GEOMETRICAL APPROACH TO PLANCK-SCALE DEFORMATIONS OF PHASE SPACES

In this thesis we describe some semi-classical properties of Quantum Gravity by the use of non-trivial geometries as a common language. We describe deformed interactions that emerge from the deformation of the Lorentz symmetry by means of curved momentum spaces. In particular, we consider the problem of how particles would interact if momentum space had an anti-de Sitter geometry: we show that some usual paths followed in the de Sitter momentum space literature could not be followed in the anti-de Sitter case due to some internal inconsistencies, and we present some alternative scenarios that could be considered. Regarding spacetime, we also exhibit a link between the disformal metrics literature and the Rainbow Gravity one, that is consistent with the deformed causal structure, presents an inherent group structure and is intrinsically covariant under coordinate transformations. Finally, we present an alternative way of describing the natural spacetime for Modified Dispersion Relations by means of a generalized Finsler geometry (which is intimately attached to variational techniques) with a well-defined massless limit, which allows us to analyze its relativistic properties and to derive observables that are compatible with this picture.