Approaches to quantization of space-time

Die Seminarsprache ist Englisch.

Content

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.

For details about the talks and for literature we refer to the webpage of the seminar.

Vorkenntnisse

Gute Kenntnisse in Differentialgeometrie, in partiellen Differentialgleichungen und in Grundlagen der mathematischen Physik.