Investigation of algebraic properties of gauge field theories on noncommutative spaces

在非常短的距离上的时空物理学是不被理解的。到目前为止，人们还没有找到一个令人满意的量子引力理论，短距离的无穷大给量子场论带来了困难。解决这些奇点的一个自然的可能性是非对易空间，因为在某种意义上它们不具有点，或者换句话说，物理系统“生活”在一个格上。过去几年的发展，特别是来自弦理论的发展，已经表明理解这些空间上的规范理论是多么重要。规范理论是物理理论，在某些对称变换下是不变的;标准模型和引力都可以用这种方式表述。本研究项目的目的就是对这些理论进行研究。特别是人们应该努力理解在一般坐标变换下具有不变性的非对易系统。

The physics of space-time at very short distances is not understood. Up to now a satisfactory theory of quantum gravitation has not bee found. Infinities at short distances lead to problems for quantum field theories. A natural possibility to resolve these singularities are noncommutative spaces, since in some sense they do not possess points, or - in other words - the physical system "lives" on a lattice. The development of the last years, in particular coming also from string theory, has shown how important is an understanding of gauge theories on these spaces. Gauge theories are physical theories, which are invariant under certain symmetry transformations; the standard model and also gravitation can be formulated in this way. Objective of this research project is the investigation of such theories. Especially one should try to understand noncommutative systems with invariance under general coordinate transformations.