**My Past and Present Professional Interests**

I am a theoretical physicist who makes unlimited use of what in my
opinion are the most appropriate mathematical concepts for the formulation
and solution of problems in local quantum physics. In this sense I
also consider myself a mathematical physicist.

As can be seen from my publication list, I started my carrier in general
(algebraic) QFT. Progress in that area in the 60's was very slow and the
ratio of tangible physical results at that time left a lot to be desired.
For this reason I worked in the late 60's and during the 70's on problems
of renormalized perturbation theory, critical phenomena and current algebras.
In particular in my collaboration with J.A. Swieca we tried to cover the
area between standard and algebraic (general) QFT with particular emphasis
on low dimensional QFT. Even though I was the first to see the relation
between euclidean zero modes and winding numbers associated with axial
anomalies (before the Atiyah-Singer index theorem became a commom place
in QFT) in exactly solvable two-dimensional models, I became gradually
disenchanted with the increasing tendency of substituting concepts of real
time local quantum physics by the mathematics of differential geometry
and algebraic topology. When I returned in the middle of the 80's to the
problems and methods of algebraic QFT, I found a different scenario with
many new structural insights and promising new concepts and methods for
a fresh start in the direction of nonperturbative and constructive insights.
My own contributions to this recent development can be found in "Motivations
and Physical Aims of Algebraic Quantum Field Theory" AOP Vol.255, No.2,
270 (1997) and my course on
"Localization and Nonperturbative Local
Quantum Physics".

**My present research activities**

During the last 3 years I worked on a new nonperturbative approach
to QFT which is based on the use of modular theory of localization.

It is well known that quantization schemes start from properties (canonical
quantization, path integral quantization) which, apart from some not very
interesting superrenormalizable cases, cannot be literally true (i.e. the
renormalized theory is neither canonical nor Feynman-Kac representable)
but only serve as a mental starter. On the other hand in Local Quantum
Physics either in the pointlike (Wightman) or in terms of local algebras
(Haag-Kastler) a property following from the physical principles is not
just a mental starter which gets repaired in the process of construction,
but remais a bona fide property in every step including the final result.
This is of course related to the fact that a commutative starting point
+ quantization needs a lot of "massaging" in order to lead to a consistent
noncommutative structure whereas the (Tomita-Takesaki) modular theory of
local algebras is maximally noncommutative from the very beginning.

Modular theory leads to a vast zoo of hidden symmetry groups which
generalize geometric situations e.g. the diffeomorphism of chiral conformal
theories to hidden symmetries, typically in higher dimensions.

What makes the modular localization approach so useful for a constructive
approach to nonperturbative interactions is the fact that the modular involution
encodes the interaction in the invariant form of an S-matrix and that an
admissable S-matrix determines the associated field theory uniquely. In
case of known admissable S-matrices (this only occurs in d=1+1) the stage
for the "formfactor program" is set by the computation of the model-dependent
modular localization spaces of the wedge region (hep-th/9805093). The last
5 papers in my publication list have created the setting for a completely
novel research program to classify and to construct nontrivial QFT's
in a totally nonperturbative manner.

**My Research Proposal and Workplan**.

In the following I will briefly sketch a research proposal which I
intend to persue for the next three years). I will schematically order
the program into several sections, but it should be clear that they are
stongly interrelated by mathematical methods and common physical concepts.
*1. Extension of the existing work on formfactors of d=1+1 factorizing
models*.

The modular localization spaces and the formfactors of fields were
only computed for an extremely simple model (the Federbush model). An application
to richer factorizing models as e.g. the Sine-Gordon model is of great
interest since the conjecture that bound states only emerge if one localizes
in compact regions beyond (noncompact) wedges could be settled by such
models.
*2. Direct construction of chiral conformal fields from plektonic
statistics*.

Free bosonic/fermionic fields in any dimension can be directly constructed
from spectral and causality properties. In some sense chiral conformal
local are like free fields (no genuine interations on one light line) but
with a more subtle charge structure. The modular localization approach
opens the possibility of using the classified families of admissable plektonic
statistics as a kinematical input for the construction of the fields as
the spacetime carriers of these charges. Such a construction would not
only serve as a test for the analytic power of the new method, but also
incorporate chiral QFT back into the conceptual framework of (general)
QFT i.e. permit to do things without using special 2-dimensional structures
which have no counterpart in higher dimensions (e.g. representation theory
of diffeomorphism and affine algebras). The techniques are expected to
be similar to the previous case but in case of nonabelian braid-group statistics
the ambient space for the modular localization method is not a Fock space
but a direct sum of multiparticle spaces which are not tensor products
of one particle spaces. Such a program has already been mentioned
(with more details) in the second to last section of chapter 6 of my CBPF
lecture notes
"Localization and Nonperturbative Local
Quantum Physics".
*3. Bona Fide Anyons and Plektons*.

The proper arena for anyons and plektons of direct physical interest
is 1+2 dimensional QFT. In this case the admissable plektonic statistics
are identical to the previous case but the charge carrying free plektonic
fields are semiinfinite string-like localized and hence analytically
more complicated. They are unknown and have resisted all attempts to determine
them by quantization (Chern-Simons) methods. A modular localization construction
looks very promising. Even though the direct physical applications are
all in the nonrelativistic quasiparticle regime, questions involving spin
and statistics must be settled in the relativistic QFT regime. Contrary
to the geometric arguments of Leinaas-Myrrheim and Wilszek one has no reason
to believe that the vacuumpolarization vanishes in the nonrelativistic
limit i.e. the desired plektonic nonrelativistic quasiparticles are not
objects of ordinary quantum mechanics.

The ultimate aim is to obtain an explicite construction of all plektonic
free fields from their known statistics (given in terms of Markov traces).
*4. Hidden Symmetries*.

The modular localization framework leads, as previously mentioned,
to a vast new world of new semi-geometric and nongeometric symmetries which
remain hidden to the "Lagrangian" eye (i.e to quantization methods). The
latter only perceives (apart from the internal group symmetries allowed
by Boson/Fermion statistics) the fully geometric symmetries as Poicare
symmetry. Whereas modular theory also contains the latter, its realm is
much larger. Most of the symmetries arising from modular theory are nongeometric
(but at least local in the sense of leaving invariant space-time regions
with a nontrivial causal complement). Even for free fields the following
problems are open:

Action of the modular group for double cone localization in massive
theories.

Modular action for disconnected and multiply connected regions.

For the second problem one has a partial answer in the context of chiral
conformal field theories: the associated modular groups generate the geometric
diffeomorphism group of the circle. The corresponding infinite dimensional
group in higher dimension is of course not expected to be geometric. The
modular origin of the diffeomorphisms is very interesting because e.g.
the modular data for two seperate intervals in the vacuum state also contains
the information (by charge split) about all other charged (nonvacuum) sectors.

The modular structure is also expected to play a pivotal role
in a deeper understanding of the role of pointlike fields in the algebraic
setting of LQP (the Fredenhagen conjecture).