Extended objects in quantum field theory in three dimensions and applications

In this thesis the systematic study of Quantum Field Theories (QFT) in various dimensions is proposed from the point of view of mathematical and theoretical physics, paying special attention to systems of one and three spatial dimensions (in addition to the temporal dimension in both cases) under the influence of some particular external conditions. These conditions vary from local interactions with other external classical fields to ideal boundary conditions in confining geometries. More specifically, the main objective of this work is the study of the spectrum of quantum fluctuations of the fields in the vacuum state subject to the external conditions indicated. This study will be applied to the calculation of several relevant parameters in three-dimensional and one-dimensional extended structures. These systems have recently received increasing interest in material physics (in micro-electromechanical devices based on the Casimir effect or topological defects in metamaterials and nanotubes) and in fundamental physics (quantum effects in modern cosmology and topological defects such as domain walls, monopoles and skyrmions). Different configurations of quantum fields both in compact domains and in open ones with boundaries will be studied: -A scalar field confined between plates mimicked by the most general type of lossless and frequently independent boundary conditions. -Scalar fields propagating at finite temperature under the influence of generalised Dirac delta lattices and Pöschl-Teller combs. -Scalar fields between two parallel plates mimicked by Dirac delta potentials in a curved background of a topological Pöschl-Teller kink. -Relativistic fermionic particles propagating in the real space under the influence of either a single and a double Dirac delta potential. Only effective theories will be considered. Here effective means that the microscopic degrees of freedom relative to the atoms and quarks of the matter composing the plates or objects between which the vacuum quantum interaction energy will be studied are not going to be taken into account. The methodology developed for the project is the following. Firstly, the spectrum of the non-relativistic Schrödinger operator or the relativistic Dirac one that will give rise to the set of one-particle states of the corresponding QFT will be characterised. Secondly, analytical and numerical results of the vacuum interaction energy between extended objects at zero temperature will be obtained. Finally, the study will be generalised to other thermodynamic magnitudes of interest such as the one loop quantum corrections to the Helmholtz free energy, the entropy and the Casimir force between objects at finite non zero temperature. Furthermore, graphical representations obtained numerically with the software Mathematica will be added. The thesis is structured in such a way that Chapter 1 gives an introduction to the work as a whole and the following chapters present the concrete results of each of the systems listed above. Finally, Chapter 6 summarises the main conclusions to give an overall view of the work carried out.El objetivo de esta tesis es el estudio, bajo el punto de vista de la física matemática, de teorías cuánticas de campos (TCC) en una y en tres dimensiones espaciales (aparte de la temporal) bajo la influencia de diversas condiciones externas. Estas condiciones comprenden tanto la interacción con otros campos clásicos externos así como condiciones de borde en geometrías confinantes. En particular, el principal interés de este trabajo es el estudio del espectro de las fluctuaciones cuánticas de los campos en el estado de vacío sujeto a las condiciones externas anteriormente indicadas. Este estudio permitirá obtener parámetros relevantes en algunas estructuras extensas en una y tres dimensiones. Este tipo de sistemas han suscitado recientemente un gran interés en la física de materiales (por ejemplo en dispositivos microelectromecánicos basados en el efecto Casimir, nanotubos y defectos topológicos en metamateriales) y en física fundamental (defectos topológicos como paredes de dominio, cuerdas cósmicas, monopolos y skyrmiones). A lo largo de la tesis se van a estudiar las diferentes configuraciones de campos cuánticos, tanto en dominios compactos como en dominios abiertos con bordes, que se enumeran a continuación: -Campos escalares confinados entre placas representadas por las condiciones de contorno independientes de las frecuencias más generales posibles. -Campos escalares que se propagan a temperatura finita bajo la influencia de redes de tipo delta de Dirac generalizadas y peines formados con potenciales Pöschl-Teller. -Campos escalares en un sistema formado por dos placas paralelas modelizadas por potenciales delta de Dirac introducidas en un potencial de fondo curvo dado por un kink topológico de tipo sine-Gordon. -Partículas fermiónicas relativistas que se propagan en el espacio real bajo la influencia de tanto uno como varios potenciales delta de Dirac. Es importante destacar que en esta tesis se van a considerar únicamente teorías ”efectivas”, en el sentido de que no se van a tener en cuenta los grados de libertad microscópicos relativos a los átomos y los quarks de la materia que compone las placas o objetos entre los cuales se va a estudiar la energía de interacción cuántica de vacío. La metodología general empleada para la obtención de estos objetivos es la siguiente: primero se caracterizará el espectro del operador de Schrödinger no relativista que dará lugar al conjunto de estados de una partícula de la teoría cuántica de campos correspondiente, después se obtendrán fórmulas analíticas para el cálculo de la energía de vacío de interacción entre los objetos extensos considerados a temperatura cero y finalmente se generalizará el estudio a otras magnitudes termodinámicas de interés como las correcciones cuánticas a un lazo de la energía libre de Helmholtz, la entropía y la fuerza de Casimir entre los objetos a temperatura finita no nula. Se obtendrán resultados analíticos cuando sea posible y además, todo ello irá acompañado de representaciones gráficas obtenidas numéricamente con ayuda del software Mathematica. La tesis está organizada de forma que el primer capítulo es una introducción bibliográfica al trabajo en su conjunto y los siguientes capítulos presentan los resultados concretos de cada uno de los sistemas anteriormente enumerados. Finalmente, el capítulo 6 resume las principales conclusiones de la tesis para dar una visión global del trabajo realizado.Escuela de DoctoradoDoctorado en Físic

Asymptotic safety in QFT: from quantum gravity to graphene

In this work we investigate properties of scale invariant theories. This kind of theories describe a variety of phenomena, and two particular examples are discussed. On the one hand, more than 100 years after the discovery of General Relativity by Einstein, we still don't know how to unify gravity and quantum mechanics. One possibility is that on very small scales, gravity could be scale invariant, allowing for a finite ultraviolet completion. On the other hand, we will study the phase diagram of graphene and related materials. Scale invariant points in phase diagrams are related to second order phase transitions, and near these universal behaviour is found. To investigate these systems, nonperturbative renormalisation group methods are used. In order to achieve trustworthy results, also technical progress, both analytical and numerical, had to be made. On the analytical side, the Mathematica package xAct is used to derive the equations underlying the scale invariance of the theories. To solve these numerically, pseudo-spectral methods are systematically introduced in the present context for the first time. The results thus obtained support the ultraviolet completion of gravity by a scale invariant point. The dependence on gauge fixing and parametrisation is investigated, and found to be reasonably small. The 2-loop counterterm, being the hallmark of the perturbative nonrenormalisability of gravity, is shown to be irrelevant at the scale invariant point. Finally, the split-Ward identities are partially solved by resolving correlation functions. Regarding graphene and similar materials, different levels of approximation show a very good convergence of results for critical exponents and anomalous dimensions at the phase transition studied. The combined power of both analytical and numerical methods excels particularly - without either, the calculations wouldn't be possible

From condensed matter to higgs physics: solving functional renormalization group equations globally in field space

By means of the functional renormalization group (FRG), systems can be described in a nonperturbative way. The derived flow equations are solved via pseudo-spectral methods. As they allow to resolve the full field dependence of the effective potential and provide highly accurate results, these numerical methods are very powerful but have hardly been used in the FRG context. We show their benefits using several examples. Moreover, we apply the pseudo-spectral methods to explore the phase diagram of a bosonic model with two coupled order parameters and to clarify the nature of a possible metastability of the Higgs-Yukawa potential.In the phase diagram of systems with two competing order parameters, fixed points govern multicritical behavior. Such systems are often discussed in the context of condensed matter. Considering the phase diagram of the bosonic model between two and three dimensions, we discover additional fixed points besides the well-known ones from studies in three dimensions. Interestingly, our findings suggest that in certain regions of the phase diagram, two universality classes coexist. To our knowledge, this is the first bosonic model where coexisting (multi-)criticalities are found. Also, the absence of nontrivial fixed points can have a physical meaning, such as in the electroweak sector of the standard model which suffers from the triviality problem. The electroweak transition giving rise to the Higgs mechanism is dominated by the Gaussian fixed point. Due to the low Higgs mass, perturbative calculations suggest a metastable potential. However, the existence of the lower Higgs-mass bound eventually is interrelated with the maximal ultraviolet extension of the standard model. A relaxation of the lower bound would mean that the standard model may be still valid to even higher scales. Within a simple Higgs-Yukawa model, we discuss the origin of metastabilities and mechanisms, which relax the Higgs-mass bound, including higher field operators

Dynamics of bright solitons in Bose–Einstein condensates: investigations of soliton behaviour in the vector Gross–Pitaevskii equation and applications to enhanced matter-wave interferometry

Bright solitons in a quasi-1D Bose–Einstein condensate can be used to enhance precision in matter-wave interferometry, due to their inherent robustness and support against dispersion. Such a soliton interferometer typically relies on a potential barrier used to split a single soliton into two smaller coherent solitons which can then be recombined on the same barrier. In this thesis we examine two extensions to this scheme. Firstly, we investigate a binary BEC system consisting of two bright solitons which are coupled through a mutual nonlinear interaction term. We derive a set of conditions under which the two components can be separated on a potential barrier and use numerical simulations to probe the regimes beyond which this mathematical treatment is applicable. We then use the numerical simulations to look at the effect of the nonlinear coupling on the dynamics of the binary solitons interacting with the barrier. We also look at the interference behaviour found by doubling the simulation time in either a ring trap or a harmonic trapping potential (to ensure recombination on the barrier); as well as the case where the solitons start spatially separated on either side of the barrier in order to find conditions under which the solitons will combine on the barrier. We find a good agreement between the analytical predictions and the results of simulations. Beyond the regions of parameter space where the predictions are expected to hold, we find complex transmission and interference behaviour as a result of nonlinear effects. The second part of this thesis consists of an examination of the prospect of using a subwavelength barrier scheme in a soliton interferometry experiment. This involves using two resonant coupling beams in a Λ-system with a spatially varying intensity. Under certain conditions, this can be used to form an effective potential barrier with a width which is not diffraction-limited. We look at suitable parameter regimes for such a barrier to split and recombine solitons in an interferometer and probe the effects of possible complications such as misalignment in the beams and different scattering lengths in the different states. We simulate the soliton interferometer using the full three component GPE as well as the single component analogue with the effective potential in order to characterise the soliton behaviour and the dependence of the interferometer sensitivity on the system parameters. We find a trend towards an idealised soliton interferometer with a decreasing value of the parameter, w, controlling the barrier characteristics. Also, we demonstrate an agreement between the relevant three component GPE and the analogous single component GPE, in the limit of strong coupling fields

Quantum Cosmology

Within the second half of the last century, quantum cosmology concretely became one of the main research lines within gravitational theory and cosmology. Substantial progress has been made. Furthermore, quantum cosmology can become a domain that will gradually develop further over the next handful of decades, perhaps assisted by technological developments. Indications for new physics (i.e., beyond the standard model of particle physics or general relativity) could emerge and then the observable universe would surely be seen from quite a new perspective. This motivates bringing quantum cosmology to more research groups and individuals.This Special Issue (SI) aims to provide a wide set of reviews, ranging from foundational issues to (very) recent advancing discussions. Concretely, we want to inspire new work proposing observational tests, providing an aggregated set of contributions, covering several lines, some of which are thoroughly explored, some allowing progress, and others much unexplored. The aim of this SI is motivate new researchers to employ and further develop quantum cosmology over the forthcoming decades. Textbooks and reviews exist on the present subject, and this SI will complementarily assist in offering open access to a set of wide-ranging reviews. Hopefully, this will assist new interested researchers, in having a single open access online volume, with reviews that can help. In particular, this will help in selecting what to explore, what to read in more detail, where to proceed, and what to investigate further within quantum cosmology

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

Asymptotics, Geometry, and Soft Matter

This dissertation is concerned with two problems that lie at the interface of soft-matter physics, geometry, and asymptotic analysis, but otherwise have no bearing on one another. In the first problem, I consider the equilibrium thermal fluctuations of deformable mechanical frameworks. These frameworks have served as highly idealized representations of mechanical structures that underlie a plethora of soft, few-body systems at the submicron scale such as colloidal clusters and DNA origami. When the holonomic constraints in a framework cease to be linearly independent, singularities can appear in its configuration space, where it becomes energetically softer. Consequently, the framework\u27s free-energy landscape becomes dominated by the neighborhoods of points corresponding to these singularities. In the second problem, I study the localization of elastic waves in thin elastic structures with spatially varying curvature profiles, using a curved rod and a uniaxially-curved shell as concrete examples. Waves propagating on such structures have multiple components owing to the curvature-mediated coupling of the tangential and normal components of the displacement field. Here, using the semiclassical approximation, I show that these waves form localized, bound states around points where the absolute curvature of the structure has a minimum. Both these problems exemplify the subtle interplay between the mechanical properties of soft materials and their geometry, which further sets the stage for many interesting consequences

Symmetry and Mesoscopic Physics

Symmetry is one of the most important notions in natural science; it lies at the heart of fundamental laws of nature and serves as an important tool for understanding the properties of complex systems, both classical and quantum. Another trend, which has in recent years undergone intensive development, is mesoscopic physics. This branch of physics also combines classical and quantum ideas and methods. Two main directions can be distinguished in mesoscopic physics. One is the study of finite quantum systems of mesoscopic sizes. Such systems, which are between the atomic and macroscopic scales, exhibit a variety of novel phenomena and find numerous applications in creating modern electronic and spintronic devices. At the same time, the behavior of large systems can be influenced by mesoscopic effects, which provides another direction within the framework of mesoscopic physics. The aim of the present book is to emphasize the phenomena that lie at the crossroads between the concept of symmetry and mesoscopic physics

Quantum Field Theories, Isomonodromic Deformations and Matrix Models

Recent years have seen a proliferation of exact results in quantum field theories, owing mostly to supersymmetric localisation. Coupled with decades of study of dualities, this ensured the development of many novel nontrivial correspondences linking seemingly disparate parts of the mathematical landscape. Among these, the link between supersymmetric gauge theories with 8 supercharges and Painlev{\'e} equations, interpreted as the exact RG flow of their codimension 2 defects and passing through a correspondence with two-dimensional conformal field theory, was highly surprising. Similarly surprising was the realisation that three-dimensional matrix models coming from M-theory compute these solutions, and provide a non-perturbative completion of the topological string. Extending these two results is the focus of my work. After giving a review of the basics, hopefully useful to researchers in the field also for uses besides understanding the thesis, two parts based on published and unpublished results follow. The first is focused on giving Painlev{\'e}-type equations for general groups and linear quivers, and the second on matrix models