Quantum cosmology, polymer matter, and modified collapse

In this dissertation I address the following three questions: 1) how do different clocks compare in regard of the avoidance of the big bang singularity, 2) what is the high energy behavior of polymer quantized matter, and 3) how do quantum gravity effects modify the formation and the properties of black holes? To discuss 1), I consider an isotropic Friedmann-Robertson-Walker cosmology sourced by a non-negative cosmological constant and a massless scalar field. I choose the scale factor as clock and, using a trick by Dirac, Schrodinger quantize the reduced square root Hamiltonian. From the resulting spinor equation I show that no semiclassical wave packet avoids the big bang. I compare this work with that in loop quantum cosmology, where the scalar field is chosen as time variable and the big bang is avoided. As for 2), I explain the details of a project with Professor Husain in which we study a scalar field on a curved background and perform a polymer Fock quantization of the matter degrees of freedom. The quantization is based on the assumption that the underlying field space is discrete and it therefore comes with a fundamental scale. This renders the usual operators on the Fock space scale dependent and causes a transition from a bosonic to a fermionic behavior in the ultraviolet regime. Finally for 3), I present work with Professor Husain and Professor Seahra on a modification of the Hamiltonian formulation of a massless scalar field minimally coupled to a spherically symmetric spacetime. The modification is designed to mimic polymer quantum effects in the region where the scalar field collapses and it is implemented in such a way that the symmetry group of general relativity is preserved. This causes a drastic change in the way black holes form and, thus, in their properties. Namely, the scalar field has to overcome a repulsion and black holes form with a finite mass. Furthermore, the modification gives rise to oscillations in the relation between the black hole mass and the initial data parameters, as well as to a universality and robustness of the findings.