Publications

We consider transition amplitudes in the coloured simplicial Boulatov model for three-dimensional Riemannian quantum gravity. First, we discuss aspects of the topology of coloured graphs with non-empty boundaries. Using a modification of the standard rooting procedure of coloured tensor models, we then write transition amplitudes systematically as topological expansions. We analyse the transition amplitudes for the simplest boundary topology, the 2-sphere, and prove that they factorize into a sum entirely given by the combinatorics of the boundary spin network state and that the leading order is given by graphs representing the closed 3-ball in the large N limit. This is the first step towards a more detailed study of the holographic nature of coloured Boulatov-type GFT models for topological field theories and quantum gravity. 

We investigate the properties of a fairly large class of boundary conditions for the linearised Einstein equations in the Riemannian setting, ones which generalise the linearised counterpart of boundary conditions proposed by Anderson. Through the prism of the quest to quantise gravitational waves in curved spacetimes, we study their properties from the point of view of ellipticity, gauge invariance, and the existence of a spectral gap.

The aim of this paper is to prove the existence of Hadamard states for the Maxwell equations on any globally hyperbolic spacetime. This will be achieved by introducing a new gauge fixing condition, the Cauchy radiation gauge, that will allow to suppress all the unphysical degrees of freedom. The key ingredient for achieving this gauge is a new Hodge decomposition for differential k-forms in Sobolev spaces on complete (possibly non-compact) Riemannian manifolds. 

This thesis is devoted to the study of 3-dimensional quantum gravity as a spin foam model and group field theory. In the first part of this thesis, we review some general physical and mathematical aspects of 3-dimensional gravity, focusing on its topological nature. Afterwards, we review some important aspects of the Ponzano-Regge spin foam model for 3-dimensional Riemannian quantum gravity and explain in some details how it is related to the discretized path integral of general relativity in its first-order formulation. Furthermore, we discuss briefly some related spin foam models and review the notion of spin network states in order to properly define transition amplitudes of these models.

The main results of this thesis are contained in the second part. We start by reviewing the Boulatov group field theory and explain how it is related to the Ponzano-Regge model and some advantages of introducing colouring. Afterwards, we give a very detailed review of the topology of coloured graphs with non-empty boundary and review techniques, which are devolved in crystallization theory, a branch of geometric topology. In the last part of this chapter, we apply these techniques in order to define suitable boundary observables and transition amplitudes of this model and in order to set up a formalism for dealing with transition amplitudes in the coloured Boulatov model in a more systematic way by writing them as topological expansions. We also apply these techniques to the simplest possible boundary state representing a 2-sphere. Last but not least, we review some results regarding quasi-local holography in the Ponzano-Regge model, construct some explicit examples of coloured graphs representing manifolds with torus boundary and discuss the transition amplitude of some fixed boundary graph representing a 2-torus. 

γγ-pair production, the process in which an electron and a positron are created from the collision of two photons, plays a major role in many questions in high-energy astrophysics. In the first part of this bachelor thesis this process is described in a mathematical way, in order to get an idea of how it works. Kinematics, consequences following from the energy and momentum conservation laws, will be discussed and the calculation of the energies of the created electron and positron will be shown. Furthermore, the cross section for γγ-pair production will be discussed. In the next chapter, three different approaches for deriving the "spectrum", which is the number of created particles per energy, are presented. The first one is an analytic derivation by Felix Aharonian et al. from 1983 for the case of isotropically distributed photon fields. A second approach is an approximation for the case of a highly energetic photon passing through a target photon field with photon energies, which are smaller than the rest energy of the electron. The last possibility of deriving the spectrum of electrons and positrons is by applying a simulation. For this, a Monte-Carlo simulation for γγ-pair production is described. In the last section of this bachelor thesis, the results of my research are presented. In this part, the approximation found by Aharonian et al. is compared with the analytic calculation and the output of the Monte-Carlo simulation. It could be shown that the approximation is also accurate for scenarios it was not made for. Furthermore, the different approaches are compared for some cases, which are relevant for the physics of relativistic jets.