For the understanding of the variable, transient and non-thermal universe, unbiased long-term monitoring is crucial. To constrain the emission mechanisms at the highest energies, it is important to characterize the very high energy emission and its correlation with observations at other wavelengths. At very high energies, only a limited number of instruments is available. This article reviews the current status of monitoring of the extra-galactic sky at TeV energies.

Single-molecule super-resolution microscopy (SMLM) techniques like dSTORM can reveal biological structures down to the nanometer scale. The achievable resolution is not only defined by the localization precision of individual fluorescent molecules, but also by their density, which becomes a limiting factor e.g., in expansion microscopy. Artificial deep neural networks can learn to reconstruct dense super-resolved structures such as microtubules from a sparse, noisy set of data points. This approach requires a robust method to assess the quality of a predicted density image and to quantitatively compare it to a ground truth image. Such a quality measure needs to be differentiable to be applied as loss function in deep learning. We developed a new trainable quality measure based on Fourier Ring Correlation (FRC) and used it to train deep neural networks to map a small number of sampling points to an underlying density. Smooth ground truth images of microtubules were generated from localization coordinates using an anisotropic Gaussian kernel density estimator. We show that the FRC criterion ideally complements the existing state-of-the-art multiscale structural similarity index, since both are interpretable and there is no trade-off between them during optimization. The TensorFlow implementation of our FRC metric can easily be integrated into existing deep learning workflows.

In this thesis, I study entanglement in quantum field theory, using methods from operator algebra theory. More precisely, the thesis covers original research on the entanglement properties of the free fermionic field. After giving a pedagogical introduction to algebraic methods in quantum field theory, as well as the modular theory of Tomita-Takesaki and its relation to entanglement, I present a coherent framework that allows to solve Tomita-Takesaki theory for free fermionic fields in any number of dimensions. Subsequently, I use the derived machinery on the free massless fermion in two dimensions, where the formulae can be evaluated analytically. In particular, this entails the derivation of the resolvent of restrictions of the propagator, by means of solving singular integral equations. In this way, I derive the modular flow, modular Hamiltonian, modular correlation function, R\'enyi entanglement entropy, von-Neumann entanglement entropy, relative entanglement entropy, and mutual information for multi-component regions. All of this is done for the vacuum and thermal states, both on the infinite line and the circle with (anti-)periodic boundary conditions. Some of these results confirm previous results from the literature, such as the modular Hamiltonian and entanglement entropy in the vacuum state. The non-universal solutions for modular flow, modular correlation function, and R\'enyi entropy, however are new, in particular at finite temperature on the circle. Additionally, I show how boundaries of spacetime affect entanglement, as well as how one can define relative (entanglement) entropy and mutual information in theories with superselection rules. The findings regarding modular flow in multi-component regions can be summarised as follows: In the non-degenerate vacuum state, modular flow is multi-local, in the sense that it mixes the field operators along multiple trajectories, with one trajectory per component. This was already known from previous literature but is presented here in a more explicit form. In particular, I present the exact solution for the dynamics of the mixing process. What was not previously known at all, is that the modular flow of the thermal state on the circle is infinitely multi-local even for a connected region, in the sense that it mixes the field along an infinite, discretely distributed set, of trajectories. In the limit of high temperatures, all trajectories but the local one are pushed towards the boundary of the region, where their amplitude is damped exponentially, leaving only the local result. At low temperatures, on the other hand, these trajectories distribute densely in the region to either---for anti-periodic boundary conditions---cancel, or---for periodic boundary conditions---recover the non-local contribution due to the degenerate vacuum state. Proceeding to spacetimes with boundaries, I show explicitly how the presence of a boundary implies entanglement between the two components of the Dirac spinor. By computing the mutual information between the components inside a connected region, I show quantitatively that this entanglement decreases as an inverse square law at large distances from the boundary. In addition, full conformal symmetry (which is explicitly broken due to the presence of a boundary) is recovered from the exact solution for modular flow, far away from the boundary. As far as I know, all of these results are new, although related results were published by another group during the final stage of this thesis. Finally, regarding relative entanglement entropy in theories with superselection sectors, I introduce charge and flux resolved relative entropies, which are novel measures for the distinguishability of states, incorporating a charge operator, central to the algebra of observables. While charge resolved relative entropy has the interpretation of being a ``distinguishability per charge sector'', I argue that it is physically meaningless without placing a cutoff, due to infinite short-distance entanglement. Flux resolved relative entropy, on the other hand, overcomes this problem by inserting an Aharonov-Bohm flux and thus passing to a variant of the grand canonical ensemble. It takes a well defined value, even without putting a cutoff, and I compute its value between various states of the free massless fermion on the line, the charge operator being the total fermion number.

In the last decade continuous-time quantum Monte Carlo in the hybridization expansion (CTHYB) was one of the most successful Monte Carlo techniques to describe correlated quantum phenomena in conjunction with dynamical mean field theory (DMFT). The first part of the thesis consists of algorithmical developments regarding CTHYB and DMFT. I provide a complete derivation and an extensive discussion of the expansion formula. We generalized it to treat spin-orbit coupling, and invented the superstate sampling algorithm to make it efficient enough for describing systems with general interactions, crystal fields and spin-orbit coupling at low temperatures. But CTHYB is known to fail in the standard implementation for equal-time correlators, certain higher-order Green’s functions and the atomic limit; we discovered that its estimator for the Greens function is also inconsistent for Anderson impurities with finite, discrete baths. I focus then on further improvements of CTHYB that we have conceived and worked on, in particular for f-orbitals and for taking physical symmetries into account in the calculation of the Monte Carlo observables. The second part of the thesis presents selected physical applications of these methods. I show DMFT calculations of highest accuracy for elemental iron and nickel and discover a new mechanism of magnetic ordering in nickel: the ordering of band structure-induced local moments. Then we analyze the stability of this phenomenon under pressure and temperatures, that characterize in the Earth’s core. We find, that the mechanism survives these conditions and may give a significant contribution to the generation of the Earth’s magnetic field. The next topic is the stability of double Dirac fermions against electronic correlations. We find, that the Coulomb interaction in the corresponding material Bi2 CuO4 are strong enough to destroy the double Dirac cone, and substantial uniform pressure is necessary to restore them. In the last chapter I derive the properties of Higgs and Goldstone bosons from Ginzburg-Landau theory, and identify these excitations in a model of an excitonic magnet.

In this review paper, we stress the importance of the higher transcendental Wright functions of the second kind in the framework of Mathematical Physics. We first start with the analytical properties of the classical Wright functions of which we distinguish two kinds. We then justify the relevance of the Wright functions of the second kind as fundamental solutions of the time-fractional diffusion-wave equations. Indeed, we think that this approach is the most accessible point of view for describing non-Gaussian stochastic processes and the transition from sub-diffusion processes to wave propagation. Through the sections of the text and suitable appendices, we plan to address the reader in this pathway towards the applications of the Wright functions of the second kind.