Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance

Abstract

This series of lectures gives an introduction to the non-perturbative and background-independent formulation for a quantum theory of gravitation which is called loop quantum gravity. The Hilbert space of kinematical quantum states is constructed and a complete basis of spin network states is introduced. Afterwards an application of the formalism is provided by the spectral analysis of the area operator, which is the quantum analogue of the classical area function. This leads to one of the key results of loop quantum gravity obtained in the last few years: the derivation of the discreteness of the geometry and the computation of the quanta of area. Special importance is attached to the role played by the diffeomorphism group in order to clarify the notion of observability in general relativity - a concept far from being trivial. Finally an outlock onto a possible dynamical extension of the theory is given, leading to a “sum over histories” approach, namely a so-called spin foam model. Throughout the whole lecture great significance is attached to conceptual and interpretational issues.