Current Research Projects of Edwin Langmann

I am working on several projects in quantum field theory which at the borderline between mathematical physics, high energy physics and solid state physics.  Below I give a more specific description of some of my recent results.
 

(1) The geometry of quantum fields.

Despite of our extensive knowledge on quantum field theory (QFT) we are still lacking a complete conceptual and mathematical understanding of it. Noncommutative geometry (NCG) is a powerful mathematical framework promising a deeper understanding of central issues such as regularization and renormalization. We have been interested in gauge theories and associated anomalies where the noncommutative nature of quantum systems and the geometric nature of gauge fields can be united into a coherent picture. Our aim is not only a better understanding of QFT but also to obtain novel results in noncommutative geometry which are interesting on their own right. In recent work we have found QFT models motivated by NCG and which are exactly solvable [1,2,3]. I also found variants of these models which describe 2D correlated fermions which can be solved exactly [4,5] - these recent results are connecting my research in NCG to my other two projects described below.

Collaborators: J. Mickelsson, R. Szabo, K. Zarembo

Keywords:
quantum field theory
renormalization
noncommutative geometry
gauge theories
 

(2) Exact results on quantum many-body systems.

This project is on an interesting class of integrable quantum systems and their relation to anyons, the fractional quantum Hall effect, and gauge theories. Exploiting these relations I recently found a novel algorithm to solve the general elliptic case by using a second quantization of this model [6,7]. I now am working on getting more explicit explicit formulas on the solution of this model. We have also recently found and solved a novel type of exactly solvable system with local momentum dependent interactions [8,9].

Collaborators: Cornelius Paufler, Martin Hallnäs, Harald Grosse

Keywords:
Calogero-Moser systems
integrable systems
quantum Hall effect
loop groups

(3) Computation methods for correlated fermion models.

In this project we combine analytic and numeric methods to develop new computation tools for strongly correlated fermion systems such as the two dimensional Hubbard model. Recently I found a class of exactly solvable models describing correlated fermions in 2D. They can be obtained by a truncation from the 2D Hubbard model which generalizes mean field theory by taking into account a particular type of spatial correlations [4,5].
 

Collaborators:  M. Wallin

Keywords:
magnetism
superconductivity
metal-insulator transition
correlated fermion systems
Hubbard model
 

My research is funded by:

VR (The Swedish Research Council)
Göran Gustafssons Foundation
 

Publications since 1994