  We explore the possibility to geometrize the interaction of massive fermions
with the quantum structure of space-time, trying to create a theoretical
background, in order to explain what some recent experimental results seem to
implicate on the propagation of Ultra High Energy Cosmic Rays (UHECR). We will
investigate part of the phenomenological implications of this approach on the
predicted effect of the UHECR suppression, in fact recent evidences seem to
involve the modification of the GZK cut-off phenomenon. The search for an
effective theory, which can explain this physical effect, is based on Lorentz
Invariance Violation (LIV), which is introduced via Modified Dispersion
Relations (MDRs). Furthermore we illustrate that this perspective implies a
more general geometry of space-time than the usual Riemannian one, indicating,
for example, the opportunity to resort to Finsler theory.
