##
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