The analysis and the results contained in this work are rooted in a first contact between the quantum theory and the general theory of relativity. By first contact it is meant that we are not considering candidates for “unified theories", but rather we focus on aspects of the full quantum theory in changing geometric backgrounds: the analysis of such an interaction already had important applications in cosmology, e.g. in the description of the evolution of fields in inflationary scenarios. Another compelling – and still growing – area of application is the study of thermodynamical properties of gravitional systems, which covers the main bulk of this thesis.

In this Thesis is presented a study on dynamical properties of mixtures of ultraold Bose gases. The behaviour of this system in different regimes is analysed: with and without coherent coupling between the two components, in homogeneous and harmonic shaped trapping potentials and in different dimensions and geometries. Most of the results presented here have been obtained by means of numerical solutions of coupled Gross-Pitaevskii equations and have been compared with theoretical predictions (and sometimes experiments), describing the same phenomena. In particualr the stability of persistent currents in a two-component Bose-Einstein condensate in a toroidal trap is studied in both the miscible and the immiscible regime. In the miscible regime we show that superflow decay is related to linear instabilities of the spin-density Bogoliubov mode. We find a region of partial stability, where the flow is stable in the majority component while it decays in the minority component. We also characterize the dynamical instability appearing for a large relative velocity between the two components. In the immiscible regime the stability criterion is modified and depends on the specific density distribution of the two components. The effect of a coherent coupling between the two components is also discussed. A study on the collective modes of the minority component of a highly unbalanced Bose-Bose mixture is also presented. In the immiscible case we find that the ground state can be a two-domain walls soliton. Although the mode frequencies are continuous at the transition, their behaviour is very different with respect to the miscible case. The dynamical behaviour of the solitonic structure and the frequency dependence on the inter- and intra-species interaction is numerically studied using coupled Gross-Pitaevskii equations. The results of the study on the static and the dynamic response of coherently coupled two component Bose-Einstein condensates due to a spin-dipole perturbation is also sown. The static dipole susceptibility is determined and is shown to be a key quantity to identify the second order ferromagnetic transition occurring at large inter-species interactions. The dynamics, which is obtained by quenching the spin-dipole perturbation, is very much affected by the system being paramagnetic or ferromagnetic and by the correlation between the motional and the internal degrees of freedom. In the paramagnetic phase the gas exhibits well defined out-of-phase dipole oscillations, whose frequency can be related to the susceptibility of the system using a sum rule approach. In particular in the interaction SU (2) symmetric case, when all the two-body interactions are the same, the external dipole oscillation coincides with the internal Rabi flipping frequency. In the ferromagnetic case, where linear response theory is not applicable, the system shows highly non-linear dynamics. In particular we observe phenomena related to ground state selection: the gas, initially trapped in a domain wall configuration, reaches a final state corresponding to the magnetic ground state plus small density ripples. Interestingly, the time during which the gas is unable to escape from its initial configuration is found to be proportional to the square root of the wall surface tension.

This thesis is devoted to the study of the conformation of monosacchrides in six-membered ring form. The main goal is to develop and apply new computational tools to investigate conformational properties and to improve the description of carbohydrates in the framework of molecular dynamics simulations. In the field of monosaccharides, modeling the system within the molecular dynamics framework presents troublesome aspects. The most important issue is that some force fields (e.g., the chosen gromos 45a4 parameter set) fail in reproducing the conformational preferences of the sugar constituents, with the appearance of unphysical conformations. This lack stems from the fact that the conformational behavior, dominated by few structures, generates a severe bottleneck: the non-ergodicity of the system by any practical means. This aspect explains the interest in free energy calculations, and methods exist, such as umbrella sampling or metadynamics, that allow to accelerate the sampling of different conformations by adding bias forces. In general, accelerated sampling methods are based on the choice of collective variables (CVs), which is of particular importance for the proper reconstruction of free energy landscapes. In the field of conformational analysis, suitable CVs have to be considered to describe non-planar, puckered conformations of cyclic structures. One of the main goals of this work is the enhancement of the gromos 45a4 force field for carbohydrates, with respect to the ability to describe ring conformation (that is, puckering) of six-membered rings. To this end, the development of efficient computational tools for the investigation of the general puckering problem are presented. In particular, we indicate how to exploit the capabilities of the metadynamics algorithm applied to the investigation of puckered ring conformers, exploring also different parametrizations of puckered structures to assess their respective advantages as collective variables for metadynamics.

Quantum Monte Carlo Methods applied to strongly correlated and highly inhomogeneous many-Fermion systems

We use mean-field theory to investigate the dynamics of superfluid fermions. This thesis includes our two works. The first one is to study Josephson oscillations and self-trapping of superfluid fermions in a double-well potential with time-dependent Bogoliubov-de Gennes equations. We investigate the behaviour of a two-component Fermi superfluid. We numerically solve the time-dependent Bogoliubov-de Gennes equations and characterize the regimes of Josephson oscillations and self-trapping for different potential barriers and initial conditions. In the weak link limit the results agree with a two-mode model where the relative population and the phase difference between the two wells obey coupled nonlinear Josephson equations. A more complex dynamics is predicted for large amplitude oscillations and large tunneling. The second one is to calculate the dynamic structure factor of unitary fermions. We have studied the dynamic structure factor of unitary fermions both at zero and finite temperature using the Bogoliubov-de Gennes theory and also Superfluid Local Density Approximation. We have derived the expression of the linear response function and the dynamic structure factor in the random phase approximation. At zero temperature, the SLDA+RPA formalism indeed provides a better accuracy at low momentum transfer and also its static structure factor is closer to quantum Monte Carlo value than that in BdG+RPA; however SLDA seems to give worse results for the molecular excitations at large momentum transfer. We have discussed the role of temperature and the comparison between SLDA and BdG, as well as with experimental data. The analysis is still at a preliminary level, but it suggests that mean-field theories can indeed be used to extract quantitative information about the order parameter and the excitations of the system by two-photon Bragg scattering experiments.

Semi-classical aspects of black hole physics are studied with particular emphasis on Hawking radiation and its derivation from tunnelling method techniques.

In this work we discuss some specific features related to the concept of singularity in the gravitational theory. We give a brief review of some various definitions for singularity, then we explore some "negative" results, in the sense they are not able to reproduce, in general, a regular solution. We present some of these approaches, namely the non-commutative geometry; the Non-Linear Electrodynamics; and the conformal approach. We later generalize these results into a no-go theorem, which is actually a fully original result. In the second part of this work, we discuss some working examples of regular solutions: we present three of them already present in literature (non-minimal Yang-Mills coupling, mimetic field approach and non-polynomial gravity), then we use such results to build up a model of a regular cosmological solution. Its generating mechanism and its main features are described, replacing the Big Bang with a bounce; the inflationary behavior at large time is also recovered. In the following two chapters, we present some different schemes to build regular solutions from the coupling between gravity and a scalar field. In particular, in chapter 7, we use a minimal coupling, while in chapter 8 we find some sufficient (though not necessary) conditions to build a regular solution, within the framework of the Horndeski theory. In both cases we are not able to find explicit results. In the ninth chapter we discuss a model of a regular black hole, coupling gravity with some fluid: in this case, an exact solution is found. We prove it is regular and we show some of its general features; we also discuss the time-dependent case, although we are only able to discuss its asymptotic behavior. We also discuss some of its problems, mainly due to instability. In the appendices we try to extend the no-go theorem to $F(R)$ theories and try to solve the instabilities of the fluid approach respectively.

Already in ancient Greece, the pre-Socratic philosophers thought that natural phenomena, although different, were homogeneous, of the same fundamental nature. In their theories can be found the search for a common reference point (arché) that puts order in the chaotic multiplicity of phenomena. After Albert Einstein’s theory of gravitation (General Relativity -GR-) was published in 1915, the search for a unified field theory that combines gravity with electromagnetism began to become serious. It seemed plausible that there were no other fundamental forces. The main contributors were Gunnar Nordstrom, Hermann Weyl, Arthur Eddington, Theodor Kaluza, Oskar Klein (See Theory of Kaluza-Klein, 1921) and most notably the many attempts by Einstein and his collaborators. No attempt went through. In the first half of the twentieth century quantum mechanics was consolidated, an instrument capable of overcoming the inadequacy of classical mechanics to explain phenomena and properties such as blackbody radiation, the photoelectric effect, the specific heat of solids, the atomic spectra, the stability of atoms, the Compton effect, .... When in the thirties Fermi and Yukawa ’s studies led to the discovery of nuclear forces, the quantum formalism proved to be appropriate for the description of the new phenomena and, in 1967-68, Sheldon Glashow, Steven Weinberg and Abdus Salam showed how the weak nuclear force and the electromagnetism were simply different manifestations of the same force (electroweak). Since then, proposals have been done to include in a single grand unification theory also the strong nuclear force, and some of them (GTU SU(5) and SO(10)) have provided testable predictions as the quantization of electric charge. At classical level there is an extension of the Kaluza-Klein theory on a 11-dimensional space M4 × S1 × S2 × CP2. It corresponds to Einstein’s General Relativity with 7 extra dimensions, and considers all four forces as different expressions of a “mega†gravitational field. The forces are unified at the classical level but, once quantized, the theory turns out to be inconsistent and therefore unusable. This is because the nuclear forces have range of 10−15 m for strong force and of 10−18 m for weak force, distances at which classical physics loses its meaning. Ultimately, it seems that quantum mechanics is compatible with electroweak and strong interactions only if we limit ourselves to spaces of dimensionality less than or equal to 4. In addition, it is inconsistent with General Relativity for spaces with more of 3 dimensions. For these reasons, the theory of Kaluza-Klein fails doubly. Really, the incompatibility is not between general relativity and quantum mechanics in its entirety, but rather between General Relativity and the method of calculation used in quantum mechanics: perturbative expansion whose terms, in the cases indicted above, become . To get around this problem two different approaches have been taken: String Theory and Loop Quantum Gravity. The first has completely changed the wording of quantum theory, from considering local interactions, where the phenomena occur at specific points (of Feynman graphs), to interactions “extended†, where the phenomena are distributed along one limited dimension (string), open or closed. This system has eliminated the divergences in the terms of perturbative expansion, but has developed other anomalies, eliminated only by building up the theory on a space of 11 dimensions. Unfortunately, the extra dimensions introduce a huge number of arbitrariness, such as the theory can predict everything and nothing. The scientific community hopes to identify some potential whose minimum make a selection between these arbitrariness, but we are still far from such a result. The alternative discussed in this thesis is the Loop Quantum Gravity. This is simply the union of GR and quantum mechanics, without modifying the basic axioms of both. It can be made only in spaces of dimensionality equal to 4 and it surrenders completely the perturbative expansion. This produces fascinating predictions, such as the inflation of early universe, and the lack of singularities in the black holes and in the big bang. It also provides the picture of a “combinatorial†universe, described by nodes connected by paths, whose only necessary variables are integer numbers associated with nodes and paths. This last point in particular escapes the string theory which, whilst losing the locality, is however concentrated within the “very small†. The Loop Quantum Gravity, by contrast, is able to describe the universe as a whole, and to deal with transitions between universes of different topology. The downside is that the calculations are so complex that they are impracticable. Strategies have been developed to introduce a different perturbative expansion that makes the calculations feasible, but this introduces important changes to the initial structure of the theory, in a way that eliminate the beautiful cosmological predictions. Nevertheless, we tried to calculate the graviton propagator in this new “modified framework†, and the result is compatible with linearized quantum GR . For this reason, this framework has not been abandoned. It also seems that this formalism can easily be extended to include extra-dimensions and adapted to the unified theory of Kaluza Klein. This thesis has been developed in an attempt to contribute to the desire for simplification and connection to the essence that has always been in the natural sciences. In particular, it was given a demonstration of how the †modified framework†of Loop Quantum Gravity is derivable from a classical formulation of the GR of Palatini type. Finally, we give suggestions for extending the model to 11 dimensions, because 11 is the number suggested by String theory, by the classical theory of Kaluza Klein, and by the GTU SO(10). Probably the truth lies somewhere in between, maybe an action of a 4-dimensional brane immersed in a 11-dimensional universe would be the right compromise between String Theory and Loop Quantum Gravity. A 4-dimensional brane represents our universe, and any contact with other branes of a much smaller scale put small pieces of it in vibration. Depending on the number of dimensions in which contact is, the part could be a vibrating string or a two- or three-brane (with probability decreasing rapidly moving from string to the three-brane). So, we even lose the distinction between the notions of particles and universes, making the first totally unnecessary. The action of a 4-brane is equivalent to the action of Loop Quantum Gravity, with the coordinate-fields which assume the role of gauge fields, and the indexes in the 11-dimensional space that would become similar to the indexes of internal gauge. This thesis focuses on two specific problems: the calculation of the graviton propagator in Loop Quantum Gravity and the derivation of the “modified framework†from the Palatini formulation of GR (Chapter 8). While the first it was simply supported with a minimum contribution, the second is a problem undertaken by the student in a completely independent way that, while waiting for more in-depth audits, has not yet shown any inconsistency and for now can be hailed a success. A small space is reserved for some inedited consideration undertaken by the student on the “physical†projector. This operator is intended to project the Hilbert space of kinematic states in the subspace of physical states. The conclusion of the argument is simple and somewhat disturbing: the Loop Quantum Gravity is not an unitary theory!

The aim of this work is to investigate the both, some mathematical and physical general aspect of modified gravity, and, more specifically, the proprieties of viable, realistic models of modified gravity which can be used to reproduce the inflation and the dark energy epoch of universe today.

One of the most interesting problems in string theory is to understand how the background space-time on which the string propagates arises in a self-consistent way. For open strings, there are two main approaches to this problem, boundary string field theory (BSFT) and cubic string field theory (CSFT). In the first part of this Thesis we deal with the construction of the spacetime tachyon effective action in BSFT. Renormalization fixed points are solutions of classical equations of motion and should be viewed as solutions of classical string field theory. We have constructed the Witten-Shatashvili (WS) space-time action S and shown that some solitonic solutions are lower dimensional D-branes for which the finite value of S provides a quite accurate prediction of the D-brane tension. We have derived the explicit relation between the CSFT and WS action as a field redefinition which is nonsingular on-shell only when the normalization factor in the WS action coincides with the tension of the D25-brane, in agreement with the conjectures involving tachyon condensation. We have also found a time-dependent solution of CSFT whose evolution is driven by a diffusion equation that makes the equations of motion local with respect to the time variable. The analysis here proposed has attracted a good deal of attention for its potential cosmological applications. The profile can be expressed in terms of a series in powers of exponentials of the time coordinate, and gives evidence of a well-defined but wildly oscillatory behavior. The tachyon rolls well past the minimum of the potential, then turns around and begins to oscillate with ever increasing amplitude. Furthermore, we have derived an analytic series solution of the elliptic equations providing the 4-tachyon off-shell amplitude. From such a solution we computed the exact coefficient of the quartic effective action relevant for time-dependent solutions and we derived the exact coefficient of the quartic tachyon coupling. We studied the rolling tachyon solution expressed as a series of exponentials of the time coordinate both using level-truncation computations and the exact 4-tachyon amplitude. The results for the level-truncated coefficients converge to those derived using the exact string amplitude and confirm the wild oscillatory behavior. In the second part of the Thesis we consider the extension of the gauge/gravity correspondence to systems with reduced and hence more realistic supersymmetry, which is one of the main steps towards a non-perturbative description of confining, QCD-like, gauge theories in terms of gravitational backgrounds. If string theory on AdS5xS5 is integrable, the theory on simple orbifolds of that space would also be expected to be integrable. We have computed the planar finite size corrections to the spectrum of the dilatation operator acting on states of a certain limit of conformal N = 2 quiver gauge field theory which is a ZM-orbifold of N = 4 SYM theory. We matched the result to the string dual, IIB superstrings on a pp-wave background with a periodically identified null coordinate. Up to two loops, we have shown that the computations done by using an effective Hamiltonian technique and a twisted Bethe Ansatz agree with each other and also agree with a computation of the analogous quantity in string theory. Our results are consistent with integrability of the N = 2 theory.