In the seminar we present and compare three different approaches to the quantization of fields in space-time and to the quantization of the underlying space-time.
In the first series of talks, we give an overview of non-commutative geometry in the sense of Alain Connes. The main objects are spectral tripels (A,D,H) which consist of an algebra A, an A-module H and an operator D acting on H. For a classical Riemannian manifold M equipped with a metric and a spin structure, one takes as A the algebra of smooth functions on M, as H the space of all spinor fields and as D the Dirac operator. This encodes the geometry of M. A non-commutative space is then a spectral triple for which A is non-commutative. In particular we reformulate the Einstein-Hilbert functional as a spectral action.
The second part is devoted to the quantization of fields in a fixed globally hyperbolic space-time. In preparation, we analyze the solutions of the classical field equations. For the quantization we introduce a nets of $C^*$-algebras. We construct CCR and CAR-representations and construct the Fock space in the bosonic and the fermionic case.
In the third part, we introduce the fermionic projector approach. After a general overview, we give the general construction of the fermionic projector in a globally hyperbolic space-time. The underlying action principle is analyzed in the setting of causal variational principles. The generalization to causal fermion systems allows to describe "quantum space-times". The connection to quantum field theory in Minkowski space is obtained by taking the so-called continuum limit.