Research

lechnerg125

As a mathematical physicist, I work on conceptual and mathematical problems arising in quantum field theory. I am particularly interested in new approaches to the problem of rigorously constructing quantum field theory models. A novel line of research addresses this question via operator-algebraic deformations of quantum field theories, a topic also connected to non-commutative geometry and quantum field theory on non-commutative spaces.

A further interest are integrable models in quantum field theory and related structures, such as the Yang-Baxter equation, braid group representations and link invariants.

Some recent research of mine is:

Yang-Baxter Representations (with Ulrich Pennig and Simon Wood)
The Yang-Baxter equation  appears in a remarkable number of different fields, such as statistical mechanics, knot theory, quantum groups. quantum field theory, quantum mechanics, quantum information theory, subfactors, … . This equation is very hard to solve directly, and the structure of the set of its solutions is insufficiently understood. We considered the case of involutive solutions $R$, i.e. satisfying $R^2=1$. In this case, we found a complete classification of all solutions up to a natural equivalence suggested by applications in quantum field theory.

Quantum-mechanical Backflow (with Henning Bostelmann and Daniela Cadamuro)
One incarnation of the uncertainty relation is the so-called backflow effect: A quantum-mechanical particle in one spatial dimension that has positive momentum (with probability 1) may still have negative probability flux, i.e. the probability can flow in the direction opposite to the momentum. We investigated the backflow effect for interacting systems governed by an arbitrary short-range potential. It turns out that backflow is a universal effect that exists in any such system, and its magnitude can be estimated by analytic methods.

Constructive Algebraic Quantum Field Theory
In 2015, I wrote a book chapter reviewing the status of constructive algebraic quantum field theory.

For more details about my research, you can have a look at my publications and talks:

Publications

Talks

 


 

Recent teaching

Winter 2017: Mathematical Foundations of Quantum Physics

Winter 2017: Foundations of Mathematics

250px-TrefoilMirrorWinter 2015: Knots (An introduction to knot theory)