Extended objects in quantum field theory in three dimensions and applications

Resumen

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 Dirac ¿-¿¿ lattices and Pöschl-Teller combs. -Scalar fields between two parallel plates mimicked by Dirac ¿ 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 ¿ 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.