Threshold Effects and Lorentz Symmetry

Evidence on the violation of Lorentz symmetry arises from the observation of cosmic rays with energies beyond the GZK cutoff, $E_{GZK} \simeq 4 \times 10^{19} eV$, from the apparent transparency of the Universe to the propagation of high energy gamma radiation and from the stability of pions in air showers. These three paradoxes can be explained through deformations of the relativistic dispersion relation. Theoretical ideas aimed to understand how Lorentz symmetry may be broken and phenomenologically interesting deformations of the relativistic dispersion relation may arise are briefly discussed.