# Seminar: Approaches to quantization of space-time

Prof. Bernd Ammann, Prof. Felix Finster

Veranstaltungsnummer

## Inhalt des Seminars

Die Seminarsprache ist Englisch.
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.

## The detailed Program

is available here as pdf.
### Vorkenntnisse

Gute Kenntnisse in Differentialgeometrie, in partiellen Differentialgleichungen und in Grundlagen der mathematischen Physik.
## Ort und Zeit:

Do 10-12 in M101

Bernd Ammann, 1.2.2012 oder später