparrow_+$ or some subgroup $\CG\subset
\CP^\uparrow_+$ the automorphisms $\alpha_{g}$ are {\it implemented} in the
representation $\pi$ if there is a continuous unitary representation $U$ of
$\CG$ on $\CH_{\pi}$ such that
\begin{equation}\label{1.1}U(g)\pi(A)U^*(g)=\pi(\alpha_gA),\qquad
A\in\CA(\BR^d),\ g\in\CG.
\end{equation}
By \eqref{1.0}, it is clear that
\begin{equation}\label{1.1a}U(g)\CM_{\pi}(O)U^*(g)=\CM_{\pi}(gO).
\end{equation}
For a classification of the representations $\pi$ one often takes
for $\CG$ the translation group $\BR^d$ or the subgroup of time
translations.
If a topological group $\CG$ acts on a $C^*$-algebra $\CA$
by automorphisms $\alpha_g$ one speaks of a {\it $C^*$ dynamical system}.
If a representation $\pi$ of $\CA$ is given,
then it is of interest to know whether or not the group action
is unitarily implemented in the representation. The answer is known up to
multiplicity
problems implying that eventually one might have to to change the
multiplicity of the representation. Representations which differ
at most in their multiplicity are called {\it quasi-equivalent}.
Such representations have the same {\it folium} of states, i.e., states
given by a density matrix in the representation. These are the
ultra-weakly continuous (normal) states on the von Neumann algebra
$\pi(\CA)^{\prime\prime}$.
For a discussion of the problem of implemented group
action and its history see \cite{Bch96},
Section II.7. The main result is:
\begin{theorem} {\rm (Borchers).}
Let $\{\CA,\CG,\alpha\}$ be a $C^*$-dynamical system, with $\CG$ a
topological group, and let $\pi$ be a representation of $\CA$.
The following are equivalent:
\begin{itemize}
\item[(i)]
There exists a representation $\widehat{\pi}$ of $\CA$ that is
quasi-equivalent to $\pi$
and such that $\alpha_g$ is implemented in the
representation
$\widehat{\pi}$ by a continuous unitary
representation of $\CG$.
\item[(ii)]
The folium $F(\pi)$ of $\pi$ is invariant
under the adjoint action $\alpha_g^*$ and the action is strongly
continuous in $g$ on $F(\pi)$ .
\end{itemize}
\end{theorem}
A {\it thermal representation} is characterized by the group of time
translations. Its representation $U(t)$ must have an invariant vector
$\Omega_\beta\in\CH_\pi$ which is cyclic and separating for the representation.
Moreover, the representation $U(t)$ shall fulfill the $\beta$-KMS condition,
which will be explained together with the Tomita-Takesaki theory.
We call a
representation $\pi$ a {\it particle representation}
if the whole translation group $\BR^d$ is implemented by a
unitary group $U(a)$ that fulfills the spectrum condition, i.e.,
the spectrum of $U$ is contained in the closed, forward light-cone
$\overline V^+$.
The name ``particle representation'' will be used in accordance with the
terminology in \cite{Bch96}.
This does not imply that the mass operator has a discrete part.
In nature probably most massive particles are infra-particles implying
that there are representations such that the mass operator has a
purely continuous spectrum.
A particle representation $\pi$ is
called a {\it vacuum representation}, if $U(a)$ has an invariant vector
$\Omega$ and if this vector is cyclic for the representation.
Moreover, we will assume that vacuum representations are factor
representations. A vacuum representation will always be denoted by $\pi_0$.
If $\pi$ is a representation with cyclic vector $\Omega$ then we say
that the {\it Reeh-Schlieder property\/} holds for
$(\pi,\Omega)$, if $\Omega$ is
cyclic for every $\CM_{\pi}(O)$ with $O$ open and nonempty. If $\pi$
is a particle representation enjoying the weak additivity property
and if $\psi\in\CH_\pi$ has compact energy support then
$(\pi,\psi)$ has the Reeh-Schlieder property. For the
vacuum vector it was proved by Reeh and Schlieder \cite{RS61}
\subsection{\bf Maximal local algebras.}
Domains of special interest in our discussion are the {\it wedges} in
$\BR^d$. They are defined in the following manner
by two non-zero light-like vectors
$\ell_1,\ell_2$, belonging to the boundary of the forward light-cone
$V^+$ :
\begin{equation}W(\ell_1,\ell_2)=\{\lambda\ell_1+\mu\ell_2+\ell^\perp:
\lambda>0, \mu<0,(\ell^\perp,\ell_i)=0, i=1,2\}. \end{equation} The
plane spanned by $\ell_1$ and $\ell_2$ will be called the {\it
characteristic two-plane} of the wedge $W(\ell_1,\ell_2)$. The
translated wedge $W(\ell_1,\ell_2)+a$ is denoted by $W(\ell_1,\ell_2,a)$.
The $C^*$-algebra $\CA(W)$ is the algebra generated by all
$\CA(O)$ with $O\subset W$. A representation $\pi$ of the theory of local
observables fulfills {\it wedge duality}, if for every wedge $W$ one has
\begin{equation}\CM_{\pi}(W)'=\CM_{\pi}(W'),\end{equation}
where $W'$ denotes the interior of the space-like complement of $W$, i.e.
$W(\ell_1,\ell_2)'=W(\ell_2,\ell_1)$
and $\CM_\pi(G)$ the von Neumann algebra generated by $\pi(\CA(G))$.
The index $\pi$ will be dropped if $\pi$ is a vacuum representation.
Another important family of domains are the double cones or other
domains which are the open interior of intersections of wedges.
Double cones, denoted by $D$, play a special role. At several occasions
one assumes or derives some properties of the algebras
$\CM_{\pi}(W)$ associated with wedges and one wants to deduce
corresponding properties of a double cone algebra. This is only
possible if the latter algebra can be expressed in terms of wedge algebras.
(For short we often use the term ``double cone algebra'' instead of
``algebra associated
with a double cone'', and analogously for other domains.)
Namely, the double cone
algebra must have the form
\begin{equation}\label{maximal}\CM_{\pi}(D)=\cap\{\CM_{\pi}(W):\ D\subset W\}.
\end{equation}
If this is the case then we call $\CM_\pi(D)$ a {\it maximal
local algebra}. It is easily checked that the theory constructed from the
maximal local algebras fulfills all requirements of the QFTLO. If
$\CM_{\pi}(D)$ is not the maximal local algebra then one can {\it
define}
$\CM_{\pi,{\rm max}}(D)$ by the right hand side of Eq. \eqref{maximal}.
If wedge duality holds, these algebras are, indeed, the maximal
extensions of $\CM_{\pi}(D)$ compatible with locality.
In general the QFTLO's $\{\CA(O)\}$ and $\{\CM_{\pi,{\rm max}}(O)\}$
will have different families of representations. We shall
consider the maximal algebras only for a vacuum representation and denote
them in this case simply by
$\CM_{{\rm max}}(O)$.
For reasons we will see later, several aspects of QFTLO can be handled
by taking as input von Neumann algebras $\CM(W)$ associated only with wedges.
Of course the algebras associated with $W$ and $W'$ have to commute.
In this case one can always define double cone algebras as on the
right hand side of \eqref{maximal} and these algebras will satisfy
locality. The only problem is that in general one does not know the
their size, in particular $\CM(W)$ need not be generated by the
$\CM(D)$'s with $D\subset W$.
\subsection{\bf Charge sectors.}
Two representations $\pi_1,\pi_2$ of $\CA(\BR^d)$ are quasi-equivalent
if they have the same kernel and if the isomorphism
$\pi_1(\CA(\BR^d))\leftrightarrow \pi_2(\CA(\BR^d))$ extends to an
isomorphism between the von Neumann algebras generated by the two
representations.
For a $C^*$-algebra as
complicated as (non-trivial) QFTLO there exists at least a continuum
of non-equivalent representations. Therefore, one is interested in
principles which group the set of representations into sub-families.
The principle mostly used is that of {\it local equivalence}. Two
representations $\pi_1,\pi_2$ are called {\it locally equivalent}, if
for every bounded open region the representations $\pi_1(\CA(O))$ and
$\pi_2(\CA(O))$ are quasi-equivalent. If one of them is a vacuum
representation then one calls the second representation {\it locally
normal}. The requirement of local normality is often used in order to
select from the set of thermal representations (defined by other
means) a suitable sub-family which, from some point of view, can be
regarded as physically acceptable.
While in the Lagrangian or in Wightman's field theory charged fields
are put in by hand, it is the philosophy of the QFTLO that
representations of the observable algebra describing a finite
number of charged particles shall be constructed from the algebra of
observables. Also the charged fields connecting the vacuum
representation with the representations describing charges should be
constructed with help of the different representations. This can be
worked out if one uses the equivalence relation introduced by Borchers
\cite{Bch65}. If $\pi_0$ is a vacuum representation, then a factor
representation $\pi_1$ is called a {\it charged sector}
if $\pi_0$ and $\pi_1$ have the
same kernel and if for every $O$ the isomorphism $\pi_0(\CA(O'))
\leftrightarrow \pi_1(\CA(O'))$ extends to an isomorphism of the
corresponding von Neumann algebras. Here $O'$ denotes the space-like
complement of $O$. With this concept Doplicher, Haag and Roberts
\cite{DHR69a,DHR69b}, \cite{DR90} have worked out the details of the mentioned program.
The algebra generated by the local observables and by the localized
charged fields is called the {\it field algebra} and the corresponding net
is usually denoted by
$\{\CF(O)\}$. Within this setting also the concept of {\it
conjugate charge sectors} has a precise meaning: If an element $F$ of the
field algebra generates a charged sector by applying it to the vacuum,
then the conjugate sector is generated in the same way by $F^{*}$.
\subsection{\bf Gauge transformations.}
We shall use the term {\it gauge transformation} for any unitary
operator $U$ on the Hilbert space of a vacuum representation
$\pi_{0}$ of a QFTLO that fulfills $U\Omega=\Omega$ and
\begin{equation} U\pi_0(\CA(O))U^*=\pi_0(\CA(O))\end{equation}
for every bounded open $O$. In Lagrangian field theory such operators
typically implement transformations of the form $\psi(x)\to
\E^{\I\varphi}\psi(x)$ and generate a compact group. Since the QFTLO
is given by a set of axioms and not by a finite number of fields the
gauge group in the sense defined is not compact in general, however.
In the last section an example of such a case will be considered.
Therefore, a possible requirement for selecting a reasonable family of
QFTLO is the assumption that the gauge group is compact.
There are
other conditions implying this property. Buchholz and Wichmann
\cite{BuW86} have introduced the concept of {\it nuclearity}. This is the
requirement that the number of states, which can be created locally,
does not increase too fast with energy. Doplicher and Longo
\cite{DL84} have introduced the {\it split property}. This is equivalent to
the following: Let $O_1\subset O_2$ be two bounded open regions such
that the closure of $O_1$ is still contained in $O_2$. Let $\omega_1$
be a vector state on $\pi_0(\CA(O_1))$ and $\omega_2$ be a vector
state on $\pi_0(\CA(O'_2))$. Then exists a vector state on $\pi_0$ of
the QFTLO which coincides with $\omega_1$ on $\pi_0(\CA(O_1))$ and
with $\omega_2$ on $\pi_0(\CA(O'_2))$. Between these concepts one has
the following relations:
\begin{equation}
\text{nuclearity property}\longrightarrow
\text{split property} \longrightarrow
\text{compact gauge group}
\end{equation}
For the first arrow see \cite{BuW86} and
\cite{BDF87}, for the second see \cite{DL84}.
\subsection{\bf Tomita-Takesaki theory.}
The main tool for handling the problems connected with the PCT-theorem
in the QFTLO is the modular theory introduced by Tomita. It is usually
called Tomita-Takesaki theory because the first presentation of this
theory, beyond a preprint, is due to Takesaki.
Let $\CH$ be a Hilbert space and $\CM$ be a von Neumann algebra,
acting on this space, with commutant $\CM'$. A vector $\Omega$ is cyclic
and separating for $\CM$, if $\CM\Omega$ and $\CM'\Omega$ are dense in $\CH$.
If these conditions are fulfilled, then a modular operator $\Delta$
and a modular conjugation $J$ are associated with the pair $(\CM,\Omega)$,
such that
\begin{itemize}
\item[(i)] $\Delta$ is self-adjoint, positive and invertible,
\[
\Delta\Omega=\Omega,\qquad J\Omega=\Omega.
\]
\item[(ii)] The operator $J$ is a conjugation, i.e., $J$ is anti-linear,
$J^*=J$, $J^2=1$, and $J$ commutes with $\Delta^{\I t}$. This implies
the relation
\[
\ad J\Delta =\Delta^{-1}.
\]
\item[(iii)] For every $A\in\CM$ the vector $A\Omega$ belongs to the domain
of $\Delta^{\frac12}$, and
\[
J\Delta^{\frac12}A\Omega=A^*\Omega=:SA\Omega.
\]
\item[(iv)] The unitary group $\Delta^{\I t}$ defines a group of automorphisms
of $\CM$,
\[
\ad \Delta^{\I t}\CM =\CM, \qquad \text{for all } t\in\BR.\]
\item[(5)] $J$ maps $\CM$ onto its commutant
\[
\ad J\CM =\CM'.\]
\end{itemize}
These results apply in particular to QFTLO in a vacuum
representation because of the Reeh-Schlieder theorem which implies that
the vacuum vector is cyclic and separating for every algebra
$\CM(G)$, where $G$ is any domain which has a space-like complement
with interior points.
The matrix elements of the
modular group have the following important analyticity properties.
If $A,B\in\CM$ and $\sigma^t(A):
=\ad\Delta^{\I t} A$ then the continuous function
\[
F(t)=(\Omega,B\sigma^t(A)\Omega)
\]
has bounded analytic continuation into the strip $S(-1,0)=\{z\in\BC:
-1<\Imt z<0\}$. At the lower boundary one finds
\begin{equation}F(t-\I)=(\Omega,\sigma^t(A)B\Omega)\end{equation}
This relation is called the {\it KMS-condition}. In QFTLO the
group $\sigma^t$
coincides for thermal states with scaled time translations. If
$\tau$ denotes the automorphism of the time translations and $\Omega$
defines a thermal state at inverse temperature $\beta$ the
equations become
\begin{equation}\label{betakms}
\begin{split}
F(t)&=(\Omega,B\tau_t(A)\Omega),\\
F(t-\I\beta)&=(\Omega,\tau_t(A)B\Omega).
\end{split}
\end{equation}
The relations
Eq.\ \eqref{betakms} are called the {\it $\beta$-KMS-condition}.
\section{Review of the PCT-theorem}
\subsection{Point-fields}
In Lagrangian or Wightman field theory the PCT-ope\-ra\-tor $\Theta$
is an anti-unitary operator implementing the PCT-symmetry (if present).
For a scalar field one has the relation
\begin{equation}\Theta\Phi(x)\Theta=\Phi^*(-x).\end{equation}
The formula for higher spin looks the same with respect to the
space-time variable $x$, but in the index space one has in general to introduce
an additional transformation independent of the variable $x$. In
a suitable spinorial basis the matrix of this transformation is
diagonal and each component simply gets multiplied by a phase factor.
In addition $\Theta$ shall fulfill
the following commutation rule with the Poincar\'e transformations
\begin{equation}\label{thetapoinc}\Theta U(\Lambda,a)\Theta=U(\Lambda,-a).\end{equation}
These requirements imply
\begin{equation}\Theta=\Theta^{-1}=\Theta^*.\end{equation}
PCT-theorems are results showing the existence of a PCT-operator
$\Theta$. The product PT represents an element of the Lorentz
group. All proofs so far assume that PT is the element $-1$
and has the determinant $+1$.
This implies that the Minkowski space must have even dimensions. In the
odd dimensional case one replaces P by the total
reflection in the space perpendicular to the one-direction, denoted
P$_1$. We will not
discuss this case and restrict ourselves to even dimensional Minkowski
spaces. As remarked in the last section, however, also in odd
dimensions, where PT has determinant $-1$, free single component
hermitian fields have full PCT symmetry, even
without Lorentz covariance.
The Lagrange function for a field theory is usually invariant under
total reflection, time-reversal, and charge conjugation. But it does not
necessarily mean that these symmetries are implemented separately by
unitary or antiunitary operators.
G. L\"uders \cite{Lue54} discovered that the product of these symmetries
is always implemented by an operator $\Theta$, the PCT-operator. The
input for this result is the implemented Poincar\'e covariance,
spectrum condition, and the existence of charge conjugate partners.
Pauli \cite{Pau55} was quite excited about this result, because it clari\-fied
the relation about spin and the commutation relations of the fields.
It also gave an understanding why for Fermi-fields one should use
a positive energy representation and not one with a "Dirac sea".
(For the discussion of spin and statistics see also G. L\"uders and
B. Zumino \cite{LZ58}).
In Wightman's field theory \cite{Wi56} the PCT-theorem was proved
by R. Jost \cite{Jo57}. This proof is based on the same assumptions as for the
Lagrangian field theory which are
\begin{itemize}
\item[{1.}] The theory is Poincar\'e covariant and the translations
fulfill the spectrum condition. Moreover, the representation space contains
a cyclic vacuum vector.
\item[{2.}] The Poincar\'e transformations of the fields induce
a finite dimensional representation of the Lorentz group in the index space.
\item[{3.}] Every field in the theory is accompanied by its charge conjugate
partner (which may be the field itself).
\end{itemize}
Jost's proof is based on a result of Hall and Wightman \cite{HW57}
which says that the analytic continuation of the Wightman functions
are invariant under the complexified connected component of the
Lorentz group. In even dimension the complex Lorentz group contains
the element $-1$. The requirement 2 does not forbid that a certain
field is infinitely degenerate. But if we deal with fields
transforming with an irreducible, infinite dimensional representation of the Lorentz group
in the index space the properties leading to the spin-statistics and PCT-theorem may be
destroyed. R. Streater \cite{Str67} constructed examples where fields
with integer spin anti-commute and those with half-integer spin
commuted. Oksak and Todorov \cite{OT68} gave other examples where
Jost's proof of the PCT theorem did not work. Requirement 3 implies
a symmetry under complex conjugation for the set of Wightman functions.
\subsection{\bf Algebraic PCT-symmetry in QFTLO}
Before discussing the PCT-theorem in the QFTLO let us assume that we have
such symmetry on the algebraic level, i.e., there exists an anti-linear
automorphism $\theta$ of $\CA(\BR^d)$ with
\begin{equation}\label{theta}
\begin{split}
\theta(AB)&=\theta(A)\theta(B),\qquad \theta(\lambda A)=
\overline{\lambda}\theta(A),\\
\theta(\CA(O))&=\CA(-O),\qquad A,B\in\CA(\BR^d).\end{split}\end{equation}
The automorphism $\theta$ should
also transform the translations in the correct manner, i.e.,
\begin{equation}\label{thetaalpha}\theta(\alpha_a(A))=\alpha_{-a}(\theta(A)).\end{equation}
Since $\theta$ maps the algebra $\CA(\BR^d)$ onto itself, its
transpose,
$\theta^*$, maps the dual space $\CA(\BR^d)^*$ onto itself. This implies
in particular that $\theta$ is represented by an anti-unitary operator
on the standard representation of the enveloping von Neumann algebra
$\CA(\BR^d)^{**}$. For later discussions it is of interest to know
sub-families of representations which are mapped by the PCT-symmetry
onto itself.
\begin{theorem} % 2.1
Let $\{\CA(O)\}$ be a QFTLO with algebraic PCT-symmetry $\theta$. Assume in addition
that the translations $\alpha_a$ act strongly continuous, i.e., the
function $a\to\alpha_a(A)$ is continuous in the norm topology of
$\CA(\BR^d)$ and this for every $A\in\CA(\BR^d)$. Then the family
of particle representations is invariant under $\theta$.
Moreover, the sub-family of vacuum representations is also
invariant.\end{theorem}
The proof will be given in the next section. The requirement that the
translations act strongly continuous is not necessary. For the proof of
the general case one would have to go into details of positive
energy representations as described in \cite{Bch96}.
But since we will look only at representations which are locally normal,
the general version of Thm. (2.1) is not needed for the following
reason:
Let $\{\CA(O)\}$ be a QFTLO and $\pi_0$ a vacuum representation.
Then $\{\CM_{\pi_{0}}(O)\}$ defines a new QFTLO, and
the locally normal representations of
the two nets are clearly in one to one correspondence with each other.
Now let $\pi_{1}$ be a locally normal representation of
$\CM_{\pi_{0}}$. The corresponding
local $C^*$ algebras, $\CA_{1}(O)$, contain dense subalgebras
$\CA_{\rm c}(O)$ on which the action of $\alpha_{a}$ is strongly
continuous in $a$. For every bounded $O$, $\CA_{\rm
c}(O)^{\prime\prime}=\CA_{1}(O)^{\prime\prime}$ is isomorphic as a
von Neumann algebra to $\CM_{\pi_{0}}(O)$ because $\pi_{1}$ is
locally normal. Hence $\{\CA(O)\}$ and
$\CA_{\rm c}(O)$, which satisfies the hypothesis of
Theorem 2.1, have the same family of locally normal
representations. For the family of these representations one has the
following result:
\begin{theorem} % 2.2
Let $\{\CA(O)\}$ be a QFTLO with algebraic PCT symmetry $\theta$. Let $\pi_0$
be a vacuum representation and assume it is invariant under $\theta$.
Then $\theta$ maps the family of locally
normal representations onto itself. Moreover, the family of
charge sectors is a $\theta$ invariant subfamily of representations.
\end{theorem}
Also the proof of this result will be postponed to the next section.
As a last point we want to look at the algebraic PCT-symmetry in thermal
representations.
\begin{theorem} % 2.3
Let $\{\CA(O)\}$ be QFTLO with PCT-symmetry $\theta$. Assume that the
time translations $\alpha_t$ act strongly continuously. Then the set
of $\beta$-KMS-states is $\theta^*$ invariant.
\end{theorem}
The proof will be given in the next section.
\subsection{\bf Preparations for the PCT-theorem}
In the QFTLO one usually looks for the PCT-symmetry only in the
vacuum representation of the local observables or of the corresponding
field algebra.
The requirements for a PCT operator $\Theta$ are:
\begin{itemize}
\item $\Theta$ is antiunitary and for all bounded $O$
\begin{equation}\label{thetacov}\Theta\pi_0(\CA(O))\Theta=\pi_0(\CA(-O)).
\end{equation}
\item The relation \eqref{thetapoinc} between $\Theta$ and the
representation of $\CP_{+}^\uparrow$ holds.
\item For field algebras one replaces
$\pi_0(\CA(O))$ by $\CF(O)$ in \eqref{thetacov}. Moreover, it is
required that $\Theta$ transforms a charge sector into its conjugate
sector.
\end{itemize}
For the discussion of the PCT-theorem we will mostly look at vacuum
representations. This is sufficient because of Thm.\ 2.2 (see also
\cite{GL92}). For the
construction of the PCT-operator on the field algebra one must go
first to the maximal local algebras and construct the charged fields
as described by Doplicher, Haag and Roberts \cite{DHR69a,DHR69b},
\cite{DR90}.
Starting from the PCT-theorem in the vacuum sector one can construct
the PCT-operator for the whole field algebra. This has been done
by Guido and Longo \cite{GL95}, see also \cite{GL92}.
One more remark is in order. Also for the vacuum representation
of the observable algebra
we will use the phrases PCT-symmetry and PCT-theorem and not simply
PT-symmetry and so on. The reason is the following: The axioms
of the QFTLO are so general that one can not exclude that the algebras
$\CA(O)$ contain also charged Bose-fields. If they contain such fields
then the group of gauge transformations contains a continuous representation of the circle
group. But we do not know any manageable condition excluding the existence
of such gauge transformations. If such a charged Bose field is present then one
has a proper PCT-symmetry also in the vacuum sector.
For a long time it was impossible to prove the PCT-theorem in QFTLO
because of the lack of proper mathematics. This changed with
the appearance of the Tomita-Takesaki theory. The basic result, where
the modular group and conjugation can be computed for some
algebra of interest in QFTLO, was given in \cite{BW75}.
\begin{theorem}{\em (Bisognano and Wichmann).\,}\label{2.4}
Assume a Wightman field theory of a scalar neutral field
is such that the smeared field operators
generate the local algebras. Then:\sabsatz
\begin{itemize}
\item[1.] The modular group of the algebra associated with a wedge and the
vacuum vector coincides
with the unitary representation of the group of Lorentz boosts which maps the wedge onto itself.
\item[2.] The modular
conjugation of the wedge $W$ is given by the formula
\begin{equation}\label{thetadef}
J_W=\Theta U(R_W(\pi)).
\end{equation}
Here $\Theta$ denotes the PCT-operator of the Wightman field theory
and $U(R_W(\pi))$ is the unitary representation of the rotation which
leaves the characteristic two-plane of the wedge invariant. The angle
of rotation is $\pi$.
\item[3.] The theory fulfills wedge duality.\end{itemize}
\end{theorem}
In case of the right wedge, $W_{\rm r}=\{x:|x^0|0$, such that
\[
D+x\subset \Lambda_{g_6 W}(t_6)\cdots\Lambda_{g_1 W}(t_1) W
\]
for $|t_i|\sum\limits_{i=1}^6 \Imt\zeta_i>-\frac12.
\end{equation}
If the elements $g_1,...,g_6$ are properly chosen
then an interior point of the $\zeta$ variables corresponds to an
interior point in the $\hat g$ variables.
In the $\zeta$--variable the domain \eqref{domain} is
convex and hence simply
connected. Since the transformation \eqref{zeta}
is bi--holomorphic, it follows
that also the image in the $t$--variables is simply connected. Hence
there are no monodromy problems in these variables.
Note that the symbol $U(\Lambda)$
denotes a representation of the Lorentz group and therefore, the first
expression of Eq.~ \eqref{prod} can be written as
\begin{equation}\label{prod2}
U(\Lambda_{g_6W}(t_6)\cdots\Lambda_{g_{1}W}(t_1))A\Omega.
\end{equation}
The arguments inside of $U$ are defined for all elements of the
complex Lorentz group. $U$ applied to this product is defined if it belongs
to the domain Eq. \eqref{domain} (transformed into the variable $z$),
eventually multiplied from the left with an element of the real
Lorentz group.
In particular we can look at the product
$U(R_{g_jW}(\pi))U(\Lambda_{g_jW}(-\frac\I2))U(g)A\Omega$
for $g$ sufficiently close to $1$ in \eqref{gamma} and obtain
\[
\begin{split}
U(R_{g_jW}(\pi)\Lambda_{g_jW}(-\mfrac\I2)&g)A\Omega=
U(gR_{g_jW}(\pi)\Lambda_{g_jW}(-\mfrac\I2))A\Omega=\\ &
U(g)U(R_{g_jW}(\pi))U(\Lambda_{g_jW}(-\mfrac\I2))A\Omega.
\end{split}
\]
This implies
$U(g)U(R_{g_jW}(\pi))U(\Lambda_{g_jW}(-\frac\I2))U(g^{-1})
=\newline U(R_{gg_jW}(\pi))U(\Lambda_{gg_jW}(-\frac\I2))$.
Choosing $\Gamma_1(D+x)$ in such a way that it contains with $g_j$
also its inverse, then the statement of the proposition is obtained by
choosing $g=g_ig_j^{-1}$.
\end{proof}
Collecting the result of the discussion we obtain with $A_x=T(x)AT(-x)$
\begin{proposition}
Let $D$ be a double cone centered at the origin and
$x$ such that the closure of $D+x$ does not
contain the origin. Then for $A\in\CM^{\rm fa}(D)$ and $g$ such that
$D+x\subset gW$ the vector function
\[
U(\Lambda_{gW}(-\mfrac\I2))U(R_{gW}(\pi))A_x\Omega
\]
is independent of $g$.
\end{proposition}
\begin{proof}
From the above discussion we know that the statement is
true for $g$ in a sufficiently small neighborhood of the identity
in $\CP^\uparrow_+$. But this implies by varying the wedges and the
neighborhoods that it is true for all $g\in \Gamma(D+x)$.
\end{proof}
Using Prop. 3.3 we find
\begin{theorem}
Let $A\in \CM^{\rm fa}(D)$ with $D$ centered at the origin, and let $D+x\subset W$. Then one has
\[
U(\Lambda_W(-\mfrac\I2))A_x\Omega=\widehat A_{P_W x}\Omega
\]
with $\widehat A_{P_W x}\in \CM(D+P_W x)$.
\end{theorem}
\begin{proof}
From Eq. (3.4) we know $U(R_{gW}(\pi))
U(\Lambda_{gW}(-\mfrac\I2))A_x\Omega$
is independent of $g$ as long as $g$ belongs to the set
$\Gamma(D+x)$ (see Eq. (3.2)). Hence we obtain
\[
U(R_{W}(\pi))U(\Lambda_{W}(-\mfrac\I2))A_x\Omega=U(R_{W}(\pi))
\widehat A_x\Omega
\]
with
\[
U(R_{W}(\pi)\widehat A_x U(R_{W}(\pi)\subset\mathop{\cap}
\limits_{g\in \Gamma(D+x)} \CM(-K_{gW}(D+x)).
\]
Next observe that a translation in the characteristic two-plane of $W$
commutes with $U(R_{W}(\pi))$ and is mapped onto its negative by
$U(\Lambda_{W}(-\frac\I2))$. This implies
\[
U(R_{W}(\pi))\widehat A_x U(R_{W}(\pi))\subset\mathop{\cap}
\limits_{D+x\subset W}\mathop{\cap}
\limits_{g\in \Gamma(D+x)} \CM(-K_{gW}(D+x)=\CM(-D-x).
\]
Using the fact that $D$ is symmetric and that $R_W(\pi)(-x)=P_Wx$ holds
we get the result of the theorem.
\end{proof}
Now we are prepared to show Thm. 2.15.
\begin{proof}
First a remark: If $\CA\in\CM(W)$ and $U(\Lambda_W(t)A\Omega$ has an
analytic continuation into the strip $S(-\frac12,0)$ then the same holds
for $V_W(t)A\Omega$ because $\Delta_W^{\I t}A\Omega$ has an analytic
continuation and the relation
\begin{equation}\label{gauge}
U(\Lambda_{W}(t))=V_W(t)\Delta_W^{\I t}
\end{equation}
holds. (For a detailed proof see \cite{Bch96}.) Since all groups in
\eqref{gauge} leave the wedge $W$ invariant it follows that they commute.
If $A\in\CM(D)^{\rm fa}$
then $U(R_{gW}(\pi))U(\Lambda_{gW}(-\frac\I2))$ acts locally on
$A_x\Omega$
i.e., it maps $A_x\Omega$ into $\CM(D-x)$
and is independent of $gW$, as long as $g\in\Gamma(D+x)$. By
Eq.\ \eqref{gauge} we can
replace $U(\Lambda_{gW}(-\frac\I2)$ by $R_{gW}(-\frac\I2)
\Delta_{gW}^{\frac12}$. This implies that $U(R_{gW}(\pi))V_{gW}(-\frac\I2)
\Delta_{gW}^{\frac12}$ acts locally on
$A_x\Omega$ and is independent of $g$, as long as $g\in\Gamma(D+x)$.
Replacing $A_x$ by its adjoint $A_x^*$ and observing that the Tomita
conjugation $S_{gW}$ acts locally on
$A_x\Omega$ and is independent of $g$ in the same range as before, we find,
that $U(R_{gW}(\pi))V_{gW}(-\frac\I2)J_{gW}$ has the good properties,
i.e., it maps $A_x\Omega$ into $\CM(D-x)\Omega$ and is independent of $g$
for $g\in\Gamma(D+x))$. We define
\[
\widehat\Theta_{W}=U(R_{W}(\pi))V_{W}(-\mfrac\I2)J_{W}=
U(R_{W}(\pi))J_{W} V_{W}(\mfrac\I2).
\]
Notice that $J_W$ commutes with $V_W(t)$. Hence we obtain
$J_W V_W(-\frac\I2)J_W=V_W(\frac\I2)\newline
=V_W(-\frac\I2)^{-1}$. This implies
${\widehat\Theta_W}^2={\bf 1}$.
By assumption the fully analytic elements applied to the vacuum are a
core for $\Delta_W^{-1/2},U(\Lambda_W(-\frac{\I}2)$ and hence also for
$V_W(-\frac\I2)$ and $\widehat\Theta_W$. (Remember that
$U(\Lambda_W(t))V_W(t)=\Delta_W^{\I t}$.)
Denoting the closure by the
same symbol, $\widehat\Theta_W$ has a polar decomposition
\[
\widehat\Theta_W=\Theta_W T_W.
\]
By the uniqueness of the decomposition and the positivity of
$V_W(\frac\I2)$ it follows that the relation $T_W=V_W(\frac\I2)$ holds.
Since $\widehat\Theta_W$
maps $\CM^{\rm fa}(D+x)$ onto $\CM^{\rm fa}(D-x)$ the same must hold for
$\Theta_W$. Since by analytic continuation these expression are
independent of $W$ we get that $\Theta_W=\Theta$ is the PCT-operator
and $V_W(\frac\I2)$
is independent of $W$ and maps $\CM^{\rm fa}(D+x)$ onto itself.
\end{proof}
\begin{proof}{\bf of theorem 2.16}
During the proof of the last theorem we had
constructed the operator
$\widehat\Theta=U(R_W(\pi))J_W V_W(\frac\I2)$. By the uniqueness of
the polar decomposition we see that the positive operator $V_W(\frac\I2)$
and the PCT-operator $\Theta=U(R_W(\pi))J_W$ act locally and are
independent of $W$. This implies that $J_W$ acts locally and maps
$\CM(D+x)$ onto $\CM(D+P_W x)$. Writing $V_W(t)= \E^{\I X_W t}$ then
since
$\E^{- X_W /2}$ is independent of $W$ we conclude that also
$V_W(t)= \E^{\I X_W t}=V(t)$ is independent of $W$. This implies
\[
[V(t),U(\Lambda)]=0
\]
for all $\Lambda$.
We know that $V(t)$ maps $\CM(K_W(D+x))$ onto itself for every $W$. Choosing
$x=0$ and varying $W$ we find that $V(t)$ maps $\CM(D)$ onto itself. Since
$V(t)$ commutes with the translations it is a local gauge. Hence
also $\Delta_W^{\I t}$ acts locally for every $W$. Therefore
$V(t)$ must define a representation of the Lorentz group. Since this group
representation is Abelian it must be trivial. This implies that $U(g)$ is
the minimal representation.
\end{proof}
\section{Examples}
Free quantum fields, i.e., fields that fulfill the Klein Gordon
equation, provide simple examples illustrating several of
the points discussed in the previous sections.
We consider first the case of a single, hermitian Bose field $\Phi$
with mass $m>0$
on $\BR^d$, $d\geq 2$, transforming covariantly with
respect to the translation group, but not necessarily w.r.t.\ the
Lorentz group. The Fock space representation of $\Phi$ is determined
by the two point function,
\begin{equation}
\mathcal W_{2}(x-y)=\langle\Omega,\Phi(x)\Phi(y)\Omega\rangle.
\end{equation}
Locality, spectrum condition and positivity imply (by the
Jost-Lehmann Dyson representation \cite{Bch96}) that its Fourier transform
can be written
\begin{equation}
\tilde {\mathcal W}_{2}(p)=M(p)\theta(p^0)\delta(p\cdot p-m^2)
\end{equation}
where $M(p)$ is a polynomial in $p$ satisfying
\begin{equation}
M(p)=M(-p)\quad\text{and}\quad M(p)\geq 0
\end{equation}
for $p$ on the positive mass shell $H_{m}^+=\{p:\ p^0>0,\ p\cdot
p=m^2\}$.
Conversely, every such $M$ defines a free field satisfying all
Wightman axioms except possibly Lorentz covariance.
The field $\Phi$ gives rise to a QFTLO where the local von Neumann
algebras $\CM(O)$ are generated by the Weyl operators
$\exp(\I\Phi(f)$ with real test functions $f$ supported in $O$.
According to the results of \cite{GY00} wedge duality is always
violated unless $M$ is constant on the mass shell, i.e., unless
$\Phi$ is the usual Poincar\'e covariant, scalar free field. In
particular, neither the condition of geometric modular action nor the
Bisognano Wichmann property hold for fields with non-constant $M$. On
the other hand, it is trivial that PCT symmetry {\it always}
holds in these examples,
irrespective of $M$ and $d$. In fact, since $\Phi(x)=\Phi(x)^*$ and
$\mathcal W_{2}$ depends only on $x-y$, we have
\begin{equation}
\langle\Omega,\Phi(x)\Phi(y)\Omega\rangle
=\langle\Omega,\Phi(-y)\Phi(-x)\Omega\rangle
\label{freepct}
\end{equation}
which is exactly PCT symmetry. Space and time inversion are only
symmetries separately, however, if $M$ is even in $p^0$. These examples thus shows
that
\begin{itemize}
\item Wedge duality is not a necessary condition for PCT symmetry.
\item Full PCT symmetry is not in conflict with odd dimensionality
of space-time.
\end{itemize}
More generally we may consider free fields $\Phi_{\alpha}$ that have
an arbitrary number of components and are not necessary hermitian. As
before we assume translation covariance and spectrum condition, but
not necessarily Lorentz covariance.
It is convenient to take the set of indices $\alpha$, that label
the components of the field including the adjoint operators, as a
basis of a complex vector space $\CK$. The field operators
$\Phi(f,\sigma)$ thus depend linearly on $\sigma\in\CK$ besides the
test function $f$, and we can write the adjoints as
$\Phi(f,\sigma)^*=\Phi(\bar f,\sigma^*)$, where $\bar f$ is the
complex conjugate test function and $\sigma\to\sigma^*$ is an
antilinear involution on $\CK$. (In the case of hermitian fields
$\sigma^*$ is simply complex conjugation of the components of $\sigma$
with respect to the basis.) The Fourier transform of the two point
function
\begin{equation}
\mathcal
W_{2}(x-y;\sigma,\rho)=\langle\Omega,\Phi(x,\sigma)\Phi(y,\rho)\Omega\rangle
\end{equation}
can now be written as
\begin{equation}
\tilde {\mathcal W}_{2}(p;\sigma,\rho)=M_{\sigma,\rho}(p)\theta(p^0)\delta(p\cdot p-m^2)
\end{equation}
where $M_{\sigma,\rho}(p)$ is a polynomial in $p$ that depends
bilinearly on $\sigma$ and $\rho$. For Bose fields locality is
equilvalent to
\begin{equation}
M_{\sigma,\rho}(p)=M_{\rho,\sigma}(-p)
\label{locality}
\end{equation}
and positivity to
\begin{equation}
M_{\sigma^*,\sigma}(p)\geq 0,\qquad\text{for}\ p\in H_{m}^+,
\label{posit}
\end{equation}
which implies in particular the hermiticity condition
\begin{equation}
M_{\sigma^*,\rho}(p)=\overline{M_{\rho^*,\sigma}(p)}
,\qquad\text{for}\ p\in H_{m}^+.
\label{hermit}
\end{equation}
PCT symmetry of the field, on the other hand, requires that
\begin{equation}
M_{\sigma,\rho}(p)=\overline{M_{\rho,\sigma}(p)}=M_{\sigma^{*},\rho^{*}}(p).
\label{pctsymmfield}
\end{equation}
It is easy to give examples that satisfy \eqref{locality} and \eqref{posit}
but not \eqref{pctsymmfield}. For instance, on can take an $n$
component
field, $n\geq 2$, with $M(p)$ given by a positive definite
$n\times n$ matrix whose diagonal elements are even polynomials in $p$
and whose off diagonal elements are odd polynomials that
are
purely imaginary for
$p\in H_{m}^+$. It has to be remarked, however, that not
all such examples violate PCT symmetry {\it for the algebra of
observables}
generated by the field. In fact, two different multicomponent fields
$\Phi_{\alpha}^{(1)}$ and
$\Phi_{\beta}^{(2)}$ may generate the same local algebras of
observables, and one of them can fulfill \eqref{pctsymmfield} while the
other
does not. To discuss this in a little more detail let
us equip $\CK$ with a scalar product $\langle\cdot,\cdot\rangle$
and write
\begin{equation}
M_{\sigma^{*},\rho}(p)=\langle\sigma, M(p)\rho\rangle
\label{moperator}
\end{equation}
where $M(p)$ is a hermitian linear operator on $\CK$. If $M^{(1)}(p)$
and $M^{(2)}(p)$ correspond to the two different fields, then the
generated local algebras are clearly equal if
\begin{equation}
M^{(1)}(p)=L(p)^{*} M^{(2)}(p)L(p)
\label{equalalgebras}
\end{equation}
where $L(p)$ is an invertible linear operator on $\CK$ for all
$p\in H_{m}^+$ whose matrix elements, together with those of
$L(p)^{-1}$, are polynomials in $p$. In fact, $L(\I\partial)$ is then
an invertible matrix of differential operators (at least when operating in
fields satisfying the Klein-Gordon equation) and the two fields are
related by
\begin{equation}
\Phi_{\alpha}^{(1)}(x)=\hbox{$\sum_{\beta}$}L_{\alpha\beta}(\I\partial)
\Phi_{\beta}^{(2)}(x),\quad
\Phi_{\beta}^{(2)}(x)=\hbox{$\sum_{\alpha}$}L^{-1}_{\beta\alpha}(\I\partial)
\Phi_{\alpha}^{(1)}(x).
\end{equation}
For a true counterexample to PCT symmetry of the QFTLO generated by a
free field $\Phi_{\alpha}^{(1)}$ one must therefore pick $M^{(1)}(p)$
in such a way that it can {\it not} be written as \eqref{equalalgebras}
with a PCT symmetric $ M^{(2)}(p)$. This, however can easily be
shown e.g.\ for $M$ defined by the matrix
\begin{equation} M(p)=\left(\begin{array}{cc}
(p^0)^2 & \I m p^0\\
- \I m p^0 & (p^0)^2
\end{array} \right). \end{equation}
It should even be possible to find within the class of Lorentz non-covariant,
finite component free
fields examples satisfying wedge
duality, but still violating PCT symmetry. In fact, it is easy to check
that wedge duality certainly
holds, if $M(p)^{-1}$ exists for all for all
$p\in H_{m}^+$ and has polynomial matrix elements. A sufficient, and
by Lemma V.2 in \cite{GY00} also necessary, condition for this is that
$\det M(p)$ is constant on the mass shell. PCT symmetry, on the other
hand, requires that $M(p)$ can be written as $L(p)^{*}
M^{\prime}(p)L(p)$ with $M^{\prime}(p)$ satisfying
\eqref{pctsymmfield} and $L(p)$ polynomial in $p$. Although an
explicit example
satisfying wedge duality without the latter property is not known to
us there is hardly a doubt that such examples exist.
If one allows infinite dimensional index spaces $\CK$ Oksak and Todorov
\cite{OT68} have given examples of free fields violating PCT symmetry, but
with Lorentz covariance, i.e., where there is a representation
$\Lambda\to V(\Lambda)$ of the Lorentz group on $\CK$ such that
\begin{equation}
M_{\sigma,\rho}(\Lambda p)=
M_{V(\Lambda^{-1})\sigma,V(\Lambda^{-1})\rho}(p).
\label{eq:covar}
\end{equation}
All examples with infinite dimensional $\CK$ violate the split
property, but the status of wedge duality in this example, i.e., the
validity of the analyticity
condition of Theorem 2.6, is not known.
The full analyticity condition of Theorem 2.15 is
certainly violated.
Another instructive example is the case of an infinite number of
copies of the neutral scalar free field. Here $\CK$ is infinite
dimensional, but $M_{\sigma,\rho}(p)$ is independent of $p$:
\begin{equation}
M_{\sigma^*,\rho}(
p)=\langle\sigma,\rho\rangle
\label{}
\end{equation}
where $\langle\sigma,\rho\rangle$ is some scalar product on $\CK$. This
theory is clearly invariant under the Poincar\'e transformations that
simply ignore the index space $\CK$:
\begin{equation}
U_1(a,\Lambda)\Phi(f,\sigma)U_1^*(a,\Lambda)=\Phi(
f_{\{a,\Lambda\}},\sigma)
\end{equation}
with $f_{\{a,\Lambda\}}(x)=f(\Lambda^{{-1}}(x-a))$. This theory is also
covariant under other representations of the Poincar\'e
group. Let $V(\Lambda)$ be a continuous unitary representation
of the Lorentz group acting on $\CK$. Then we may define a
representation $U_{2}$ by $U_{2}(a,\Lambda)\Omega=\Omega$ and
\begin{equation}
U_2(a,\Lambda)\Phi(f,\sigma)U_2^*(a,\Lambda)
=\Phi(f_{\{a,\Lambda\}},V(\Lambda^{-1})\sigma)
\end{equation}
In this situation the cocycle linking $U_{1}$ and $U_{2}$
is itself a group representation and
$U_1$ is the minimal representation. This follows from the fact
that $U_1$ acts only on the test-functions which implies that
the Wightman functions fulfill the Hall--Wightman analyticity [HW57] on the
complex Lorentz group.
This is an example with wedge duality and the existence of two
representations of the Poincar\'e group. One of them is the minimal
one so that the Bisogano Wichmann property holds.
For the second representation one has the partial
analyticity property for $U_2(\Lambda_W(t))$ required for wedge duality,
but not the full analyticity
property required for Theorem 2.15. The analyticity for a fixed wedge $W$
holds because $V(\Lambda_W(t))$ is a one-parametric
group, which has sufficiently many analytic elements. On the other hand, since
$V(\Lambda)$ is a non-trivial unitary representation of the Lorentz
group it does not have an analytic continuation onto the complex
Lorentz group and therefore there are no fully analytic elements.
Finally we mention an example given in \cite{BDFS00}, Section 5.3,
where the Bisognano-Wichmann property
is violated, but the conditions of Theorem 2.18 are fulfilled so that
Poinca\'e
covariance (without spectrum condition) and PCT symmetry hold.
In this example the split property is fulfilled. One may
ask whether Poincar\'e covariance, spectrum condition and nuclearity
(which implies the split property) are sufficient to derive
the PCT theorem. No counterexamples are known and
Theorems 2.12 and 2.13 may be regarded as a step in this
direction, but this question is otherwise open.
%\bibliographystyle{plain}
%\bibliography{fic-art}
%\nocite{*}
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\end{document}
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