Courant Research Centres ´´Higher Order Structures in Mathematics´´

Mathematical Physics and Noncommutative Geometry (Dorothea Bahns)

The main area of this group's research is the application and generalization of rigorous methods of perturbative quantum field theory to noncommutative spaces. Such 'spaces' are given by associative algebras; a simple model that already exibits a wealth of properties is the Weyl algebra ('Moyal space'). One motivation to study noncommutative spaces is the attempt to implement the idea that, from a physical point of view, there should be fundamental obstructions as to the resolvability of an event in spacetime. The idea of pointlike events has to be given up, when gravitational forces and quantum effects at small distances are simultaneously taken into account, and one way to implement this idea is to consider noncommutative spaces instead of the smooth spacetimes of general relativity.

Studying quantum fields on noncommutative spaces is moreover a means to gain a better understanding of the structure and the setup of quantum field theory itself. For instance, two very prominent features of noncommutative field theory, the alleged violation of unitarity and the ultraviolet-infrared mixing property turn out to be artefacts of the formalism employed in most of the literature on the subject - a formalism which - in ordinary quantum field theory - makes sense due to the remarkable theorem of Osterwalder and Schrader, but does not seem to be a suitable approach in a noncommutative setting. To understand what the underlying structural reasons for this apparent breakdown of the formalism are, and how the problem can be solved, is currently one of our most active lines of research. Confronting these features of noncommutative field theories with ordinary quantum field theory, insight regarding the latter can be gained.

Mathematically, these investigations require methods of applied (combinatorial) algebra as well as applied (functional) analysis (keywords: Hopf algebras, microlocal analysis).

Algebraic String Quantization

Another field of research we pursue is the quantization of strings in an approach based on the so-called invariant charges. In this approach, the shape of a string's 'worldsheet' (a surface which is immersed in a higher dimensional vector space), is encoded in terms of a set of functionals which are independent of the surface's parametrization. This set is endowed with a natural Poisson structure and deforming the resulting Poisson algebra into an associative algebra yields a reparametrization invariant quantization of string theory. This setting is inequivalent to canonical string quantization and in particular, in contrast to canonical string quantization, no obstructions regarding the dimension of the sourrounding vector space have appeared so far.