Topological fermion systems: Geometry and analysis on quantum spaces

Topological fermion systems provide a general framework for desribing and analyzing non-smooth geometries. They can also be used to describe "quantum spaces" or "quantum space-times" as considered in quantum gravity. Moreover, they set the stage for the so-called fermionic projector formulation of relativistic quantum field theory.

The aim of the talk is to give a simple introduction, with an emphasis on conceptual issues. Starting from a collection of functions on $\R^3$ (which can be thought of as Schrödinger wave functions), we ask the question whether the geometry of the Euclidean space is encoded in these functions. Bringing this question into a precise mathematical form leads us to the abstract definition of topological fermion systems. This definition will be illustrated by the examples of vector fields on the sphere, a vector bundle over a manifold, and a lattice system. As an example motivated from physics, we briefly consider Dirac spinors on a globally hyperbolic Lorentzian manifold and introduce the setting of causal fermion systems. The inherent geometric and analytic structures on a topological fermion system are introduced and explained. A brief outlook on the applications to quantum field theory is given.