Quantum Gravity

Prof. Dr. Dr. h.c. Eberhard Zeidler works at the Max Planck Institute for Mathematics in the Sciences in Leipzig (Germany). In 1996 he was one of the founding directors of this institute. He is a member of the Academy of Natural Scientists Leopoldina. In 2006 he was awarded the 'Alfried Krupp Wissenschaftspreis' of the Alfried Krupp von Bohlen und Halbach-Stiftung.

This Edited Volume is based on a workshop on "Mathematical and Physical - pects of Quantum Gravity" held at the Heinrich-Fabri Institute in Blaubeuren st (Germany) from July 28th to August 1 , 2005. This workshop was the succ- sor of a similar workshop held at the same place in September 2003 on the issue of "Mathematical and Physical Aspects of Quantum Field Theories". Both wo- shops were intended to bring together mathematicians and physicists to discuss profoundquestionswithin the non-emptyintersectionofmathematics andphysics. The basic idea of this series of workshops is to cover a broad range of di?erent approaches (both mathematical and physical) to a speci?c subject in mathema- cal physics. The series of workshops is intended, in particular, to discuss the basic conceptual ideas behind di?erent mathematical and physical approaches to the subject matter concerned. The workshop on which this volume is based was devoted to what is c- monly regarded as the biggest challenge in mathematical physics: the "quanti- tion of gravity". The gravitational interaction is known to be very di?erent from the known interactions like, for instance, the electroweak or strong interaction of elementary particles. First of all, to our knowledge, any kind of energy has a gravitational coupling. Second, since Einstein it is widely accepted that gravity is intimately related to the structure of space-time. Both facts have far reaching consequences for any attempt to develop a quantum theory of gravity.

This volume presents different approaches to quantum gravity, including loop quantum gravity, string theory, noncommutative geometry, canonical quantum gravity, and more advanced approaches. Coverage includes a discussion of the recent experimental status of the search for quantum gravity effects and is accessible to non-experts in the field. Extensive indexing of all articles ensures accessibility to keywords and phrases. The construction of a quantum theory of gravity is the most fundamental challenge confronting contemporary theoretical physics. The different physical ideas evolved in developing a theory of quantum gravity require highly advanced mathematical methods. Written by leading experts in this field, this book provides the reader with an overview of the different mathematical attempts to quantize gravity.

Quantum Gravity - A Short Overview.- The Search for Quantum Gravity Effects.- Time Paradox in Quantum Gravity.- Differential Geometry in Non-Commutative Worlds.- Algebraic Approach to Quantum Gravity III: Non-Commmutative Riemannian Geometry.- Quantum Gravity as a Quantum Field Theory of Simplicial Geometry.- An Essay on the Spectral Action and its Relation to Quantum Gravity.- Towards a Background Independent Formulation of Perturbative Quantum Gravity.- Mapping-Class Groups of 3-Manifolds in Canonical Quantum Gravity.- Kinematical Uniqueness of Loop Quantum Gravity.- Topological Quantum Field Theory as Topological Quantum Gravity.- Strings, Higher Curvature Corrections, and Black Holes.- The Principle of the Fermionic Projector: An Approach for Quantum Gravity?.- Gravitational Waves and Energy Momentum Quanta.- Asymptotic Safety in Quantum Einstein Gravity: Nonperturbative Renormalizability and Fractal Spacetime Structure.- Noncommutative QFT and Renormalization.