Topics in quantum gravity

Quantum gravity is an attempt to unify general relativity with quantum mechanics which are the two highly successful fundamental theories of theoretical physics. The main difficulty in this unification arises from the fact that, while general relativity describes gravity as a macroscopic geometrical theory, quantum mechanics explains microscopic phenomena. As a further complication, not only do both theories describe different scales but also their philosophical ramifications and the mathematics used to describe them differ in a dramatic way. Consequently, one possible starting point of an attempt at a unification is quantum mechanics, i.e. particle physics, and try to incorporate gravitation. This pathway has been chosen by particle physicists which led to string theory. On the other hand, loop quantum gravity (LQG) chooses the other possibility, i.e. it takes the geometrical aspects of gravity seriously and quantizes geometry. The first part of this thesis deals with a generalization of loop quantum cosmology (LQC) to toroidal topologies. LQC is a quantization of homogenous solutions of Einstein's field equations using tools from LQG. First the general concepts of closed topologies is introduced with special emphasis on Thurston's theorem and its consequences. It is shown that new degrees of freedom called Teichmueller parameters come into play and their dynamics can be described by a Hamiltonian. Several numerical solutions for a toroidal universe are presented and discussed. Following the guidelines of LQG this dynamics are rewritten using the Ashtekar variables and numerical solutions are shown. However, in order to find a suitable Hilbert space a canonical transformation must be performed. On the other hand this transformation makes the quantization of geometrical quantities less tractable such that two different ways are presented. It is shown that in both cases the spectrum of such geometrical operators depends on the initial value problem. Furthermore, we succeed in solving the quantum Gauss constraint. In the second part of the thesis we introduce some aspects of phenomenological quantum gravity and their possible detectable signatures. The goal of phenomenological quantum gravity is to derive conclusions and make predictions from expected characteristics of a full theory of quantum gravity. One possibility is an energy-dependent speed of light arising from a quantized space such that the propagation time of two photons differs. However, the amount of these corrections is very small such that only cosmological distances can be considered. Gamma-ray bursts (GRB) are ideal candidates as they are short but very luminous bursts of gamma-rays taking place at distances billions of light-years away. We study GRBs detected by the European satellite INTEGRAL and develop a new method to analyze unbinned data. A χ2-test will provide a lower bound for quantum gravity corrections, which will be nevertheless well below the Planck mass. Then we shall study the sensibility of NASA's new satellite Fermi Gamma-ray Space Telescope and conclude that it is well suited to detect corrections. This prediction has just been confirmed when Fermi detected a very energetic photon emanating from GRB 090510 which highly constrains models with linear corrections to the speed of light. However, as it is shown at the end of this thesis, more bursts are needed in order to definitely falsify such models. (orig.)