Classical and Quantum Formulations of S1xS2 and S3 Gowdy Models Coupled with Matter

Abstract

We explore the features of gravity in its quantum regime in a mathematically rigorous way by carrying out the classical and quantum formulations of some two-Killing symmetry reductions of general relativity which admit an exact Fock/Schr\"{o}dinger quantization. Concretely, we analyze in depth the linearly polarized 3-handle (S1xS2) and 3-sphere (S3) Gowdy cosmological models coupled to massless scalar fields. Within the classical formulation, this study requires a careful application of modern differential geometry techniques and the Dirac-Bergmann theory of constrained systems. Regarding the quantization process, we analyze the construction of both Fock and Schr\"{o}dinger representations, the unitarity of the time evolution, and the existence of semiclassical states for these systems by making use of the algebraic formalism of quantum theory and the theory of unitary implementation of symplectic transformations. Several appendices give additional information on the results attained in the main body of the thesis, going deeply into the relevant mathematical aspects of the text.