Quantum Gravity Phenomenology: Thermal Dimension of Quantum Spacetime, Casuality and Momentum Conservation from Relative Locality

The original results presented in this thesis regard two very common topics of discussion in the quantum gravity debate: the dynamical dimensional reduction of spacetime and locality in quantum gravity regime. The dimensionality of quantum spacetime is often understood in terms of the spectral dimension; here, a different notion of dimensionality, the thermal dimension, is proposed. I discuss its physical properties in relation to those of the spectral dimension through the study of specific models of quantum gravity, including preliminary results obtained in the case of models with relative locality. I show that, in those cases where the spectral dimension has puzzling properties, the thermal dimension gives a different and more meaningful picture. The statistical mechanics developed to define the thermal dimension is applied also to the study of the production of primordial cosmological perturbations assuming a running Newton constant and Rainbow gravity. Concerning locality, I study in particular the theory of Relative Locality, a theoretical framework in which different observers may describe the same event as being local or non-local, depending whether it happens in the origin of their reference frame or far away from it, respectively. I show that requiring that locality is relative is enough to guarantee the objectivity of cause-effect relation in chains of events, the absence of causality-violating loops and processes violating the law of conservation of momentum.