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In this thesis we investigate the importance of causality in non-perturbative approaches to quantum gravity. Firstly, causal sets are introduced as a simple kinematical model for causal geometry. It is shown how causal sets could account for the microscopic origin of the Bekenstein entropy bound. Holography and finite entropy emerge naturally from the interplay between causality and discreteness. Going beyond causal set kinematics is problematic however. It is a hard problem to find the right amplitude to attach to each causal set that one needs to define the non-perturbative quantum dynamics of gravity. One approach which is ideally suited to define the non-perturbative gravitational path integral is dynamical triangulation. Without causality this method leads to unappealing features of the quantum geometry though. It is shown how causality is instrumental in regulating this pathological behavior. In two dimensions this approach of causal dynamical triangulations has been analytically solved by transfer matrix methods. In this thesis considerable progress has been made in the development of more powerful techniques for this approach. The formulation through matrix models and a string field theory allow us to study interesting generalizations. Particularly, it has become possible to define the topological expansion. A surprising twist of the new matrix model is that it partially disentangles the large-N and continuum limit. This makes our causal model much closer in spirit to the original idea by 't Hooft than the conventional matrix models of non-critical string theory.

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The standard approach in quantum chemistry is to expand the eigenfunctions of the non relativistic Born Oppenheimer Hamiltonian in terms of Slater determinants. The quality improvements of such wavefunctions in terms of the underlying one electron basis is frustratingly slow. The error in the correlation energy decreases only with L 3 where L is the maximum angular momentum present in the basis. The integral evaluation effort that grows with 0(N4) prevents the use of ever larger bases for obtaining more accurate results. Most of the developments are therefore focused on wavefunction models with explicit correlation to get faster convergence. Although highly successful these approaches are computationally very demanding. A different solution might be provided by constructing new operators which take care of the information loss introduced by truncating the basis. In this thesis different routes towards such new operators are investigated.

This thesis is a theoretical investigation into the properties of indirect excitons in coupled quantum wells at low lattice temperatures. The relaxational thermodynamics, optical decay and diffusion of the statistically-degenerate excitons are modelled theoretically and numerically. The excitons' thermalization from an initial energy, EJkB = 20-300 K, to the lattice temperature, Th 1 K, is investigated. The exciton optical creation and decay mechanisms are then included, as well as the change in the exciton effective temperature due to these mechanisms. While the optical creation heats the excitons, their optical decay produces an effect called 'recombination heating and cooling', and whether it produces a net cooling or a net heating of the exciton system depends on the exciton effective temperature, T. The system of excitons is also studied in two dimensions by using a quantum diffusion equation. The excitons are created by a laser pump with a cylindrically-symmetric spatial intensity profile. The created excitons move outwards from the excitation spot by drift and diffusion, and cool down while doing so. They become more optically- active as they cool, creating a ring of photoluminescence around the excitation spot. This ring was also seen in experiments of this kind. Theoretical results are fitted to experimental results, and the diffusion coefficient for exciton concentrations in the range of 0 < n2D < 2.5 x 1010cm-2 varies from 0.06 to 25cm2/s when Tb = 1.5 K, and the disorder amplitude in the sample is U 0.9 eV. Finally, a novel kind of laser trap used in experiments to spatially confine the excitons is modelled theoretically. While the experiments were carried out at Th = 1.5 K giving an occupation number of the ground state of 8, theoretical simulations show that for a lattice temperature of Tb = 0.4 K the occupation number of the ground state is 500. The trap is also modelled as a homogeneous trap, and simulations show that when Tb is decreased further the fraction of excitons in the ground state increases dramatically.

The work described in this thesis is a theoretical investigation of the properties of exciton-polaritons in quantum wells (QWs). The polariton effect is first studied in the case of a completely coherent interaction between QW excitons and bulk photons, i.e. in the so called strong coupling limit. Then, an incoherent damping rate for the exciton states is included and the resulting modifications in the polariton dispersion are analyzed. A microscopic model which accounts for the scattering of QW excitons by random impurities is also proposed. In the strong coupling limit, a definitive and correct description of the QW polariton dispersion, for both confined and radiative modes, is obtained when the exciton- photon coupling is treated non perturbatively. A self-consistent perturbation theory which qualitatively agrees with the obtained results is also formulated. With increasing the incoherent damping, the orthogonality between radiative and confined polariton states is not affected, but a phase transition from the strong coupling regime to a weak coupling one occurs for both modes. The crossover between the two regimes is attributed to a topological change of the polariton dispersion curves when the damping rate reaches a critical value. A microscopic approach dealing with scattering of excitons by random impurities is formulated in terms of a quadratic Hamiltonian for QW excitons, bulk photons and localised impurities. By analyzing the preliminary results based on the calculation of the relevant eigenstates, the mixing between radiative and confined modes is observed.

Toposes and quasi-toposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the different ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding "good" quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the "best" regular category (called its regular completion) that embeds it. The second assigns to a regular category the "best" exact category (called its ex/reg completion) that embeds it. These two constructions are of independent interest. There are quasi-toposes that arise as regular completions and toposes, such as those of sheaves on a locale, that arise as ex/reg completions but which are not exact completions. We give a characterization of the categories with finite limits whose exact completions are toposes. This provides a very simple way of presenting realizability toposes, it allows us to give a very simple characterization of the presheaf toposes whose exact completions are themselves toposes and also to find new examples of toposes arising as exact completions. We also characterize universal closure operators in exact completions in terms of topologies, in a way analogous to the case of presheaf toposes and Grothendieck topologies. We then identify two "extreme" topologies in our sense and give simple conditions which ensure that the regular completion of a category is the category of separated objects for one of these topologies. This connection allows us to derive good properties of regular completions such as local cartesian closure. This, in turn, is part of our study of when a regular completion is a quasi-topos. The second extreme topology gives rise, as its category of sheaves, to the category of what we call complete equivalence relations. We then characterize the locally cartesian closed regular categories whose associated category of complete equivalence relations is a topos. Moreover, we observe that in this case the topos is nothing but the ex/reg completion of the original category.