The Friedrichs Model and its use in resonance phenomena

We present here a relation of different types of Friedrichs models and their use in the description and comprehension of resonance phenomena. We first discuss the basic Friedrichs model and obtain its resonance in the case that this is simple or doubly degenerated. Next, we discuss the model with $N$ levels and show how the probability amplitude has an oscillatory behavior. Two generalizations of the Friedrichs model are suitable to introduce resonance behavior in quantum field theory. We also discuss a discrete version of the Friedrichs model and also a resonant interaction between two systems both with continuous spectrum. In an Appendix, we review the mathematics of rigged Hilbert spaces.Comment: 105 page

Classical and quantum perturbations to the primordial universe

In this Ph.D. thesis we analyse both classical and quantum effects relevant for the study of cosmological perturbations. We choose this particular topic because, through the analysis of cosmological perturbations, it is possible to explore a wide range of different physical phenomena. Moreover, they are a central and important piece in the puzzle of the history of the universe. The most obvious relevance of cosmological perturbations is the study of structure formation and the large scale structure of the universe. In this regard, such perturbations are related to primordial gravitational waves and primordial magnetic fields. Given their dependence on pre-recombination phenomena, they could give us some information on the universe before hydrogen recombination. Classical perturbations have been widely studied in literature, with the main focus on isotropic cosmological models. While this is usually a good approximation, the presence of a primordial magnetic field causes a coupling between different algebraic modes of the usual decomposition, connecting density perturbations, primordial magnetic fields and primordial gravitational waves. Moreover, the presence of the magnetic field requires the use of an anisotropic cosmological model. While small, these relations are important in the evolution of anisotropic structures. Furthermore, such primordial seeds of the magnetic fields are widely believed to be the origin of the magnetic fields measured today in galaxies. In the first part of this thesis, we analyse these relations, together with the possible effects that a non ideal, i.e. viscous, cosmological fluid could have on the growth of perturbations. We focus our attention to a Bianchi I model, improving the results of some preceding papers. The second part of the thesis focuses on the semiclassical approximation of quantum gravity. Quantum effects are believed to influence the birth and dynamics of perturbation seeds and, in general, the dynamics of the primordial universe. This way, the mathematical scheme used to represent these effects is a central point in the description of quantum gravity regarding such seeds. Furthermore, even more care is required to split the WKB action between embedding variables and physical degrees of freedom, and in many models the quantum gravity corrections to the Schrödinger equation violate the unitarity of the system evolution. This decomposition shares some similarities with the Born-Oppenheimer approximation of molecular physics. We perform a critical analysis of two different ways to apply this decomposition. In particular, we analyse limits and perspectives of the different proposals to solve the non unitarity problem, even comparing expansions in different fundamental physical constants (Planck constant and mass). We find the source of non-unitary effects in a common assumption in the definition of WKB time, and we propose an alternative formulation. Also, we show how the usual assumptions of classicality of the physical quantities must be handled with care, focusing our attention to the implementation of the classical background in the perturbation scheme. Studies in this research field are very important because they could bind CMB measurements and primordial gravitational waves to quantum gravity, bringing us finally an experimental playground

Field theoretic formulation and empirical tracking of spatial processes

Spatial processes are attacked on two fronts. On the one hand, tools from theoretical and statistical physics can be used to understand behaviour in complex, spatially-extended multi-body systems. On the other hand, computer vision and statistical analysis can be used to study 4D microscopy data to observe and understand real spatial processes in vivo. On the rst of these fronts, analytical models are developed for abstract processes, which can be simulated on graphs and lattices before considering real-world applications in elds such as biology, epidemiology or ecology. In the eld theoretic formulation of spatial processes, techniques originating in quantum eld theory such as canonical quantisation and the renormalization group are applied to reaction-di usion processes by analogy. These techniques are combined in the study of critical phenomena or critical dynamics. At this level, one is often interested in the scaling behaviour; how the correlation functions scale for di erent dimensions in geometric space. This can lead to a better understanding of how macroscopic patterns relate to microscopic interactions. In this vein, the trace of a branching random walk on various graphs is studied. In the thesis, a distinctly abstract approach is emphasised in order to support an algorithmic approach to parts of the formalism. A model of self-organised criticality, the Abelian sandpile model, is also considered. By exploiting a bijection between recurrent con gurations and spanning trees, an e cient Monte Carlo algorithm is developed to simulate sandpile processes on large lattices. On the second front, two case studies are considered; migratory patterns of leukaemia cells and mitotic events in Arabidopsis roots. In the rst case, tools from statistical physics are used to study the spatial dynamics of di erent leukaemia cell lineages before and after a treatment. One key result is that we can discriminate between migratory patterns in response to treatment, classifying cell motility in terms of sup/super/di usive regimes. For the second case study, a novel algorithm is developed to processes a 4D light-sheet microscopy dataset. The combination of transient uorescent markers and a poorly localised specimen in the eld of view leads to a challenging tracking problem. A fuzzy registration-tracking algorithm is developed to track mitotic events so as to understand their spatiotemporal dynamics under normal conditions and after tissue damage.Open Acces

Theoretical Concepts of Quantum Mechanics

Quantum theory as a scientific revolution profoundly influenced human thought about the universe and governed forces of nature. Perhaps the historical development of quantum mechanics mimics the history of human scientific struggles from their beginning. This book, which brought together an international community of invited authors, represents a rich account of foundation, scientific history of quantum mechanics, relativistic quantum mechanics and field theory, and different methods to solve the Schrodinger equation. We wish for this collected volume to become an important reference for students and researchers

An Efficient Hemodynamic Workflow in Computational Surgery

For few decades, it has been shown that atherosclerosis is the cause of the majority of clinical cardiovascular diseases including peripheral arterial diseases. The diagnosis and treatment for vascular disease has evolved significantly over the past years considering the rapid advances in imaging technologies. In recent years, computational fluid dynamics has been increasingly used as a simulation tool for blood flows. Numerous researches connect wall shear stress quantities to endovascular diseases such as stenosis, aneurism, and atherosclerosis. A thorough knowledge of vascular anatomy and hemodynamic would be beneficial for understanding the development and progression of the disease, the therapeutic decision process and follow up. The objective of this dissertation is to propose a computational fluid dynamic framework that includes: Understanding how streamline efficiently hemodynamic simulation for main arteries to produce database for clinical study/Providing some confidence estimate on numerical results/Extending the state of the art of clinical study by including motion and particles analysis.Computer Science, Department o