Renormalization and phenomenology of quantum electrodynamics with high-energy Lorentz violation

Lorentz invariance is experimentally verified with a high degree of precision, so that to date it appears to be one of the most precise symmetries in nature. Nevertheless, Lorentz breaking at high energies is widely explored as a possible feature of new physics beyond the Standard Model. The point of view adopted in this thesis is that, if one relaxes the assumption of Lorentz invariance, the set of renormalizable quantum field theories can be enlarged. In fact the ultraviolet behaviour of propagators can be improved by means of higher space derivatives, preserving locality, causality and unitarity. We study explicitly the renormalization of the electromagnetic sector of this Lorentz violating extended Standard Model as well as the interplay between the high-energy and the low-energy theory. Doing so we realize that the power-like divergences of the low-energy theory become arbitrary, as they are multiplied by coefficients that incorporate an arbitrary renormalization scheme choice inherited from the high-energy theory. Consequently, if the elementary Higgs field is present, this arbitrariness makes the hierarchy problem disappear. Moreover, if one assume that Lorentz symmetry is not exact, several phenomena that are otherwise forbidden can occur, such as the Cherenkov radiation in vacuo. Comparing the predictions of the theory with experimental data and observations, we put bounds on the values of Lorentz-violating parameters and especially on the magnitude of the typical energy scale of Lorentz violation. Indeed, we argue that the scale of Lorentz violation may be smaller than the Planck scale, and if this were true, the understanding of physics around the Planck scale, in particular the formulation of gravity, should be completely reconsidered.