Research: Working Groups

Emerging Field Project: Quantum Geometry

Prof. Dr. Thomas Thiemann

Institut für Theoretische Physik

Universität Erlangen-Nürnberg

Staudtstraße 7

91058 Erlangen

Germany

Prof. Dr. Klaus Mecke

Institut für Theoretische Physik

Staudtstraße 7

91058 Erlangen

Germany

Prof. Dr. Catherine Meusburger

Department Mathematik

Cauerstr. 11

91058 Erlangen

Germany

Prof. Dr. Karl-Hermann Neeb

Department Mathematik

Cauerstr. 11

91058 Erlangen

Germany

The reason why a consistent quantum theory of gravity is so difficult to construct is that the principles of Quantum Field Theory and General Relativity in their current formulation contradict each other: While Quantum Field Theory rests on a background geometry in its very definition, General Relativity forbids such a background structure, also by definition. In more detail, Quantum Field Theory in its current formulation describes the propagation of matter in a given spacetime. One has to assume a fixed spacetime geometry in order to define a theory that describes quantum matter. In contrast, General Relativity does not allow for a fixed spacetime geometry. Instead, the geometry of the spacetime arises dynamically from the matter content of the theory. Matter and geometry interact, and their interaction is governed by the Einstein equations.

It transpires that the current formalism of Quantum Field Theory is insufficient for Quantum Geometry. In the latter there is no given spacetime geometry on which the gravitational field can propagate, the gravitational field itself determines the geometry of the spacetime.

To make progress, new ideas are needed, both from physics and mathematics. On the side of mathematics, similar structures and problems emerge naturally in the representation theory of Lie groups, Lie algebras and their deformations (quantum groups), in particular when they are infinite dimensional. Here the group itself is the source of various natural geometric structures such as symplectic manifolds or Poisson spaces. Now the principal problem is to associate representations to these geometric structures in a natural way which can be viewed as a passage to quantum geometric structures. This is a mathematical variant of a "quantisation process" similar to the passage from classical models to quantum models in physics. Ideally, one would like to have a dictionary translating between spectral properties, decomposition of representations etc. on the quantum side and corresponding geometric properties in the classical context.

Although large parts of this can be realised for various classes of finite dimensional groups, most of the presently known methods break down for infinite dimensional ones, so that completely new approaches have to be explored. Two of the most important classes of infinite dimensional groups are groups of gauge transformations and of diffeomorphisms, and these arise as symmetry groups in Quantum Gravity. Therefore it is natural to join forces in exploring the representation theoretic interface between infinite dimensional Lie groups and Quantum Gravity.

- I.
**Choice of Poisson algebra**

(Meusburger, Thiemann) - II.
**Representation Theory**

(Neeb, Thiemann) - III.
**New geometric quantization schemes**

(Neeb, Meusburger) - IV.
**3 dimensional and 4 dimensional General Relativity**(Meusburger, Thiemann) - V.
**Quantum Dynamics**

(Mecke, Thiemann) - VI.
**Classical Limit**

(Mecke, Neeb, Thiemann) - VII.
**Hyperbolic Geometry and Regge Calculus**

(Mecke, Meusburger) - VIII.
**Conformal Field Theory and Black Holes**

(Mecke, Meusburger, Neeb)