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\begin{document}

\title{Uniqueness of Inverse Scattering Problem in Local Quantum Physics\thanks{%
This work was supported by the CNPq}}
\author{Bert Schroer \\
%EndAName
FU-Berlin, Institut f\"{u}r Theoretische Physik, \\
Arnimallee 14, 14195 Berlin, Germany \\
presently: CBPF Rio de Janeiro \ \\
email: schroer@cbpf.br}
\date{June 2001}
\maketitle

\begin{abstract}
It is shown that the a Bisognano-Wichmann-Unruh inspired formulation of
local quantum physics which starts from wedge-localized algebras, leads to a
uniqueness proof for the scattering problem. The important mathematical tool
is the thermal KMS aspect of localization and its strengthening by the
requirement of crossing symmetry for generalized formfactors.
\end{abstract}

\section{Inverse Problem in LQP}

Most inverse problems have their origin in classical physics where they
result from the question to what extend scattering (asymptotic) data allow a
reconstruction of local data. The universality and importance of the problem
inspired Marc Kac to the aphorism ``How to hear the shape of a drum'' which
refers to Weyl's problem associated with geometric reconstructions from the
asymptotic distribution of eigenvalues of the Laplace-Beltrami operator. In
its present use it incorporates a wide range of problems concerning the
partial or complete determination of local data from a seemingly weaker
asymptotic input. In local quantum physics also the problem of how to
re-construct the full content of QFT from its ``observable shadow'' i.e. the
so-called DR-theory \cite{DR} may be viewed as an inverse problem.

The inverse problem in the present context is the question to what extends
the data contained in a physically admissible (unitary, crossing
symmetry,...) S-matrix determines a quantum field theory. Since it is easy
to see that an S-matrix cannot be uniquely related to one field but rather
is shared by very \ big equivalence classes of local fields, it is clear
that the first step in such an investigation is to formulate QFT in a way
that two isomorphic theories whose different appearance is only due to the
use of different ``field-coordinatizations'' are easily recognizable as
being one and the same. This is not possible or rather extremely cumbersome
in the standard approach based on pointlike fields. Fortunately there exists
such a framework which pays due attention to fields belonging to the same
local equivalence class (fields from the same Borchers class) and which
therefore generate the same system of local algebras as the given field:
algebraic quantum field theory (AQFT) or local quantum physics (LQP) \cite
{Haag}. Its relation to standard QFT is similar to that of coordinate-based
differential geometry to its more modern coordinate-free intrinsic
formulation. Here as there one retains all the underlying principles and
only introduces additional more elegant concepts to implement them. If one
wants to emphasize the different methods one often uses the AQFT or LQP
instead of QFT (a name as ``intrinsic'' or ``field-coordinatization-free''
QFT would appear a bit clumsy).

After this reformulation one may hope that the inverse problem is
well-defined and admits at most one solution in terms of nets of algebras
i.e. only one LQP-model. Using the powerful mathematical tool of
Tomita-Takesaki theory (and the closely related crossing symmetry) as
adapted in recent years to LQP \cite{Bo}, we show that if a solution exists
at all, it is necessarily unique.

This result generalizes previous special findings concerning the modular
derivation of the wedge localization interpretation of the
Zamolodchikov-Faddeev algebra in d=1+1 in which case one also is able to
control a good part of the existence problem \cite{S1}\cite{S2}. The
rapidity-dependent Z-F operators are a special case of ``(vacuum) \textbf{p}%
olarization-\textbf{f}ree-\textbf{g}enerators'' (PFG) for the
wedge-localized algebra and in this special setting the crossing symmetry as
formulated in the LSZ theory is a consequence of the more fundamental
thermal aspect of wedge localization. In the general case the PFG's have
very unwieldy domain properties \cite{BBS} and we are presently only able to
show uniqueness of the associated field theory by using both modular
localization properties of wedge algebras as well as momentum space crossing
properties of the generalized formfactors.

\section{The use of modular theory}

The new tool on which the proof relies in an essential way is modular
theory. Since even in the setting of QFT ``modular'' occurs with different
meanings, we will briefly define its present use in the sequel. The more
common use is that of modular invariance in chiral conformal field theory.
Although this is not its present meaning, a future connection to this
causality-related classification tool for certain families of 2-dimensional
local models to the present also locality-based use of the (Tomita-Takesaki)
modular theory in local quantum physics is by no means ruled out.

Consider the $x_{0}$-$x_{1}$ wedge $W_{0}$ in d-dimensional Minkowski
spacetime which is defined by taking convex combination of the two light
rays generators ($\pm $1,1,0..0) and on which the affiliated Lorentz boost $%
\Lambda _{W_{0}}(\chi )$ acts as an automorphism. Let us change the
parametrization by the factor $2\pi $ and write the unitary representation
in QFT in terms of a selfadjoint $K$ boost generator as 
\begin{equation}
\Delta ^{it}\equiv U(\Lambda _{W_{0}}(\chi =2\pi t))=e^{2\pi itK}  \label{K}
\end{equation}
We then may introduce an unbounded positive operator $\Delta ^{\frac{1}{2}}$%
by ``analytic continuation in t'' which in functional calculus terms means
that we are restricting the Hilbert space to those vectors $\psi $ which
upon action with $\Delta ^{it}$ lead to a vector valued function $\psi
(t)\equiv \Delta ^{it}\psi $ which is continuous in $-i\pi \leq Imt\leq 0$
and analytic on the open strip $-i\pi <\theta <0.$ In addition to the wedge
related boost, we also consider the antiunitary (since it involves time
reversal) reflection along the edge 
\begin{align}
J& \equiv U(r) \\
r& :x_{0},x_{1}\rightarrow -x_{0},-x_{1}  \notag
\end{align}
which up to a $\pi $-rotation around the $e_{1}$-axis is identical to the
TCP transformation. Since this transformation commutes with the boost $%
\Delta ^{it},$ its antiunitarity leads to the commutation relation (always
on the relevant domains) 
\begin{align}
JK& =-KJ  \label{J} \\
J\Delta ^{\frac{1}{2}}& =\Delta ^{-\frac{1}{2}}J  \notag
\end{align}
on the respective domains of definition. This in turn yields 
\begin{align}
S^{2}& \subset 1 \\
for\,\,S& \equiv J\Delta ^{\frac{1}{2}}  \notag
\end{align}
i.e. we encounter the rare case (not even to be found in extensive textbooks
on mathematical physics as that by Reed-Simon) of an unbounded antilinear
operator which is involutive on its domain. In a moment we will see that it
is just this somewhat exotic property which enables the encoding of
spacetime geometric information concerning quantum localization into
abstract domain properties.

It is the content of a theorem (the Bisognano-Wichmann theorem \cite{Haag})
that this operator is Tomita's famous S-involution for the operator algebra
which the local fields generate if one restricts the smearing functions to
have support in the wedge. Namely in terms of the affiliated von Neumann
algebra (which is the setting of the Tomita-Takesaki theory) $\mathcal{A}%
(W_{0}),$ the $S$ fulfills \cite{Haag}\cite{Bo} 
\begin{equation}
SA\Omega =A^{\ast }\Omega   \label{equ}
\end{equation}
where $\Omega $ is the vacuum vector and the star denotes the hermitian
adjoint in operator algebras. This is a special case of the Tomita Takesaki
modular theory whose prerequisite is the existence of a von Neumann algebra
in ``general position'' i.e. a pair ($\mathcal{A}$,$\Omega $) with $\Omega $
being a cyclic and seperating vector for $\mathcal{A}\footnote{%
Physicist, who independently developped these concepts, often (especially in
chiral conformal field theory) talk about the unique ``operator-statevector
relation'' $A\longleftrightarrow A\Omega ;$ the correct mathematical backing
is the reference to the Reeh-Schlieder theorem \cite{St-Wi}. It is often
overlooked that the relation is not universal but depends on the chosen $%
\mathcal{A}(\mathcal{O}).$}.$ From this input Tomita and Takesaki derive:

\begin{itemize}
\item  S defined by (\ref{equ}) is a densely defined closed antilinear
involution whose polar decomposition $S=J\Delta ^{\frac{1}{2}}$ leads to an
antiunitary reflection $J$ (abstract generalization of a TCP reflection) and
modular dynamics $\Delta ^{it}$ (abstract generalization of a Hamiltonian)

\item  The $J,\Delta ^{it}$ have the following significance with respect to
the operator algebra 
\begin{align}
AdJ\mathcal{A}& =\mathcal{A}^{\prime } \\
Ad\Delta ^{it}\mathcal{A}& =\sigma _{t}(\mathcal{A})  \notag
\end{align}
here as in the sequel the upper dash on an operator algebra denotes its
commutant and the modular unitary $\Delta ^{it}$ implements the modular
group $\sigma _{t}(\cdot )$ which different from the former only depends on
the state $\omega (A)=\left( \Omega ,A\Omega \right) ,\,A\in \mathcal{A}$
and not its implementing vector $\Omega .$

\item  A necessary and sufficient condition for the standardness
(cyclicity+separating property) of the pair ($\mathcal{A},\Omega $) is the
thermal KMS property in terms of the state $\omega $ is: there exists a $%
2\pi $-open-strip analytic function (continuous in the closed strip ) $%
F_{A,B}(z)$ with 
\begin{align*}
F_{A,B}(t)& \equiv \omega (\sigma _{t}(A)B) \\
\omega (B\sigma _{t}(A))& =lim_{z\rightarrow t+i}F_{A,B}(z)
\end{align*}
\end{itemize}

The above theorem of B-W may now be rephrased as saying that those operator
wedge algebras which are generated by covariant fields do have a geometric
modular theory. In more recent times there have been successful attempts to
establish these geometric modular aspects of wedge algebras directly in the
seemingly more general setting of algebraic QFT which avoids the use of
fields already at the start \cite{Bo-Yn}. The validity of the KMS condition
with the modular group acting geometrically as the Lorentz-boost is
sufficient for establishing that also $J$ acts geometrically i.e. that the
von Neumann commutant is localized in the geometrically opposite wedge $%
W^{\prime}$ (Haag duality) $\mathcal{A}(W)=\mathcal{A}(W^{\prime})^{\prime}$.

The Poincar\'{e} group generates from one standard wedge algebra $(\mathcal{A%
}(W_{0}),\Omega)$ a net of wedge algebras ($\mathcal{A}(W),\Omega $)$%
_{W\subset\mathcal{W}}.$ In order to extract sufficient physical
informations one needs nets for smaller compact causally closed regions. A
net of double cones $D$ may be defined in terms of intersections 
\begin{equation}
\mathcal{A}(D)\equiv\cap_{D\subset W}\mathcal{A}(W)
\end{equation}
In order to achieve our goal we must be able to relate the wedge algebra
with the scattering operator $S_{sc}.$ This is possible in the LSZ framework
of QFT because although the representation theory of the connected
Poincare-group for the incoming (outgoing) free fields is the same as for
the interacting Heisenberg fields, this is not so for the reflections
involving time reversal. In particular the $J$ in (\ref{J}) which represents
the wedge reflection in the presence of interactions is different from its
interaction-free asymptotic counterpart \cite{S1} $J_{in}$%
\begin{equation}
J=S_{sc}J_{in}
\end{equation}
This implies that in the characterization of the wedge-localized (dense)
subspace: 
\begin{align}
H(W) & =H_{R}(W)+iH_{R}(W) \\
H_{R}(W) & =real\text{\thinspace\thinspace}subspace\left\{ \psi|S\psi
=\psi\right\}  \notag \\
S\left( \psi_{1}+i\psi_{2}\right) & =\psi_{1}-i\psi_{2},\,\,S=S_{sc}S_{in} 
\notag
\end{align}
the position of the dense subspace $H(W)$ inside the total Hilbert space
depends in a subtle way on the interaction through $S_{sc}.$ The domain of $%
\Delta^{\frac{1}{2}}(\ref{K})$ is now encoded more concretely in terms of a
complex dense space $H(W)$ whose real and imaginary part are vectors in a
closed real subspace $H_{R}(W).$ These real closed subspaces encode the full
spatial aspect of wedge localization. With the help of the graph of $S$\ one
may even introduce a topology in terms of which the dense subspace becomes a
Hilbert space in its own right\footnote{$H(W)$ with the $S$-graph norm may
be called the thermal Hilbert space, because it offers a natural description
of the (Hawking-Unruh) thermal aspects of the vacuum upon its restriction to
the wedge algebra.}, but all these spatial concepts are still removed from
the task of characterizing a wedge algebra uniquely in terms of the
scattering matrix. The reason is the following. The algebra-state vector
relation $A\longleftrightarrow A\Omega$ is not universal but changes with
the algebra, even in the family of wedges. In particular the spatial modular
theory permits the existence of two different algebras with the same wedge
as long as they remain indistinguishable in their one-time action on the
vacuum.

Connes has given a criterion \cite{Connes} which allows to obtain from the
spatial modular theory to an algebra with the same modular objects. This is
achieved by controlling certain properties of so-called facial subcones of a
natural cone associated $\mathcal{P}(\mathcal{A}(W))$ with $H_{R}(W).$ But
one presently lacks a physical foundation and control for such a procedure.
Fortunately for our interest in uniqueness (falling short of an actual
construction), such a difficult mathematical road can be bypassed in terms
of an additional physical assumption: crossing symmetry of particle
matrixelements (formfactors) of wedge localized operators. For this we need
to remind the reader of a bit of scattering theory adapted to the algebraic
framework.

\section{Uniqueness from KMS-thermality and crossing}

It has been shown that any vector $\psi$ which is in the domain of the
positive analytically contimued standard L-boost (\ref{K}) \ $\Delta _{W}^{%
\frac{1}{2}}$ has a unique relation to an (generally unbounded) operator $%
F_{\psi,\mathcal{A}(W)}$ affiliated with $\mathcal{A}(W)$ with 
\begin{equation*}
F_{\psi,\mathcal{A}(W)}\Omega=\Psi,\,\,F_{\psi,\mathcal{A}%
(W)}^{\ast}\Omega=S_{W}\Psi
\end{equation*}
But this famous state-vector-operator relation depends crucially on the
standard pair ($\mathcal{A}(W),\Omega$). If the same scattering data would
allow for another wedge algebra $\mathcal{B}(W)\neq\mathcal{A}(W),$ the same
vector $\Psi\in H(W)$ is associated with another operator $\Psi=F_{\psi ,%
\mathcal{B}(W)}\Omega$. $H(W)$ contains all those in- or out- n-particle
vectors which are in the domain of $\Delta_{W}^{\frac{1}{2}}$ which form a
dense set.

Let us assume that we are dealing with a state-vector of the special form $%
\Psi =A\Omega ,A\in \mathcal{A}(W).$ With respect to the $\mathcal{B}(W)$
algebra there exists a unique affiliated densely defined closed operator $F$
with 
\begin{align}
& A\Omega =F\Omega  \\
& F\eta \mathcal{B}(W)  \notag
\end{align}
where in the last line we used the standard $\eta $ notation for an operator
affiliated with $\mathcal{B}(W).$ This forces in particular the inner
products with the n-particle out state vectors to be the same 
\begin{equation}
^{out}\left\langle p_{n}...p_{1}\left| A\right| \Omega \right\rangle
=\,^{out}\left\langle p_{n}...p_{1}\left| F\right| \Omega \right\rangle 
\label{vac}
\end{equation}
For those operators $A\in \mathcal{A}(W)$ which are localized in a double
cone $\mathcal{A}(\mathcal{O})\subset \mathcal{A}(W)$ the LSZ-formalism and
onshell analytic continuation lead to the crossing symmetry (see appendix)

\begin{align}
& ^{out}\left\langle p_{1},p_{2},...p_{l}\left| A\right|
q_{1},q_{2}...q_{k}\right\rangle ^{in}=  \label{cro} \\
& \underset{z\rightarrow \theta -i\pi }{a.c.}^{out}\left\langle
p_{1},p_{2},...p_{l-1}\left| A\right| q_{1},q_{2}...q_{k},\bar{p}_{l}(\theta
)\right\rangle ^{in}+c.t=  \notag \\
:& ^{out}\left\langle p_{1},p_{2},...p_{l-1}\left| A\right|
q_{1},q_{2}...q_{k},-\bar{p}_{l}\right\rangle ^{in}+c.t.  \notag
\end{align}
whereb the last equality defines the meaning of a formfactor with a negative
momentum. Crossing symmetry is a momentum space property within LSZ
scattering theory. It says that in multiparticle matrix elements of
operators between outgoing bra- and incoming ket- particle states as above,
one can flip a particle in the incoming ket state to the outgoing bra if one
also converts it into an antiparticle (charge conservation) at an
analytically continued real point $p\rightarrow -p$ on the complex physical
mass shell (energy-momentum conservation). The bar on the momentum denotes
the antiparticle and in order to indicate the analytic continuation we have
chosen the momentum p to be of the form $p=m(ch\theta ,sh\theta ,0,0)$ (i.e.
affiliated with the standard wedge, which can always be achieved by
L-covariance), so that the analytic continuation corresponds to $\theta
\rightarrow i\pi -\theta $. The $c.t.$ contraction terms consist of a $%
\left\langle p_{l}|q_{i}\right\rangle $ $\delta $-function multiplied with a
lower $l+k-2$ particle formfactor. Note that the crossing of the S-matrix
itself arises for $A=1.$ The most prominent crossing relation is that for
elastic scattering which involves the simultaneous flip of two particles
with one from the incoming and one from the outgoing configuration (the flip
of only one is not compatible with the onshell energy-momentum conservation)
The last line is an abuse of notation since an analytic continuation is
generally not implementable as an operator or Hilbert space operations.
Crossing is not a Wigner symmetry in state space but rather a property which
involves analytic continuation. In order to justify it, there has to be at
least strip analyticity in the rapidity $\theta $ of the momentum to be
crossed so that the forward particle mass shell can be analytically
connected with the antiparticle backward shell. Starting from the
matrixelement (\ref{vac}) the successive application of the crossing
property (\ref{cro}) allows to obtain arbitrary matrixelements of $A$
between bra-out and ket-in particle states by starting from a special
situation with only the vacuum on one side.

The analytic aspects are evidently inherited by the right hand side in (\ref
{vac}). But we have to justify the interpretation in terms of crossing
particles from ket into bra's. For this purpose we imagine the vector $%
F\Omega $ to be weakly approximated by a sequence of $F_{n}\Omega $ with $%
F_{n}$ being operators in the $\mathcal{B}$-algebra which are localized in
an increasing family of double cones approaching the standard wedge.
According to LSZ scattering theory the crossing relations apply to compactly
localized operators in the $\mathcal{B}$-net. Hence the interpretation of
the limiting analyticity relation as a particle crossing relation is
unavoidable. But this means that crossing symmetry generalizes the equality
of $A\in \mathcal{A}(W)$ with $F\eta \mathcal{B}(W)$ to that of $A/F$
formfactors on a dense set of out-in states. This is only possible if $F$ is
also bounded and equal to an $A\in \mathcal{A}(W).$

A direct attempt of a proof using only the KMS property of the $\mathcal{A}%
(W)$ and $\mathcal{B}(W)$ algebras remains inconclusive because the known
domain properties of the afilliated operators $F\eta\mathcal{B}(W)$ are too
weak to secure the algebraic aspects of the $B-$KMS property. Although it is
easy to choose a dense set of asymptotic n-particle states which are created
by $\mathcal{A}(W)$ or $\mathcal{B}(W)$ affiliated operators, the domain
justification for applying KMS property to those operators fails. Domain
properties of $F^{\prime}s$ are very different from those of smeared
pointlike fields, except for the special case of ``tempered PFG's'' \cite
{BBS}\footnote{%
Temperedness restrictions for nonlocal creation anyonic operators actually
appeared first in \cite{Mund}.}. The temperedness restriction allows only
elastic interactions in d=1+1. In fact the only known models are those where
the Fouriertransforms of the PFG's $G_{W}(x)$ fulfill a
Zamolodchikov-Faddeev algebra which in the simplest case of a selfconjugate
particle reads

\begin{align}
G_{W}(x) & =\frac{1}{\sqrt{2\pi}}\int(e^{-ipx}Z(\theta)+h.c.)d\theta
,\,\,\,p=m(ch\theta,sh\theta) \\
Z(\theta)Z(\theta^{\prime}) & =S(\theta-\theta^{\prime})Z(\theta^{\prime
})Z(\theta)  \notag \\
Z(\theta)Z^{\ast}(\theta^{\prime}) & =S^{-1}(\theta-\theta^{\prime})Z^{\ast
}(\theta^{\prime})Z(\theta)+\delta(\theta-\theta^{\prime})  \notag
\end{align}

The unitarity of the structure functions $S(\theta)$ is a consequence of the 
$^{\ast}$-algebra property of the $Z^{\prime}s$ and the crossing symmetry of
the mixed $\ A\in\mathcal{A(W)-}\,\,G_{W}$-correlation functions follows
from the wedge localization and the ensuing KMS property. The $Z^{\prime}s$
have a very simple representation in a bosonic/fermionic Fock space. Each
operator $A$ affiliated with $\mathcal{A}(W)$ has a formal power series
expansion

\begin{equation}
A=\sum \frac{1}{n!}\int_{C}...\int_{C}a_{n}(\theta _{1},...\theta
_{n}):Z(\theta _{1})...Z(\theta _{n}):  \label{series}
\end{equation}
where $Z(\theta -i\pi )=Z(\theta )^{\ast }$ and each integration path $C$
extends over the upper and lower part of the rim of the strip. The
strip-analyticity of the coefficient functions $a_{n}$ expresses the wedge
localization of $A.$ The sharpening to double cone localization by the
intersection of wedges leads to meromorphic functions which obey the
kinematical pole condition of Smirnov \cite{Smir}. Expansions like (\ref
{series}) are nothing more than a generating operator for the formfactors
i.e. bilinear forms which fall short of being genuine operators with domains
and closures. They are analogous to the LSZ expansions of Heisenberg fields
into (asymptotic) free fields. For our above uniqueness argument this is
enough, for a constructive approach this is insufficient.

It is also interesting to note that despite the close relation between the
onshell incoming fields and the onshell $Z(\theta)$, the latter share some
features with local Heisenberg fields namely one $Z(\theta)$ can have
several particle states (alias bound states) whereas each type of particle
requires the introduction of one incoming field.

Although the coefficient functions $S(\theta)$ of the $Z$-algebra turn out
to be the 2-particle scattering matrix, there is no need to know this for
the calculations: absence of real particle production, wedge-localitazion
and the related KMS property (i.e. spacetime properties) are enough.

The case without the temperedness restriction starts also from formfactors
between the dense set of wedge affiliated n-particle-ket-states and the
bra-vacuum which according to modular theory must be equal for the two
putatively theories. It then bypasses domain issues by using for the
apparently stronger assumption of crossing symmetry for the successive
movement of particles from the ket to the bra state. The asymptotic states
in the $\Delta ^{\frac{1}{2}}$ domain have a tensor product structure. If
their wedge representatives $F_{n}\Omega $ would inherit this factorization
structure in the form $F_{n}\Omega =F_{n-1}G\Omega =F_{n-1}\Omega \times
G\Omega $ with $G$ being a PFG, then the use of the crossing property would
not be necessary since the result would follow from the KMS formula of wedge
localization. But I have not been able to derive such a factorization from
the known domain properties of wedge affiliated $F^{\prime }s$ $\ $in the
general nontempered case. The use of PFG's for the construction of wedge
algebras seems to be restricted to factorizing models, in more realistic
interacting theories they do not seem to be useful generators of wedge
algebras.

\textit{\ }For a recent discussion of how the wedge algebras $\mathcal{A}(W)$
are related to their holographic projections onto the (upper) horizon $%
\mathcal{A}(R_{+})$ we refer to \cite{FS}.

\section{Related problems, outlook}

We have seen that by combining modular theory (which gives mathematical
precision to the state-vector-operator relation) with crossing symmetry
(which permits to elevate relations involving the vacuum vector to relations
between dense sets of scattering states), one obtains a uniqueness argument
for the inverse problem in QFT: a physically admissible S-matrix has inspite
of the myriads of interpolating fields at most one system of local algebras
i.e. at most one field-coordinatization-independent algebraic QFT. For the
solution of the existence problem i.e. the explicite construction of a
system of algebras (and if desired their possible pointlike generators) from
scattering concepts one presently has to assume the temperedness of wedge
algebra generating PFG's which limits the constructive approach to d=1+1
factorizing models with the additional benefit of a spacetime interpretation
of the ensuing Zamolodchikov-Faddeev algebraic structure.

These uniqueness and existence aspects touch upon age old problems of
particle physics which despite the passage of time have lost nothing of
their importance. Beginning as far back as Heisenberg's S-matrix proposal 
\cite{Heisenberg}, there was the desire to avoid the short distance problems
of pointlike field theory by advocating a pure S-matrix theory. The main aim
was to transfer as much ``physical blood'' from what one has learned about
perturbation of free fields but to avoid that offshell short distance region
which relates to coalescing or lightlike spacetime arguments of fields. With
the overwhelming success of renormalized QED and the deep conceptual gains
in the particle/field realm (LSZ, Haag-Ruelle, Wightman,..), the motivation
for a pure S-matrix theory subsided only to reappear in a strengthened form
(enriched by analytic properties and crossing) under the heading of ``the
S-matrix bootstrap''\footnote{%
For a recent review see \cite{White}.} in the 60ies (also the cradle of the
Veneziano's dual model and string theory).

The present modular enrichment subjects this old approach to a sympathetic
but critical review. Although there is agreement with its basic premise that
onshell concepts should play an important role right from the beginning and
that locally coupling free fields is not really an intrinsic God-given way
of introducing interactions enforced by the underlying principles, it would
not subscribe to an abandoning of the causality and spectral principles of
QFT. The message would rather be that one should avoid the use of
(inevitably singular) pointlike correlation functions in products and
integrals over products of correlations (which are the intermediate steps of
standard perturbation theory) in favour of multiparticle formfactors of one
individual localized operator and postpone the operator control of these
bilinear forms (particle matric elements) up to the end. After the structure
of some wedge localized algebras has been understood in a constructive
manner, there is no harm to sharpen the localization of these algebras by
forming intersections and even to use pointlike generators. The short
distance singularities of pointlike generators belonging to limits of
sequences of formfactors with improved localizations are of no harm \cite
{BFKZ}; their distributional aspects are governed by Wightman's theory of
vacuum correlations.

The bootstrap formfactor approach to factorizing models may serve as an
excellent illustration of this new way of thinking about QFT. Here one
avoids the technical frontiers between renormalizable/nonrenormalizable
interactions by totally bypassing Lagrangian quantization or causal
perturbation of free fields. This is achieved for those models by starting
with an algebraic structure which avoids pointlike fields in favor of the
above Z-operators which are only consistent with wedge-like localization.
With other words, the system of wedge algebra $\mathcal{A}(W)$ is
constructed before any pointlike field appears on the scene. The next step
namly the formation of double cone intersection algebras leads to the
so-called kinematical pole relation which relates the lower with the higher
formfactors and defines the formfactor spaces for double-cone localized
objects. Whereas the Z-algebra generators were PFGs, the double-cone
localization leads to the vacuum polarization clouds in form of Z-expansions
(\ref{series}) which extend to infinity. The finite size of the spacetime
extension of the double-cone localized \cite{S2} operators shows up in form
of a Payley-Wiener asymptotic behavior of the meromorphic formfactors and
the pointlike fields with their spacetime point in the double cone
correspond to a polynomial behavior in certain reduced formfactors \cite
{BFKZ}. The point which needs to be emphasized here is there is no
ultraviolet limitation coming from power-counting and leading to the
standard separation into renormalizable and nonrenormalizable coupling;
every admissable factorizing S-Matrix leads to power bounded formfactors in
terms of a few physical parameters (which are already preempted in the
S-matrix). This is in my view the most valuable message of that theoretical
laboratory called ``factorizing (integrable) theories''.

A limitation as that of a Lagrangian field having to carry an operator
dimension near the the canonical (free field) dimension does not appear and
with it the threat of nonrenormalizability in the sense of too many
(infinite) renormalization parameters has disappeared. So the standard
separation according to short distance behavior becomes meaningless in this
new framework; short distance properties (of what? there is no preferred
Lagrangian field coordinate!) are simply not part of the modular program.
Instead the remaining question about existence is whether the double cone
intersection algebras are nontrivial in the sense of formfactors and whether
these formfactors are really coming from closed or bounded operators.

In view of these facts it would appear to be somewhat unreasonable (though
presently it cannot be ruled out) to believe that all this happens as a
accident of the speciality of the models and that beyond factorization (or
beyond tempered PFG's) the Lagrangian point of view continues to define the
intrinsic (i.e. fixed by the principles of local quantum physics) borderline
of ultraviolet good/bad QFT. The present uniqueness argument of the inverse
problem strengthens this suspicion. The existence of a field-coordinate free
construction of local quantum physics apparently diminishes the physical
role of short distance properties of individual fields by trading the
nontrivial existence of a model against the ultraviolet odds with the
nontriviality of intersections of wedge algebras. But for a constructive
approach beyond factorizing S-matrices based on these new concepts one needs
better generators than the nontempered PFGs. Whatever will be the final
answer about the intrinsic role (if any) of the short distance problems in
the standard approach, the algebraic field-free formalism (which is expected
to allow also for perturbative solutions) should eventually teach us whether
the power counting in couplings of free fields (augmented by cohomological
tricks which lower the short distance powers of interaction polynomials in
the intermediate computational steps and bring them into the renormalizable
range\footnote{%
We refer to the BRS-like cohomological representation of massive spin Wigner
one particles spaces which allows an alternative presentation of massive
selfinteracting vectormesons to the standard Higgs mechanism \cite{D-S}.})
is the one set by the physical principles or not. In the latter case one
would have to blame the restrictions of the standard approach on the the
inapropiate and premature use of ``singular field coordinatizations''. In
view of the often made claim that string theory could be ultraviolett
finite, it may be interesting to review these problems in the algebraic
setting of QFT. We expect that the results of perturbatively renormalizable
fields will be reproduced in this setting and that it will lead to an
enlargement similar to what happened in d=1+1 with factorizing models).

It has been known that without the presence of interaction terms i.e. for
free higher spin wave equations their does indeed exist a modular approach
which, if combined with the Wigner group theoretic characterization of
massive and massless particles in terms of irreducible positive energy
representations, allows for a powerful connection of group theoretic induced
localization with real modular subspaces of the Wigner representation space 
\cite{S1}\cite{BGL}. In this way one may bypass the standard method of first
converting the content of Wigners intrinsic description into nonunique
covariant wave functions and then using the Cauchy initial value approach in
order to characterize the localization-subspaces, which is particularly
helpful with increasing spin. In the presence of local perturbations one
looses the functorial characterization of nets of local algebras in terms of
localized real subspaces. Nevertheless one may view e.g. the constructive
approach to factorizable d=1+1 models as an extension of the Wigner program
of coming from particles to local observables in the presence of
interactions. I believe that it is fruitfull to view the general modular
approach as an extension of Wigner's program.

In this paper we have concentrated on the easier uniqueness problem, leaving
the more difficult construction problem (outside the temperedness) for the
future. Of course we hope that we still are able to avoid the explicit use
of crossing symmetry and will be able to use instead the spacetime related
KMS condition. We nourish the hope that a better understood modular
construction program may reveal whether or not the present borderlines
between physically useful and less useful (renormalizable/nonrenormalizable)
theories is really the true frontier marked by the underlying principles or
only the limitation of the ad hoc implementation of interactions by locally
coupling free fields in Fockspace.

\subsection{Appendix: Crossing symmetry with LSZ reduction}

Crossing has been first observed in Feynman perturbation before it was
derived in the LSZ scattering theory. Its formal aspects are easily obtained
from the LSZ asymptotic convergence 
\begin{align}
& lim_{t\rightarrow \mp \infty }A^{\#}(f_{t})\Phi =A^{\#}(f)_{in/out}\Phi
,\,\,A^{\#}=A\,\,or\,\,A^{\ast } \\
& A(f_{t})=\int f_{t}(x)U(x)AU^{\ast }(x)d^{4}x,\,\,A\in \mathcal{A}(%
\mathcal{O})  \notag \\
f_{t}(x)& =\frac{1}{\left( 2\pi \right) ^{2}}\int e^{i\left( p_{0}-\omega
(p)\right) t-ipx}f(\vec{p})d^{4}x,\,\,\omega (p)=\sqrt{\vec{p}^{2}+m^{2}} 
\notag
\end{align}
which can be derived on a dense set of states and shown to lead to the
well-known reduction formulas . 
\begin{align}
& ^{out}\left\langle q_{1},q_{2},...q_{m}\left| F\right|
p_{1},p_{2}...p_{n}\right\rangle ^{in}|_{conn}= \\
& -i\int \,^{out}\left\langle q_{2},...q_{m}\left| K_{y}TFA^{\ast
}(y)\right| p_{1},p_{2}...p_{n}\right\rangle ^{in}d^{4}ye^{-iq_{1}y}=  \notag
\\
& -i\int \,^{out}\left\langle q_{1},q_{2},...q_{m}\left| K_{y}TFA(y)\right|
p_{2}...p_{n}\right\rangle ^{in}d^{4}ye^{ip_{1}y}\,=  \notag
\end{align}
Here the time-ordering $T$ between the original operator $F\in \mathcal{A}(%
\mathcal{O})$ and the interpolating Heisenberg field $A(x)$ resp. $A^{\ast
}(x)$ appears if one reduces a particle from the bra- or ket state \cite
{Araki}. For the definition of the time ordering of a fixed finitely
localized operator $F$ and a field with variable localization $y$ we may use 
$TFA(y)=\theta (-y)FA(y)+\theta (y)A(y)F,$ however as we place the momenta
on-shell, the definition of time ordering for $y$ near $locF$ is irrelevant.
Each such reduction is accompanied by another disconnected contribution in
which the creation operator of an outgoing particle say $a_{out}^{\ast
}(q_{1})$ changes to an incoming $a_{in}(q_{1})$ acting on the incoming
configuration (and the opposite situation i.e. $a_{in}^{\ast
}(p_{1})\rightarrow a_{out}(p_{1})$). These terms (which contain formfactors
with one particle less in the bra- and ket- vektors) have been omitted since
they do not contribute to generic nonoverlapping momentum contributions and
to the analytic continuations. Under the assumption that there is an
analytic path from $p\rightarrow -p\,$\ (or $\theta \rightarrow \theta -i\pi 
$ in the wedge adapted rapidity parametrization) the comparison between the
two expressions gives the desired crossing symmetry: a particle of momentum
p in the ket state within a formfactor is indistinguishable from a bra
antiparticle at momentum -p (here denoted as -\={p}).

In order to obtain that required analytic path it is convenient to pass from
time ordering to retardation 
\begin{equation}
TFA(y)=RFA(y)+\left\{ F,A(y)\right\} 
\end{equation}
The unordered (anticommutator) term does not have the pole structure on
which the Klein-Gordon operator $K_{y}$ can have a nontrivial on-shell
action and therefore drops out. The application of the JLD spectral
representation puts the p-dependence into the denominator of the integrand
of an integral representation where the construction of the analytic path
proceeds in a completely analog fashion to the derivation of crossing for
the S-matrix \cite{BLOT}\cite{Araki}.

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\end{document}

