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).