Aspects of fermion dynamics from Lorentz symmetry violation

In this thesis we are interested in understanding how Lorentz symmetry violation can affect some features of fermion dynamics and, perhaps, help to solve some well-known problems in particle physics, such as the origin of neutrino masses and oscillations. Firstly, we consider two Lorentz-Invariance-Violating (LIV) models and investigate the possibility of generating masses and oscillations dynamically for both Dirac and Majorana neutrinos, using non-perturbative methods, such as the Schwinger- Dyson and the effective potential approaches. In our studies, Lorentz symmetric models are extended by the inclusion of higher-order LIV operators, which improve the convergence of loop integrals and introduce a natural mass scale to the theories. We then present how Lorentz invariance can be recovered, for both models, after quantisation, in such a way that the dynamical masses and mixing are the only quantum effects that remain finite. Additionally, we study how matter fields, especially fermions, behave when coupled to two modifed gravity models. Such modifed gravity models break the 4-dimensional diffeomorphism invariance and, consequently, induce local Lorentz violation. In particular, we consider Horava-Lifshitz gravity, which presents an improved ultraviolet behaviour when compared to General Relativity (GR), and thus addresses a fundamental problem in physics: the non-(perturbative-)renormalisability of the theory of GR. We calculate the LIV one-loop corrections to the matter sector dispersion relations, after integration over graviton components, and show that, by imposing reasonable constraints on the energy scales of our gravity models, our results are consistent with the current bounds on Lorentz symmetry violation