About

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 study points of geometry has always been one of the most fruitful interfaces between mathematics and physics. One if its culmination is Einstein's theory of General Relativity which describes the gravitational interaction in physics and is formulated in terms of Lorentzian geometry. General Relativity is widely accepted as the theory which best describes classical gravity and has been tested with high experimental precision.

However, the world is not classical but quantum: observable quantities in physics underly fluctuations governed by Heisenberg's uncertainty principle and Planck's constant. The consistent unification of General Relativity with Quantum Theory is considered to be one of the most challenging research topics of modern fundamental physics and requires new input from various disciplines of mathematics and physics.

Quantum geometry tries to describe what happened at the origin of space and time, to improve our knowledge of the evolution of the cosmos and to gain new insights into the puzzles connected with inflation. A success of this approach would deepen our understanding of what happens inside a black hole and would be able to interpret current cosmological and astrophysical data more accurately, which might shed light on the mysterious dark energy. It would also enable us to compute quantum geometric corrections to elementary particle processes and much more. In other words: Textbooks of physics would need to be rewritten.

Erlangen has a long and famous tradition in the foundations of geometry and its connections to algebra and physics. Among Erlangen's achievements are the following:

- Emmy Noether's und Paul Gordan's Invariantentheorie,
- Felix Klein's `Erlanger Programm', which defines geometry in terms of symmetry groups,
- Karl von Staudt's `Geometrie der Lage', which formulates geometric concepts without a background metric.

All of these are milestones in the development of geometry. The EFP 'Quantum Geometry' continues this tradition and extends these classical aspects of geometry and symmetries to the quantum realm.

The EFP 'Quantum Geometry' aims to become the first interdisciplinary center for Quantum Geometry in Europe in which mathematicians and physicists work together on a long-term basis and in which the research is complemented by an ambitious teaching and training programme for students and young researchers. It rests on the following research pillars:

- Statistical and Computational Physics
- Infinite dimensional Lie theory
- Representation theory (of groups, algebras and quantum groups)
- General Relativity and Quantum Field Theory

The interaction between these four fields and the combination of the relevant expertise is motivated as follows: The development of a quantum theory of geometry based on the classical field theory of General Relativity requires crucial input from all other research pillars on various stages. Conversely, we expect it to have a substantial impact on the other pillars arising from constructions motivated by physics. For instance, they could give rise to completely new quantisation procedures for Lie group representations or Poisson spaces and to new manifold invariants obtained from new quantum geometric objects.

In Hamiltonian quantisation approaches the passage from General Relativity to a quantum field theory can be accomplished in two steps: First, one selects a sub *-algebra of the classical Poisson algebra and then promotes it to an abstract operator algebra. In a second step, one searches for representations of this algebra through operators on Hilbert spaces which have the right properties with respect to physics. In path integral approaches to quantisation, the spacetime manifold is discretised, for instance via a triangulation or cell decomposition. The edges, faces and volumes of this cell complex are decorated with representation theoretical data. By summing over all such assignments, one obtains a state sum model that resembles models from statistical physics and quantum field theory. To qualify as a quantum theory of gravity, a model must pass the classical limit criterion: When the quantum fluctuations of geometry are small, the theory must reduce to Einstein's classical General Relativity. The construction of the classical limit and the corresponding minimal uncertainty states require methods from statistical physics, in particular, random and stochastic geometry.

Another connection between the fields is linked to the fact that General Relativity can be viewed as a gauge theory. Its gauge freedom encodes the invariance of the theory under coordinate transformations, and the relevant gauge group includes all spacetime diffeomorphisms. The associated redundancy in the description is encoded mathematically in the corresponding infinite dimensional Lie algebras and Lie groups. This requires extracting the physically relevant (observable) invariant information by corresponding mathematical procedures.

Finally, once the mathematical foundations have been established rigorously, the implications and predictions of the theory for physics must be extracted from the description. As it can be expected that computations in Quantum Geometry will be even more complex than those arising in non perturbative Quantum Chromodynamics (QCD), this will require numerical tools as well as analytical methods.