Publications

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. 

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.

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 develop a geometric framework for Weyl quantization on pseudo-Riemannian manifolds, in which pseudodifferential operators act on sections of vector bundles equipped with arbitrary connections. We construct the associated star product and compute its semiclassical expansion up to third order in the expansion parameter. A central feature of our approach is a one-to-one correspondence between formally self-adjoint symbols and formally self-adjoint operators, extending known results from flat space to curved geometries. In addition, we analyze the Moyal equation satisfied by the Wigner function in this setting and provide explicit computations of Weyl symbols for several physically significant operators, including the Dirac, Maxwell, linearized Yang-Mills, and linearized Einstein operators. Our results lay the foundation for future developments in quantum field theory on curved spacetimes, semiclassical analysis, and chiral kinetic theory. 

In this paper, we investigate the initial value problem for symmetric hyperbolic systems on globally hyperbolic Lorentzian manifolds with potentials that are both nonlocal in time and space. When the potential is retarded and uniformly bounded in time, we establish well-posedness of the Cauchy problem on a time strip, proving existence, uniqueness, and regularity of solutions. If the potential is not retarded but has only short time range, we show that strong solutions still exist, under the additional assumptions that the uniform bound in time is sufficiently small compared to the range in time and that its kernel decays sufficiently fast in time with respect to the zero-order terms of the system. Furthermore, we present a counterexample demonstrating that when the uniform bound is too large compared to the time range, solutions may fail to exist. As an application, we discuss Maxwell's equations in linear dispersive media on ultrastatic spacetimes, as well as the Dirac equation with nonlocal potential naturally arising in the theory of causal fermion systems. Our paper aims to represent the starting point for a rigorous study for the Cauchy problem for the semiclassical Einstein equations. 

This thesis is devoted to the study of hyperbolic differential operators on globally hyperbolic

Lorentzian manifolds, linear field-theoretic models exhibiting a gauge symmetry, and their

quantisation.

In the first part, we treat the Cauchy problem for symmetric hyperbolic systems and normally

hyperbolic operators on globally hyperbolic manifolds from first principles, complemented

by many examples and a discussion of Green hyperbolicity. Although hyperbolic equations are

usually studied in the context of local interactions, there are strong motivations from several

areas of mathematical physics to consider also nonlocal interactions. As an intermezzo, we

therefore take a small deviation from the classical local theory and prove well-posedness of

the Cauchy problem for symmetric hyperbolic systems coupled to a broad class of nonlocal

potentials, with applications, for instance, to the Maxwell equations in linear dispersive media.

The subsequent part presents a detailed exposition of linear gauge theories in globally hyperbolic

spacetimes. Linear gauge theories are yet another deviation from the concept of hyperbolicity:

the corresponding equations of motion are generically non-hyperbolic, however, can

always be reduced to a constrained hyperbolic dynamics once an appropriate gauge fixing procedure

has been applied. In particular, we give a thorough analysis of their Cauchy problems

and construct the corresponding classical phase space. A central part of this section is the presentation

of several examples of direct physical interest, some of which have not appeared yet

in the literature in this context. Moreover, we explain how linear gauge theories are quantised

following the algebraic approach to quantum field theory, which offers a mathematically rigorous

quantisation scheme, ideally suited for field theories defined on Lorentzian backgrounds. In particular,

we introduce the notion of Hadamard states, which are physically distinguished states

whose two-point function has a specific singularity structure as a bidistribution, and provide a

detailed bibliographic review of existence results for such states.

The final chapter of this thesis is devoted to the quantisation of Maxwell’s theory on globally

hyperbolic spacetimes. After a detailed discussion of the Cauchy and gauge problems

for Maxwell’s theory on Lorentzian manifolds, the central goal is to prove the existence of

Hadamard states for Maxwell’s theory on any globally hyperbolic spacetime. The novelty of

our approach lies in a new gauge-fixing procedure at the level of initial data, which allows us to

suppress all the unphysical degrees of freedom. This gauge is achieved by means of a new Hodge

decomposition for differential k-forms in Sobolev spaces on complete (possibly non-compact)

Riemannian manifolds. Using tools from microlocal analysis, we explicitly construct Hadamard

states on ultrastatic spacetimes for the completely gauge-fixed theory and define states obtained

as the pull-back thereof along the gauge-fixing projector, while ensuring that this construction

preserves the Hadamard property. For general spacetimes, we employ a deformation argument.

This thesis is complemented by an appendix that, alongside other supplementary materials,

provides a self-contained introduction to microlocal analysis and pseudodifferential calculus.

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.